--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Quickcheck.thy Wed Mar 12 19:38:14 2008 +0100
@@ -0,0 +1,383 @@
+(* 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 = type +
+ fixes random :: "index \<Rightarrow> seed \<Rightarrow> 'a \<times> seed"
+
+print_classes
+
+instantiation itself :: (type) random
+begin
+
+definition
+ "random _ = return TYPE('a)"
+
+instance ..
+
+end
+
+lemma random_aux_if:
+ fixes random' :: "index \<Rightarrow> index \<Rightarrow> seed \<Rightarrow> 'a \<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;
+ fun random_inst tyco thy =
+ let
+ val { descr, index, ... } = DatatypePackage.the_datatype thy tyco;
+ val _ = if length descr > 1 then raise REC else ();
+ 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 ty = Type (tyco, map TFree vs);
+ val typ_of = DatatypeAux.typ_of_dtyp descr vs;
+ val SOME (_, _, constrs) = AList.lookup (op =) descr index;
+ val randomN = NameSpace.base @{const_name random};
+ val random_aux_name = randomN ^ "_" ^ Class.type_name tyco ^ "'";
+ fun lift_ty ty = StateMonad.liftT ty @{typ seed};
+ val ty_aux = @{typ index} --> @{typ index} --> lift_ty ty;
+ fun random ty =
+ Const (@{const_name random}, @{typ index} --> lift_ty ty);
+ val random_aux = Free (random_aux_name, ty_aux);
+ fun add_cons_arg dty (is_rec, t) =
+ let
+ val ty' = typ_of dty;
+ val rec_call = case try DatatypeAux.dest_DtRec dty
+ of SOME index' => index = index'
+ | NONE => false
+ val random' = if rec_call
+ then random_aux $ @{term "i\<Colon>index"} $ @{term "j\<Colon>index"}
+ else random ty' $ @{term "j\<Colon>index"}
+ val is_rec' = is_rec orelse DatatypeAux.is_rec_type dty;
+ val t' = StateMonad.mbind ty' ty @{typ seed} random' (Abs ("", ty', t))
+ in (is_rec', t') end;
+ fun mk_cons_t (c, dtys) =
+ let
+ val ty' = map typ_of dtys ---> ty;
+ val t = StateMonad.return ty @{typ seed} (list_comb (Const (c, ty'),
+ map Bound (length dtys - 1 downto 0)));
+ val (is_rec, t') = fold_rev add_cons_arg dtys (false, t);
+ in (is_rec, StateMonad.run ty @{typ seed} t') end;
+ fun check_empty [] = NONE
+ | check_empty xs = SOME xs;
+ fun bundle_cons_ts cons_ts =
+ let
+ val ts = map snd cons_ts;
+ val t = HOLogic.mk_list (lift_ty ty) ts;
+ val t' = Const (@{const_name select}, HOLogic.listT (lift_ty ty) --> lift_ty (lift_ty ty)) $ t;
+ val t'' = Const (@{const_name collapse}, lift_ty (lift_ty ty) --> lift_ty ty) $ t';
+ in t'' end;
+ fun bundle_conss (some_rec_t, nonrec_t) =
+ let
+ val t = case some_rec_t
+ of SOME rec_t => Const (@{const_name collapse}, lift_ty (lift_ty ty) --> lift_ty ty)
+ $ (Const (@{const_name select_default},
+ @{typ index} --> lift_ty ty --> lift_ty ty --> lift_ty (lift_ty ty))
+ $ @{term "i\<Colon>index"} $ rec_t $ nonrec_t)
+ | NONE => nonrec_t
+ in t end;
