src/HOL/Random.thy
 author haftmann Tue May 19 16:54:55 2009 +0200 (2009-05-19) changeset 31205 98370b26c2ce parent 31203 5c8fb4fd67e0 child 31261 900ebbc35e30 permissions -rw-r--r--
String.literal replaces message_string, code_numeral replaces (code_)index
```     1 (* Author: Florian Haftmann, TU Muenchen *)
```
```     2
```
```     3 header {* A HOL random engine *}
```
```     4
```
```     5 theory Random
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```     6 imports Code_Numeral List
```
```     7 begin
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```     8
```
```     9 notation fcomp (infixl "o>" 60)
```
```    10 notation scomp (infixl "o\<rightarrow>" 60)
```
```    11
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```    12
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```    13 subsection {* Auxiliary functions *}
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```    14
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```    15 definition inc_shift :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
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```    16   "inc_shift v k = (if v = k then 1 else k + 1)"
```
```    17
```
```    18 definition minus_shift :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
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```    19   "minus_shift r k l = (if k < l then r + k - l else k - l)"
```
```    20
```
```    21 fun log :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
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```    22   "log b i = (if b \<le> 1 \<or> i < b then 1 else 1 + log b (i div b))"
```
```    23
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```    24
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```    25 subsection {* Random seeds *}
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```    26
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```    27 types seed = "code_numeral \<times> code_numeral"
```
```    28
```
```    29 primrec "next" :: "seed \<Rightarrow> code_numeral \<times> seed" where
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```    30   "next (v, w) = (let
```
```    31      k =  v div 53668;
```
```    32      v' = minus_shift 2147483563 (40014 * (v mod 53668)) (k * 12211);
```
```    33      l =  w div 52774;
```
```    34      w' = minus_shift 2147483399 (40692 * (w mod 52774)) (l * 3791);
```
```    35      z =  minus_shift 2147483562 v' (w' + 1) + 1
```
```    36    in (z, (v', w')))"
```
```    37
```
```    38 lemma next_not_0:
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```    39   "fst (next s) \<noteq> 0"
```
```    40   by (cases s) (auto simp add: minus_shift_def Let_def)
```
```    41
```
```    42 primrec seed_invariant :: "seed \<Rightarrow> bool" where
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```    43   "seed_invariant (v, w) \<longleftrightarrow> 0 < v \<and> v < 9438322952 \<and> 0 < w \<and> True"
```
```    44
```
```    45 definition split_seed :: "seed \<Rightarrow> seed \<times> seed" where
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```    46   "split_seed s = (let
```
```    47      (v, w) = s;
```
```    48      (v', w') = snd (next s);
```
```    49      v'' = inc_shift 2147483562 v;
```
```    50      s'' = (v'', w');
```
```    51      w'' = inc_shift 2147483398 w;
```
```    52      s''' = (v', w'')
```
```    53    in (s'', s'''))"
```
```    54
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```    55
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```    56 subsection {* Base selectors *}
```
```    57
```
```    58 fun iterate :: "code_numeral \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a" where
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```    59   "iterate k f x = (if k = 0 then Pair x else f x o\<rightarrow> iterate (k - 1) f)"
```
```    60
```
```    61 definition range :: "code_numeral \<Rightarrow> seed \<Rightarrow> code_numeral \<times> seed" where
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```    62   "range k = iterate (log 2147483561 k)
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```    63       (\<lambda>l. next o\<rightarrow> (\<lambda>v. Pair (v + l * 2147483561))) 1
```
```    64     o\<rightarrow> (\<lambda>v. Pair (v mod k))"
```
```    65
```
```    66 lemma range:
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```    67   "k > 0 \<Longrightarrow> fst (range k s) < k"
```
```    68   by (simp add: range_def scomp_apply split_def del: log.simps iterate.simps)
```
```    69
```
```    70 definition select :: "'a list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" where
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```    71   "select xs = range (Code_Numeral.of_nat (length xs))
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```    72     o\<rightarrow> (\<lambda>k. Pair (nth xs (Code_Numeral.nat_of k)))"
```
```    73
```
```    74 lemma select:
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```    75   assumes "xs \<noteq> []"
```
```    76   shows "fst (select xs s) \<in> set xs"
```
```    77 proof -
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```    78   from assms have "Code_Numeral.of_nat (length xs) > 0" by simp
```
```    79   with range have
```
```    80     "fst (range (Code_Numeral.of_nat (length xs)) s) < Code_Numeral.of_nat (length xs)" by best
```
```    81   then have
```
```    82     "Code_Numeral.nat_of (fst (range (Code_Numeral.of_nat (length xs)) s)) < length xs" by simp
```
```    83   then show ?thesis
```
```    84     by (simp add: scomp_apply split_beta select_def)
```
```    85 qed
```
```    86
```
```    87 primrec pick :: "(code_numeral \<times> 'a) list \<Rightarrow> code_numeral \<Rightarrow> 'a" where
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```    88   "pick (x # xs) i = (if i < fst x then snd x else pick xs (i - fst x))"
```
```    89
```
```    90 lemma pick_member:
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```    91   "i < listsum (map fst xs) \<Longrightarrow> pick xs i \<in> set (map snd xs)"
```
```    92   by (induct xs arbitrary: i) simp_all
```
```    93
```
```    94 lemma pick_drop_zero:
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```    95   "pick (filter (\<lambda>(k, _). k > 0) xs) = pick xs"
```
```    96   by (induct xs) (auto simp add: expand_fun_eq)
```
```    97
```
```    98 lemma pick_same:
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```    99   "l < length xs \<Longrightarrow> Random.pick (map (Pair 1) xs) (Code_Numeral.of_nat l) = nth xs l"
```
```   100 proof (induct xs arbitrary: l)
```
```   101   case Nil then show ?case by simp
```
```   102 next
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```   103   case (Cons x xs) then show ?case by (cases l) simp_all
```
```   104 qed
```
```   105
```
```   106 definition select_weight :: "(code_numeral \<times> 'a) list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" where
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```   107   "select_weight xs = range (listsum (map fst xs))
```
```   108    o\<rightarrow> (\<lambda>k. Pair (pick xs k))"
```
```   109
```
```   110 lemma select_weight_member:
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```   111   assumes "0 < listsum (map fst xs)"
```
```   112   shows "fst (select_weight xs s) \<in> set (map snd xs)"
```
```   113 proof -
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```   114   from range assms
```
```   115     have "fst (range (listsum (map fst xs)) s) < listsum (map fst xs)" .
