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(* Author: Tobias Nipkow *)
section \<open>Bubblesort\<close>
theory Bubblesort
imports "HOL-Library.Multiset"
begin
text\<open>This is \emph{a} version of bubblesort.\<close>
context linorder
begin
fun bubble_min where
"bubble_min [] = []" |
"bubble_min [x] = [x]" |
"bubble_min (x#xs) =
(case bubble_min xs of y#ys \<Rightarrow> if x>y then y#x#ys else x#y#ys)"
lemma size_bubble_min: "size(bubble_min xs) = size xs"
by(induction xs rule: bubble_min.induct) (auto split: list.split)
lemma bubble_min_eq_Nil_iff[simp]: "bubble_min xs = [] \ xs = []"
by (metis length_0_conv size_bubble_min)
lemma bubble_minD_size: "bubble_min (xs) = ys \ size xs = size ys"
by(auto simp: size_bubble_min)
function (sequential) bubblesort where
"bubblesort [] = []" |
"bubblesort [x] = [x]" |
"bubblesort xs = (case bubble_min xs of y#ys \ y # bubblesort ys)"
by pat_completeness auto
termination
proof
show "wf(measure size)" by simp
next
fix x1 x2 y :: 'a fix xs ys :: "'a list"
show "bubble_min(x1#x2#xs) = y#ys \ (ys, x1#x2#xs) \ measure size"
by(auto simp: size_bubble_min dest!: bubble_minD_size split: list.splits if_splits)
qed
lemma mset_bubble_min: "mset (bubble_min xs) = mset xs"
apply(induction xs rule: bubble_min.induct)
apply simp
apply simp
apply (auto split: list.split)
done
lemma bubble_minD_mset:
"bubble_min (xs) = ys \ mset xs = mset ys"
by(auto simp: mset_bubble_min)
lemma mset_bubblesort:
"mset (bubblesort xs) = mset xs"
apply(induction xs rule: bubblesort.induct)
apply simp
apply simp
by(auto split: list.splits if_splits dest: bubble_minD_mset)
lemma set_bubblesort: "set (bubblesort xs) = set xs"
by(rule mset_bubblesort[THEN mset_eq_setD])
lemma bubble_min_min: "bubble_min xs = y#ys \ z \ set ys \ y \ z"
apply(induction xs arbitrary: y ys z rule: bubble_min.induct)
apply simp
apply simp
apply (fastforce split: list.splits if_splits dest!: sym[of "a#b" for a b])
done
lemma sorted_bubblesort: "sorted(bubblesort xs)"
apply(induction xs rule: bubblesort.induct)
apply simp
apply simp
apply (fastforce simp: set_bubblesort split: list.split if_splits dest: bubble_min_min)
done
end
end
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