src/HOL/Library/Tree_Multiset.thy
author nipkow
Tue Sep 22 14:31:22 2015 +0200 (2015-09-22)
changeset 61225 1a690dce8cfc
parent 60808 fd26519b1a6a
child 63861 90360390a916
permissions -rw-r--r--
tuned references
nipkow@59928
     1
(* Author: Tobias Nipkow *)
nipkow@59928
     2
wenzelm@60500
     3
section \<open>Multiset of Elements of Binary Tree\<close>
nipkow@59928
     4
nipkow@59928
     5
theory Tree_Multiset
nipkow@59928
     6
imports Multiset Tree
nipkow@59928
     7
begin
nipkow@59928
     8
wenzelm@60500
     9
text\<open>Kept separate from theory @{theory Tree} to avoid importing all of
nipkow@59928
    10
theory @{theory Multiset} into @{theory Tree}. Should be merged if
wenzelm@60500
    11
@{theory Multiset} ever becomes part of @{theory Main}.\<close>
nipkow@59928
    12
nipkow@59928
    13
fun mset_tree :: "'a tree \<Rightarrow> 'a multiset" where
nipkow@59928
    14
"mset_tree Leaf = {#}" |
nipkow@59928
    15
"mset_tree (Node l a r) = {#a#} + mset_tree l + mset_tree r"
nipkow@59928
    16
nipkow@60495
    17
lemma set_mset_tree[simp]: "set_mset (mset_tree t) = set_tree t"
nipkow@59928
    18
by(induction t) auto
nipkow@59928
    19
nipkow@59928
    20
lemma size_mset_tree[simp]: "size(mset_tree t) = size t"
nipkow@59928
    21
by(induction t) auto
nipkow@59928
    22
nipkow@59928
    23
lemma mset_map_tree: "mset_tree (map_tree f t) = image_mset f (mset_tree t)"
nipkow@59928
    24
by (induction t) auto
nipkow@59928
    25
nipkow@60505
    26
lemma mset_iff_set_tree: "x \<in># mset_tree t \<longleftrightarrow> x \<in> set_tree t"
nipkow@60505
    27
by(induction t arbitrary: x) auto
nipkow@60505
    28
nipkow@60515
    29
lemma mset_preorder[simp]: "mset (preorder t) = mset_tree t"
nipkow@59928
    30
by (induction t) (auto simp: ac_simps)
nipkow@59928
    31
nipkow@60515
    32
lemma mset_inorder[simp]: "mset (inorder t) = mset_tree t"
nipkow@59928
    33
by (induction t) (auto simp: ac_simps)
nipkow@59928
    34
nipkow@59928
    35
lemma map_mirror: "mset_tree (mirror t) = mset_tree t"
nipkow@59928
    36
by (induction t) (simp_all add: ac_simps)
nipkow@59928
    37
nipkow@59928
    38
end