src/HOL/Tools/inductive_set_package.ML

+(*  Title:      HOL/Tools/inductive_set_package.ML
+    ID:         $Id$
+    Author:     Stefan Berghofer, TU Muenchen
+
+Wrapper for defining inductive sets using package for inductive predicates,
+including infrastructure for converting between predicates and sets.
+*)
+
+signature INDUCTIVE_SET_PACKAGE =
+sig
+  val to_set_att: thm list -> attribute
+  val to_pred_att: thm list -> attribute
+  val pred_set_conv_att: attribute
+  val add_inductive_i: bool -> bstring -> bool -> bool -> bool ->
+    (string * typ option * mixfix) list ->
+    (string * typ option) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
+      local_theory -> InductivePackage.inductive_result * local_theory
+  val add_inductive: bool -> bool -> (string * string option * mixfix) list ->
+    (string * string option * mixfix) list ->
+    ((bstring * Attrib.src list) * string) list -> (thmref * Attrib.src list) list ->
+    local_theory -> InductivePackage.inductive_result * local_theory
+  val setup: theory -> theory
+end;
+
+structure InductiveSetPackage: INDUCTIVE_SET_PACKAGE =
+struct
+
+val note_theorem = LocalTheory.note Thm.theoremK;
+
+
+(**** simplify {(x1, ..., xn). (x1, ..., xn) : S} to S ****)
+
+val subset_antisym = thm "subset_antisym";
+
+val collect_mem_simproc =
+  Simplifier.simproc (theory "Set") "Collect_mem" ["Collect t"] (fn thy => fn ss =>
+    fn S as Const ("Collect", Type ("fun", [_, T])) $ t =>
+         let val (u, Ts, ps) = HOLogic.strip_split t
+         in case u of
+           (c as Const ("op :", _)) $ q $ S' =>
+             (case try (HOLogic.dest_tuple' ps) q of
+                NONE => NONE
+              | SOME ts =>
+                  if not (loose_bvar (S', 0)) andalso
+                    ts = map Bound (length ps downto 0)
+                  then
+                    let val simp = full_simp_tac (Simplifier.inherit_context ss
+                      (HOL_basic_ss addsimps [split_paired_all, split_conv])) 1
+                    in
+                      SOME (Goal.prove (Simplifier.the_context ss) [] []
+                        (Const ("==", T --> T --> propT) $ S $ S')
+                        (K (EVERY
+                          [rtac eq_reflection 1, rtac subset_antisym 1,
+                           rtac subsetI 1, dtac CollectD 1, simp,
+                           rtac subsetI 1, rtac CollectI 1, simp])))
+                    end
+                  else NONE)
+         | _ => NONE
+         end
+     | _ => NONE);
+
+(***********************************************************************************)
+(* simplifies (%x y. (x, y) : S & P x y) to (%x y. (x, y) : S Int {(x, y). P x y}) *)
+(* and        (%x y. (x, y) : S | P x y) to (%x y. (x, y) : S Un {(x, y). P x y})  *)
+(* used for converting "strong" (co)induction rules                                *)
+(***********************************************************************************)
+
+val strong_ind_simproc =
+  Simplifier.simproc HOL.thy "strong_ind" ["t"] (fn thy => fn ss => fn t =>
+    let
+      val xs = strip_abs_vars t;
+      fun close t = fold (fn x => fn u => all (fastype_of x) $ lambda x u)
+        (term_vars t) t;
+      fun mkop "op &" T x = SOME (Const ("op Int", T --> T --> T), x)
+        | mkop "op |" T x = SOME (Const ("op Un", T --> T --> T), x)
+        | mkop _ _ _ = NONE;
+      fun mk_collect p T t =
+        let val U = HOLogic.dest_setT T
+        in HOLogic.Collect_const U $
+          HOLogic.ap_split' (HOLogic.prod_factors p) U HOLogic.boolT t
+        end;
+      fun decomp (Const (s, _) $ ((m as Const ("op :",
+            Type (_, [_, Type (_, [T, _])]))) $ p $ S) $ u) =
+              mkop s T (m, p, S, mk_collect p T (head_of u))
+        | decomp (Const (s, _) $ u $ ((m as Const ("op :",
+            Type (_, [_, Type (_, [T, _])]))) $ p $ S)) =
