src/HOL/Random.thy
author wenzelm
Thu Feb 11 23:00:22 2010 +0100 (2010-02-11)
changeset 35115 446c5063e4fd
parent 33236 ea75c6ea643e
child 35266 07a56610c00b
permissions -rw-r--r--
modernized translations;
formal markup of @{syntax_const} and @{const_syntax};
minor tuning;
haftmann@29815
     1
(* Author: Florian Haftmann, TU Muenchen *)
haftmann@22528
     2
haftmann@26265
     3
header {* A HOL random engine *}
haftmann@22528
     4
haftmann@22528
     5
theory Random
haftmann@31205
     6
imports Code_Numeral List
haftmann@22528
     7
begin
haftmann@22528
     8
haftmann@29823
     9
notation fcomp (infixl "o>" 60)
haftmann@29823
    10
notation scomp (infixl "o\<rightarrow>" 60)
haftmann@29823
    11
haftmann@29823
    12
haftmann@26265
    13
subsection {* Auxiliary functions *}
haftmann@26265
    14
haftmann@33236
    15
fun log :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
haftmann@33236
    16
  "log b i = (if b \<le> 1 \<or> i < b then 1 else 1 + log b (i div b))"
haftmann@33236
    17
haftmann@31205
    18
definition inc_shift :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
haftmann@26265
    19
  "inc_shift v k = (if v = k then 1 else k + 1)"
haftmann@26265
    20
haftmann@31205
    21
definition minus_shift :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
haftmann@26265
    22
  "minus_shift r k l = (if k < l then r + k - l else k - l)"
haftmann@26265
    23
haftmann@30495
    24
haftmann@26265
    25
subsection {* Random seeds *}
haftmann@26038
    26
haftmann@31205
    27
types seed = "code_numeral \<times> code_numeral"
haftmann@22528
    28
haftmann@31205
    29
primrec "next" :: "seed \<Rightarrow> code_numeral \<times> seed" where
haftmann@26265
    30
  "next (v, w) = (let
haftmann@26265
    31
     k =  v div 53668;
haftmann@33236
    32
     v' = minus_shift 2147483563 ((v mod 53668) * 40014) (k * 12211);
haftmann@26265
    33
     l =  w div 52774;
haftmann@33236
    34
     w' = minus_shift 2147483399 ((w mod 52774) * 40692) (l * 3791);
haftmann@26265
    35
     z =  minus_shift 2147483562 v' (w' + 1) + 1
haftmann@26265
    36
   in (z, (v', w')))"
haftmann@26265
    37
haftmann@29823
    38
definition split_seed :: "seed \<Rightarrow> seed \<times> seed" where
haftmann@26038
    39
  "split_seed s = (let
haftmann@26038
    40
     (v, w) = s;
haftmann@26038
    41
     (v', w') = snd (next s);
haftmann@26265
    42
     v'' = inc_shift 2147483562 v;
haftmann@33236
    43
     w'' = inc_shift 2147483398 w
haftmann@33236
    44
   in ((v'', w'), (v', w'')))"
haftmann@26038
    45
haftmann@26038
    46
haftmann@26265
    47
subsection {* Base selectors *}
haftmann@22528
    48
haftmann@31205
    49
fun iterate :: "code_numeral \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a" where
haftmann@30495
    50
  "iterate k f x = (if k = 0 then Pair x else f x o\<rightarrow> iterate (k - 1) f)"
haftmann@22528
    51
haftmann@31205
    52
definition range :: "code_numeral \<Rightarrow> seed \<Rightarrow> code_numeral \<times> seed" where
haftmann@30495
    53
  "range k = iterate (log 2147483561 k)
haftmann@30495
    54
      (\<lambda>l. next o\<rightarrow> (\<lambda>v. Pair (v + l * 2147483561))) 1
haftmann@29823
    55
