(* 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)
use "Tools/quickcheck_generators.ML"
setup Quickcheck_Generators.setup
subsection {* Code 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
no_notation fcomp (infixl "o>" 60)
no_notation scomp (infixl "o\<rightarrow>" 60)
subsection {* The Random-Predicate Monad *}
fun iter' ::
"'a itself => code_numeral => code_numeral => code_numeral * code_numeral => ('a::random) Predicate.pred"
where
"iter' T nrandom sz seed = (if nrandom = 0 then bot_class.bot else
let ((x, _), seed') = random sz seed
in Predicate.Seq (%u. Predicate.Insert x (iter' T (nrandom - 1) sz seed')))"
definition iter :: "code_numeral => code_numeral => code_numeral * code_numeral => ('a::random) Predicate.pred"
where
"iter nrandom sz seed = iter' (TYPE('a)) nrandom sz seed"
lemma [code]:
"iter nrandom sz seed = (if nrandom = 0 then bot_class.bot else
let ((x, _), seed') = random sz seed
in Predicate.Seq (%u. Predicate.Insert x (iter (nrandom - 1) sz seed')))"
unfolding iter_def iter'.simps[of _ nrandom] ..
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 (semilattice_sup_class.sup P1 P2, s''))"
definition if_randompred :: "bool \<Rightarrow> unit randompred"
where
"if_randompred b = (if b then single () else empty)"
definition iterate_upto :: "(code_numeral => 'a) => code_numeral => code_numeral => 'a randompred"
where
"iterate_upto f n m = Pair (Code_Numeral.iterate_upto f n m)"
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)"
hide_fact (open) iter'.simps iter_def empty_def single_def bind_def union_def if_randompred_def iterate_upto_def not_randompred_def Random_def map_def
hide_type (open) randompred
hide_const (open) random collapse beyond random_fun_aux random_fun_lift
iter' iter empty single bind union if_randompred iterate_upto not_randompred Random map
end