src/HOL/Random.thy
 author haftmann Mon, 18 May 2009 15:45:42 +0200 changeset 31196 82ff416d7d66 parent 31186 b458b4ac570f child 31203 5c8fb4fd67e0 permissions -rw-r--r--
hide fact log_def -- should not shadow regular log definition
```
(* Author: Florian Haftmann, TU Muenchen *)

header {* A HOL random engine *}

theory Random
imports Code_Index
begin

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

subsection {* Auxiliary functions *}

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

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

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

subsection {* Random seeds *}

types seed = "index \<times> index"

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

lemma next_not_0:
"fst (next s) \<noteq> 0"
by (cases s) (auto simp add: minus_shift_def Let_def)

primrec seed_invariant :: "seed \<Rightarrow> bool" where
"seed_invariant (v, w) \<longleftrightarrow> 0 < v \<and> v < 9438322952 \<and> 0 < w \<and> True"

lemma if_same: "(if b then f x else f y) = f (if b then x else y)"
by (cases b) simp_all

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;
s'' = (v'', w');
w'' = inc_shift 2147483398 w;
s''' = (v', w'')
in (s'', s'''))"

subsection {* Base selectors *}

fun iterate :: "index \<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 :: "index \<Rightarrow> seed \<Rightarrow> index \<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_Index.of_nat (length xs))
o\<rightarrow> (\<lambda>k. Pair (nth xs (Code_Index.nat_of k)))"

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

primrec pick :: "(index \<times> 'a) list \<Rightarrow> index \<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)

definition select_weight :: "(index \<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

definition select_default :: "index \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> seed \<Rightarrow> 'a \<times> seed" where
[code del]: "select_default k x y = range k
o\<rightarrow> (\<lambda>l. Pair (if l + 1 < k then x else y))"

lemma select_default_zero:
"fst (select_default 0 x y s) = y"
by (simp add: scomp_apply split_beta select_default_def)

lemma select_default_code [code]:
"select_default k x y = (if k = 0
then range 1 o\<rightarrow> (\<lambda>_. Pair y)
else range k o\<rightarrow> (\<lambda>l. Pair (if l + 1 < k then x else y)))"
proof
fix s
have "snd (range (Code_Index.of_nat 0) s) = snd (range (Code_Index.of_nat 1) s)"
by (simp add: range_def scomp_Pair scomp_apply split_beta)
then show "select_default k x y s = (if k = 0
then range 1 o\<rightarrow> (\<lambda>_. Pair y)
else range k o\<rightarrow> (\<lambda>l. Pair (if l + 1 < k then x else y))) s"
by (cases "k = 0") (simp_all add: select_default_def scomp_apply split_beta)
qed

subsection {* @{text ML} interface *}

ML {*
structure Random_Engine =
struct

type seed = int * int;

local

val seed = 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" seed_invariant split_seed
iterate range select pick select_weight select_default
hide (open) fact log_def

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

end

```