    o\<rightarrow> (\<lambda>v. Pair (v mod k))"
haftmann@26265
    56
haftmann@26265
    57
lemma range:
haftmann@30495
    58
  "k > 0 \<Longrightarrow> fst (range k s) < k"
haftmann@30495
    59
  by (simp add: range_def scomp_apply split_def del: log.simps iterate.simps)
haftmann@26038
    60
haftmann@29823
    61
definition select :: "'a list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" where
haftmann@31205
    62
  "select xs = range (Code_Numeral.of_nat (length xs))
haftmann@31205
    63
    o\<rightarrow> (\<lambda>k. Pair (nth xs (Code_Numeral.nat_of k)))"
haftmann@29823
    64
     
haftmann@26265
    65
lemma select:
haftmann@26265
    66
  assumes "xs \<noteq> []"
haftmann@26265
    67
  shows "fst (select xs s) \<in> set xs"
haftmann@26265
    68
proof -
haftmann@31205
    69
  from assms have "Code_Numeral.of_nat (length xs) > 0" by simp
haftmann@26265
    70
  with range have
haftmann@31205
    71
    "fst (range (Code_Numeral.of_nat (length xs)) s) < Code_Numeral.of_nat (length xs)" by best
haftmann@26265
    72
  then have
haftmann@31205
    73
    "Code_Numeral.nat_of (fst (range (Code_Numeral.of_nat (length xs)) s)) < length xs" by simp
haftmann@26265
    74
  then show ?thesis
haftmann@29823
    75
    by (simp add: scomp_apply split_beta select_def)
haftmann@26265
    76
qed
haftmann@22528
    77
haftmann@31205
    78
primrec pick :: "(code_numeral \<times> 'a) list \<Rightarrow> code_numeral \<Rightarrow> 'a" where
haftmann@31180
    79
  "pick (x # xs) i = (if i < fst x then snd x else pick xs (i - fst x))"
haftmann@31180
    80
haftmann@31180
    81
lemma pick_member:
haftmann@31180
    82
  "i < listsum (map fst xs) \<Longrightarrow> pick xs i \<in> set (map snd xs)"
haftmann@31180
    83
  by (induct xs arbitrary: i) simp_all
haftmann@31180
    84
haftmann@31180
    85
lemma pick_drop_zero:
haftmann@31180
    86
  "pick (filter (\<lambda>(k, _). k > 0) xs) = pick xs"
haftmann@31180
    87
  by (induct xs) (auto simp add: expand_fun_eq)
haftmann@31180
    88
haftmann@31203
    89
lemma pick_same:
haftmann@31205
    90
  "l < length xs \<Longrightarrow> Random.pick (map (Pair 1) xs) (Code_Numeral.of_nat l) = nth xs l"
haftmann@31203
    91
proof (induct xs arbitrary: l)
haftmann@31203
    92
  case Nil then show ?case by simp
haftmann@31203
    93
next
haftmann@31203
    94
  case (Cons x xs) then show ?case by (cases l) simp_all
haftmann@31203
    95
qed
haftmann@31203
    96
haftmann@31205
    97
definition select_weight :: "(code_numeral \<times> 'a) list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" where
haftmann@31180
    98
  "select_weight xs = range (listsum (map fst xs))
haftmann@31180
    99
   o\<rightarrow> (\<lambda>k. Pair (pick xs k))"
haftmann@31180
   100
haftmann@31180
   101
lemma select_weight_member:
haftmann@31180
   102
  assumes "0 < listsum (map fst xs)"
haftmann@31180
   103
  shows "fst (select_weight xs s) \<in> set (map snd xs)"
haftmann@31180
   104
proof -
haftmann@31180
   105
  from range assms
haftmann@31180
   106
    have "fst (range (listsum (map fst xs)) s) < listsum (map fst xs)" .
haftmann@31180
   107
  with pick_member
haftmann@31180
   108
    have "pick xs (fst (range (listsum (map fst xs)) s)) \<in> set (map snd xs)" .
