src/HOL/ex/Random.thy
 changeset 26265 4b63b9e9b10d parent 26261 b6a103ace4db child 26589 43cb72871897
--- a/src/HOL/ex/Random.thy	Wed Mar 12 19:38:13 2008 +0100
+++ b/src/HOL/ex/Random.thy	Wed Mar 12 19:38:14 2008 +0100
@@ -2,30 +2,75 @@
Author:     Florian Haftmann, TU Muenchen
*)

-header {* A simple term generator *}
+header {* A HOL random engine *}

theory Random
begin

-subsection {* The random engine *}
+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)"
+
+function
+  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))"
+by pat_completeness auto
+termination
+  by (relation "measure (nat_of_index o snd)")
+
+
+subsection {* Random seeds *}

types seed = "index \<times> index"

-definition
+primrec
"next" :: "seed \<Rightarrow> index \<times> seed"
where
-  "next s = (let
-     (v, w) = s;
-     k = v div 53668;
-     v' = 40014 * (v - k * 53668) - k * 12211;
-     v'' = (if v' < 0 then v' + 2147483563 else v');
-     l = w div 52774;
-     w' = 40692 * (w - l * 52774) - l * 3791;
-     w'' = (if w' < 0 then w' + 2147483399 else w');
-     z = v'' - w'';
-     z' = (if z < 1 then z + 2147483562 else z)
-   in (z',  (v'', w'')))"
+  "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
+
+(*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: 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 \<Rightarrow> seed \<times> seed"
@@ -33,27 +78,14 @@
"split_seed s = (let
(v, w) = s;
(v', w') = snd (next s);
-     v'' = (if v = 2147483562 then 1 else v + 1);
+     v'' = inc_shift 2147483562 v;
s'' = (v'', w');
-     w'' = (if w = 2147483398 then 1 else w + 1);
+     w'' = inc_shift 2147483398 w;
s''' = (v', w'')
in (s'', s'''))"

-text {* Base selectors *}

-primrec
-  pick :: "(index \<times> 'a) list \<Rightarrow> index \<Rightarrow> 'a"
-where
-  "pick (x#xs) n = (if n < fst x then snd x else pick xs (n - fst x))"
-
-function
-  iLogBase :: "index \<Rightarrow> index \<Rightarrow> index"
-where
-  "iLogBase b i = (if b \<le> 1 \<or> i < b then 1 else 1 + iLogBase b (i div b))"
-by pat_completeness auto
-termination
-  by (relation "measure (nat_of_index o snd)")
+subsection {* Base selectors *}

function
range_aux :: "index \<Rightarrow> index \<Rightarrow> seed \<Rightarrow> index \<times> seed"
@@ -69,122 +101,94 @@
definition
range :: "index \<Rightarrow> seed \<Rightarrow> index \<times> seed"
where
-  "range k s = (let
-    b = 2147483561;
-    n = iLogBase b k;
-    (v, s') = range_aux n 1 s
-  in (v mod k, s'))"
+  "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 mbind_def split_def del: range_aux.simps log.simps)
+qed

definition
select :: "'a list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed"
where
-  [simp]: "select xs s = (let
-    (k, s') = range (index_of_nat (length xs)) s
-  in (nth xs (nat_of_index k), s'))"
+  "select xs = (do
+     k \<leftarrow> range (index_of_nat (length xs));
+     return (nth xs (nat_of_index k))
+   done)"
+
+lemma select:
+  assumes "xs \<noteq> []"
+  shows "fst (select xs s) \<in> 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 mbind_def split_def)
+qed

definition
-  select_weight :: "(index \<times> 'a) list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed"
+  select_default :: "index \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> seed \<Rightarrow> 'a \<times> seed"
where
-  [simp]: "select_weight xs s = (let
-    (k, s') = range (foldr (op + \<circ> fst) xs 0) s
-  in (pick xs k, s'))"
+  [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 mbind_def split_def select_default_def)