+              mkop s T (m, p, mk_collect p T (head_of u), S)
+        | decomp _ = NONE;
+      val simp = full_simp_tac (Simplifier.inherit_context ss
+        (HOL_basic_ss addsimps [mem_Collect_eq, split_conv])) 1;
+    in
+      if null xs then NONE
+      else case decomp (strip_abs_body t) of
+        NONE => NONE
+      | SOME (bop, (m, p, S, S')) =>
+          SOME (mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
+            (close (HOLogic.mk_Trueprop (HOLogic.mk_eq
+              (t, list_abs (xs, m $ p $ (bop $ S $ S'))))))
+            (K (EVERY
+              [REPEAT (rtac ext 1), rtac iffI 1,
+               EVERY [etac conjE 1, rtac IntI 1, simp, simp,
+                 etac IntE 1, rtac conjI 1, simp, simp] ORELSE
+               EVERY [etac disjE 1, rtac UnI1 1, simp, rtac UnI2 1, simp,
+                 etac UnE 1, rtac disjI1 1, simp, rtac disjI2 1, simp]]))))
+            handle ERROR _ => NONE
+    end);
+
+(* only eta contract terms occurring as arguments of functions satisfying p *)
+fun eta_contract p =
+  let
+    fun eta b (Abs (a, T, body)) =
+          (case eta b body of
+             body' as (f $ Bound 0) =>
+               if loose_bvar1 (f, 0) orelse not b then Abs (a, T, body')
+               else incr_boundvars ~1 f
+           | body' => Abs (a, T, body'))
+      | eta b (t $ u) = eta b t $ eta (p (head_of t)) u
+      | eta b t = t
+  in eta false end;
+
+fun eta_contract_thm p =
+  Conv.fconv_rule (Conv.then_conv (Thm.beta_conversion true, fn ct =>
+    Thm.transitive (Thm.eta_conversion ct)
+      (Thm.symmetric (Thm.eta_conversion
+        (cterm_of (theory_of_cterm ct) (eta_contract p (term_of ct)))))));
+
+
+(***********************************************************)
+(* rules for converting between predicate and set notation *)
+(*                                                         *)
+(* rules for converting predicates to sets have the form   *)
+(* P (%x y. (x, y) : s) = (%x y. (x, y) : S s)             *)
+(*                                                         *)
+(* rules for converting sets to predicates have the form   *)
+(* S {(x, y). p x y} = {(x, y). P p x y}                   *)
+(*                                                         *)
+(* where s and p are parameters                            *)
+(***********************************************************)
+
+structure PredSetConvData = GenericDataFun
+(
+  type T =
+    {(* rules for converting predicates to sets *)
+     to_set_simps: thm list,
+     (* rules for converting sets to predicates *)
+     to_pred_simps: thm list,
+     (* arities of functions of type t set => ... => u set *)
+     set_arities: (typ * (int list list option list * int list list option)) list Symtab.table,
+     (* arities of functions of type (t => ... => bool) => u => ... => bool *)
+     pred_arities: (typ * (int list list option list * int list list option)) list Symtab.table};
+  val empty = {to_set_simps = [], to_pred_simps = [],
+    set_arities = Symtab.empty, pred_arities = Symtab.empty};
+  val extend = I;
+  fun merge _
+    ({to_set_simps = to_set_simps1, to_pred_simps = to_pred_simps1,
+      set_arities = set_arities1, pred_arities = pred_arities1},
+     {to_set_simps = to_set_simps2, to_pred_simps = to_pred_simps2,
+      set_arities = set_arities2, pred_arities = pred_arities2}) =
+    {to_set_simps = Drule.merge_rules (to_set_simps1, to_set_simps2),
+     to_pred_simps = Drule.merge_rules (to_pred_simps1, to_pred_simps2),
+     set_arities = Symtab.merge_list op = (set_arities1, set_arities2),
+     pred_arities = Symtab.merge_list op = (pred_arities1, pred_arities2)};
+);
+
+fun name_type_of (Free p) = SOME p
+  | name_type_of (Const p) = SOME p
+  | name_type_of _ = NONE;
+
+fun map_type f (Free (s, T)) = Free (s, f T)
