src/HOL/Library/Tree_Multiset.thy
author wenzelm
Wed Jun 17 11:03:05 2015 +0200 (2015-06-17)
changeset 60500 903bb1495239
parent 59928 b9b7f913a19a
child 60502 aa58872267ee
permissions -rw-r--r--
isabelle update_cartouches;
     1 (* Author: Tobias Nipkow *)
     2 
     3 section \<open>Multiset of Elements of Binary Tree\<close>
     4 
     5 theory Tree_Multiset
     6 imports Multiset Tree
     7 begin
     8 
     9 text\<open>Kept separate from theory @{theory Tree} to avoid importing all of
    10 theory @{theory Multiset} into @{theory Tree}. Should be merged if
    11 @{theory Multiset} ever becomes part of @{theory Main}.\<close>
    12 
    13 fun mset_tree :: "'a tree \<Rightarrow> 'a multiset" where
    14 "mset_tree Leaf = {#}" |
    15 "mset_tree (Node l a r) = {#a#} + mset_tree l + mset_tree r"
    16 
    17 lemma set_of_mset_tree[simp]: "set_of (mset_tree t) = set_tree t"
    18 by(induction t) auto
    19 
    20 lemma size_mset_tree[simp]: "size(mset_tree t) = size t"
    21 by(induction t) auto
    22 
    23 lemma mset_map_tree: "mset_tree (map_tree f t) = image_mset f (mset_tree t)"
    24 by (induction t) auto
    25 
    26 lemma multiset_of_preorder[simp]: "multiset_of (preorder t) = mset_tree t"
    27 by (induction t) (auto simp: ac_simps)
    28 
    29 lemma multiset_of_inorder[simp]: "multiset_of (inorder t) = mset_tree t"
    30 by (induction t) (auto simp: ac_simps)
    31 
    32 lemma map_mirror: "mset_tree (mirror t) = mset_tree t"
    33 by (induction t) (simp_all add: ac_simps)
    34 
    35 lemma del_rightmost_mset_tree:
    36   "\<lbrakk> del_rightmost t = (t',x);  t \<noteq> \<langle>\<rangle> \<rbrakk> \<Longrightarrow> mset_tree t = {#x#} + mset_tree t'"
    37 apply(induction t arbitrary: t' rule: del_rightmost.induct)
    38 by (auto split: prod.splits) (auto simp: ac_simps)
    39 
    40 end