src/HOL/ex/MergeSort.thy
 author krauss Tue Feb 03 11:16:28 2009 +0100 (2009-02-03) changeset 29780 1df0e5af40b9 parent 19872 1b53b196f85f child 30661 54858c8ad226 permissions -rw-r--r--
mergesort example: recdef->fun, localized
```     1 (*  Title:      HOL/ex/Merge.thy
```
```     2     ID:         \$Id\$
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```     3     Author:     Tobias Nipkow
```
```     4     Copyright   2002 TU Muenchen
```
```     5 *)
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```     6
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```     7 header{*Merge Sort*}
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```     8
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```     9 theory MergeSort
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```    10 imports Sorting
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```    11 begin
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```    12
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```    13 context linorder
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```    14 begin
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```    15
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```    16 fun merge :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```    17 where
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```    18   "merge (x#xs) (y#ys) =
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```    19          (if x \<le> y then x # merge xs (y#ys) else y # merge (x#xs) ys)"
```
```    20 | "merge xs [] = xs"
```
```    21 | "merge [] ys = ys"
```
```    22
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```    23 lemma multiset_of_merge[simp]:
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```    24      "multiset_of (merge xs ys) = multiset_of xs + multiset_of ys"
```
```    25 apply(induct xs ys rule: merge.induct)
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```    26 apply (auto simp: union_ac)
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```    27 done
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```    28
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```    29 lemma set_merge[simp]: "set (merge xs ys) = set xs \<union> set ys"
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```    30 apply(induct xs ys rule: merge.induct)
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```    31 apply auto
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```    32 done
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```    33
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```    34 lemma sorted_merge[simp]:
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```    35      "sorted (op \<le>) (merge xs ys) = (sorted (op \<le>) xs & sorted (op \<le>) ys)"
```
```    36 apply(induct xs ys rule: merge.induct)
```
```    37 apply(simp_all add: ball_Un not_le less_le)
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```    38 apply(blast intro: order_trans)
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```    39 done
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```    40
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```    41 fun msort :: "'a list \<Rightarrow> 'a list"
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```    42 where
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```    43   "msort [] = []"
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```    44 | "msort [x] = [x]"
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```    45 | "msort xs = merge (msort (take (size xs div 2) xs))
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```    46 	                  (msort (drop (size xs div 2) xs))"
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```    47
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```    48 theorem sorted_msort: "sorted (op \<le>) (msort xs)"
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```    49 by (induct xs rule: msort.induct) simp_all
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```    50
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```    51 theorem multiset_of_msort: "multiset_of (msort xs) = multiset_of xs"
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```    52 apply (induct xs rule: msort.induct)
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```    53   apply simp_all
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```    54 apply (subst union_commute)
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```    55 apply (simp del:multiset_of_append add:multiset_of_append[symmetric] union_assoc)
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```    56 apply (simp add: union_ac)
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```    57 done
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```    58
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```    59 end
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```    60
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```    61
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```    62 end
```