src/HOL/ex/MergeSort.thy
 author nipkow Thu Apr 14 17:57:23 2005 +0200 (2005-04-14) changeset 15732 faa48c5b1402 parent 15631 cbee04ce413b child 15815 62854cac5410 permissions -rw-r--r--
```     1 (*  Title:      HOL/ex/Merge.thy
```
```     2     ID:         \$Id\$
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```     3     Author:     Tobias Nipkow
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```     4     Copyright   2002 TU Muenchen
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```     5
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```     6 Merge sort
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```     7 *)
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```     8
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```     9 theory MergeSort = Sorting:
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```    10
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```    11 consts merge :: "('a::linorder)list * 'a list \<Rightarrow> 'a list"
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```    12
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```    13 recdef merge "measure(%(xs,ys). size xs + size ys)"
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```    14 "merge(x#xs,y#ys) =
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```    15  (if x <= y then x # merge(xs,y#ys) else y # merge(x#xs,ys))"
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```    16 "merge(xs,[]) = xs"
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```    17 "merge([],ys) = ys"
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```    18
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```    19 lemma multiset_of_merge[simp]:
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```    20  "multiset_of (merge(xs,ys)) = multiset_of xs + multiset_of ys"
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```    21 apply(induct xs ys rule: merge.induct)
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```    22 apply (auto simp: union_ac)
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```    23 done
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```    24
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```    25 lemma set_merge[simp]: "set(merge(xs,ys)) = set xs \<union> set ys"
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```    26 apply(induct xs ys rule: merge.induct)
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```    27 apply auto
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```    28 done
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```    29
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```    30 lemma sorted_merge[simp]:
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```    31  "sorted (op <=) (merge(xs,ys)) = (sorted (op <=) xs & sorted (op <=) ys)"
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```    32 apply(induct xs ys rule: merge.induct)
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```    33 apply(simp_all add:ball_Un linorder_not_le order_less_le)
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```    34 apply(blast intro: order_trans)
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```    35 done
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```    36
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```    37 consts msort :: "('a::linorder) list \<Rightarrow> 'a list"
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```    38 recdef msort "measure size"
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```    39 "msort [] = []"
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```    40 "msort [x] = [x]"
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```    41 "msort xs = merge(msort(take (size xs div 2) xs),
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```    42                   msort(drop (size xs div 2) xs))"
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```    43
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```    44 lemma sorted_msort: "sorted op <= (msort xs)"
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```    45 by (induct xs rule: msort.induct) simp_all
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```    46
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```    47 lemma multiset_of_msort: "multiset_of (msort xs) = multiset_of xs"
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```    48 apply (induct xs rule: msort.induct)
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```    49   apply simp
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```    50  apply simp
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```    51 apply simp
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```    52 apply (subst union_commute)
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```    53 apply (simp del:multiset_of_append add:multiset_of_append[symmetric] union_assoc)
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```    54 apply (simp add: union_ac)
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```    55 done
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```    56
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```    57 end
```