haftmann@31180
   109
  then show ?thesis by (simp add: select_weight_def scomp_def split_def) 
haftmann@31180
   110
qed
haftmann@31180
   111
haftmann@31268
   112
lemma select_weight_cons_zero:
haftmann@31268
   113
  "select_weight ((0, x) # xs) = select_weight xs"
haftmann@31268
   114
  by (simp add: select_weight_def)
haftmann@31268
   115
haftmann@31203
   116
lemma select_weigth_drop_zero:
haftmann@31261
   117
  "select_weight (filter (\<lambda>(k, _). k > 0) xs) = select_weight xs"
haftmann@31203
   118
proof -
haftmann@31203
   119
  have "listsum (map fst [(k, _)\<leftarrow>xs . 0 < k]) = listsum (map fst xs)"
haftmann@31203
   120
    by (induct xs) auto
haftmann@31203
   121
  then show ?thesis by (simp only: select_weight_def pick_drop_zero)
haftmann@31203
   122
qed
haftmann@31203
   123
haftmann@31203
   124
lemma select_weigth_select:
haftmann@31203
   125
  assumes "xs \<noteq> []"
haftmann@31261
   126
  shows "select_weight (map (Pair 1) xs) = select xs"
haftmann@31203
   127
proof -
haftmann@31261
   128
  have less: "\<And>s. fst (range (Code_Numeral.of_nat (length xs)) s) < Code_Numeral.of_nat (length xs)"
haftmann@31203
   129
    using assms by (intro range) simp
haftmann@31205
   130
  moreover have "listsum (map fst (map (Pair 1) xs)) = Code_Numeral.of_nat (length xs)"
haftmann@31203
   131
    by (induct xs) simp_all
haftmann@31203
   132
  ultimately show ?thesis
haftmann@31203
   133
    by (auto simp add: select_weight_def select_def scomp_def split_def
haftmann@31203
   134
      expand_fun_eq pick_same [symmetric])
haftmann@31203
   135
qed
haftmann@31203
   136
haftmann@26265
   137
haftmann@26265
   138
subsection {* @{text ML} interface *}
haftmann@22528
   139
haftmann@22528
   140
ML {*
haftmann@26265
   141
structure Random_Engine =
haftmann@22528
   142
struct
haftmann@22528
   143
haftmann@26038
   144
type seed = int * int;
haftmann@22528
   145
haftmann@22528
   146
local
haftmann@26038
   147
wenzelm@32740
   148
val seed = Unsynchronized.ref 
haftmann@26265
   149
  (let
haftmann@26265
   150
    val now = Time.toMilliseconds (Time.now ());
haftmann@26038
   151
    val (q, s1) = IntInf.divMod (now, 2147483562);
haftmann@26038
   152
    val s2 = q mod 2147483398;
haftmann@26265
   153
  in (s1 + 1, s2 + 1) end);
haftmann@26265
   154
haftmann@22528
   155
in
haftmann@26038
   156
haftmann@26038
   157
fun run f =
haftmann@26038
   158
  let
haftmann@26265
   159
    val (x, seed') = f (! seed);
haftmann@26038
   160
    val _ = seed := seed'
haftmann@26038
   161
  in x end;
haftmann@26038
   162
haftmann@22528
   163
end;
haftmann@22528
   164
haftmann@22528
   165
end;
haftmann@22528
   166
*}
haftmann@22528
   167
haftmann@31180
   168
hide (open) type seed
haftmann@33236
   169
hide (open) const inc_shift minus_shift log "next" split_seed
haftmann@31636
   170
  iterate range select pick select_weight
haftmann@31180
   171
haftmann@29823
   172
no_notation fcomp (infixl "o>" 60)
haftmann@29823
   173
no_notation scomp (infixl "o\<rightarrow>" 60)
haftmann@29823
   174
haftmann@26038
   175
end
haftmann@28145
   176