src/HOL/Library/Tree_Multiset.thy
 author wenzelm Wed Mar 08 10:50:59 2017 +0100 (2017-03-08) changeset 65151 a7394aa4d21c parent 63861 90360390a916 child 66556 2d24e2c02130 permissions -rw-r--r--
tuned proofs;
```     1 (* Author: Tobias Nipkow *)
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```     2
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```     3 section \<open>Multiset of Elements of Binary Tree\<close>
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```     4
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```     5 theory Tree_Multiset
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```     6 imports Multiset Tree
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```     7 begin
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```     8
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```     9 text\<open>Kept separate from theory @{theory Tree} to avoid importing all of
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```    10 theory @{theory Multiset} into @{theory Tree}. Should be merged if
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```    11 @{theory Multiset} ever becomes part of @{theory Main}.\<close>
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```    12
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```    13 fun mset_tree :: "'a tree \<Rightarrow> 'a multiset" where
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```    14 "mset_tree Leaf = {#}" |
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```    15 "mset_tree (Node l a r) = {#a#} + mset_tree l + mset_tree r"
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```    16
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```    17 fun subtrees_mset :: "'a tree \<Rightarrow> 'a tree multiset" where
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```    18 "subtrees_mset Leaf = {#Leaf#}" |
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```    19 "subtrees_mset (Node l x r) = add_mset (Node l x r) (subtrees_mset l + subtrees_mset r)"
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```    20
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```    21
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```    22 lemma set_mset_tree[simp]: "set_mset (mset_tree t) = set_tree t"
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```    23 by(induction t) auto
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```    24
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```    25 lemma size_mset_tree[simp]: "size(mset_tree t) = size t"
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```    26 by(induction t) auto
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```    27
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```    28 lemma mset_map_tree: "mset_tree (map_tree f t) = image_mset f (mset_tree t)"
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```    29 by (induction t) auto
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```    30
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```    31 lemma mset_iff_set_tree: "x \<in># mset_tree t \<longleftrightarrow> x \<in> set_tree t"
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```    32 by(induction t arbitrary: x) auto
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```    33
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```    34 lemma mset_preorder[simp]: "mset (preorder t) = mset_tree t"
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```    35 by (induction t) (auto simp: ac_simps)
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```    36
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```    37 lemma mset_inorder[simp]: "mset (inorder t) = mset_tree t"
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```    38 by (induction t) (auto simp: ac_simps)
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```    39
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```    40 lemma map_mirror: "mset_tree (mirror t) = mset_tree t"
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```    41 by (induction t) (simp_all add: ac_simps)
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```    42
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```    43
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```    44 lemma in_subtrees_mset_iff[simp]: "s \<in># subtrees_mset t \<longleftrightarrow> s \<in> subtrees t"
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```    45 by(induction t) auto
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```    46
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```    47 end
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