+ val random_rhs = constrs
+ |> map mk_cons_t
+ |> List.partition fst
+ |> apfst (Option.map bundle_cons_ts o check_empty)
+ |> apsnd bundle_cons_ts
+ |> bundle_conss;
+ val random_aux_undef = (HOLogic.mk_Trueprop o HOLogic.mk_eq)
+ (random_aux $ @{term "0\<Colon>index"} $ @{term "j\<Colon>index"}, Const (@{const_name undefined}, lift_ty ty))
+ val random_aux_eq = (HOLogic.mk_Trueprop o HOLogic.mk_eq)
+ (random_aux $ @{term "Suc_index i"} $ @{term "j\<Colon>index"}, random_rhs);
+ val random_eq = (HOLogic.mk_Trueprop o HOLogic.mk_eq) (Const (@{const_name random},
+ @{typ index} --> lift_ty ty) $ @{term "i\<Colon>index"},
+ random_aux $ @{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_aux_if}
+ |> Drule.instantiate' [SOME (Thm.ctyp_of thy ty)] [SOME (Thm.cterm_of thy random_aux)]
+ |> (fn thm => thm OF simps)
+ |> singleton (ProofContext.export lthy (ProofContext.init thy))
+ in
+ lthy
+ |> LocalTheory.theory (PureThy.note Thm.internalK (random_aux_name ^ "_code", thm)
+ #-> Code.add_func)
+ end;
+ in
+ thy
+ |> TheoryTarget.instantiation ([tyco], vs, @{sort random})
+ |> PrimrecPackage.add_primrec [(random_aux_name, SOME ty_aux, NoSyn)]
+ [(("", [del_func]), random_aux_undef), (("", [del_func]), random_aux_eq)]
+ |-> add_code
+ |> `(fn lthy => Syntax.check_term lthy random_eq)
+ |-> (fn eq => Specification.definition (NONE, (("", []), eq)))
+ |> snd
+ |> Class.prove_instantiation_instance (K (Class.intro_classes_tac []))
+ |> LocalTheory.exit
+ |> ProofContext.theory_of
+ end;
+ fun add_random_inst [tyco] = (fn thy => random_inst tyco thy handle REC =>
+ (warning ("Will not generated random elements for mutual recursive type " ^ quote tyco); thy))
+ | add_random_inst tycos = tap (fn _ => warning
+ ("Will not generated random elements for mutual recursive type(s) " ^ commas (map quote tycos)));
+in DatatypePackage.interpretation add_random_inst end
+*}
+
+instantiation int :: random
+begin
+
+definition
+ "random n = (do
+ (b, m) \<leftarrow> random n;
+ return (if b then int m else - int m)
+ done)"
+
+instance ..
+
+end
+
+instantiation set :: (random) random
+begin
+
+primrec random_set' :: "index \<Rightarrow> index \<Rightarrow> seed \<Rightarrow> 'a set \<times> seed" where
+ "random_set' 0 j = undefined"
+ | "random_set' (Suc_index i) j = collapse (select_default i
+ (do x \<leftarrow> random i; xs \<leftarrow> random_set' i j; return (insert x xs) done)
+ (return {}))"
+
+lemma random_set'_code [code func]:
+ "random_set' i j s = (if i = 0 then undefined else collapse (select_default (i - 1)
+ (do x \<leftarrow> random (i - 1); xs \<leftarrow> random_set' (i - 1) j; return (insert x xs) done)
+ (return {})) s)"
+ by (rule random_aux_if random_set'.simps)+
+
+definition
+ "random i = random_set' i i"
+
+instance ..
