src/HOL/Quickcheck_Exhaustive.thy

order and graph for code symbols

+
−(* Author: Lukas Bulwahn, TU Muenchen *)
+
−
+
−header {* A simple counterexample generator performing exhaustive testing *}
+
−
+
−theory Quickcheck_Exhaustive
+
−imports Quickcheck_Random
+
−keywords "quickcheck_generator" :: thy_decl
+
−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 \<Rightarrow> (bool * term list) option) \<Rightarrow> natural \<Rightarrow> (bool * term list) option"
+
−  
+
−class full_exhaustive = term_of +
+
−  fixes full_exhaustive :: "('a * (unit => term) \<Rightarrow> (bool * term list) option) \<Rightarrow> natural \<Rightarrow> (bool * term list) option"
+
−
+
−instantiation natural :: full_exhaustive
+
−begin
+
−
+
−function full_exhaustive_natural' :: "(natural * (unit => term) => (bool * term list) option) => natural => natural => (bool * term list) option"
+
−  where "full_exhaustive_natural' f d i =
+
−    (if d < i then None
+
−    else (f (i, %_. Code_Evaluation.term_of i)) orelse (full_exhaustive_natural' f d (i + 1)))"
+
−by pat_completeness auto
+
−
+
−termination
+
−  by (relation "measure (%(_, d, i). nat_of_natural (d + 1 - i))")
+
−    (auto simp add: less_natural_def)
+
−
+
−definition "full_exhaustive f d = full_exhaustive_natural' f d 0"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation natural :: exhaustive
+
−begin
+
−
+
−function exhaustive_natural' :: "(natural => (bool * term list) option) => natural => natural => (bool * term list) option"
+
−  where "exhaustive_natural' f d i =
+
−    (if d < i then None
+
−    else (f i orelse exhaustive_natural' f d (i + 1)))"
+
−by pat_completeness auto
+
−
+
−termination
+
−  by (relation "measure (%(_, d, i). nat_of_natural (d + 1 - i))")
+
−    (auto simp add: less_natural_def)
+
−
+
−definition "exhaustive f d = exhaustive_natural' f d 0"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation integer :: exhaustive
+
−begin
+
−
+
−function exhaustive_integer' :: "(integer => (bool * term list) option) => integer => integer => (bool * term list) option"
+
−  where "exhaustive_integer' f d i = (if d < i then None else (f i orelse exhaustive_integer' f d (i + 1)))"
+
−by pat_completeness auto
+
−
+
−termination 
+
−  by (relation "measure (%(_, d, i). nat_of_integer (d + 1 - i))")
+
−    (auto simp add: less_integer_def nat_of_integer_def)
+
−
+
−definition "exhaustive f d = exhaustive_integer' f (integer_of_natural d) (- (integer_of_natural d))"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation integer :: full_exhaustive
+
−begin
+
−
+
−function full_exhaustive_integer' :: "(integer * (unit => term) => (bool * term list) option) => integer => integer => (bool * term list) option"
+
−  where "full_exhaustive_integer' 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_integer' f d (i + 1)))"
+
−by pat_completeness auto
+
−
+
−termination 
+
−  by (relation "measure (%(_, d, i). nat_of_integer (d + 1 - i))")
+
−    (auto simp add: less_integer_def nat_of_integer_def)
+
−
+
−definition "full_exhaustive f d = full_exhaustive_integer' f (integer_of_natural d) (- (integer_of_natural d))"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation nat :: exhaustive
+
−begin
+
−
+
−definition "exhaustive f d = exhaustive (%x. f (nat_of_natural x)) d"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation nat :: full_exhaustive
+
−begin
+
−
+
−definition "full_exhaustive f d = full_exhaustive (%(x, xt). f (nat_of_natural x, %_. Code_Evaluation.term_of (nat_of_natural x))) d"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation int :: exhaustive
