(* 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 *}
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 = range_aux (log 2147483561 k) 1
o\<rightarrow> (\<lambda>v. Pair (v mod k))"
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: scomp_apply range_def del: range_aux.simps log.simps)
qed
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
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;
*}
no_notation fcomp (infixl "o>" 60)
no_notation scomp (infixl "o\<rightarrow>" 60)
end