+  | map_type f (Var (ixn, T)) = Var (ixn, f T)
+  | map_type f _ = error "map_type";
+
+fun find_most_specific is_inst f eq xs T =
+  find_first (fn U => is_inst (T, f U)
+    andalso forall (fn U' => eq (f U, f U') orelse not
+      (is_inst (T, f U') andalso is_inst (f U', f U)))
+        xs) xs;
+
+fun lookup_arity thy arities (s, T) = case Symtab.lookup arities s of
+    NONE => NONE
+  | SOME xs => find_most_specific (Sign.typ_instance thy) fst (op =) xs T;
+
+fun lookup_rule thy f rules = find_most_specific
+  (swap #> Pattern.matches thy) (f #> fst) (op aconv) rules;
+
+fun infer_arities thy arities (optf, t) fs = case strip_comb t of
+    (Abs (s, T, u), []) => infer_arities thy arities (NONE, u) fs
+  | (Abs _, _) => infer_arities thy arities (NONE, Envir.beta_norm t) fs
+  | (u, ts) => (case Option.map (lookup_arity thy arities) (name_type_of u) of
+      SOME (SOME (_, (arity, _))) =>
+        (fold (infer_arities thy arities) (arity ~~ List.take (ts, length arity)) fs
+           handle Subscript => error "infer_arities: bad term")
+    | _ => fold (infer_arities thy arities) (map (pair NONE) ts)
+      (case optf of
+         NONE => fs
+       | SOME f => AList.update op = (u, the_default f
+           (Option.map (curry op inter f) (AList.lookup op = fs u))) fs));
+
+
+(**************************************************************)
+(*    derive the to_pred equation from the to_set equation    *)
+(*                                                            *)
+(* 1. instantiate each set parameter with {(x, y). p x y}     *)
+(* 2. apply %P. {(x, y). P x y} to both sides of the equation *)
+(* 3. simplify                                                *)
+(**************************************************************)
+
+fun mk_to_pred_inst thy fs =
+  map (fn (x, ps) =>
+    let
+      val U = HOLogic.dest_setT (fastype_of x);
+      val x' = map_type (K (HOLogic.prodT_factors' ps U ---> HOLogic.boolT)) x
+    in
+      (cterm_of thy x,
+       cterm_of thy (HOLogic.Collect_const U $
+         HOLogic.ap_split' ps U HOLogic.boolT x'))
+    end) fs;
+
+fun mk_to_pred_eq p fs optfs' T thm =
+  let
+    val thy = theory_of_thm thm;
+    val insts = mk_to_pred_inst thy fs;
+    val thm' = Thm.instantiate ([], insts) thm;
+    val thm'' = (case optfs' of
+        NONE => thm' RS sym
+      | SOME fs' =>
+          let
+            val U = HOLogic.dest_setT (body_type T);
+            val Ts = HOLogic.prodT_factors' fs' U;
+            (* FIXME: should cterm_instantiate increment indexes? *)
+            val arg_cong' = Thm.incr_indexes (Thm.maxidx_of thm + 1) arg_cong;
+            val (arg_cong_f, _) = arg_cong' |> cprop_of |> Drule.strip_imp_concl |>
+              Thm.dest_comb |> snd |> Drule.strip_comb |> snd |> hd |> Thm.dest_comb
+          in
+            thm' RS (Drule.cterm_instantiate [(arg_cong_f,
+              cterm_of thy (Abs ("P", Ts ---> HOLogic.boolT,
+                HOLogic.Collect_const U $ HOLogic.ap_split' fs' U
+                  HOLogic.boolT (Bound 0))))] arg_cong' RS sym)
+          end)
+  in
+    Simplifier.simplify (HOL_basic_ss addsimps [mem_Collect_eq, split_conv]
+      addsimprocs [collect_mem_simproc]) thm'' |>
+        zero_var_indexes |> eta_contract_thm (equal p)
+  end;
+
+
+(**** declare rules for converting predicates to sets ****)
+
+fun add ctxt thm {to_set_simps, to_pred_simps, set_arities, pred_arities} =
+  case prop_of thm of
+    Const ("Trueprop", _) $ (Const ("op =", Type (_, [T, _])) $ lhs $ rhs) =>
+      (case body_type T of
+         Type ("bool", []) =>
+           let
+             val thy = Context.theory_of ctxt;
+             fun factors_of t fs = case strip_abs_body t of
+                 Const ("op :", _) $ u $ S =>
+                   if is_Free S orelse is_Var S then
+                     let val ps = HOLogic.prod_factors u