+
−begin
+
−
+
−function exhaustive_int' :: "(int => (bool * term list) option) => int => int => (bool * term list) option"
+
−  where "exhaustive_int' f d i = (if d < i then None else (f i orelse exhaustive_int' f d (i + 1)))"
+
−by pat_completeness auto
+
−
+
−termination 
+
−  by (relation "measure (%(_, d, i). nat (d + 1 - i))") auto
+
−
+
−definition "exhaustive f d = exhaustive_int' f (int_of_integer (integer_of_natural d))
+
−  (- (int_of_integer (integer_of_natural d)))"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation int :: full_exhaustive
+
−begin
+
−
+
−function full_exhaustive_int' :: "(int * (unit => term) => (bool * term list) option) => int => int => (bool * term list) option"
+
−  where "full_exhaustive_int' 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_int' 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_int' f (int_of_integer (integer_of_natural d))
+
−  (- (int_of_integer (integer_of_natural d)))"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation prod :: (exhaustive, exhaustive) exhaustive
+
−begin
+
−
+
−definition
+
−  "exhaustive f d = exhaustive (%x. exhaustive (%y. f ((x, y))) d) d"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−definition (in term_syntax) [code_unfold]: "valtermify_pair x y = Code_Evaluation.valtermify (Pair :: 'a :: typerep => 'b :: typerep => 'a * 'b) {\<cdot>} x {\<cdot>} y"
+
−
+
−instantiation prod :: (full_exhaustive, full_exhaustive) full_exhaustive
+
−begin
+
−
+
−definition
+
−  "full_exhaustive f d = full_exhaustive (%x. full_exhaustive (%y. f (valtermify_pair x y)) d) d"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation set :: (exhaustive) exhaustive
+
−begin
+
−
+
−fun exhaustive_set
+
−where
+
−  "exhaustive_set f i = (if i = 0 then None else (f {} orelse exhaustive_set (%A. f A orelse exhaustive (%x. if x \<in> A then None else f (insert x A)) (i - 1)) (i - 1)))"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation set :: (full_exhaustive) full_exhaustive
+
−begin
+
−
+
−fun full_exhaustive_set 
+
−where
+
−  "full_exhaustive_set f i = (if i = 0 then None else (f valterm_emptyset orelse full_exhaustive_set (%A. f A orelse Quickcheck_Exhaustive.full_exhaustive (%x. if fst x \<in> fst A then None else f (valtermify_insert x A)) (i - 1)) (i - 1)))"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation "fun" :: ("{equal, exhaustive}", exhaustive) exhaustive
+
−begin
+
−
+
−fun exhaustive_fun' :: "(('a => 'b) => (bool * term list) option) => natural => natural => (bool * 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) => (bool * term list) option) => natural => (bool * term list) option"
+
−where
+
−  "exhaustive_fun f d = exhaustive_fun' f d d" 
+
−
+
−instance ..
+
−
+
−end
+
−
+
−definition [code_unfold]: "valtermify_absdummy = (%(v, t). (%_::'a. v, %u::unit. Code_Evaluation.Abs (STR ''x'') (Typerep.typerep TYPE('a::typerep)) (t ())))"
+
−
+
−definition (in term_syntax) [code_unfold]: "valtermify_fun_upd g a b = Code_Evaluation.valtermify (fun_upd :: ('a :: typerep => 'b :: typerep) => 'a => 'b => 'a => 'b) {\<cdot>} g {\<cdot>} a {\<cdot>} b"
+
−
+
−instantiation "fun" :: ("{equal, full_exhaustive}", full_exhaustive) full_exhaustive
+
−begin
+
−
+
−fun full_exhaustive_fun' :: "(('a => 'b) * (unit => term) => (bool * term list) option) => natural => natural => (bool * term list) option"
+
−where
+
−  "full_exhaustive_fun' f i d = (full_exhaustive (%v. f (valtermify_absdummy v)) d)
+
−   orelse (if i > 1 then
+
−     full_exhaustive_fun' (%g. full_exhaustive (%a. full_exhaustive (%b.
