src/HOL/Imperative_HOL/ex/Imperative_Reverse.thy
 author haftmann Tue Jul 13 12:01:34 2010 +0200 (2010-07-13 ago) changeset 37796 08bd610b2583 parent 37771 1bec64044b5e child 37797 96551d6b1414 permissions -rw-r--r--
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```     1 (*  Title:      HOL/Imperative_HOL/ex/Imperative_Reverse.thy
```
```     2     Author:     Lukas Bulwahn, TU Muenchen
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```     3 *)
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```     4
```
```     5 header {* An imperative in-place reversal on arrays *}
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```     6
```
```     7 theory Imperative_Reverse
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```     8 imports Imperative_HOL Subarray
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```     9 begin
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```    10
```
```    11 fun swap :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap" where
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```    12   "swap a i j = (do
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```    13      x \<leftarrow> Array.nth a i;
```
```    14      y \<leftarrow> Array.nth a j;
```
```    15      Array.upd i y a;
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```    16      Array.upd j x a;
```
```    17      return ()
```
```    18    done)"
```
```    19
```
```    20 fun rev :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap" where
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```    21   "rev a i j = (if (i < j) then (do
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```    22      swap a i j;
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```    23      rev a (i + 1) (j - 1)
```
```    24    done)
```
```    25    else return ())"
```
```    26
```
```    27 notation (output) swap ("swap")
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```    28 notation (output) rev ("rev")
```
```    29
```
```    30 declare swap.simps [simp del] rev.simps [simp del]
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```    31
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```    32 lemma swap_pointwise: assumes "crel (swap a i j) h h' r"
```
```    33   shows "get_array a h' ! k = (if k = i then get_array a h ! j
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```    34       else if k = j then get_array a h ! i
```
```    35       else get_array a h ! k)"
```
```    36 using assms unfolding swap.simps
```
```    37 by (elim crel_elims)
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```    38  (auto simp: length_def)
```
```    39
```
```    40 lemma rev_pointwise: assumes "crel (rev a i j) h h' r"
```
```    41   shows "get_array a h' ! k = (if k < i then get_array a h ! k
```
```    42       else if j < k then get_array a h ! k
```
```    43       else get_array a h ! (j - (k - i)))" (is "?P a i j h h'")
```
```    44 using assms proof (induct a i j arbitrary: h h' rule: rev.induct)
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```    45   case (1 a i j h h'')
```
```    46   thus ?case
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```    47   proof (cases "i < j")
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```    48     case True
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```    49     with 1[unfolded rev.simps[of a i j]]
```
```    50     obtain h' where
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```    51       swp: "crel (swap a i j) h h' ()"
```
```    52       and rev: "crel (rev a (i + 1) (j - 1)) h' h'' ()"
```
```    53       by (auto elim: crel_elims)
```
```    54     from rev 1 True
```
```    55     have eq: "?P a (i + 1) (j - 1) h' h''" by auto
```
```    56
```
```    57     have "k < i \<or> i = k \<or> (i < k \<and> k < j) \<or> j = k \<or> j < k" by arith
```
```    58     with True show ?thesis
```
```    59       by (elim disjE) (auto simp: eq swap_pointwise[OF swp])
```
```    60   next
```
```    61     case False
```
```    62     with 1[unfolded rev.simps[of a i j]]
```
```    63     show ?thesis
```
```    64       by (cases "k = j") (auto elim: crel_elims)
```
```    65   qed
```
```    66 qed
```
```    67
```
```    68 lemma rev_length:
```
```    69   assumes "crel (rev a i j) h h' r"
```
```    70   shows "Array.length a h = Array.length a h'"
```
```    71 using assms
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```    72 proof (induct a i j arbitrary: h h' rule: rev.induct)
```
```    73   case (1 a i j h h'')
```
```    74   thus ?case
```
```    75   proof (cases "i < j")
```
```    76     case True
```
```    77     with 1[unfolded rev.simps[of a i j]]
```
```    78     obtain h' where
```
```    79       swp: "crel (swap a i j) h h' ()"
```
```    80       and rev: "crel (rev a (i + 1) (j - 1)) h' h'' ()"
```
```    81       by (auto elim: crel_elims)
```
```    82     from swp rev 1 True show ?thesis
```
```    83       unfolding swap.simps
```
```    84       by (elim crel_elims) fastsimp
```
```    85   next
```
```    86     case False
```
```    87     with 1[unfolded rev.simps[of a i j]]
```
```    88     show ?thesis
```
```    89       by (auto elim: crel_elims)
```
```    90   qed
```
```    91 qed
```
```    92
```
```    93 lemma rev2_rev': assumes "crel (rev a i j) h h' u"
```
```    94   assumes "j < Array.length a h"
```
```    95   shows "subarray i (j + 1) a h' = List.rev (subarray i (j + 1) a h)"
```
```    96 proof -
```
```    97   {
```
```    98     fix k
```
```    99     assume "k < Suc j - i"
```
```   100     with rev_pointwise[OF assms(1)] have "get_array a h' ! (i + k) = get_array a h ! (j - k)"
```
```   101       by auto
```
```   102   }
```
```   103   with assms(2) rev_length[OF assms(1)] show ?thesis
```
```   104   unfolding subarray_def Array.length_def
```
```   105   by (auto simp add: length_sublist' rev_nth min_def nth_sublist' intro!: nth_equalityI)
```
```   106 qed
```
```   107
```
```   108 lemma rev2_rev:
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```   109   assumes "crel (rev a 0 (Array.length a h - 1)) h h' u"
```
```   110   shows "get_array a h' = List.rev (get_array a h)"
```
```   111   using rev2_rev'[OF assms] rev_length[OF assms] assms
```
```   112     by (cases "Array.length a h = 0", auto simp add: Array.length_def
```
```   113       subarray_def sublist'_all rev.simps[where j=0] elim!: crel_elims)
```
```   114   (drule sym[of "List.length (get_array a h)"], simp)
```
```   115
```
```   116 end
```