(* 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

 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 (Code_Index.nat_of 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: monad_collapse range_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 (Code_Index.of_nat (length xs));

      return (nth xs (Code_Index.nat_of k))

    done)"

 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 (auto simp add: monad_collapse select_def)

 qed

 definition

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

 where

   [code 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: monad_collapse 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: monad_collapse select_default_def 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