+
−       f (valtermify_fun_upd g a b)) d) d) (i - 1) d else None)"
+
−
+
−definition full_exhaustive_fun :: "(('a => 'b) * (unit => term) => (bool * term list) option) => natural => (bool * 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 \<Rightarrow> term) \<Rightarrow> (bool * term list) option) \<Rightarrow> (bool * term list) option"
+
−  fixes enum_term_of :: "'a itself \<Rightarrow> unit \<Rightarrow> term list"
+
−  
+
−fun check_all_n_lists :: "(('a :: check_all) list * (unit \<Rightarrow> term list) \<Rightarrow> (bool * term list) option) \<Rightarrow> natural \<Rightarrow> (bool * 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 (in term_syntax) [code_unfold]: "termify_fun_upd g a b = (Code_Evaluation.termify (fun_upd :: ('a :: typerep => 'b :: typerep) => 'a => 'b => 'a => 'b) <\<cdot>> g <\<cdot>> a <\<cdot>> b)"
+
−
+
−definition mk_map_term :: " (unit \<Rightarrow> typerep) \<Rightarrow> (unit \<Rightarrow> typerep) \<Rightarrow> (unit \<Rightarrow> term list) \<Rightarrow> (unit \<Rightarrow> term list) \<Rightarrow> unit \<Rightarrow> 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 (\<lambda>(ys, yst). f (the o map_of (zip enum ys), mk_term yst)) (natural_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) ()) (List.n_lists (length (enum_term_of_a ())) (enum_term_of (TYPE('b)) ())))"
+
− 
+
−instance ..
+
−
+
−end
+
−
+
−fun (in term_syntax) check_all_subsets :: "(('a :: typerep) set * (unit => term) => (bool * term list) option) => ('a * (unit => term)) list => (bool * term list) option"
+
−where
+
−  "check_all_subsets f [] = f valterm_emptyset"
+
−| "check_all_subsets f (x # xs) = check_all_subsets (%s. case f s of Some ts => Some ts | None => f (valtermify_insert x s)) xs"
+
−
+
−
+
−definition (in term_syntax) [code_unfold]: "term_emptyset = Code_Evaluation.termify ({} :: ('a :: typerep) set)"
+
−definition (in term_syntax) [code_unfold]: "termify_insert x s = Code_Evaluation.termify (insert :: ('a::typerep) => 'a set => 'a set)  <\<cdot>> x <\<cdot>> s"
+
−
+
−definition (in term_syntax) setify :: "('a::typerep) itself => term list => term"
+
−where
+
−  "setify T ts = foldr (termify_insert T) ts (term_emptyset T)" 
+
−
+
−instantiation set :: (check_all) check_all
+
−begin
+
−
+
−definition
+
−  "check_all_set f =
+
−     check_all_subsets f (zip (Enum.enum :: 'a list) (map (%a. %u :: unit. a) (Quickcheck_Exhaustive.enum_term_of (TYPE ('a)) ())))"
+
−
+
−definition enum_term_of_set :: "'a set itself => unit => term list"
+
−where
+
−  "enum_term_of_set _ _ = map (setify (TYPE('a))) (sublists (Quickcheck_Exhaustive.enum_term_of (TYPE('a)) ()))"
+
−
+
−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' \<Rightarrow> Some x' | None \<Rightarrow> 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
+
−
+
−definition (in term_syntax) [code_unfold]: "termify_pair x y = Code_Evaluation.termify (Pair :: 'a :: typerep => 'b :: typerep => 'a * 'b) <\<cdot>> x <\<cdot>> y"
+
−
+
−instantiation prod :: (check_all, check_all) check_all
+
−begin
+
−
+
−definition
+
−  "check_all f = check_all (%x. check_all (%y. f (valtermify_pair x y)))"
+
−
+
−definition enum_term_of_prod :: "('a * 'b) itself => unit => term list"
+
−where
+
−  "enum_term_of_prod = (%_ _. map (%(x, y). termify_pair TYPE('a) TYPE('b) x y)
+
−     (List.product (enum_term_of (TYPE('a)) ()) (enum_term_of (TYPE('b)) ())))"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−definition (in term_syntax) [code_unfold]: "valtermify_Inl x = Code_Evaluation.valtermify (Inl :: 'a :: typerep => 'a + 'b :: typerep) {\<cdot>} x"
+
−definition (in term_syntax) [code_unfold]: "valtermify_Inr x = Code_Evaluation.valtermify (Inr :: 'b :: typerep => 'a ::typerep + 'b) {\<cdot>} x"
