# Theory Tree_Multiset

theory Tree_Multiset
imports Multiset Tree
```(* Author: Tobias Nipkow *)

section ‹Multiset of Elements of Binary Tree›

theory Tree_Multiset
imports Multiset Tree
begin

text ‹
Kept separate from theory @{theory "HOL-Library.Tree"} to avoid importing all of theory @{theory
"HOL-Library.Multiset"} into @{theory "HOL-Library.Tree"}. Should be merged if @{theory
"HOL-Library.Multiset"} ever becomes part of @{theory Main}.
›

fun mset_tree :: "'a tree ⇒ 'a multiset" where
"mset_tree Leaf = {#}" |
"mset_tree (Node l a r) = {#a#} + mset_tree l + mset_tree r"

fun subtrees_mset :: "'a tree ⇒ 'a tree multiset" where
"subtrees_mset Leaf = {#Leaf#}" |
"subtrees_mset (Node l x r) = add_mset (Node l x r) (subtrees_mset l + subtrees_mset r)"

lemma mset_tree_empty_iff[simp]: "mset_tree t = {#} ⟷ t = Leaf"
by (cases t) auto

lemma set_mset_tree[simp]: "set_mset (mset_tree t) = set_tree t"
by(induction t) auto

lemma size_mset_tree[simp]: "size(mset_tree t) = size t"
by(induction t) auto

lemma mset_map_tree: "mset_tree (map_tree f t) = image_mset f (mset_tree t)"
by (induction t) auto

lemma mset_iff_set_tree: "x ∈# mset_tree t ⟷ x ∈ set_tree t"
by(induction t arbitrary: x) auto

lemma mset_preorder[simp]: "mset (preorder t) = mset_tree t"
by (induction t) (auto simp: ac_simps)

lemma mset_inorder[simp]: "mset (inorder t) = mset_tree t"
by (induction t) (auto simp: ac_simps)

lemma map_mirror: "mset_tree (mirror t) = mset_tree t"
by (induction t) (simp_all add: ac_simps)

lemma in_subtrees_mset_iff[simp]: "s ∈# subtrees_mset t ⟷ s ∈ subtrees t"
by(induction t) auto

end
```