(* ID: $Id: Random.thy,v 1.13 2008/04/09 15:46:19 haftmann Exp $ 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 => index => index" where "inc_shift v k = (if v = k then 1 else k + 1)" definition minus_shift :: "index => index => index => index" where "minus_shift r k l = (if k < l then r + k - l else k - l)" function log :: "index => index => index" where "log b i = (if b ≤ 1 ∨ i < b then 1 else 1 + log b (i div b))" by pat_completeness auto termination by (relation "measure (nat_of_index o snd)") (auto simp add: index) subsection {* Random seeds *} types seed = "index × index" primrec "next" :: "seed => index × 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) ≠ 0" apply (cases s) apply (auto simp add: minus_shift_def Let_def) done primrec seed_invariant :: "seed => bool" where "seed_invariant (v, w) <-> 0 < v ∧ v < 9438322952 ∧ 0 < w ∧ 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 => seed × 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 => index => seed => index × 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 => seed => index × 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 => seed => 'a × 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 ≠ []" shows "fst (select xs s) ∈ 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 => 'a => 'a => seed => 'a × 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
lemma next_not_0:
fst (next s) ≠ 0
lemma if_same:
(if b then f x else f y) = f (if b then x else y)
lemma range:
0 < k ==> fst (Random.range k s) < k
lemma select:
xs ≠ [] ==> fst (select xs s) ∈ set xs
lemma select_default_zero:
fst (select_default 0 x y s) = y
lemma select_default_code:
select_default k x y =
(if k = 0 then do uu_ <- Random.range 1;
return y done
else do l <- Random.range k;
return (if l + 1 < k then x else y) done)