(* Author: Florian Haftmann, TU Muenchen *)

header {* A simple counterexample generator *}

theory Quickcheck

imports Random Code_Evaluation

uses ("Tools/quickcheck_generators.ML")

begin

notation fcomp (infixl "o>" 60)

notation scomp (infixl "o\<rightarrow>" 60)

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

class random = typerep +

  fixes random :: "code_numeral \<Rightarrow> Random.seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> Random.seed"

subsection {* Fundamental and numeric types*}

instantiation bool :: random

begin

definition

  "random i = Random.range 2 o\<rightarrow>

    (\<lambda>k. Pair (if k = 0 then Code_Evaluation.valtermify False else Code_Evaluation.valtermify True))"

instance ..

end

instantiation itself :: (typerep) random

begin

definition random_itself :: "code_numeral \<Rightarrow> Random.seed \<Rightarrow> ('a itself \<times> (unit \<Rightarrow> term)) \<times> Random.seed" where

  "random_itself _ = Pair (Code_Evaluation.valtermify TYPE('a))"

instance ..

end

instantiation char :: random

begin

definition

  "random _ = Random.select chars o\<rightarrow> (\<lambda>c. Pair (c, \<lambda>u. Code_Evaluation.term_of c))"

instance ..

end

instantiation String.literal :: random

begin

definition 

  "random _ = Pair (STR '''', \<lambda>u. Code_Evaluation.term_of (STR ''''))"

instance ..

end

instantiation nat :: random

begin

definition random_nat :: "code_numeral \<Rightarrow> Random.seed \<Rightarrow> (nat \<times> (unit \<Rightarrow> Code_Evaluation.term)) \<times> Random.seed" where

  "random_nat i = Random.range (i + 1) o\<rightarrow> (\<lambda>k. Pair (

     let n = Code_Numeral.nat_of k

     in (n, \<lambda>_. Code_Evaluation.term_of n)))"

instance ..

end

instantiation int :: random

begin

definition

  "random i = Random.range (2 * i + 1) o\<rightarrow> (\<lambda>k. Pair (

     let j = (if k \<ge> i then Code_Numeral.int_of (k - i) else - Code_Numeral.int_of (i - k))

     in (j, \<lambda>_. Code_Evaluation.term_of j)))"

instance ..

end

subsection {* Complex generators *}

text {* Towards @{typ "'a \<Rightarrow> 'b"} *}

axiomatization random_fun_aux :: "typerep \<Rightarrow> typerep \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> term)

  \<Rightarrow> (Random.seed \<Rightarrow> ('b \<times> (unit \<Rightarrow> term)) \<times> Random.seed) \<Rightarrow> (Random.seed \<Rightarrow> Random.seed \<times> Random.seed)

  \<Rightarrow> Random.seed \<Rightarrow> (('a \<Rightarrow> 'b) \<times> (unit \<Rightarrow> term)) \<times> Random.seed"

definition random_fun_lift :: "(Random.seed \<Rightarrow> ('b \<times> (unit \<Rightarrow> term)) \<times> Random.seed)

  \<Rightarrow> Random.seed \<Rightarrow> (('a\<Colon>term_of \<Rightarrow> 'b\<Colon>typerep) \<times> (unit \<Rightarrow> term)) \<times> Random.seed" where

  "random_fun_lift f = random_fun_aux TYPEREP('a) TYPEREP('b) (op =) Code_Evaluation.term_of f Random.split_seed"

instantiation "fun" :: ("{eq, term_of}", random) random

begin

definition random_fun :: "code_numeral \<Rightarrow> Random.seed \<Rightarrow> (('a \<Rightarrow> 'b) \<times> (unit \<Rightarrow> term)) \<times> Random.seed" where

  "random i = random_fun_lift (random i)"

instance ..

end

text {* Towards type copies and datatypes *}

definition collapse :: "('a \<Rightarrow> ('a \<Rightarrow> 'b \<times> 'a) \<times> 'a) \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'a" where

  "collapse f = (f o\<rightarrow> id)"

definition beyond :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where

  "beyond k l = (if l > k then l else 0)"

lemma beyond_zero:

  "beyond k 0 = 0"

  by (simp add: beyond_def)

lemma random_aux_rec:

  fixes random_aux :: "code_numeral \<Rightarrow> 'a"

  assumes "random_aux 0 = rhs 0"

    and "\<And>k. random_aux (Suc_code_numeral k) = rhs (Suc_code_numeral k)"

  shows "random_aux k = rhs k"

  using assms by (rule code_numeral.induct)

setup {* Quickcheck.setup *}

subsection {* the Random-Predicate Monad *} 

types 'a randompred = "Random.seed \<Rightarrow> ('a Predicate.pred \<times> Random.seed)"

definition empty :: "'a randompred"

  where "empty = Pair (bot_class.bot)"

definition single :: "'a => 'a randompred"

  where "single x = Pair (Predicate.single x)"

definition bind :: "'a randompred \<Rightarrow> ('a \<Rightarrow> 'b randompred) \<Rightarrow> 'b randompred"

  where

    "bind R f = (\<lambda>s. let

       (P, s') = R s;

       (s1, s2) = Random.split_seed s'

     in (Predicate.bind P (%a. fst (f a s1)), s2))"

definition union :: "'a randompred \<Rightarrow> 'a randompred \<Rightarrow> 'a randompred"

where

  "union R1 R2 = (\<lambda>s. let

     (P1, s') = R1 s; (P2, s'') = R2 s'

   in (upper_semilattice_class.sup P1 P2, s''))"

definition if_randompred :: "bool \<Rightarrow> unit randompred"

where

  "if_randompred b = (if b then single () else empty)"

definition not_randompred :: "unit randompred \<Rightarrow> unit randompred"

where

  "not_randompred P = (\<lambda>s. let

     (P', s') = P s

   in if Predicate.eval P' () then (Orderings.bot, s') else (Predicate.single (), s'))"

definition Random :: "(Random.seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> Random.seed) \<Rightarrow> 'a randompred"

  where "Random g = scomp g (Pair o (Predicate.single o fst))"

definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a randompred \<Rightarrow> 'b randompred)"

  where "map f P = bind P (single o f)"

subsection {* Code setup *}

use "Tools/quickcheck_generators.ML"

setup {* Quickcheck_Generators.setup *}

code_const random_fun_aux (Quickcheck "Quickcheck'_Generators.random'_fun")

  -- {* With enough criminal energy this can be abused to derive @{prop False};

  for this reason we use a distinguished target @{text Quickcheck}

  not spoiling the regular trusted code generation *}

code_reserved Quickcheck Quickcheck_Generators

hide (open) fact empty_def single_def bind_def union_def if_randompred_def not_randompred_def Random_def map_def

hide (open) type randompred

hide (open) const random collapse beyond random_fun_aux random_fun_lift

  empty single bind union if_randompred not_randompred Random map

no_notation fcomp (infixl "o>" 60)

no_notation scomp (infixl "o\<rightarrow>" 60)

end