
haftmann@22528: (*  ID:         $Id$
haftmann@22528:     Author:     Florian Haftmann, TU Muenchen
haftmann@22528: *)
haftmann@22528: 
haftmann@22528: header {* A simple random engine *}
haftmann@22528: 
haftmann@22528: theory Random
haftmann@22528: imports State_Monad
haftmann@22528: begin
haftmann@22528: 
haftmann@22528: fun
haftmann@22528:   pick :: "(nat \<times> 'a) list \<Rightarrow> nat \<Rightarrow> 'a"
haftmann@22528: where
haftmann@22528:   pick_undef: "pick [] n = undefined"
haftmann@22528:   | pick_simp: "pick ((k, v)#xs) n = (if n < k then v else pick xs (n - k))"
haftmann@22528: lemmas [code nofunc] = pick_undef
haftmann@22528: 
haftmann@22528: typedecl randseed
haftmann@22528: 
haftmann@22528: axiomatization
haftmann@22528:   random_shift :: "randseed \<Rightarrow> randseed"
haftmann@22528: 
haftmann@22528: axiomatization
haftmann@22528:   random_seed :: "randseed \<Rightarrow> nat"
haftmann@22528: 
haftmann@22528: definition
haftmann@22528:   random :: "nat \<Rightarrow> randseed \<Rightarrow> nat \<times> randseed" where
haftmann@22528:   "random n s = (random_seed s mod n, random_shift s)"
haftmann@22528: 
haftmann@22528: lemma random_bound:
haftmann@22528:   assumes "0 < n"
haftmann@22528:   shows "fst (random n s) < n"
haftmann@22528: proof -
haftmann@22528:   from prems mod_less_divisor have "!!m .m mod n < n" by auto
haftmann@22528:   then show ?thesis unfolding random_def by simp 
haftmann@22528: qed
haftmann@22528: 
haftmann@22528: lemma random_random_seed [simp]:
haftmann@22528:   "snd (random n s) = random_shift s" unfolding random_def by simp
haftmann@22528: 
haftmann@22528: definition
haftmann@22528:   select :: "'a list \<Rightarrow> randseed \<Rightarrow> 'a \<times> randseed" where
haftmann@22528:   [simp]: "select xs = (do
haftmann@22528:       n \<leftarrow> random (length xs);
haftmann@22528:       return (nth xs n)
haftmann@22528:     done)"
haftmann@22528: definition
haftmann@22528:   select_weight :: "(nat \<times> 'a) list \<Rightarrow> randseed \<Rightarrow> 'a \<times> randseed" where
haftmann@22528:   [simp]: "select_weight xs = (do
haftmann@22528:       n \<leftarrow> random (foldl (op +) 0 (map fst xs));
haftmann@22528:       return (pick xs n)
haftmann@22528:     done)"
haftmann@22528: 
haftmann@22528: lemma
haftmann@22528:   "select (x#xs) s = select_weight (map (Pair 1) (x#xs)) s"
haftmann@22528: proof (induct xs)
haftmann@22528:   case Nil show ?case by (simp add: monad_collapse random_def)
haftmann@22528: next
haftmann@22528:   have map_fst_Pair: "!!xs y. map fst (map (Pair y) xs) = replicate (length xs) y"
haftmann@22528:   proof -
haftmann@22528:     fix xs
haftmann@22528:     fix y
haftmann@22528:     show "map fst (map (Pair y) xs) = replicate (length xs) y"
haftmann@22528:       by (induct xs) simp_all
haftmann@22528:   qed
haftmann@22528:   have pick_nth: "!!xs n. n < length xs \<Longrightarrow> pick (map (Pair 1) xs) n = nth xs n"
haftmann@22528:   proof -
haftmann@22528:     fix xs
haftmann@22528:     fix n
haftmann@22528:     assume "n < length xs"
haftmann@22528:     then show "pick (map (Pair 1) xs) n = nth xs n"
haftmann@22528:     proof (induct xs arbitrary: n)
haftmann@22528:       case Nil then show ?case by simp
haftmann@22528:     next
haftmann@22528:       case (Cons x xs) show ?case
haftmann@22528:       proof (cases n)
haftmann@22528:         case 0 then show ?thesis by simp
haftmann@22528:       next
haftmann@22528:         case (Suc _)
haftmann@22528:     from Cons have "n < length (x # xs)" by auto
haftmann@22528:         then have "n < Suc (length xs)" by simp
haftmann@22528:         with Suc have "n - 1 < Suc (length xs) - 1" by auto
haftmann@22528:         with Cons have "pick (map (Pair (1\<Colon>nat)) xs) (n - 1) = xs ! (n - 1)" by auto
haftmann@22528:         with Suc show ?thesis by auto
haftmann@22528:       qed
haftmann@22528:     qed
haftmann@22528:   qed
haftmann@22528:   have sum_length: "!!xs. foldl (op +) 0 (map fst (map (Pair 1) xs)) = length xs"
haftmann@22528:   proof -
haftmann@22528:     have replicate_append:
haftmann@22528:       "!!x xs y. replicate (length (x # xs)) y = replicate (length xs) y @ [y]"
haftmann@22528:       by (simp add: replicate_app_Cons_same)
haftmann@22528:     fix xs
haftmann@22528:     show "foldl (op +) 0 (map fst (map (Pair 1) xs)) = length xs"
haftmann@22528:     unfolding map_fst_Pair proof (induct xs)
haftmann@22528:       case Nil show ?case by simp
haftmann@22528:     next
haftmann@22528:       case (Cons x xs) then show ?case unfolding replicate_append by simp
haftmann@22528:     qed
haftmann@22528:   qed
haftmann@22528:   have pick_nth_random:
haftmann@22528:     "!!x xs s. pick (map (Pair 1) (x#xs)) (fst (random (length (x#xs)) s)) = nth (x#xs) (fst (random (length (x#xs)) s))"
haftmann@22528:   proof -
haftmann@22528:     fix s
haftmann@22528:     fix x
haftmann@22528:     fix xs
haftmann@22528:     have bound: "fst (random (length (x#xs)) s) < length (x#xs)" by (rule random_bound) simp
haftmann@22528:     from pick_nth [OF bound] show
haftmann@22528:       "pick (map (Pair 1) (x#xs)) (fst (random (length (x#xs)) s)) = nth (x#xs) (fst (random (length (x#xs)) s))" .
