(* 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 *}

 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)"

 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))"

 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 (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"

 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 :: "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_weigth_drop_zero:

   "Random.select_weight (filter (\<lambda>(k, _). k > 0) xs) = Random.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 "Random.select_weight (map (Pair 1) xs) = Random.select xs"

 proof -

   have less: "\<And>s. fst (Random.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

 definition select_default :: "code_numeral \<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_Numeral.of_nat 0) s) = snd (range (Code_Numeral.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

 no_notation fcomp (infixl "o>" 60)

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

 end