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theory Quickcheck_Generators(* Author: Florian Haftmann, TU Muenchen *) header {* Experimental counterexample generators *} theory Quickcheck_Generators imports Quickcheck State_Monad begin subsection {* Type @{typ "'a => '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 undefined}, 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 :: "typerep => typerep => ('a => 'a => bool) => ('a => term) => (seed => ('b × (unit => term)) × seed) => (seed => seed × seed) => seed => (('a => 'b) × (unit => term)) × seed" code_const random_fun_aux (SML "Random'_Engine.random'_fun") instantiation "fun" :: ("{eq, term_of}", "{type, random}") random begin definition random_fun :: "index => seed => (('a => 'b) × (unit => term)) × seed" where "random n = random_fun_aux TYPEREP('a) TYPEREP('b) (op =) Code_Eval.term_of (random n) split_seed" instance .. end code_reserved SML Random_Engine subsection {* Datatypes *} definition collapse :: "('a => ('a => 'b × 'a) × 'a) => 'a => 'b × 'a" where "collapse f = (do g \<leftarrow> f; g done)" ML {* structure StateMonad = struct fun liftT T sT = sT --> HOLogic.mk_prodT (T, sT); fun liftT' sT = sT --> sT; fun return T sT x = Const (@{const_name return}, T --> liftT T sT) $ x; fun scomp T1 T2 sT f g = Const (@{const_name scomp}, liftT T1 sT --> (T1 --> liftT T2 sT) --> liftT T2 sT) $ f $ g; end; *} lemma random'_if: fixes random' :: "index => index => seed => ('a × (unit => term)) × seed" assumes "random' 0 j = (λs. undefined)" 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) setup {* let exception REC of string; exception TYP 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 => term"}); fun mk_split thy ty ty' = Sign.mk_const thy (@{const_name split}, [ty, @{typ "unit => 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 => 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 Typerep.typerep (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, 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::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 = Long_Name.base_name @{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 = if Sign.of_sort thy (ty, @{sort random}) then ((ty, false), random ty $ @{term "j::index"}) else raise TYP ("Will not generate random elements for type(s) " ^ quote (hd tycos)); fun dtyp tyco = ((this_ty, true), random' $ @{term "i::index"} $ @{term "j::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::index"} $ @{term "j::index"}, Abs ("s", @{typ seed}, Const (@{const_name undefined}, HOLogic.mk_prodT (term_ty this_ty, @{typ seed})))), (random' $ @{term "Suc_index i"} $ @{term "j::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 ((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::index"}, random' $ @{term "i::index"} $ @{term "i::index"}) val del_func = Attrib.internal (fn _ => Thm.declaration_attribute (fn thm => Context.mapping (Code.del_eqn 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)); val c = (fst o dest_Const o fst o strip_comb o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of) thm; in lthy |> LocalTheory.theory (Code.del_eqns c #> PureThy.add_thm ((Binding.name (fst (dest_Free random') ^ "_code"), thm), [Thm.kind_internal]) #-> Code.add_eqn) end; in thy |> TheoryTarget.instantiation ([tyco], vs, @{sort random}) |> PrimrecPackage.add_primrec [(Binding.name (fst (dest_Free random')), SOME (snd (dest_Free random')), NoSyn)] (map (fn eq => ((Binding.empty, [del_func]), eq)) eqs') |-> add_code |> `(fn lthy => Syntax.check_term lthy eq) |-> (fn eq => Specification.definition (NONE, (Attrib.empty_binding, eq))) |> snd |> Class.prove_instantiation_instance (K (Class.intro_classes_tac [])) |> LocalTheory.exit_global 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) | TYP msg => (warning msg; thy) in DatatypePackage.interpretation add_random_inst end *} subsection {* 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, λu. Code_Eval.App (Code_Eval.Const (STR ''Int.int'') TYPEREP(nat => int)) (t ())) else (- int m, λu. Code_Eval.App (Code_Eval.Const (STR ''HOL.uminus_class.uminus'') TYPEREP(int => int)) (Code_Eval.App (Code_Eval.Const (STR ''Int.int'') TYPEREP(nat => int)) (t ())))) done)" instance .. end subsection {* Examples *} theorem "map g (map f xs) = map (g o f) xs" quickcheck [generator = code] by (induct xs) simp_all theorem "map g (map f xs) = map (f o g) xs" quickcheck [generator = code] oops theorem "rev (xs @ ys) = rev ys @ rev xs" quickcheck [generator = code] by simp theorem "rev (xs @ ys) = rev xs @ rev ys" quickcheck [generator = code] oops theorem "rev (rev xs) = xs" quickcheck [generator = code] by simp theorem "rev xs = xs" quickcheck [generator = code] oops primrec app :: "('a => 'a) list => 'a => 'a" where "app [] x = x" | "app (f # fs) x = app fs (f x)" lemma "app (fs @ gs) x = app gs (app fs x)" quickcheck [generator = code] by (induct fs arbitrary: x) simp_all lemma "app (fs @ gs) x = app fs (app gs x)" quickcheck [generator = code] oops primrec occurs :: "'a => 'a list => nat" where "occurs a [] = 0" | "occurs a (x#xs) = (if (x=a) then Suc(occurs a xs) else occurs a xs)" primrec del1 :: "'a => 'a list => 'a list" where "del1 a [] = []" | "del1 a (x#xs) = (if (x=a) then xs else (x#del1 a xs))" lemma "Suc (occurs a (del1 a xs)) = occurs a xs" -- {* Wrong. Precondition needed.*} quickcheck [generator = code] oops lemma "xs ~= [] --> Suc (occurs a (del1 a xs)) = occurs a xs" quickcheck [generator = code] -- {* Also wrong.*} oops lemma "0 < occurs a xs --> Suc (occurs a (del1 a xs)) = occurs a xs" quickcheck [generator = code] by (induct xs) auto primrec replace :: "'a => 'a => 'a list => 'a list" where "replace a b [] = []" | "replace a b (x#xs) = (if (x=a) then (b#(replace a b xs)) else (x#(replace a b xs)))" lemma "occurs a xs = occurs b (replace a b xs)" quickcheck [generator = code] -- {* Wrong. Precondition needed.*} oops lemma "occurs b xs = 0 ∨ a=b --> occurs a xs = occurs b (replace a b xs)" quickcheck [generator = code] by (induct xs) simp_all subsection {* Trees *} datatype 'a tree = Twig | Leaf 'a | Branch "'a tree" "'a tree" primrec leaves :: "'a tree => 'a list" where "leaves Twig = []" | "leaves (Leaf a) = [a]" | "leaves (Branch l r) = (leaves l) @ (leaves r)" primrec plant :: "'a list => 'a tree" where "plant [] = Twig " | "plant (x#xs) = Branch (Leaf x) (plant xs)" primrec mirror :: "'a tree => 'a tree" where "mirror (Twig) = Twig " | "mirror (Leaf a) = Leaf a " | "mirror (Branch l r) = Branch (mirror r) (mirror l)" theorem "plant (rev (leaves xt)) = mirror xt" quickcheck [generator = code] --{* Wrong! *} oops theorem "plant (leaves xt @ leaves yt) = Branch xt yt" quickcheck [generator = code] --{* Wrong! *} oops datatype 'a ntree = Tip "'a" | Node "'a" "'a ntree" "'a ntree" primrec inOrder :: "'a ntree => 'a list" where "inOrder (Tip a)= [a]" | "inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)" primrec root :: "'a ntree => 'a" where "root (Tip a) = a" | "root (Node f x y) = f" theorem "hd (inOrder xt) = root xt" quickcheck [generator = code] --{* Wrong! *} oops lemma "int (f k) = k" quickcheck [generator = code] oops end