-(*lemma
-  "select (x#xs) s = select_weight (map (Pair 1) (x#xs)) s"
-proof (induct xs)
-  case Nil show ?case apply (auto simp add: Let_def split_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
-  have map_fst_Pair: "!!xs y. map fst (map (Pair y) xs) = replicate (length xs) y"
-  proof -
-    fix xs
-    fix y
-    show "map fst (map (Pair y) xs) = replicate (length xs) y"
-      by (induct xs) simp_all
-  qed
-  have pick_nth: "!!xs n. n < length xs \<Longrightarrow> pick (map (Pair 1) xs) n = nth xs n"
-  proof -
-    fix xs
-    fix n
-    assume "n < length xs"
-    then show "pick (map (Pair 1) xs) n = nth xs n"
-    proof (induct xs arbitrary: n)
-      case Nil then show ?case by simp
-    next
-      case (Cons x xs) show ?case
-      proof (cases n)
-        case 0 then show ?thesis by simp
-      next
-        case (Suc _)
-    from Cons have "n < length (x # xs)" by auto
-        then have "n < Suc (length xs)" by simp
-        with Suc have "n - 1 < Suc (length xs) - 1" by auto
-        with Cons have "pick (map (Pair (1\<Colon>nat)) xs) (n - 1) = xs ! (n - 1)" by auto
-        with Suc show ?thesis by auto
-      qed
-    qed
-  qed
-  have sum_length: "!!xs. foldl (op +) 0 (map fst (map (Pair 1) xs)) = length xs"
-  proof -
-    have replicate_append:
-      "!!x xs y. replicate (length (x # xs)) y = replicate (length xs) y @ [y]"
-    fix xs
-    show "foldl (op +) 0 (map fst (map (Pair 1) xs)) = length xs"
-    unfolding map_fst_Pair proof (induct xs)
-      case Nil show ?case by simp
-    next
-      case (Cons x xs) then show ?case unfolding replicate_append by simp
-    qed
-  qed
-  have pick_nth_random:
-    "!!x xs s. pick (map (Pair 1) (x#xs)) (fst (random (length (x#xs)) s)) = nth (x#xs) (fst (random (length (x#xs)) s))"
-  proof -
-    fix s
-    fix x
-    fix xs
-    have bound: "fst (random (length (x#xs)) s) < length (x#xs)" by (rule random_bound) simp
-    from pick_nth [OF bound] show
-      "pick (map (Pair 1) (x#xs)) (fst (random (length (x#xs)) s)) = nth (x#xs) (fst (random (length (x#xs)) s))" .
-  qed
-  have pick_nth_random_do:
-    "!!x xs s. (do n \<leftarrow> random (length (x#xs)); return (pick (map (Pair 1) (x#xs)) n) done) s =
-      (do n \<leftarrow> random (length (x#xs)); return (nth (x#xs) n) done) s"
-  unfolding monad_collapse split_def unfolding pick_nth_random ..
-  case (Cons x xs) then show ?case
-    unfolding select_weight_def sum_length pick_nth_random_do
-    by simp
-qed*)
+  case True then show ?thesis
+    by (simp add: run_def mbind_def split_def select_default_def expand_fun_eq range_def)
+qed

-text {* The @{text ML} interface *}
+
+subsection {* @{text ML} interface *}

ML {*
-structure Quickcheck =
+structure Random_Engine =
struct

type seed = int * int;

local

-val seed = ref (0, 0);
-
-fun init () =
-  let
-    val now = Time.toNanoseconds (Time.now ());
+val seed = ref
+  (let
+    val now = Time.toMilliseconds (Time.now ());
val (q, s1) = IntInf.divMod (now, 2147483562);
val s2 = q mod 2147483398;
-  in seed := (s1 + 1, s2 + 1) end;
-
+  in (s1 + 1, s2 + 1) end);
+
in

-val seed = seed; (* FIXME *)
-
fun run f =
let
-    val (x, seed') = f (!seed);
+    val (x, seed') = f (! seed);
val _ = seed := seed'
-    val _ = if fst (! seed) = 0 orelse snd (! seed) = 0 then
-      (warning "broken random seed"; init ())
-      else ()
in x end;

