(* ID: $Id$
Author: Florian Haftmann, TU Muenchen
*)
header {* A simple counterexample generator *}
theory Quickcheck
imports Random Eval
begin
subsection {* The @{text random} class *}
class random = rtype +
fixes random :: "index \<Rightarrow> seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> seed"
text {* Type @{typ "'a itself"} *}
instantiation itself :: ("{type, rtype}") random
begin
definition
"random _ = return (TYPE('a), \<lambda>u. Eval.Const (STR ''TYPE'') RTYPE('a))"
instance ..
end
text {* Datatypes *}
lemma random'_if:
fixes random' :: "index \<Rightarrow> index \<Rightarrow> seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<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
exception REC of string;
fun mk_collapse thy ty = Sign.mk_const thy
(@{const_name collapse}, [@{typ seed}, ty]);
fun term_ty ty = HOLogic.mk_prodT (ty, @{typ "unit \<Rightarrow> term"});
fun mk_split thy ty ty' = Sign.mk_const thy
(@{const_name split}, [ty, @{typ "unit \<Rightarrow> term"}, StateMonad.liftT (term_ty ty') @{typ seed}]);
fun mk_scomp_split thy ty ty' t t' =
StateMonad.scomp (term_ty ty) (term_ty ty') @{typ seed} t
(mk_split thy ty ty' $ Abs ("", ty, Abs ("", @{typ "unit \<Rightarrow> term"}, t')))
fun mk_cons thy this_ty (c, args) =
let
val tys = map (fst o fst) args;
val c_ty = tys ---> this_ty;
val c = Const (c, tys ---> this_ty);
val t_indices = map (curry ( op * ) 2) (length tys - 1 downto 0);
val c_indices = map (curry ( op + ) 1) t_indices;
val c_t = list_comb (c, map Bound c_indices);
val t_t = Abs ("", @{typ unit}, Eval.mk_term Free RType.rtype
(list_comb (c, map (fn k => Bound (k + 1)) t_indices))
|> map_aterms (fn t as Bound _ => t $ @{term "()"} | t => t));
val return = StateMonad.return (term_ty this_ty) @{typ seed}
(HOLogic.mk_prod (c_t, t_t));
val t = fold_rev (fn ((ty, _), random) =>
mk_scomp_split thy ty this_ty random)
args return;
val is_rec = exists (snd o fst) args;
in (is_rec, StateMonad.run (term_ty this_ty) @{typ seed} t) end;
fun mk_conss thy ty [] = NONE
| mk_conss thy ty [(_, t)] = SOME t
| mk_conss thy ty ts = SOME (mk_collapse thy (term_ty ty) $
(Sign.mk_const thy (@{const_name select}, [StateMonad.liftT (term_ty ty) @{typ seed}]) $
HOLogic.mk_list (StateMonad.liftT (term_ty ty) @{typ seed}) (map snd ts)));
fun mk_clauses thy ty (tyco, (ts_rec, ts_atom)) =
let
val SOME t_atom = mk_conss thy ty ts_atom;
in case mk_conss thy ty ts_rec
of SOME t_rec => mk_collapse thy (term_ty ty) $
(Sign.mk_const thy (@{const_name select_default}, [StateMonad.liftT (term_ty ty) @{typ seed}]) $
@{term "i\<Colon>index"} $ t_rec $ t_atom)
| NONE => t_atom
end;
fun mk_random_eqs thy vs tycos =
let
val this_ty = Type (hd tycos, map TFree vs);
val this_ty' = StateMonad.liftT (term_ty this_ty) @{typ seed};
val random_name = NameSpace.base @{const_name random};
val random'_name = random_name ^ "_" ^ Class.type_name (hd tycos) ^ "'";
fun random ty = Sign.mk_const thy (@{const_name random}, [ty]);
val random' = Free (random'_name,
@{typ index} --> @{typ index} --> this_ty');
fun atom ty = ((ty, false), random ty $ @{term "j\<Colon>index"});
fun dtyp tyco = ((this_ty, true), random' $ @{term "i\<Colon>index"} $ @{term "j\<Colon>index"});
fun rtyp tyco tys = raise REC
("Will not generate random elements for mutual recursive type " ^ quote (hd tycos));
val rhss = DatatypePackage.construction_interpretation thy
{ atom = atom, dtyp = dtyp, rtyp = rtyp } vs tycos
|> (map o apsnd o map) (mk_cons thy this_ty)
|> (map o apsnd) (List.partition fst)
|> map (mk_clauses thy this_ty)
val eqss = map ((apsnd o map) (HOLogic.mk_Trueprop o HOLogic.mk_eq) o (fn rhs => ((this_ty, random'), [
(random' $ @{term "0\<Colon>index"} $ @{term "j\<Colon>index"}, Const (@{const_name undefined}, this_ty')),