haftmann@22528:   qed
haftmann@22528:   have pick_nth_random_do:
haftmann@22528:     "!!x xs s. (do n \<leftarrow> random (length (x#xs)); return (pick (map (Pair 1) (x#xs)) n) done) s =
haftmann@22528:       (do n \<leftarrow> random (length (x#xs)); return (nth (x#xs) n) done) s"
haftmann@22528:   unfolding monad_collapse split_def unfolding pick_nth_random ..
haftmann@22528:   case (Cons x xs) then show ?case
haftmann@22528:     unfolding select_weight_def sum_length pick_nth_random_do
haftmann@22528:     by simp
haftmann@22528: qed
haftmann@22528: 
haftmann@22528: definition
haftmann@22528:   random_int :: "int \<Rightarrow> randseed \<Rightarrow> int * randseed" where
haftmann@22528:   "random_int k = (do n \<leftarrow> random (nat k); return (int n) done)"
haftmann@22528: 
haftmann@22528: lemma random_nat [code]:
haftmann@22528:   "random n = (do k \<leftarrow> random_int (int n); return (nat k) done)"
haftmann@22528: unfolding random_int_def by simp
haftmann@22528: 
haftmann@22528: axiomatization
haftmann@22528:   run_random :: "(randseed \<Rightarrow> 'a * randseed) \<Rightarrow> 'a"
haftmann@22528: 
haftmann@22528: ML {*
haftmann@22528: signature RANDOM =
haftmann@22528: sig
haftmann@22528:   type seed = IntInf.int;
haftmann@22528:   val seed: unit -> seed;
haftmann@22528:   val value: IntInf.int -> seed -> IntInf.int * seed;
haftmann@22528: end;
haftmann@22528: 
haftmann@22528: structure Random : RANDOM =
haftmann@22528: struct
haftmann@22528: 
haftmann@22528: open IntInf;
haftmann@22528: 
haftmann@22528: exception RANDOM;
haftmann@22528: 
haftmann@22528: type seed = int;
haftmann@22528: 
haftmann@22528: local
haftmann@22528:   val a = fromInt 16807;
haftmann@22528:     (*greetings to SML/NJ*)
haftmann@22528:   val m = (the o fromString) "2147483647";
haftmann@22528: in
haftmann@22528:   fun next s = (a * s) mod m;
haftmann@22528: end;
haftmann@22528: 
haftmann@22528: local
haftmann@22528:   val seed_ref = ref (fromInt 1);
haftmann@22528: in
haftmann@22528:   fun seed () =
haftmann@22528:     let
haftmann@22528:       val r = next (!seed_ref)
haftmann@22528:     in
haftmann@22528:       (seed_ref := r; r)
haftmann@22528:     end;
haftmann@22528: end;
haftmann@22528: 
haftmann@22528: fun value h s =
haftmann@22528:   if h < 1 then raise RANDOM
haftmann@22528:   else (s mod (h - 1), seed ());
haftmann@22528: 
haftmann@22528: end;
haftmann@22528: *}
haftmann@22528: 
haftmann@22528: code_type randseed
haftmann@22528:   (SML "Random.seed")
haftmann@22528: 
haftmann@22528: code_const random_int
haftmann@22528:   (SML "Random.value")
haftmann@22528: 
haftmann@22528: code_const run_random
haftmann@22528:   (SML "case _ (Random.seed ()) of (x, '_) => x")
haftmann@22528: 
haftmann@22528: code_gen select select_weight
haftmann@22528:   (SML #)
haftmann@22528: 
haftmann@22528: end