+                     in (SOME ps, (S, ps) :: fs) end
+                   else (NONE, fs)
+               | _ => (NONE, fs);
+             val (h, ts) = strip_comb lhs
+             val (pfs, fs) = fold_map factors_of ts [];
+             val ((h', ts'), fs') = (case rhs of
+                 Abs _ => (case strip_abs_body rhs of
+                     Const ("op :", _) $ u $ S =>
+                       (strip_comb S, SOME (HOLogic.prod_factors u))
+                   | _ => error "member symbol on right-hand side expected")
+               | _ => (strip_comb rhs, NONE))
+           in
+             case (name_type_of h, name_type_of h') of
+               (SOME (s, T), SOME (s', T')) =>
+                 (case Symtab.lookup set_arities s' of
+                    NONE => ()
+                  | SOME xs => if exists (fn (U, _) =>
+                        Sign.typ_instance thy (T', U) andalso
+                        Sign.typ_instance thy (U, T')) xs
+                      then
+                        error ("Clash of conversion rules for operator " ^ s')
+                      else ();
+                  {to_set_simps = thm :: to_set_simps,
+                   to_pred_simps =
+                     mk_to_pred_eq h fs fs' T' thm :: to_pred_simps,
+                   set_arities = Symtab.insert_list op = (s',
+                     (T', (map (AList.lookup op = fs) ts', fs'))) set_arities,
+                   pred_arities = Symtab.insert_list op = (s,
+                     (T, (pfs, fs'))) pred_arities})
+             | _ => error "set / predicate constant expected"
+           end
+       | _ => error "equation between predicates expected")
+  | _ => error "equation expected";
+
+val pred_set_conv_att = Thm.declaration_attribute
+  (fn thm => fn ctxt => PredSetConvData.map (add ctxt thm) ctxt);
+
+
+(**** convert theorem in set notation to predicate notation ****)
+
+fun is_pred tab t =
+  case Option.map (Symtab.lookup tab o fst) (name_type_of t) of
+    SOME (SOME _) => true | _ => false;
+
+fun to_pred_simproc rules =
+  let val rules' = map mk_meta_eq rules
+  in
+    Simplifier.simproc HOL.thy "to_pred" ["t"]
+      (fn thy => K (lookup_rule thy (prop_of #> Logic.dest_equals) rules'))
+  end;
+
+fun to_pred_proc thy rules t = case lookup_rule thy I rules t of
+    NONE => NONE
+  | SOME (lhs, rhs) =>
+      SOME (Envir.subst_vars
+        (Pattern.match thy (lhs, t) (Vartab.empty, Vartab.empty)) rhs);
+
+fun to_pred thms ctxt thm =
+  let
+    val thy = Context.theory_of ctxt;
+    val {to_pred_simps, set_arities, pred_arities, ...} =
+      fold (add ctxt) thms (PredSetConvData.get ctxt);
+    val fs = filter (is_Var o fst)
+      (infer_arities thy set_arities (NONE, prop_of thm) []);
+    (* instantiate each set parameter with {(x, y). p x y} *)
+    val insts = mk_to_pred_inst thy fs
+  in
+    thm |>
+    Thm.instantiate ([], insts) |>
+    Simplifier.full_simplify (HOL_basic_ss addsimprocs
+      [to_pred_simproc (mem_Collect_eq :: split_conv :: to_pred_simps)]) |>
+    eta_contract_thm (is_pred pred_arities)
+  end;
+
+val to_pred_att = Thm.rule_attribute o to_pred;
+    
+
+(**** convert theorem in predicate notation to set notation ****)
+
+fun to_set thms ctxt thm =
+  let
+    val thy = Context.theory_of ctxt;
+    val {to_set_simps, pred_arities, ...} =
+      fold (add ctxt) thms (PredSetConvData.get ctxt);
+    val fs = filter (is_Var o fst)
+      (infer_arities thy pred_arities (NONE, prop_of thm) []);
+    (* instantiate each predicate parameter with %x y. (x, y) : s *)
+    val insts = map (fn (x, ps) =>
+      let
+        val Ts = binder_types (fastype_of x);
+        val T = HOLogic.mk_tupleT ps Ts;
+        val x' = map_type (K (HOLogic.mk_setT T)) x
+      in
+        (cterm_of thy x,
+         cterm_of thy (list_abs (map (pair "x") Ts, HOLogic.mk_mem