end;
@@ -192,325 +196,4 @@
end;
*}

-
-subsection {* The @{text random} class *}
-
-class random =
-  fixes random :: "index \<Rightarrow> seed \<Rightarrow> 'a\<Colon>{} \<times> seed"
-
-instantiation itself :: (type) random
-begin
-
-definition
-  "random _ = return TYPE('a)"
-
-instance ..
-
end
-
-lemma random_aux_if:
-  fixes random' :: "index \<Rightarrow> index \<Rightarrow> seed \<Rightarrow> 'a \<times> seed"
-  assumes "random' 0 j = undefined"
-    and "\<And>i. random' (Suc_index i) j = rhs2 i"
-  shows "random' i j s = (if i = 0 then undefined else rhs2 (i - 1) s)"
-  by (cases i rule: index.exhaust) (insert assms, simp_all add: undefined_fun)
-
-setup {*
-let
-  fun random_inst tyco thy =
-    let
-      val { descr, index, ... } = DatatypePackage.the_datatype thy tyco;
-      val (raw_vs, _) = DatatypePackage.the_datatype_spec thy tyco;
-      val vs = (map o apsnd)
-        (curry (Sorts.inter_sort (Sign.classes_of thy)) @{sort random}) raw_vs;
-      val ty = Type (tyco, map TFree vs);
-      val typ_of = DatatypeAux.typ_of_dtyp descr vs;
-      val SOME (_, _, constrs) = AList.lookup (op =) descr index;
-      val randomN = NameSpace.base @{const_name random};
-      val random_aux_name = randomN ^ "_" ^ Class.type_name tyco ^ "'";
-      fun lift_ty ty = StateMonad.liftT ty @{typ seed};
-      val ty_aux = @{typ index} --> @{typ index} --> lift_ty ty;
-      fun random ty =
-        Const (@{const_name random}, @{typ index} --> lift_ty ty);
-      val random_aux = Free (random_aux_name, ty_aux);
-      fun add_cons_arg dty (is_rec, t) =
-        let
-          val ty' = typ_of dty;
-          val random' = if can DatatypeAux.dest_DtRec dty
-            then random_aux \$ @{term "i\<Colon>index"} \$ @{term "j\<Colon>index"}
-            else random ty' \$ @{term "j\<Colon>index"}
-          val is_rec' = is_rec orelse DatatypeAux.is_rec_type dty;
-          val t' = StateMonad.mbind ty' ty @{typ seed} random' (Abs ("", ty', t))
-        in (is_rec', t') end;
-      fun mk_cons_t (c, dtys) =
-        let
-          val ty' = map typ_of dtys ---> ty;
-          val t = StateMonad.return ty @{typ seed} (list_comb (Const (c, ty'),
-            map Bound (length dtys - 1 downto 0)));
-          val (is_rec, t') = fold_rev add_cons_arg dtys (false, t);
-        in (is_rec, StateMonad.run ty @{typ seed} t') end;
-      fun check_empty [] = NONE
-        | check_empty xs = SOME xs;
-      fun bundle_cons_ts cons_ts =
-        let
-          val ts = map snd cons_ts;
-          val t = HOLogic.mk_list (lift_ty ty) ts;
-          val t' = Const (@{const_name select}, HOLogic.listT (lift_ty ty) --> lift_ty (lift_ty ty)) \$ t;
-          val t'' = Const (@{const_name collapse}, lift_ty (lift_ty ty) --> lift_ty ty) \$ t';
-        in t'' end;
-      fun bundle_conss (some_rec_t, nonrec_t) =
-        let
-          val rec' = case some_rec_t
-           of SOME rec_t => SOME (HOLogic.mk_prod (@{term "i\<Colon>index"}, rec_t))
-            | NONE => NONE;
-          val nonrec' = HOLogic.mk_prod (@{term "1\<Colon>index"}, nonrec_t);
-          val i_ty = HOLogic.mk_prodT (@{typ index}, lift_ty ty);
-          val t = HOLogic.mk_list i_ty (the_list rec' @ single nonrec');
-          val t' = Const (@{const_name select_weight}, HOLogic.listT i_ty --> lift_ty (lift_ty ty)) \$ t;
-          val t'' = Const (@{const_name collapse}, lift_ty (lift_ty ty) --> lift_ty ty) \$ t';
-        in t'' end;
-      val random_rhs = constrs
-        |> map mk_cons_t
-        |> List.partition fst
-        |> apfst (Option.map bundle_cons_ts o check_empty)
-        |> apsnd bundle_cons_ts
-        |> bundle_conss;
