
diff -r 07f53494cf20 -r ab8c54355f2e src/HOL/ex/Quickcheck.thy
--- a/src/HOL/ex/Quickcheck.thy	Fri Feb 06 13:43:19 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,413 +0,0 @@
-(* Author: Florian Haftmann, TU Muenchen *)
-
-header {* A simple counterexample generator *}
-
-theory Quickcheck
-imports Random Code_Eval Map
-begin
-
-subsection {* The @{text random} class *}
-
-class random = typerep +
-  fixes random :: "index \<Rightarrow> seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> seed"
-
-text {* Type @{typ "'a itself"} *}
-
-instantiation itself :: ("{type, typerep}") random
-begin
-
-definition
-  "random _ = return (TYPE('a), \<lambda>u. Code_Eval.Const (STR ''TYPE'') TYPEREP('a))"
-
-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 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 \<Rightarrow> typerep \<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 TYPEREP('a) TYPEREP('b) (op =) Code_Eval.term_of (random n) split_seed"
-
-instance ..
-
-end
-
-code_reserved SML Random_Engine
-
-text {* Datatypes *}
-
-definition
-  collapse :: "('a \<Rightarrow> ('a \<Rightarrow> 'b \<times> 'a) \<times> 'a) \<Rightarrow> 'a \<Rightarrow> 'b \<times> '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 \<Rightarrow> index \<Rightarrow> seed \<Rightarrow> ('a \<times> (unit \<Rightarrow> term)) \<times> seed"
-  assumes "random' 0 j = (\<lambda>s. 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)
-
-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 \<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 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\<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 = if Sign.of_sort thy (ty, @{sort random})
-        then ((ty, false), random ty $ @{term "j\<Colon>index"})
-        else raise TYP
-          ("Will not generate random elements for type(s) " ^ quote (hd tycos));
-      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"}, Abs ("s", @{typ seed},
-            Const (@{const_name undefined}, HOLogic.mk_prodT (term_ty this_ty, @{typ seed})))),
-          (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 ((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_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
-*}
-
-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. Code_Eval.App (Code_Eval.Const (STR ''Int.int'') TYPEREP(nat \<Rightarrow> int)) (t ()))
-       else (- int m, \<lambda>u. Code_Eval.App (Code_Eval.Const (STR ''HOL.uminus_class.uminus'') TYPEREP(int \<Rightarrow> int))
-         (Code_Eval.App (Code_Eval.Const (STR ''Int.int'') TYPEREP(nat \<Rightarrow> int)) (t ()))))
-   done)"
-
-instance ..
-
-end
-
-
-subsection {* Quickcheck generator *}
-
-ML {*
-structure Quickcheck =
-struct
-
-open Quickcheck;
-
-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 t =
-  let
-    val tys = (map snd o fst o strip_abs) t;
-    val t' = mk_generator_expr thy t tys;
-    val f = Code_ML.eval_term ("Quickcheck.eval_ref", eval_ref) thy t' [];
-  in f #> Random_Engine.run #> (Option.map o map) (Code.postprocess_term thy) end;
-
-end
-*}
-
-setup {*
-  Quickcheck.add_generator ("code", Quickcheck.compile_generator_expr o ProofContext.theory_of)
-*}
-
-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 \<Rightarrow> 'a) list \<Rightarrow> 'a \<Rightarrow> '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 \<Rightarrow> 'a list \<Rightarrow> nat" where
-  "occurs a [] = 0"
-  | "occurs a (x#xs) = (if (x=a) then Suc(occurs a xs) else occurs a xs)"
-
-primrec del1 :: "'a \<Rightarrow> 'a list \<Rightarrow> '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 ~= [] \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs"
-  quickcheck [generator = code]
-    -- {* Also wrong.*}
-  oops
-
-lemma "0 < occurs a xs \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs"
-  quickcheck [generator = code]
-  by (induct xs) auto
-
-primrec replace :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> '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 \<or> a=b \<longrightarrow> 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 \<Rightarrow> 'a list" where
-  "leaves Twig = []"
-  | "leaves (Leaf a) = [a]"
-  | "leaves (Branch l r) = (leaves l) @ (leaves r)"
-
-primrec plant :: "'a list \<Rightarrow> 'a tree" where
-  "plant [] = Twig "
-  | "plant (x#xs) = Branch (Leaf x) (plant xs)"
-
-primrec mirror :: "'a tree \<Rightarrow> '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 \<Rightarrow> 'a list" where
-  "inOrder (Tip a)= [a]"
-  | "inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)"
-
-primrec root :: "'a ntree \<Rightarrow> '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