+           (HOLogic.mk_tuple' ps T (map Bound (length ps downto 0)), x'))))
+      end) fs
+  in
+    Simplifier.full_simplify (HOL_basic_ss addsimps to_set_simps
+        addsimprocs [strong_ind_simproc])
+      (Thm.instantiate ([], insts) thm)
+  end;
+
+val to_set_att = Thm.rule_attribute o to_set;
+
+
+(**** preprocessor for code generator ****)
+
+fun codegen_preproc thy =
+  let
+    val {to_pred_simps, set_arities, pred_arities, ...} =
+      PredSetConvData.get (Context.Theory thy);
+    fun preproc thm =
+      if exists_Const (fn (s, _) => case Symtab.lookup set_arities s of
+          NONE => false
+        | SOME arities => exists (fn (_, (xs, _)) =>
+            forall is_none xs) arities) (prop_of thm)
+      then
+        thm |>
+        Simplifier.full_simplify (HOL_basic_ss addsimprocs
+          [to_pred_simproc (mem_Collect_eq :: split_conv :: to_pred_simps)]) |>
+        eta_contract_thm (is_pred pred_arities)
+      else thm
+  in map preproc end;
+
+fun code_ind_att optmod = to_pred_att [] #> InductiveCodegen.add optmod NONE;
+
+
+(**** definition of inductive sets ****)
+
+fun add_ind_set_def verbose alt_name coind no_elim no_ind cs
+    intros monos params cnames_syn ctxt =
+  let
+    val thy = ProofContext.theory_of ctxt;
+    val {set_arities, pred_arities, to_pred_simps, ...} =
+      PredSetConvData.get (Context.Proof ctxt);
+    fun infer (Abs (_, _, t)) = infer t
+      | infer (Const ("op :", _) $ t $ u) =
+          infer_arities thy set_arities (SOME (HOLogic.prod_factors t), u)
+      | infer (t $ u) = infer t #> infer u
+      | infer _ = I;
+    val new_arities = filter_out
+      (fn (x as Free (_, Type ("fun", _)), _) => x mem params
+        | _ => false) (fold (snd #> infer) intros []);
+    val params' = map (fn x => (case AList.lookup op = new_arities x of
+        SOME fs =>
+          let
+            val T = HOLogic.dest_setT (fastype_of x);
+            val Ts = HOLogic.prodT_factors' fs T;
+            val x' = map_type (K (Ts ---> HOLogic.boolT)) x
+          in
+            (x, (x',
+              (HOLogic.Collect_const T $
+                 HOLogic.ap_split' fs T HOLogic.boolT x',
+               list_abs (map (pair "x") Ts, HOLogic.mk_mem
+                 (HOLogic.mk_tuple' fs T (map Bound (length fs downto 0)),
+                  x)))))
+          end
+       | NONE => (x, (x, (x, x))))) params;
+    val (params1, (params2, params3)) =
+      params' |> map snd |> split_list ||> split_list;
+
+    (* equations for converting sets to predicates *)
+    val ((cs', cs_info), eqns) = cs |> map (fn c as Free (s, T) =>
+      let
+        val fs = the_default [] (AList.lookup op = new_arities c);
+        val U = HOLogic.dest_setT (body_type T);
+        val Ts = HOLogic.prodT_factors' fs U;
+        val c' = Free (s ^ "p",
+          map fastype_of params1 @ Ts ---> HOLogic.boolT)
+      in
+        ((c', (fs, U, Ts)),
+         (list_comb (c, params2),
+          HOLogic.Collect_const U $ HOLogic.ap_split' fs U HOLogic.boolT
+            (list_comb (c', params1))))
+      end) |> split_list |>> split_list;
+    val eqns' = eqns @
+      map (prop_of #> HOLogic.dest_Trueprop #> HOLogic.dest_eq)
+        (mem_Collect_eq :: split_conv :: to_pred_simps);
+
+    (* predicate version of the introduction rules *)
+    val intros' =
+      map (fn (name_atts, t) => (name_atts,
+        t |>
+        map_aterms (fn u =>
+          (case AList.lookup op = params' u of
+             SOME (_, (u', _)) => u'
+           | NONE => u)) |>
+        Pattern.rewrite_term thy [] [to_pred_proc thy eqns'] |>
+        eta_contract (member op = cs' orf is_pred pred_arities))) intros;
+    val cnames_syn' = map (fn (s, _) => (s ^ "p", NoSyn)) cnames_syn;
+    val monos' = map (to_pred [] (Context.Proof ctxt)) monos;
