Quelle  Tree_Multiset.thy   Sprache: Isabelle

(* Author: Tobias Nipkow *)

section ‹Multiset of Elements of Binary Tree›

theory Tree_Multiset
imports Multiset Tree
begin

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