(*  ID:         $Id$

    Author:     Florian Haftmann, TU Muenchen

*)

header {* A HOL random engine *}

theory Random

imports State_Monad Code_Index

begin

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"

apply (cases s)

apply (auto simp add: minus_shift_def Let_def)

done

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

(*lemma seed_invariant:

  assumes "seed_invariant (index_of_nat v, index_of_nat w)"

    and "(index_of_nat z, (index_of_nat v', index_of_nat w')) = next (index_of_nat v, index_of_nat w)"

  shows "seed_invariant (index_of_nat v', index_of_nat w')"

using assms

apply (auto simp add: seed_invariant_def)

apply (auto simp add: minus_shift_def Let_def)

apply (simp_all add: if_same cong del: if_cong)

apply safe

unfolding not_less

oops*)

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 *}

function

  range_aux :: "index \<Rightarrow> index \<Rightarrow> seed \<Rightarrow> index \<times> seed"

where

  "range_aux k l s = (if k = 0 then (l, s) else

    let (v, s') = next s

  in range_aux (k - 1) (v + l * 2147483561) s')"

by pat_completeness auto

termination

  by (relation "measure (nat_of_index o fst)")

    (auto simp add: index)

definition

  range :: "index \<Rightarrow> seed \<Rightarrow> index \<times> seed"

where

  "range k = (do

     v \<leftarrow> range_aux (log 2147483561 k) 1;

     return (v mod k)

   done)"

lemma range:

  assumes "k > 0"

  shows "fst (range k s) < k"

proof -

  obtain v w where range_aux:

    "range_aux (log 2147483561 k) 1 s = (v, w)"

    by (cases "range_aux (log 2147483561 k) 1 s")

  with assms show ?thesis

    by (simp add: range_def run_def scomp_def split_def del: range_aux.simps log.simps)

qed

definition

  select :: "'a list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed"

where

  "select xs = (do

     k \<leftarrow> range (index_of_nat (length xs));

     return (nth xs (nat_of_index k))

   done)"

lemma select:

  assumes "xs \<noteq> []"

  shows "fst (select xs s) \<in> set xs"

proof -

  from assms have "index_of_nat (length xs) > 0" by simp

  with range have

    "fst (range (index_of_nat (length xs)) s) < index_of_nat (length xs)" by best

  then have

    "nat_of_index (fst (range (index_of_nat (length xs)) s)) < length xs" by simp

  then show ?thesis

    by (auto simp add: select_def run_def scomp_def split_def)

qed

definition

  select_default :: "index \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> seed \<Rightarrow> 'a \<times> seed"

where

  [code func del]: "select_default k x y = (do

     l \<leftarrow> range k;

     return (if l + 1 < k then x else y)

   done)"

lemma select_default_zero:

  "fst (select_default 0 x y s) = y"

  by (simp add: run_def scomp_def split_def select_default_def)

lemma select_default_code [code]:

  "select_default k x y = (if k = 0 then do

     _ \<leftarrow> range 1;

     return y

   done else do

     l \<leftarrow> range k;

     return (if l + 1 < k then x else y)

   done)"

proof (cases "k = 0")

  case False then show ?thesis by (simp add: select_default_def)

next

  case True then show ?thesis

    by (simp add: run_def scomp_def split_def select_default_def expand_fun_eq range_def)

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;

*}

end