+    val ({preds, intrs, elims, raw_induct, ...}, ctxt1) =
+      InductivePackage.add_ind_def verbose "" coind
+        no_elim no_ind cs' intros' monos' params1 cnames_syn' ctxt;
+
+    (* define inductive sets using previously defined predicates *)
+    val (defs, ctxt2) = LocalTheory.defs Thm.internalK
+      (map (fn ((c_syn, (fs, U, _)), p) => (c_syn, (("", []),
+         fold_rev lambda params (HOLogic.Collect_const U $
+           HOLogic.ap_split' fs U HOLogic.boolT (list_comb (p, params3))))))
+         (cnames_syn ~~ cs_info ~~ preds)) ctxt1;
+
+    (* prove theorems for converting predicate to set notation *)
+    val ctxt3 = fold
+      (fn (((p, c as Free (s, _)), (fs, U, Ts)), (_, (_, def))) => fn ctxt =>
+        let val conv_thm =
+          Goal.prove ctxt (map (fst o dest_Free) params) []
+            (HOLogic.mk_Trueprop (HOLogic.mk_eq
+              (list_comb (p, params3),
+               list_abs (map (pair "x") Ts, HOLogic.mk_mem
+                 (HOLogic.mk_tuple' fs U (map Bound (length fs downto 0)),
+                  list_comb (c, params))))))
+            (K (REPEAT (rtac ext 1) THEN simp_tac (HOL_basic_ss addsimps
+              [def, mem_Collect_eq, split_conv]) 1))
+        in
+          ctxt |> note_theorem ((s ^ "p_" ^ s ^ "_eq",
+            [Attrib.internal (K pred_set_conv_att)]),
+              [conv_thm]) |> snd
+        end) (preds ~~ cs ~~ cs_info ~~ defs) ctxt2;
+
+    (* convert theorems to set notation *)
+    val rec_name = if alt_name = "" then
+      space_implode "_" (map fst cnames_syn) else alt_name;
+    val cnames = map (Sign.full_name (ProofContext.theory_of ctxt3) o #1) cnames_syn;  (* FIXME *)
+    val (intr_names, intr_atts) = split_list (map fst intros);
+    val raw_induct' = to_set [] (Context.Proof ctxt3) raw_induct;
+    val (intrs', elims', induct, ctxt4) =
+      InductivePackage.declare_rules rec_name coind no_ind cnames
+      (map (to_set [] (Context.Proof ctxt3)) intrs) intr_names intr_atts
+      (map (fn th => (to_set [] (Context.Proof ctxt3) th,
+         map fst (fst (RuleCases.get th)))) elims)
+      raw_induct' ctxt3
+  in
+    ({intrs = intrs', elims = elims', induct = induct,
+      raw_induct = raw_induct', preds = map fst defs},
+     ctxt4)
+  end;
+
+val add_inductive_i = InductivePackage.gen_add_inductive_i add_ind_set_def;
+val add_inductive = InductivePackage.gen_add_inductive add_ind_set_def;
+
+val mono_add_att = to_pred_att [] #> InductivePackage.mono_add;
+val mono_del_att = to_pred_att [] #> InductivePackage.mono_del;
+
+
+(** package setup **)
+
+(* setup theory *)
+
+val setup =
+  Attrib.add_attributes
+    [("pred_set_conv", Attrib.no_args pred_set_conv_att,
+      "declare rules for converting between predicate and set notation"),
+     ("to_set", Attrib.syntax (Attrib.thms >> to_set_att),
+      "convert rule to set notation"),
+     ("to_pred", Attrib.syntax (Attrib.thms >> to_pred_att),
+      "convert rule to predicate notation")] #>
+  Codegen.add_attribute "ind_set"
+    (Scan.option (Args.$$$ "target" |-- Args.colon |-- Args.name) >> code_ind_att) #>
+  Codegen.add_preprocessor codegen_preproc #>
+  Attrib.add_attributes [("mono_set", Attrib.add_del_args mono_add_att mono_del_att,
+    "declaration of monotonicity rule for set operators")] #>
+  Context.theory_map (Simplifier.map_ss (fn ss =>
+    ss addsimprocs [collect_mem_simproc]));
+
+(* outer syntax *)
+
+local structure P = OuterParse and K = OuterKeyword in
+
+val ind_set_decl = InductivePackage.gen_ind_decl add_ind_set_def;
+
+val inductive_setP =
+  OuterSyntax.command "inductive_set" "define inductive sets" K.thy_decl (ind_set_decl false);
+
+val coinductive_setP =
+  OuterSyntax.command "coinductive_set" "define coinductive sets" K.thy_decl (ind_set_decl true);
+
+val _ = OuterSyntax.add_parsers [inductive_setP, coinductive_setP];
+
+end;
+
+end;