src/HOL/Random.thy
 author wenzelm Thu, 11 Feb 2010 23:00:22 +0100 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;
```
(* Author: Florian Haftmann, TU Muenchen *)

header {* A HOL random engine *}

theory Random
imports Code_Numeral List
begin

notation fcomp (infixl "o>" 60)
notation scomp (infixl "o\<rightarrow>" 60)

subsection {* Auxiliary functions *}

fun log :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
"log b i = (if b \<le> 1 \<or> i < b then 1 else 1 + log b (i div b))"

definition inc_shift :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
"inc_shift v k = (if v = k then 1 else k + 1)"

definition minus_shift :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
"minus_shift r k l = (if k < l then r + k - l else k - l)"

subsection {* Random seeds *}

types seed = "code_numeral \<times> code_numeral"

primrec "next" :: "seed \<Rightarrow> code_numeral \<times> seed" where
"next (v, w) = (let
k =  v div 53668;
v' = minus_shift 2147483563 ((v mod 53668) * 40014) (k * 12211);
l =  w div 52774;
w' = minus_shift 2147483399 ((w mod 52774) * 40692) (l * 3791);
z =  minus_shift 2147483562 v' (w' + 1) + 1
in (z, (v', w')))"

definition split_seed :: "seed \<Rightarrow> seed \<times> seed" where
"split_seed s = (let
(v, w) = s;
(v', w') = snd (next s);
v'' = inc_shift 2147483562 v;
w'' = inc_shift 2147483398 w
in ((v'', w'), (v', w'')))"

subsection {* Base selectors *}

fun iterate :: "code_numeral \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a" where
"iterate k f x = (if k = 0 then Pair x else f x o\<rightarrow> iterate (k - 1) f)"

definition range :: "code_numeral \<Rightarrow> seed \<Rightarrow> code_numeral \<times> seed" where
"range k = iterate (log 2147483561 k)
(\<lambda>l. next o\<rightarrow> (\<lambda>v. Pair (v + l * 2147483561))) 1
o\<rightarrow> (\<lambda>v. Pair (v mod k))"

lemma range:
"k > 0 \<Longrightarrow> fst (range k s) < k"
by (simp add: range_def scomp_apply split_def del: log.simps iterate.simps)

definition select :: "'a list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" where
"select xs = range (Code_Numeral.of_nat (length xs))
o\<rightarrow> (\<lambda>k. Pair (nth xs (Code_Numeral.nat_of k)))"

lemma select:
assumes "xs \<noteq> []"
shows "fst (select xs s) \<in> set xs"
proof -
from assms have "Code_Numeral.of_nat (length xs) > 0" by simp
with range have
"fst (range (Code_Numeral.of_nat (length xs)) s) < Code_Numeral.of_nat (length xs)" by best
then have
"Code_Numeral.nat_of (fst (range (Code_Numeral.of_nat (length xs)) s)) < length xs" by simp
then show ?thesis
by (simp add: scomp_apply split_beta select_def)
qed

primrec pick :: "(code_numeral \<times> 'a) list \<Rightarrow> code_numeral \<Rightarrow> 'a" where
"pick (x # xs) i = (if i < fst x then snd x else pick xs (i - fst x))"

lemma pick_member:
"i < listsum (map fst xs) \<Longrightarrow> pick xs i \<in> set (map snd xs)"
by (induct xs arbitrary: i) simp_all

lemma pick_drop_zero:
"pick (filter (\<lambda>(k, _). k > 0) xs) = pick xs"
by (induct xs) (auto simp add: expand_fun_eq)

lemma pick_same:
"l < length xs \<Longrightarrow> Random.pick (map (Pair 1) xs) (Code_Numeral.of_nat l) = nth xs l"
proof (induct xs arbitrary: l)
case Nil then show ?case by simp
next
case (Cons x xs) then show ?case by (cases l) simp_all
qed

definition select_weight :: "(code_numeral \<times> 'a) list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" where
"select_weight xs = range (listsum (map fst xs))
o\<rightarrow> (\<lambda>k. Pair (pick xs k))"

lemma select_weight_member:
assumes "0 < listsum (map fst xs)"
shows "fst (select_weight xs s) \<in> set (map snd xs)"
proof -
from range assms
have "fst (range (listsum (map fst xs)) s) < listsum (map fst xs)" .
with pick_member
have "pick xs (fst (range (listsum (map fst xs)) s)) \<in> set (map snd xs)" .
then show ?thesis by (simp add: select_weight_def scomp_def split_def)
qed

lemma select_weight_cons_zero:
"select_weight ((0, x) # xs) = select_weight xs"

lemma select_weigth_drop_zero:
"select_weight (filter (\<lambda>(k, _). k > 0) xs) = select_weight xs"
proof -
have "listsum (map fst [(k, _)\<leftarrow>xs . 0 < k]) = listsum (map fst xs)"
by (induct xs) auto
then show ?thesis by (simp only: select_weight_def pick_drop_zero)
qed

lemma select_weigth_select:
assumes "xs \<noteq> []"
shows "select_weight (map (Pair 1) xs) = select xs"
proof -
have less: "\<And>s. fst (range (Code_Numeral.of_nat (length xs)) s) < Code_Numeral.of_nat (length xs)"
using assms by (intro range) simp
moreover have "listsum (map fst (map (Pair 1) xs)) = Code_Numeral.of_nat (length xs)"
by (induct xs) simp_all
ultimately show ?thesis
by (auto simp add: select_weight_def select_def scomp_def split_def
expand_fun_eq pick_same [symmetric])
qed

subsection {* @{text ML} interface *}

ML {*
structure Random_Engine =
struct

type seed = int * int;

local

val seed = Unsynchronized.ref
(let
val now = Time.toMilliseconds (Time.now ());
val (q, s1) = IntInf.divMod (now, 2147483562);
val s2 = q mod 2147483398;
in (s1 + 1, s2 + 1) end);

in

fun run f =
let
val (x, seed') = f (! seed);
val _ = seed := seed'
in x end;

end;

end;
*}

hide (open) type seed
hide (open) const inc_shift minus_shift log "next" split_seed
iterate range select pick select_weight

no_notation fcomp (infixl "o>" 60)
no_notation scomp (infixl "o\<rightarrow>" 60)

end

```