(*  ID:         $Id$

    Author:     Florian Haftmann, TU Muenchen

*)

header {* A simple term generator *}

theory Random

imports State_Monad Code_Index List Eval

begin

subsection {* The random engine *}

types seed = "index \<times> index"

definition

  "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'')))"

definition

  split_seed :: "seed \<Rightarrow> seed \<times> seed"

where

  "split_seed s = (let

     (v, w) = s;

     (v', w') = snd (next s);

     v'' = (if v = 2147483562 then 1 else v + 1);

     s'' = (v'', w');

     w'' = (if w = 2147483398 then 1 else w + 1);

     s''' = (v', w'')

   in (s'', s'''))"

text {* Base selectors *}

primrec

  pick :: "(nat \<times> 'a) list \<Rightarrow> nat \<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)")

    (auto simp add: index)

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 (nat_of_index o fst)")

    (auto simp add: index)

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'))"

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'))"

definition

  select_weight :: "(nat \<times> 'a) list \<Rightarrow> seed \<Rightarrow> 'a \<times> seed"

where

  [simp]: "select_weight xs s = (let

    (k, s') = range (foldr (op + \<circ> index_of_nat \<circ> fst) xs 0) s

  in (pick xs (nat_of_index k), s'))"

(*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) 

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]"

      by (simp add: replicate_app_Cons_same)

    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*)

text {* The @{text ML} interface *}

ML {*

structure Quickcheck =

struct

type seed = int * int;

local

val seed = ref (0, 0);

fun init () =

  let

    val now = Time.toNanoseconds (Time.now ());

    val (q, s1) = IntInf.divMod (now, 2147483562);

    val s2 = q mod 2147483398;

  in seed := (s1 + 1, s2 + 1) end;

in

val seed = seed; (* FIXME *)

fun run f =

  let

    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;

end;

*}

subsection {* The @{text random} class *}

class random = type +

  fixes random :: "index \<Rightarrow> seed \<Rightarrow> 'a \<times> seed"

-- {* FIXME dummy implementations *}

instantiation nat :: random

begin

definition

  "random k = apfst nat_of_index \<circ> range k"

instance ..

end

instantiation bool :: random

begin

definition

  "random _ = apfst (op = 0) \<circ> range 2"

instance ..

end

instantiation unit :: random

begin

definition

  "random _ = Pair ()"

instance ..

end

instantiation "+" :: (random, random) random

begin

definition

  "random n = (do

     k \<leftarrow> next;

     let l = k div 2;

     (if k mod 2 = 0 then do 

       x \<leftarrow> random l;

       return (Inl x)

     done else do

       x \<leftarrow> random l;

       return (Inr x)

     done)

   done)"

instance ..

end

instantiation "*" :: (random, random) random

begin

definition

  "random n = (do

     x \<leftarrow> random n;

     y \<leftarrow> random n;

     return (x, y)

   done)"

instance ..

end

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

instantiation list :: (random) random

begin

primrec

  random_list_aux

where

  "random_list_aux 0 = Pair []"

  | "random_list_aux (Suc n) = (do

       x \<leftarrow> random (index_of_nat (Suc n));

       xs \<leftarrow> random_list_aux n;

       return (x#xs)

     done)"

definition

  [code func del]: "random n = random_list_aux (nat_of_index n)"

lemma [code func]:

  "random n = (if n = 0 then return ([]::'a list)

    else do

       x \<leftarrow> random n;

       xs \<leftarrow> random (n - 1);

       return (x#xs)

     done)"

unfolding random_list_def

by (cases "nat_of_index n", auto)

  (cases n, auto)+

(*fun

  random_list

where

  "random_list n = (if n = 0 then (\<lambda>_. ([]::'a list)) 

    else random n \<guillemotright>=\<^isub>R (\<lambda>x::'a. random_list (n - 1) \<guillemotright>=\<^isub>R (\<lambda>(xs::'a list) _. x#xs)))"*)

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;

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}] *}

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}) *}

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, 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