-      val random_aux_undef = (HOLogic.mk_Trueprop o HOLogic.mk_eq)
-        (random_aux \$ @{term "0\<Colon>index"} \$ @{term "j\<Colon>index"}, Const (@{const_name undefined}, lift_ty ty))
-      val random_aux_eq = (HOLogic.mk_Trueprop o HOLogic.mk_eq)
-        (random_aux \$ @{term "Suc_index i"} \$ @{term "j\<Colon>index"}, random_rhs);
-      val random_eq = (HOLogic.mk_Trueprop o HOLogic.mk_eq) (Const (@{const_name random},
-        @{typ index} --> lift_ty ty) \$ @{term "i\<Colon>index"},
-          random_aux \$ @{term "i\<Colon>index"} \$ @{term "i\<Colon>index"});
-      val del_func = Attrib.internal (fn _ => Thm.declaration_attribute
-        (fn thm => Context.mapping (Code.del_func thm) I));
-      fun add_code simps lthy =
-        let
-          val thy = ProofContext.theory_of lthy;
-          val thm = @{thm random_aux_if}
-            |> Drule.instantiate' [SOME (Thm.ctyp_of thy ty)] [SOME (Thm.cterm_of thy random_aux)]
-            |> (fn thm => thm OF simps)
-            |> singleton (ProofContext.export lthy (ProofContext.init thy))
-        in
-          lthy
-          |> LocalTheory.theory (PureThy.note Thm.internalK (random_aux_name ^ "_code", thm)
-        end;
-    in
-      thy
-      |> TheoryTarget.instantiation ([tyco], vs, @{sort random})
-      |> PrimrecPackage.add_primrec [(random_aux_name, SOME ty_aux, NoSyn)]
-           [(("", [del_func]), random_aux_undef), (("", [del_func]), random_aux_eq)]
-      |> `(fn lthy => Syntax.check_term lthy random_eq)
-      |-> (fn eq => Specification.definition (NONE, (("", []), eq)))
-      |> snd
-      |> Class.prove_instantiation_instance (K (Class.intro_classes_tac []))
-      |> LocalTheory.exit
-      |> ProofContext.theory_of
-    end;
-  fun add_random_inst [tyco] = (fn thy => case try (random_inst tyco) thy
-       of SOME thy => thy
-        | NONE => (warning ("Failed to generate random elements for type" ^ tyco); thy))
-    | add_random_inst tycos = tap (fn _ => warning
-        ("Will not generated random elements for mutual recursive types " ^ commas (map quote tycos)));
-*}
-
-instantiation int :: random
-begin
-
-definition
-  "random n = (do
-     (b, m) \<leftarrow> random n;
-     return (if b then int m else - int m)
-   done)"
-
-instance ..
-
-end
-
-code_reserved SML Quickcheck
-
-
-subsection {* Quickcheck generator *}
-
-ML {*
-structure Quickcheck =
-struct
-
-open Quickcheck;
-
-val eval_ref : (unit -> int -> int * int -> term list option * (int * int)) option ref = ref NONE;
-
-fun mk_generator_expr prop tys =
-  let
-    val bounds = map_index (fn (i, ty) => (i, ty)) tys;
-    val result = list_comb (prop, map (fn (i, _) => Bound (length tys - i - 1)) bounds);
-    val terms = map (fn (i, ty) => Const (@{const_name Eval.term_of}, ty --> @{typ term}) \$ Bound (length tys - i - 1)) bounds;
-    val check = @{term "If \<Colon> bool \<Rightarrow> term list option \<Rightarrow> term list option \<Rightarrow> term list option"}
-      \$ result \$ @{term "None \<Colon> term list option"} \$ (@{term "Some \<Colon> term list \<Rightarrow> term list option "} \$ HOLogic.mk_list @{typ term} terms);
-    val return = @{term "Pair \<Colon> term list option \<Rightarrow> seed \<Rightarrow> term list option \<times> seed"};
-    fun mk_bindtyp ty = @{typ seed} --> HOLogic.mk_prodT (ty, @{typ seed});
-    fun mk_bindclause (i, ty) t = Const (@{const_name mbind}, mk_bindtyp ty
-      --> (ty --> mk_bindtyp @{typ "term list option"}) --> mk_bindtyp @{typ "term list option"})
-      \$ (Const (@{const_name random}, @{typ index} --> mk_bindtyp ty)
-        \$ Bound i) \$ Abs ("a", ty, t);
-    val t = fold_rev mk_bindclause bounds (return \$ check);
-  in Abs ("n", @{typ index}, t) end;
