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(* Author: Tobias Nipkow *)
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section \<open>Multiset of Elements of Binary Tree\<close>
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theory Tree_Multiset
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imports Multiset Tree
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begin
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text\<open>Kept separate from theory @{theory Tree} to avoid importing all of
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theory @{theory Multiset} into @{theory Tree}. Should be merged if
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@{theory Multiset} ever becomes part of @{theory Main}.\<close>
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fun mset_tree :: "'a tree \<Rightarrow> 'a multiset" where
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"mset_tree Leaf = {#}" |
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"mset_tree (Node l a r) = {#a#} + mset_tree l + mset_tree r"
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lemma set_mset_tree[simp]: "set_mset (mset_tree t) = set_tree t"
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by(induction t) auto
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lemma size_mset_tree[simp]: "size(mset_tree t) = size t"
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by(induction t) auto
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lemma mset_map_tree: "mset_tree (map_tree f t) = image_mset f (mset_tree t)"
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by (induction t) auto
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lemma mset_iff_set_tree: "x \<in># mset_tree t \<longleftrightarrow> x \<in> set_tree t"
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by(induction t arbitrary: x) auto
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lemma mset_preorder[simp]: "mset (preorder t) = mset_tree t"
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by (induction t) (auto simp: ac_simps)
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lemma mset_inorder[simp]: "mset (inorder t) = mset_tree t"
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by (induction t) (auto simp: ac_simps)
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lemma map_mirror: "mset_tree (mirror t) = mset_tree t"
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by (induction t) (simp_all add: ac_simps)
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end
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