(random' $ @{term "Suc_index i"} $ @{term "j\<Colon>index"}, rhs)
]))) rhss;
in eqss end;
fun random_inst [tyco] thy =
let
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 { descr, index, ... } = DatatypePackage.the_datatype thy tyco;
val ((this_ty, random'), eqs') = singleton (mk_random_eqs thy vs) tyco;
val eq = (HOLogic.mk_Trueprop o HOLogic.mk_eq)
(Sign.mk_const thy (@{const_name random}, [this_ty]) $ @{term "i\<Colon>index"},
random' $ @{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'_if}
|> Drule.instantiate' [SOME (Thm.ctyp_of thy this_ty)] [SOME (Thm.cterm_of thy random')]
|> (fn thm => thm OF simps)
|> singleton (ProofContext.export lthy (ProofContext.init thy))
in
lthy
|> LocalTheory.theory (PureThy.add_thm ((fst (dest_Free random') ^ "_code", thm), [Thm.kind_internal])
#-> Code.add_func)
end;
in
thy
|> TheoryTarget.instantiation ([tyco], vs, @{sort random})
|> PrimrecPackage.add_primrec
[(Name.binding (fst (dest_Free random')), SOME (snd (dest_Free random')), NoSyn)]
(map (fn eq => ((Name.no_binding, [del_func]), eq)) eqs')
|-> add_code
|> `(fn lthy => Syntax.check_term lthy eq)
|-> (fn eq => Specification.definition (NONE, ((Name.no_binding, []), eq)))
|> snd
|> Class.prove_instantiation_instance (K (Class.intro_classes_tac []))
|> LocalTheory.exit
|> ProofContext.theory_of
end
| random_inst tycos thy = raise REC
("Will not generate random elements for mutual recursive type(s) " ^ commas (map quote tycos));
fun add_random_inst tycos thy = random_inst tycos thy
handle REC msg => (warning msg; thy);
in DatatypePackage.interpretation add_random_inst end
*}
text {* Type @{typ int} *}
instantiation int :: random
begin
definition
"random n = (do
(b, _) \<leftarrow> random n;
(m, t) \<leftarrow> random n;
return (if b then (int m, \<lambda>u. Eval.App (Eval.Const (STR ''Int.int'') RTYPE(nat \<Rightarrow> int)) (t ()))
else (- int m, \<lambda>u. Eval.App (Eval.Const (STR ''HOL.uminus_class.uminus'') RTYPE(int \<Rightarrow> int))
(Eval.App (Eval.Const (STR ''Int.int'') RTYPE(nat \<Rightarrow> int)) (t ()))))
done)"
instance ..
end
text {* Type @{typ "'a \<Rightarrow> 'b"} *}
ML {*
structure Random_Engine =
struct
open Random_Engine;
fun random_fun (T1 : typ) (T2 : typ) (eq : 'a -> 'a -> bool) (term_of : 'a -> term)
(random : Random_Engine.seed -> ('b * (unit -> term)) * Random_Engine.seed)
(random_split : Random_Engine.seed -> Random_Engine.seed * Random_Engine.seed)
(seed : Random_Engine.seed) =
let
val (seed', seed'') = random_split seed;
val state = ref (seed', [], Const (@{const_name arbitrary}, T1 --> T2));
val fun_upd = Const (@{const_name fun_upd},
(T1 --> T2) --> T1 --> T2 --> T1 --> T2);
fun random_fun' x =
let
val (seed, fun_map, f_t) = ! state;
in case AList.lookup (uncurry eq) fun_map x
of SOME y => y
| NONE => let
val t1 = term_of x;
val ((y, t2), seed') = random seed;
val fun_map' = (x, y) :: fun_map;
val f_t' = fun_upd $ f_t $ t1 $ t2 ();
val _ = state := (seed', fun_map', f_t');
in y end
end;
fun term_fun' () = #3 (! state);
in ((random_fun', term_fun'), seed'') end;
end
*}
axiomatization
random_fun_aux :: "rtype \<Rightarrow> rtype \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> term)
\<Rightarrow> (seed \<Rightarrow> ('b \<times> (unit \<Rightarrow> term)) \<times> seed) \<Rightarrow> (seed \<Rightarrow> seed \<times> seed)
\<Rightarrow> seed \<Rightarrow> (('a \<Rightarrow> 'b) \<times> (unit \<Rightarrow> term)) \<times> seed"
code_const random_fun_aux (SML "Random'_Engine.random'_fun")
instantiation "fun" :: ("{eq, term_of}", "{type, random}") random
begin
definition random_fun :: "index \<Rightarrow> seed \<Rightarrow> (('a \<Rightarrow> 'b) \<times> (unit \<Rightarrow> term)) \<times> seed" where
"random n = random_fun_aux RTYPE('a) RTYPE('b) (op =) Eval.term_of (random n) split_seed"
instance ..
