(* Author: Lukas Bulwahn, TU Muenchen *) header {* A simple counterexample generator performing exhaustive testing *} theory Quickcheck_Exhaustive imports Quickcheck uses ("Tools/Quickcheck/exhaustive_generators.ML") begin subsection {* basic operations for exhaustive generators *} definition orelse :: "'a option => 'a option => 'a option" (infixr "orelse" 55) where [code_unfold]: "x orelse y = (case x of Some x' => Some x' | None => y)" subsection {* exhaustive generator type classes *} class exhaustive = term_of + fixes exhaustive :: "('a \ term list option) \ code_numeral \ term list option" class full_exhaustive = term_of + fixes full_exhaustive :: "('a * (unit => term) \ term list option) \ code_numeral \ term list option" instantiation code_numeral :: full_exhaustive begin function full_exhaustive_code_numeral' :: "(code_numeral * (unit => term) => term list option) => code_numeral => code_numeral => term list option" where "full_exhaustive_code_numeral' f d i = (if d < i then None else (f (i, %_. Code_Evaluation.term_of i)) orelse (full_exhaustive_code_numeral' f d (i + 1)))" by pat_completeness auto termination by (relation "measure (%(_, d, i). Code_Numeral.nat_of (d + 1 - i))") auto definition "full_exhaustive f d = full_exhaustive_code_numeral' f d 0" instance .. end instantiation code_numeral :: exhaustive begin function exhaustive_code_numeral' :: "(code_numeral => term list option) => code_numeral => code_numeral => term list option" where "exhaustive_code_numeral' f d i = (if d < i then None else (f i orelse exhaustive_code_numeral' f d (i + 1)))" by pat_completeness auto termination by (relation "measure (%(_, d, i). Code_Numeral.nat_of (d + 1 - i))") auto definition "exhaustive f d = exhaustive_code_numeral' f d 0" instance .. end instantiation nat :: exhaustive begin definition "exhaustive f d = exhaustive (%x. f (Code_Numeral.nat_of x)) d" instance .. end instantiation nat :: full_exhaustive begin definition "full_exhaustive f d = full_exhaustive (%(x, xt). f (Code_Numeral.nat_of x, %_. Code_Evaluation.term_of (Code_Numeral.nat_of x))) d" instance .. end instantiation int :: exhaustive begin function exhaustive' :: "(int => term list option) => int => int => term list option" where "exhaustive' f d i = (if d < i then None else (f i orelse exhaustive' f d (i + 1)))" by pat_completeness auto termination by (relation "measure (%(_, d, i). nat (d + 1 - i))") auto definition "exhaustive f d = exhaustive' f (Code_Numeral.int_of d) (- (Code_Numeral.int_of d))" instance .. end instantiation int :: full_exhaustive begin function full_exhaustive' :: "(int * (unit => term) => term list option) => int => int => term list option" where "full_exhaustive' f d i = (if d < i then None else (case f (i, %_. Code_Evaluation.term_of i) of Some t => Some t | None => full_exhaustive' f d (i + 1)))" by pat_completeness auto termination by (relation "measure (%(_, d, i). nat (d + 1 - i))") auto definition "full_exhaustive f d = full_exhaustive' f (Code_Numeral.int_of d) (- (Code_Numeral.int_of d))" instance .. end instantiation prod :: (exhaustive, exhaustive) exhaustive begin definition "exhaustive f d = exhaustive (%x. exhaustive (%y. f ((x, y))) d) d" instance .. end instantiation prod :: (full_exhaustive, full_exhaustive) full_exhaustive begin definition "full_exhaustive f d = full_exhaustive (%(x, t1). full_exhaustive (%(y, t2). f ((x, y), %u. let T1 = (Typerep.typerep (TYPE('a))); T2 = (Typerep.typerep (TYPE('b))) in Code_Evaluation.App (Code_Evaluation.App ( Code_Evaluation.Const (STR ''Product_Type.Pair'') (Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Product_Type.prod'') [T1, T2]]])) (t1 ())) (t2 ()))) d) d" instance .. end instantiation "fun" :: ("{equal, exhaustive}", exhaustive) exhaustive begin fun exhaustive_fun' :: "(('a => 'b) => term list option) => code_numeral => code_numeral => term list option" where "exhaustive_fun' f i d = (exhaustive (%b. f (%_. b)) d) orelse (if i > 1 then exhaustive_fun' (%g. exhaustive (%a. exhaustive (%b. f (g(a := b))) d) d) (i - 1) d else None)" definition exhaustive_fun :: "(('a => 'b) => term list option) => code_numeral => term list option" where "exhaustive_fun f d = exhaustive_fun' f d d" instance .. end instantiation "fun" :: ("{equal, full_exhaustive}", full_exhaustive) full_exhaustive begin fun full_exhaustive_fun' :: "(('a => 'b) * (unit => term) => term list option) => code_numeral => code_numeral => term list option" where "full_exhaustive_fun' f i d = (full_exhaustive (%(b, t). f (%_. b, %_. Code_Evaluation.Abs (STR ''x'') (Typerep.typerep TYPE('a)) (t ()))) d) orelse (if i > 1 then full_exhaustive_fun' (%(g, gt). full_exhaustive (%(a, at). full_exhaustive (%(b, bt). f (g(a := b), (%_. let A = (Typerep.typerep (TYPE('a))); B = (Typerep.typerep (TYPE('b))); fun = (%T U. Typerep.Typerep (STR ''fun'') [T, U]) in Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.Const (STR ''Fun.fun_upd'') (fun (fun A B) (fun A (fun B (fun A B))))) (gt ())) (at ())) (bt ())))) d) d) (i - 1) d else None)" definition full_exhaustive_fun :: "(('a => 'b) * (unit => term) => term list option) => code_numeral => term list option" where "full_exhaustive_fun f d = full_exhaustive_fun' f d d" instance .. end subsubsection {* A smarter enumeration scheme for functions over finite datatypes *} class check_all = enum + term_of + fixes check_all :: "('a * (unit \ term) \ term list option) \ term list option" fixes enum_term_of :: "'a itself \ unit \ term list" fun check_all_n_lists :: "(('a :: check_all) list * (unit \ term list) \ term list option) \ code_numeral \ term list option" where "check_all_n_lists f n = (if n = 0 then f ([], (%_. [])) else check_all (%(x, xt). check_all_n_lists (%(xs, xst). f ((x # xs), (%_. (xt () # xst ())))) (n - 1)))" definition mk_map_term :: " (unit \ typerep) \ (unit \ typerep) \ (unit \ term list) \ (unit \ term list) \ unit \ term" where "mk_map_term T1 T2 domm rng = (%_. let T1 = T1 (); T2 = T2 (); update_term = (%g (a, b). Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.Const (STR ''Fun.fun_upd'') (Typerep.Typerep (STR ''fun'') [Typerep.Typerep (STR ''fun'') [T1, T2], Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''fun'') [T1, T2]]]])) g) a) b) in List.foldl update_term (Code_Evaluation.Abs (STR ''x'') T1 (Code_Evaluation.Const (STR ''HOL.undefined'') T2)) (zip (domm ()) (rng ())))" instantiation "fun" :: ("{equal, check_all}", check_all) check_all begin definition "check_all f = (let mk_term = mk_map_term (%_. Typerep.typerep (TYPE('a))) (%_. Typerep.typerep (TYPE('b))) (enum_term_of (TYPE('a))); enum = (Enum.enum :: 'a list) in check_all_n_lists (\(ys, yst). f (the o map_of (zip enum ys), mk_term yst)) (Code_Numeral.of_nat (length enum)))" definition enum_term_of_fun :: "('a => 'b) itself => unit => term list" where "enum_term_of_fun = (%_ _. let enum_term_of_a = enum_term_of (TYPE('a)); mk_term = mk_map_term (%_. Typerep.typerep (TYPE('a))) (%_. Typerep.typerep (TYPE('b))) enum_term_of_a in map (%ys. mk_term (%_. ys) ()) (Enum.n_lists (length (enum_term_of_a ())) (enum_term_of (TYPE('b)) ())))" instance .. end instantiation unit :: check_all begin definition "check_all f = f (Code_Evaluation.valtermify ())" definition enum_term_of_unit :: "unit itself => unit => term list" where "enum_term_of_unit = (%_ _. [Code_Evaluation.term_of ()])" instance .. end instantiation bool :: check_all begin definition "check_all f = (case f (Code_Evaluation.valtermify False) of Some x' \ Some x' | None \ f (Code_Evaluation.valtermify True))" definition enum_term_of_bool :: "bool itself => unit => term list" where "enum_term_of_bool = (%_ _. map Code_Evaluation.term_of (Enum.enum :: bool list))" instance .. end instantiation prod :: (check_all, check_all) check_all begin definition "check_all f = check_all (%(x, t1). check_all (%(y, t2). f ((x, y), %u. let T1 = (Typerep.typerep (TYPE('a))); T2 = (Typerep.typerep (TYPE('b))) in Code_Evaluation.App (Code_Evaluation.App ( Code_Evaluation.Const (STR ''Product_Type.Pair'') (Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Product_Type.prod'') [T1, T2]]])) (t1 ())) (t2 ()))))" definition enum_term_of_prod :: "('a * 'b) itself => unit => term list" where "enum_term_of_prod = (%_ _. map (%(x, y). let T1 = (Typerep.typerep (TYPE('a))); T2 = (Typerep.typerep (TYPE('b))) in Code_Evaluation.App (Code_Evaluation.App ( Code_Evaluation.Const (STR ''Product_Type.Pair'') (Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Product_Type.prod'') [T1, T2]]])) x) y) (Enum.product (enum_term_of (TYPE('a)) ()) (enum_term_of (TYPE('b)) ()))) " instance .. end instantiation sum :: (check_all, check_all) check_all begin definition "check_all f = (case check_all (%(a, t). f (Inl a, %_. let T1 = (Typerep.typerep (TYPE('a))); T2 = (Typerep.typerep (TYPE('b))) in Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inl'') (Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]])) (t ()))) of Some x' => Some x' | None => check_all (%(b, t). f (Inr b, %_. let T1 = (Typerep.typerep (TYPE('a))); T2 = (Typerep.typerep (TYPE('b))) in Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inr'') (Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]])) (t ()))))" definition enum_term_of_sum :: "('a + 'b) itself => unit => term list" where "enum_term_of_sum = (%_ _. let T1 = (Typerep.typerep (TYPE('a))); T2 = (Typerep.typerep (TYPE('b))) in map (Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inl'') (Typerep.Typerep (STR ''fun'') [T1, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]]))) (enum_term_of (TYPE('a)) ()) @ map (Code_Evaluation.App (Code_Evaluation.Const (STR ''Sum_Type.Inr'') (Typerep.Typerep (STR ''fun'') [T2, Typerep.Typerep (STR ''Sum_Type.sum'') [T1, T2]]))) (enum_term_of (TYPE('b)) ()))" instance .. end instantiation nibble :: check_all begin definition "check_all f = f (Code_Evaluation.valtermify Nibble0) orelse f (Code_Evaluation.valtermify Nibble1) orelse f (Code_Evaluation.valtermify Nibble2) orelse f (Code_Evaluation.valtermify Nibble3) orelse f (Code_Evaluation.valtermify Nibble4) orelse f (Code_Evaluation.valtermify Nibble5) orelse f (Code_Evaluation.valtermify Nibble6) orelse f (Code_Evaluation.valtermify Nibble7) orelse f (Code_Evaluation.valtermify Nibble8) orelse f (Code_Evaluation.valtermify Nibble9) orelse f (Code_Evaluation.valtermify NibbleA) orelse f (Code_Evaluation.valtermify NibbleB) orelse f (Code_Evaluation.valtermify NibbleC) orelse f (Code_Evaluation.valtermify NibbleD) orelse f (Code_Evaluation.valtermify NibbleE) orelse f (Code_Evaluation.valtermify NibbleF)" definition enum_term_of_nibble :: "nibble itself => unit => term list" where "enum_term_of_nibble = (%_ _. map Code_Evaluation.term_of (Enum.enum :: nibble list))" instance .. end instantiation char :: check_all begin definition "check_all f = check_all (%(x, t1). check_all (%(y, t2). f (Char x y, %_. Code_Evaluation.App (Code_Evaluation.App (Code_Evaluation.term_of Char) (t1 ())) (t2 ()))))" definition enum_term_of_char :: "char itself => unit => term list" where "enum_term_of_char = (%_ _. map Code_Evaluation.term_of (Enum.enum :: char list))" instance .. end instantiation option :: (check_all) check_all begin definition "check_all f = f (Code_Evaluation.valtermify (None :: 'a option)) orelse check_all (%(x, t). f (Some x, %_. Code_Evaluation.App (Code_Evaluation.Const (STR ''Option.option.Some'') (Typerep.Typerep (STR ''fun'') [Typerep.typerep TYPE('a), Typerep.Typerep (STR ''Option.option'') [Typerep.typerep TYPE('a)]])) (t ())))" definition enum_term_of_option :: "'a option itself => unit => term list" where "enum_term_of_option = (% _ _. (Code_Evaluation.term_of (None :: 'a option)) # (map (Code_Evaluation.App (Code_Evaluation.Const (STR ''Option.option.Some'') (Typerep.Typerep (STR ''fun'') [Typerep.typerep TYPE('a), Typerep.Typerep (STR ''Option.option'') [Typerep.typerep TYPE('a)]]))) (enum_term_of (TYPE('a)) ())))" instance .. end instantiation Enum.finite_1 :: check_all begin definition "check_all f = f (Code_Evaluation.valtermify Enum.finite_1.a\<^isub>1)" definition enum_term_of_finite_1 :: "Enum.finite_1 itself => unit => term list" where "enum_term_of_finite_1 = (%_ _. [Code_Evaluation.term_of Enum.finite_1.a\<^isub>1])" instance .. end instantiation Enum.finite_2 :: check_all begin definition "check_all f = (case f (Code_Evaluation.valtermify Enum.finite_2.a\<^isub>1) of Some x' \ Some x' | None \ f (Code_Evaluation.valtermify Enum.finite_2.a\<^isub>2))" definition enum_term_of_finite_2 :: "Enum.finite_2 itself => unit => term list" where "enum_term_of_finite_2 = (%_ _. map Code_Evaluation.term_of (Enum.enum :: Enum.finite_2 list))" instance .. end instantiation Enum.finite_3 :: check_all begin definition "check_all f = (case f (Code_Evaluation.valtermify Enum.finite_3.a\<^isub>1) of Some x' \ Some x' | None \ (case f (Code_Evaluation.valtermify Enum.finite_3.a\<^isub>2) of Some x' \ Some x' | None \ f (Code_Evaluation.valtermify Enum.finite_3.a\<^isub>3)))" definition enum_term_of_finite_3 :: "Enum.finite_3 itself => unit => term list" where "enum_term_of_finite_3 = (%_ _. map Code_Evaluation.term_of (Enum.enum :: Enum.finite_3 list))" instance .. end subsection {* Bounded universal quantifiers *} class bounded_forall = fixes bounded_forall :: "('a \ bool) \ code_numeral \ bool" subsection {* Fast exhaustive combinators *} class fast_exhaustive = term_of + fixes fast_exhaustive :: "('a \ unit) \ code_numeral \ unit" consts throw_Counterexample :: "term list => unit" consts catch_Counterexample :: "unit => term list option" code_const throw_Counterexample (Quickcheck "raise (Exhaustive'_Generators.Counterexample _)") code_const catch_Counterexample (Quickcheck "(((_); NONE) handle Exhaustive'_Generators.Counterexample ts => SOME ts)") subsection {* Continuation passing style functions as plus monad *} type_synonym 'a cps = "('a => term list option) => term list option" definition cps_empty :: "'a cps" where "cps_empty = (%cont. None)" definition cps_single :: "'a => 'a cps" where "cps_single v = (%cont. cont v)" definition cps_bind :: "'a cps => ('a => 'b cps) => 'b cps" where "cps_bind m f = (%cont. m (%a. (f a) cont))" definition cps_plus :: "'a cps => 'a cps => 'a cps" where "cps_plus a b = (%c. case a c of None => b c | Some