```
```   116   with pick_member
```
```   117     have "pick xs (fst (range (listsum (map fst xs)) s)) \<in> set (map snd xs)" .
```
```   118   then show ?thesis by (simp add: select_weight_def scomp_def split_def)
```
```   119 qed
```
```   120
```
```   121 lemma select_weigth_drop_zero:
```
```   122   "Random.select_weight (filter (\<lambda>(k, _). k > 0) xs) = Random.select_weight xs"
```
```   123 proof -
```
```   124   have "listsum (map fst [(k, _)\<leftarrow>xs . 0 < k]) = listsum (map fst xs)"
```
```   125     by (induct xs) auto
```
```   126   then show ?thesis by (simp only: select_weight_def pick_drop_zero)
```
```   127 qed
```
```   128
```
```   129 lemma select_weigth_select:
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```   130   assumes "xs \<noteq> []"
```
```   131   shows "Random.select_weight (map (Pair 1) xs) = Random.select xs"
```
```   132 proof -
```
```   133   have less: "\<And>s. fst (Random.range (Code_Numeral.of_nat (length xs)) s) < Code_Numeral.of_nat (length xs)"
```
```   134     using assms by (intro range) simp
```
```   135   moreover have "listsum (map fst (map (Pair 1) xs)) = Code_Numeral.of_nat (length xs)"
```
```   136     by (induct xs) simp_all
```
```   137   ultimately show ?thesis
```
```   138     by (auto simp add: select_weight_def select_def scomp_def split_def
```
```   139       expand_fun_eq pick_same [symmetric])
```
```   140 qed
```
```   141
```
```   142 definition select_default :: "code_numeral \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" where
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```   143   [code del]: "select_default k x y = range k
```
```   144      o\<rightarrow> (\<lambda>l. Pair (if l + 1 < k then x else y))"
```
```   145
```
```   146 lemma select_default_zero:
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```   147   "fst (select_default 0 x y s) = y"
```
```   148   by (simp add: scomp_apply split_beta select_default_def)
```
```   149
```
```   150 lemma select_default_code [code]:
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```   151   "select_default k x y = (if k = 0
```
```   152     then range 1 o\<rightarrow> (\<lambda>_. Pair y)
```
```   153     else range k o\<rightarrow> (\<lambda>l. Pair (if l + 1 < k then x else y)))"
```
```   154 proof
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```   155   fix s
```
```   156   have "snd (range (Code_Numeral.of_nat 0) s) = snd (range (Code_Numeral.of_nat 1) s)"
```
```   157     by (simp add: range_def scomp_Pair scomp_apply split_beta)
```
```   158   then show "select_default k x y s = (if k = 0
```
```   159     then range 1 o\<rightarrow> (\<lambda>_. Pair y)
```
```   160     else range k o\<rightarrow> (\<lambda>l. Pair (if l + 1 < k then x else y))) s"
```
```   161     by (cases "k = 0") (simp_all add: select_default_def scomp_apply split_beta)
```
```   162 qed
```
```   163
```
```   164
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```   165 subsection {* @{text ML} interface *}
```
```   166
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```   167 ML {*
```
```   168 structure Random_Engine =
```
```   169 struct
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```   170
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```   171 type seed = int * int;
```
```   172
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```   173 local
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```   174
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```   175 val seed = ref
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```   176   (let
```
```   177     val now = Time.toMilliseconds (Time.now ());
```
```   178     val (q, s1) = IntInf.divMod (now, 2147483562);
```
```   179     val s2 = q mod 2147483398;
```
```   180   in (s1 + 1, s2 + 1) end);
```
```   181
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```   182 in
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```   183
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```   184 fun run f =
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```   185   let
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```   186     val (x, seed') = f (! seed);
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```   187     val _ = seed := seed'
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```   188   in x end;
```
```   189
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```   190 end;
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```   191
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```   192 end;
```
```   193 *}
```
```   194
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```   195 hide (open) type seed
```
```   196 hide (open) const inc_shift minus_shift log "next" seed_invariant split_seed
```
```   197   iterate range select pick select_weight select_default
```
```   198
```
```   199 no_notation fcomp (infixl "o>" 60)
```
```   200 no_notation scomp (infixl "o\<rightarrow>" 60)
```
```   201
```
```   202 end
```
```   203
```