+
+end
+
+code_reserved SML Quickcheck
+
+
+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 prop tys =
+ let
+ val bounds = map_index (fn (i, ty) => (i, ty)) tys;
+ val result = list_comb (prop, map (fn (i, _) => Bound (length tys - i - 1)) bounds);
+ val terms = map (fn (i, ty) => Const (@{const_name Eval.term_of}, ty --> @{typ term}) $ Bound (length tys - i - 1)) 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 "} $ HOLogic.mk_list @{typ term} terms);
+ val return = @{term "Pair \<Colon> term list option \<Rightarrow> seed \<Rightarrow> term list option \<times> seed"};
+ fun mk_bindtyp ty = @{typ seed} --> HOLogic.mk_prodT (ty, @{typ seed});
+ fun mk_bindclause (i, ty) t = Const (@{const_name mbind}, mk_bindtyp ty
+ --> (ty --> mk_bindtyp @{typ "term list option"}) --> mk_bindtyp @{typ "term list option"})
+ $ (Const (@{const_name random}, @{typ index} --> mk_bindtyp ty)
+ $ Bound i) $ Abs ("a", ty, t);
+ 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 = CodePackage.eval_term ("Quickcheck.eval_ref", eval_ref) thy
+ (mk_generator_expr 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 prop tys;
+ val eq = Logic.mk_equals (Free ("VALUE", fastype_of t), t)
+ in
+ thy
+ |> TheoryTarget.init NONE
+ |> Specification.definition (NONE, (("", []), eq))
+ |> snd
+ |> LocalTheory.exit
+ |> ProofContext.theory_of
+ end;
+
+end
+*}
+
+(*setup {* Quickcheck.VALUE @{term "\<lambda>(n::nat) (m::nat) (q::nat). n = m + q + 1"} [@{typ nat}, @{typ nat}, @{typ nat}] *}
+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) *}*)
+
+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) (Sign.string_of_term @{theory}) *}
+ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 25 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+
+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}] *}
+
+(*definition "FOO = (True, Suc 0)"
+
+code_module (test) QuickcheckExample
+ file "~~/../../gen_code/quickcheck'.ML"
+ contains FOO*)
+
+ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 25 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 3 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+
+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) (Sign.string_of_term @{theory}) *}
+ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 25 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 8 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 8 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 8 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 88 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+
+subsection {* Incremental function generator *}
+
+ML {*
+structure Quickcheck =
+struct
+
+open Quickcheck;
+
+fun random_fun (eq : 'a -> 'a -> bool)
+ (random : Random_Engine.seed -> 'b * 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', []);
+ fun random_fun' x =
+ let
+ val (seed, fun_map) = ! state;
+ in case AList.lookup (uncurry eq) fun_map x
+ of SOME y => y
+ | NONE => let
+ val (y, seed') = random seed;
+ val _ = state := (seed', (x, y) :: fun_map);
+ in y end
+ end;
+ in (random_fun', seed'') end;
+
+end
+*}
+
+axiomatization
+ random_fun_aux :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> (seed \<Rightarrow> 'b \<times> seed)
+ \<Rightarrow> (seed \<Rightarrow> seed \<times> seed) \<Rightarrow> seed \<Rightarrow> ('a \<Rightarrow> 'b) \<times> seed"
+
+code_const random_fun_aux (SML "Quickcheck.random'_fun")
+
+instantiation "fun" :: (term_of, term_of) term_of
+begin
+
+instance ..
+
+end
+
+code_const "Eval.term_of :: ('a\<Colon>term_of \<Rightarrow> 'b\<Colon>term_of) \<Rightarrow> _"
+ (SML "(fn '_ => Const (\"arbitrary\", dummyT))")
+
+instantiation "fun" :: (eq, "{type, random}") random
+begin
+
+definition
+ "random n = random_fun_aux (op =) (random n) split_seed"
+
+instance ..
+
+end
+
+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) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 20 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+
+ML {* val f = Quickcheck.compile_generator_expr @{theory}
+ @{term "\<lambda>(A \<Colon> nat set) B. card (A \<union> B) = card A + card B"} [@{typ "nat set"}, @{typ "nat set"}] *}
+
+ML {* f 1 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 2 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 3 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 5 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 6 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 10 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 10 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+
+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) (Sign.string_of_term @{theory}) *}
+ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 4 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 10 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 10 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 10 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 10 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 10 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 8 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+ML {* f 8 |> (Option.map o map) (Sign.string_of_term @{theory}) *}
+
+end