-
-fun compile_generator_expr thy prop tys =
-  let
-    val f = CodePackage.eval_term ("Quickcheck.eval_ref", eval_ref) thy
-      (mk_generator_expr prop tys) [];
-  in f #> run #> (Option.map o map) (Code.postprocess_term thy) end;
-
-fun VALUE prop tys thy =
-  let
-    val t = mk_generator_expr prop tys;
-    val eq = Logic.mk_equals (Free ("VALUE", fastype_of t), t)
-  in
-    thy
-    |> TheoryTarget.init NONE
-    |> Specification.definition (NONE, (("", []), eq))
-    |> snd
-    |> LocalTheory.exit
-    |> ProofContext.theory_of
-  end;
-
-end
-*}
-
-ML {* val f = Quickcheck.compile_generator_expr @{theory}
-  @{term "\<lambda>(n::nat) (m::nat) (q::nat). n = m + q + 1"} [@{typ nat}, @{typ nat}, @{typ nat}] *}
-
-ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 25 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-
-ML {* val f = Quickcheck.compile_generator_expr @{theory}
-  @{term "\<lambda>(n::int) (m::int) (q::int). n = m + q + 1"} [@{typ int}, @{typ int}, @{typ int}] *}
-
-(*setup {* Quickcheck.VALUE @{term "\<lambda>(n::int) (m::int) (q::int). n = m + q + 1"} [@{typ int}, @{typ int}, @{typ int}] *}
-
-export_code VALUE in SML module_name QuickcheckExample file "~~/../../gen_code/quickcheck.ML"
-
-definition "FOO = (True, Suc 0)"
-
-code_module (test) QuickcheckExample
-  file "~~/../../gen_code/quickcheck'.ML"
-  contains FOO*)
-
-ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 25 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 3 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-
-ML {* val f = Quickcheck.compile_generator_expr @{theory}
-  @{term "\<lambda>(xs\<Colon>nat list) ys. rev (xs @ ys) = rev xs @ rev ys"}
-  [@{typ "nat list"}, @{typ "nat list"}] *}
-
-ML {* f 15 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 25 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 8 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 8 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 8 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 88 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-
-subsection {* Incremental function generator *}
-
-ML {*
-structure Quickcheck =
-struct
-
-open Quickcheck;
-
-fun random_fun (eq : 'a -> 'a -> bool)
-    (random : seed -> 'b * seed)
-    (random_split : seed -> seed * seed)
-    (seed : seed) =
-  let
-    val (seed', seed'') = random_split seed;
-    val state = ref (seed', []);
-    fun random_fun' x =
-      let
-        val (seed, fun_map) = ! state;
-      in case AList.lookup (uncurry eq) fun_map x
-       of SOME y => y
-        | NONE => let
-              val (y, seed') = random seed;
-              val _ = state := (seed', (x, y) :: fun_map);
-            in y end
-      end;
-  in (random_fun', seed'') end;
-
-end
-*}
-
-axiomatization
-  random_fun_aux :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> (seed \<Rightarrow> 'b \<times> seed)
-    \<Rightarrow> (seed \<Rightarrow> seed \<times> seed) \<Rightarrow> seed \<Rightarrow> ('a \<Rightarrow> 'b) \<times> seed"
-
-code_const random_fun_aux (SML "Quickcheck.random'_fun")
-
-instantiation "fun" :: (term_of, term_of) term_of
-begin
-
-instance ..
-
-end
-
-code_const "Eval.term_of :: ('a\<Colon>term_of \<Rightarrow> 'b\<Colon>term_of) \<Rightarrow> _"
-  (SML "(fn '_ => Const (\"arbitrary\", dummyT))")
-
-instantiation "fun" :: (eq, "{type, random}") random
-begin
-
-definition
-  "random n = random_fun_aux (op =) (random n) split_seed"
-
-instance ..
-
-end
-
-ML {* val f = Quickcheck.compile_generator_expr @{theory}
-  @{term "\<lambda>f k. int (f k) = k"} [@{typ "int \<Rightarrow> nat"}, @{typ int}] *}
-
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
-
-end