src/HOL/ex/MergeSort.thy
 author kleing Sat Mar 26 00:01:56 2005 +0100 (2005-03-26) changeset 15631 cbee04ce413b parent 13201 3cc108872aca child 15732 faa48c5b1402 permissions -rw-r--r--
use Library/Multiset instead of own definition
```     1 (*  Title:      HOL/ex/Merge.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow
```
```     4     Copyright   2002 TU Muenchen
```
```     5
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```     6 Merge sort
```
```     7 *)
```
```     8
```
```     9 theory MergeSort = Sorting:
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```    10
```
```    11 consts merge :: "('a::linorder)list * 'a list \<Rightarrow> 'a list"
```
```    12
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```    13 recdef merge "measure(%(xs,ys). size xs + size ys)"
```
```    14 "merge(x#xs,y#ys) =
```
```    15  (if x <= y then x # merge(xs,y#ys) else y # merge(x#xs,ys))"
```
```    16 "merge(xs,[]) = xs"
```
```    17 "merge([],ys) = ys"
```
```    18
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```    19 lemma [simp]: "multiset_of (merge(xs,ys)) = multiset_of xs + multiset_of ys"
```
```    20 apply(induct xs ys rule: merge.induct)
```
```    21 apply (auto simp: union_ac)
```
```    22 done
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```    23
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```    24 lemma [simp]: "set(merge(xs,ys)) = set xs \<union> set ys"
```
```    25 apply(induct xs ys rule: merge.induct)
```
```    26 apply auto
```
```    27 done
```
```    28
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```    29 lemma [simp]:
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```    30  "sorted (op <=) (merge(xs,ys)) = (sorted (op <=) xs & sorted (op <=) ys)"
```
```    31 apply(induct xs ys rule: merge.induct)
```
```    32 apply(simp_all add:ball_Un linorder_not_le order_less_le)
```
```    33 apply(blast intro: order_trans)
```
```    34 done
```
```    35
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```    36 consts msort :: "('a::linorder) list \<Rightarrow> 'a list"
```
```    37 recdef msort "measure size"
```
```    38 "msort [] = []"
```
```    39 "msort [x] = [x]"
```
```    40 "msort xs = merge(msort(take (size xs div 2) xs),
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```    41                   msort(drop (size xs div 2) xs))"
```
```    42
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```    43 lemma "sorted op <= (msort xs)"
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```    44 by (induct xs rule: msort.induct) simp_all
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```    45
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```    46 lemma "multiset_of (msort xs) = multiset_of xs"
```
```    47 apply (induct xs rule: msort.induct)
```
```    48   apply simp
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```    49  apply simp
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```    50 apply simp
```
```    51 apply (subst union_commute)
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```    52 apply (simp del:multiset_of_append add:multiset_of_append[symmetric] union_assoc)
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```    53 apply (simp add: union_ac)
```
```    54 done
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```    55
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```    56 end
```