+
−
+
−instantiation sum :: (check_all, check_all) check_all
+
−begin
+
−
+
−definition
+
−  "check_all f = check_all (%a. f (valtermify_Inl a)) orelse check_all (%b. f (valtermify_Inr b))"
+
−
+
−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\<^sub>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\<^sub>1])"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation Enum.finite_2 :: check_all
+
−begin
+
−
+
−definition
+
−  "check_all f = (f (Code_Evaluation.valtermify Enum.finite_2.a\<^sub>1)
+
−    orelse f (Code_Evaluation.valtermify Enum.finite_2.a\<^sub>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 = (f (Code_Evaluation.valtermify Enum.finite_3.a\<^sub>1)
+
−    orelse f (Code_Evaluation.valtermify Enum.finite_3.a\<^sub>2)
+
−    orelse f (Code_Evaluation.valtermify Enum.finite_3.a\<^sub>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
+
−
+
−instantiation Enum.finite_4 :: check_all
+
−begin
+
−
+
−definition
+
−  "check_all f = (f (Code_Evaluation.valtermify Enum.finite_4.a\<^sub>1)
+
−    orelse f (Code_Evaluation.valtermify Enum.finite_4.a\<^sub>2)
+
−    orelse f (Code_Evaluation.valtermify Enum.finite_4.a\<^sub>3)
+
−    orelse f (Code_Evaluation.valtermify Enum.finite_4.a\<^sub>4))"
+
−
+
−definition enum_term_of_finite_4 :: "Enum.finite_4 itself => unit => term list"
+
−where
+
−  "enum_term_of_finite_4 = (%_ _. map Code_Evaluation.term_of (Enum.enum :: Enum.finite_4 list))"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−subsection {* Bounded universal quantifiers *}
+
−
+
−class bounded_forall =
+
−  fixes bounded_forall :: "('a \<Rightarrow> bool) \<Rightarrow> natural \<Rightarrow> bool"
+
−
+
−subsection {* Fast exhaustive combinators *}
+
−
+
−class fast_exhaustive = term_of +
+
−  fixes fast_exhaustive :: "('a \<Rightarrow> unit) \<Rightarrow> natural \<Rightarrow> unit"
+
−
+
−axiomatization throw_Counterexample :: "term list => unit"
+
−axiomatization catch_Counterexample :: "unit => term list option"
+
−
+
−code_printing
+
−  constant throw_Counterexample \<rightharpoonup>
+
−    (Quickcheck) "raise (Exhaustive'_Generators.Counterexample _)"
+
−| constant catch_Counterexample \<rightharpoonup>
+
−    (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 => (bool * term list) option) => natural => (bool * 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) => natural => 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 (True, [])) 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 generators for any first-order data type *}
+
−
+
−axiomatization unknown :: 'a
+
−
+
−notation (output) unknown  ("?")
+
−
+
−ML_file "Tools/Quickcheck/exhaustive_generators.ML"
+
−
+
−setup {* Exhaustive_Generators.setup *}
+
−
+
−declare [[quickcheck_batch_tester = exhaustive]]
+
−
+
−subsection {* Defining generators for abstract types *}
+
−
+
−ML_file "Tools/Quickcheck/abstract_generators.ML"
+
−
+
−hide_fact (open) orelse_def
+
−no_notation orelse (infixr "orelse" 55)
+
−
+
−hide_fact
+
−  exhaustive_int'_def
+
−  exhaustive_integer'_def
+
−  exhaustive_natural'_def
+
−
+
−hide_const valtermify_absdummy valtermify_fun_upd valterm_emptyset valtermify_insert valtermify_pair
+
−  valtermify_Inl valtermify_Inr
+
−  termify_fun_upd term_emptyset termify_insert termify_pair setify
+
−
+
−hide_const (open)
+
−  exhaustive full_exhaustive
+
−  exhaustive_int' full_exhaustive_int'
+
−  exhaustive_integer' full_exhaustive_integer'
+
−  exhaustive_natural' full_exhaustive_natural'
+
−  throw_Counterexample catch_Counterexample
+
−  check_all enum_term_of
+
−  orelse unknown mk_map_term check_all_n_lists check_all_subsets
+
−
+
−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