x => Some x)" definition cps_if :: "bool => unit cps" where "cps_if b = (if b then cps_single () else cps_empty)" definition cps_not :: "unit cps => unit cps" where "cps_not n = (%c. case n (%u. Some []) of None => c () | Some _ => None)" type_synonym 'a pos_bound_cps = "('a => term list option) => code_numeral => term list option" definition pos_bound_cps_empty :: "'a pos_bound_cps" where "pos_bound_cps_empty = (%cont i. None)" definition pos_bound_cps_single :: "'a => 'a pos_bound_cps" where "pos_bound_cps_single v = (%cont i. cont v)" definition pos_bound_cps_bind :: "'a pos_bound_cps => ('a => 'b pos_bound_cps) => 'b pos_bound_cps" where "pos_bound_cps_bind m f = (%cont i. if i = 0 then None else (m (%a. (f a) cont i) (i - 1)))" definition pos_bound_cps_plus :: "'a pos_bound_cps => 'a pos_bound_cps => 'a pos_bound_cps" where "pos_bound_cps_plus a b = (%c i. case a c i of None => b c i | Some x => Some x)" definition pos_bound_cps_if :: "bool => unit pos_bound_cps" where "pos_bound_cps_if b = (if b then pos_bound_cps_single () else pos_bound_cps_empty)" datatype 'a unknown = Unknown | Known 'a datatype 'a three_valued = Unknown_value | Value 'a | No_value type_synonym 'a neg_bound_cps = "('a unknown => term list three_valued) => code_numeral => term list three_valued" definition neg_bound_cps_empty :: "'a neg_bound_cps" where "neg_bound_cps_empty = (%cont i. No_value)" definition neg_bound_cps_single :: "'a => 'a neg_bound_cps" where "neg_bound_cps_single v = (%cont i. cont (Known v))" definition neg_bound_cps_bind :: "'a neg_bound_cps => ('a => 'b neg_bound_cps) => 'b neg_bound_cps" where "neg_bound_cps_bind m f = (%cont i. if i = 0 then cont Unknown else m (%a. case a of Unknown => cont Unknown | Known a' => f a' cont i) (i - 1))" definition neg_bound_cps_plus :: "'a neg_bound_cps => 'a neg_bound_cps => 'a neg_bound_cps" where "neg_bound_cps_plus a b = (%c i. case a c i of No_value => b c i | Value x => Value x | Unknown_value => (case b c i of No_value => Unknown_value | Value x => Value x | Unknown_value => Unknown_value))" definition neg_bound_cps_if :: "bool => unit neg_bound_cps" where "neg_bound_cps_if b = (if b then neg_bound_cps_single () else neg_bound_cps_empty)" definition neg_bound_cps_not :: "unit pos_bound_cps => unit neg_bound_cps" where "neg_bound_cps_not n = (%c i. case n (%u. Some []) i of None => c (Known ()) | Some _ => No_value)" definition pos_bound_cps_not :: "unit neg_bound_cps => unit pos_bound_cps" where "pos_bound_cps_not n = (%c i. case n (%u. Value []) i of No_value => c () | Value _ => None | Unknown_value => None)" subsection {* Defining combinators for any first-order data type *} definition catch_match :: "term list option => term list option => term list option" where [code del]: "catch_match t1 t2 = (SOME t. t = t1 \ t = t2)" code_const catch_match (Quickcheck "(_) handle Match => _") use "Tools/Quickcheck/exhaustive_generators.ML" setup {* Exhaustive_Generators.setup *} declare [[quickcheck_batch_tester = exhaustive]] hide_fact orelse_def catch_match_def no_notation orelse (infixr "orelse" 55) hide_const (open) orelse catch_match mk_map_term check_all_n_lists hide_type (open) cps pos_bound_cps neg_bound_cps unknown three_valued hide_const (open) cps_empty cps_single cps_bind cps_plus cps_if cps_not pos_bound_cps_empty pos_bound_cps_single pos_bound_cps_bind pos_bound_cps_plus pos_bound_cps_if pos_bound_cps_not neg_bound_cps_empty neg_bound_cps_single neg_bound_cps_bind neg_bound_cps_plus neg_bound_cps_if neg_bound_cps_not Unknown Known Unknown_value Value No_value end