end
code_reserved SML Random_Engine
subsection {* Quickcheck generator *}
ML {*
structure Quickcheck =
struct
val eval_ref : (unit -> int -> int * int -> term list option * (int * int)) option ref = ref NONE;
fun mk_generator_expr thy prop tys =
let
val bound_max = length tys - 1;
val bounds = map_index (fn (i, ty) =>
(2 * (bound_max - i) + 1, 2 * (bound_max - i), 2 * i, ty)) tys;
val result = list_comb (prop, map (fn (i, _, _, _) => Bound i) bounds);
val terms = HOLogic.mk_list @{typ term} (map (fn (_, i, _, _) => Bound i $ @{term "()"}) 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 "} $ terms);
val return = @{term "Pair \<Colon> term list option \<Rightarrow> seed \<Rightarrow> term list option \<times> seed"};
fun mk_termtyp ty = HOLogic.mk_prodT (ty, @{typ "unit \<Rightarrow> term"});
fun mk_split ty = Sign.mk_const thy
(@{const_name split}, [ty, @{typ "unit \<Rightarrow> term"}, StateMonad.liftT @{typ "term list option"} @{typ seed}]);
fun mk_scomp_split ty t t' =
StateMonad.scomp (mk_termtyp ty) @{typ "term list option"} @{typ seed} t (*FIXME*)
(mk_split ty $ Abs ("", ty, Abs ("", @{typ "unit \<Rightarrow> term"}, t')));
fun mk_bindclause (_, _, i, ty) = mk_scomp_split ty
(Sign.mk_const thy (@{const_name random}, [ty]) $ Bound i)
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 = Code_ML.eval_term ("Quickcheck.eval_ref", eval_ref) thy
(mk_generator_expr thy prop tys) [];
in f #> Random_Engine.run #> (Option.map o map) (Code.postprocess_term thy) end;
fun VALUE prop tys thy =
let
val t = mk_generator_expr thy prop tys;
val eq = Logic.mk_equals (Free ("VALUE", fastype_of t), t)
in
thy
|> TheoryTarget.init NONE
|> Specification.definition (NONE, ((Name.no_binding, []), eq))
|> snd
|> LocalTheory.exit
|> ProofContext.theory_of
end;
end
*}
subsection {* Examples *}
(*export_code "random :: index \<Rightarrow> seed \<Rightarrow> ((_ \<Rightarrow> _) \<times> (unit \<Rightarrow> term)) \<times> seed"
in SML file -*)
(*setup {* Quickcheck.VALUE
@{term "\<lambda>f k. int (f k) = k"} [@{typ "int \<Rightarrow> nat"}, @{typ int}] *}
export_code VALUE in SML module_name QuickcheckExample file "~~/../../gen_code/quickcheck.ML"
use "~~/../../gen_code/quickcheck.ML"
ML {* Random_Engine.run (QuickcheckExample.range 1) *}*)
(*definition "FOO = (True, Suc 0)"
code_module (test) QuickcheckExample
file "~~/../../gen_code/quickcheck'.ML"
contains FOO*)
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) (Syntax.string_of_term @{context}) *}
ML {* f 5 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 25 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 1 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 1 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 2 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 2 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
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) (Syntax.string_of_term @{context}) *}
ML {* f 5 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 25 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 1 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 1 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 2 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 2 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 3 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 4 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 4 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* val f = Quickcheck.compile_generator_expr @{theory}
@{term "\<lambda>(xs\<Colon>int list) ys. rev (xs @ ys) = rev xs @ rev ys"}
[@{typ "int list"}, @{typ "int list"}] *}
ML {* f 15 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 5 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 25 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 1 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 1 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 2 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 2 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 4 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 4 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 5 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 8 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 8 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 8 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 88 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 1 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 2 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 3 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 4 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 5 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 6 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 10 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 10 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* val f = Quickcheck.compile_generator_expr @{theory}
@{term "\<lambda>(s \<Colon> string). s \<noteq> rev s"} [@{typ string}] *}
ML {* f 4 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 4 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 4 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 4 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 10 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 10 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 10 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 10 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 10 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 8 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 8 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
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) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
ML {* f 20 |> (Option.map o map) (Syntax.string_of_term @{context}) *}
end