src/HOL/Random.thy
 author haftmann Tue Oct 27 15:32:21 2009 +0100 (2009-10-27) changeset 33236 ea75c6ea643e parent 32740 9dd0a2f83429 child 35266 07a56610c00b permissions -rw-r--r--
tuned
1 (* Author: Florian Haftmann, TU Muenchen *)
3 header {* A HOL random engine *}
5 theory Random
6 imports Code_Numeral List
7 begin
9 notation fcomp (infixl "o>" 60)
10 notation scomp (infixl "o\<rightarrow>" 60)
13 subsection {* Auxiliary functions *}
15 fun log :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
16   "log b i = (if b \<le> 1 \<or> i < b then 1 else 1 + log b (i div b))"
18 definition inc_shift :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
19   "inc_shift v k = (if v = k then 1 else k + 1)"
21 definition minus_shift :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
22   "minus_shift r k l = (if k < l then r + k - l else k - l)"
25 subsection {* Random seeds *}
27 types seed = "code_numeral \<times> code_numeral"
29 primrec "next" :: "seed \<Rightarrow> code_numeral \<times> seed" where
30   "next (v, w) = (let
31      k =  v div 53668;
32      v' = minus_shift 2147483563 ((v mod 53668) * 40014) (k * 12211);
33      l =  w div 52774;
34      w' = minus_shift 2147483399 ((w mod 52774) * 40692) (l * 3791);
35      z =  minus_shift 2147483562 v' (w' + 1) + 1
36    in (z, (v', w')))"
38 definition split_seed :: "seed \<Rightarrow> seed \<times> seed" where
39   "split_seed s = (let
40      (v, w) = s;
41      (v', w') = snd (next s);
42      v'' = inc_shift 2147483562 v;
43      w'' = inc_shift 2147483398 w
44    in ((v'', w'), (v', w'')))"
47 subsection {* Base selectors *}
49 fun iterate :: "code_numeral \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a" where
50   "iterate k f x = (if k = 0 then Pair x else f x o\<rightarrow> iterate (k - 1) f)"
52 definition range :: "code_numeral \<Rightarrow> seed \<Rightarrow> code_numeral \<times> seed" where
53   "range k = iterate (log 2147483561 k)
54       (\<lambda>l. next o\<rightarrow> (\<lambda>v. Pair (v + l * 2147483561))) 1
55     o\<rightarrow> (\<lambda>v. Pair (v mod k))"
57 lemma range:
58   "k > 0 \<Longrightarrow> fst (range k s) < k"
59   by (simp add: range_def scomp_apply split_def del: log.simps iterate.simps)
61 definition select :: "'a list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" where
62   "select xs = range (Code_Numeral.of_nat (length xs))
63     o\<rightarrow> (\<lambda>k. Pair (nth xs (Code_Numeral.nat_of k)))"
65 lemma select:
66   assumes "xs \<noteq> []"
67   shows "fst (select xs s) \<in> set xs"
68 proof -
69   from assms have "Code_Numeral.of_nat (length xs) > 0" by simp
70   with range have
71     "fst (range (Code_Numeral.of_nat (length xs)) s) < Code_Numeral.of_nat (length xs)" by best
72   then have
73     "Code_Numeral.nat_of (fst (range (Code_Numeral.of_nat (length xs)) s)) < length xs" by simp
74   then show ?thesis
75     by (simp add: scomp_apply split_beta select_def)
76 qed
78 primrec pick :: "(code_numeral \<times> 'a) list \<Rightarrow> code_numeral \<Rightarrow> 'a" where
79   "pick (x # xs) i = (if i < fst x then snd x else pick xs (i - fst x))"
81 lemma pick_member:
82   "i < listsum (map fst xs) \<Longrightarrow> pick xs i \<in> set (map snd xs)"
83   by (induct xs arbitrary: i) simp_all
85 lemma pick_drop_zero:
86   "pick (filter (\<lambda>(k, _). k > 0) xs) = pick xs"
87   by (induct xs) (auto simp add: expand_fun_eq)
89 lemma pick_same:
90   "l < length xs \<Longrightarrow> Random.pick (map (Pair 1) xs) (Code_Numeral.of_nat l) = nth xs l"
91 proof (induct xs arbitrary: l)
92   case Nil then show ?case by simp
93 next
94   case (Cons x xs) then show ?case by (cases l) simp_all
95 qed
97 definition select_weight :: "(code_numeral \<times> 'a) list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" where
98   "select_weight xs = range (listsum (map fst xs))
99    o\<rightarrow> (\<lambda>k. Pair (pick xs k))"
101 lemma select_weight_member:
102   assumes "0 < listsum (map fst xs)"
103   shows "fst (select_weight xs s) \<in> set (map snd xs)"
104 proof -
105   from range assms
106     have "fst (range (listsum (map fst xs)) s) < listsum (map fst xs)" .
107   with pick_member
108     have "pick xs (fst (range (listsum (map fst xs)) s)) \<in> set (map snd xs)" .
109   then show ?thesis by (simp add: select_weight_def scomp_def split_def)
110 qed
112 lemma select_weight_cons_zero:
113   "select_weight ((0, x) # xs) = select_weight xs"
114   by (simp add: select_weight_def)
116 lemma select_weigth_drop_zero:
117   "select_weight (filter (\<lambda>(k, _). k > 0) xs) = select_weight xs"
118 proof -
119   have "listsum (map fst [(k, _)\<leftarrow>xs . 0 < k]) = listsum (map fst xs)"
120     by (induct xs) auto
121   then show ?thesis by (simp only: select_weight_def pick_drop_zero)
122 qed
124 lemma select_weigth_select:
125   assumes "xs \<noteq> []"
126   shows "select_weight (map (Pair 1) xs) = select xs"
127 proof -
128   have less: "\<And>s. fst (range (Code_Numeral.of_nat (length xs)) s) < Code_Numeral.of_nat (length xs)"
129     using assms by (intro range) simp
130   moreover have "listsum (map fst (map (Pair 1) xs)) = Code_Numeral.of_nat (length xs)"
131     by (induct xs) simp_all
132   ultimately show ?thesis
133     by (auto simp add: select_weight_def select_def scomp_def split_def
134       expand_fun_eq pick_same [symmetric])
135 qed
138 subsection {* @{text ML} interface *}
140 ML {*
141 structure Random_Engine =
142 struct
144 type seed = int * int;
146 local
148 val seed = Unsynchronized.ref
149   (let
150     val now = Time.toMilliseconds (Time.now ());
151     val (q, s1) = IntInf.divMod (now, 2147483562);
152     val s2 = q mod 2147483398;
153   in (s1 + 1, s2 + 1) end);
155 in
157 fun run f =
158   let
159     val (x, seed') = f (! seed);
160     val _ = seed := seed'
161   in x end;
163 end;
165 end;
166 *}
168 hide (open) type seed
169 hide (open) const inc_shift minus_shift log "next" split_seed
170   iterate range select pick select_weight
172 no_notation fcomp (infixl "o>" 60)
173 no_notation scomp (infixl "o\<rightarrow>" 60)
175 end