File inductive_package.ML (Isabelle2005: October 2005)

# File inductive_package.ML

    (*  Title:      ZF/Tools/inductive_package.ML
        ID:         $Id: inductive_package.ML,v 1.33 2005/09/08 14:18:09 haftmann Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1994  University of Cambridge

    Fixedpoint definition module -- for Inductive/Coinductive Definitions

    The functor will be instantiated for normal sums/products (inductive defs)
                             and non-standard sums/products (coinductive defs)

    Sums are used only for mutual recursion;
    Products are used only to derive "streamlined" induction rules for relations
    *)

    type inductive_result =
       {defs       : thm list,             (*definitions made in thy*)
        bnd_mono   : thm,                  (*monotonicity for the lfp definition*)
        dom_subset : thm,                  (*inclusion of recursive set in dom*)
        intrs      : thm list,             (*introduction rules*)
        elim       : thm,                  (*case analysis theorem*)
        mk_cases   : string -> thm,        (*generates case theorems*)
        induct     : thm,                  (*main induction rule*)
        mutual_induct : thm};              (*mutual induction rule*)

    (*Functor's result signature*)
    signature INDUCTIVE_PACKAGE =
    sig
      (*Insert definitions for the recursive sets, which
         must *already* be declared as constants in parent theory!*)
      val add_inductive_i: bool -> term list * term ->
        ((bstring * term) * theory attribute list) list ->
        thm list * thm list * thm list * thm list -> theory -> theory * inductive_result
      val add_inductive_x: string list * string -> ((bstring * string) * theory attribute list) list
        -> thm list * thm list * thm list * thm list -> theory -> theory * inductive_result
      val add_inductive: string list * string ->
        ((bstring * string) * Attrib.src list) list ->
        (thmref * Attrib.src list) list * (thmref * Attrib.src list) list *
        (thmref * Attrib.src list) list * (thmref * Attrib.src list) list ->
        theory -> theory * inductive_result
    end;

    (*Declares functions to add fixedpoint/constructor defs to a theory.
      Recursive sets must *already* be declared as constants.*)
    functor Add_inductive_def_Fun
        (structure Fp: FP and Pr : PR and CP: CARTPROD and Su : SU val coind: bool)
     : INDUCTIVE_PACKAGE =
    struct

    open Ind_Syntax;

    val co_prefix = if coind then "co" else "";

    (* utils *)

    (*make distinct individual variables a1, a2, a3, ..., an. *)
    fun mk_frees a [] = []
      | mk_frees a (T::Ts) = Free(a,T) :: mk_frees (Symbol.bump_string a) Ts;

    (* add_inductive(_i) *)

    (*internal version, accepting terms*)
    fun add_inductive_i verbose (rec_tms, dom_sum)
      intr_specs (monos, con_defs, type_intrs, type_elims) thy =
    let
      val _ = Theory.requires thy "Inductive" "(co)inductive definitions";
      val sign = sign_of thy;

      val (intr_names, intr_tms) = split_list (map fst intr_specs);
      val case_names = RuleCases.case_names intr_names;

      (*recT and rec_params should agree for all mutually recursive components*)
      val rec_hds = map head_of rec_tms;

      val dummy = assert_all is_Const rec_hds
              (fn t => "Recursive set not previously declared as constant: " ^
                       Sign.string_of_term sign t);

      (*Now we know they are all Consts, so get their names, type and params*)
      val rec_names = map (#1 o dest_Const) rec_hds
      and (Const(_,recT),rec_params) = strip_comb (hd rec_tms);

      val rec_base_names = map Sign.base_name rec_names;
      val dummy = assert_all Syntax.is_identifier rec_base_names
        (fn a => "Base name of recursive set not an identifier: " ^ a);

      local (*Checking the introduction rules*)
        val intr_sets = map (#2 o rule_concl_msg sign) intr_tms;
        fun intr_ok set =
            case head_of set of Const(a,recT) => a mem rec_names | _ => false;
      in
        val dummy =  assert_all intr_ok intr_sets
           (fn t => "Conclusion of rule does not name a recursive set: " ^
                    Sign.string_of_term sign t);
      end;

      val dummy = assert_all is_Free rec_params
          (fn t => "Param in recursion term not a free variable: " ^
                   Sign.string_of_term sign t);

      (*** Construct the fixedpoint definition ***)
      val mk_variant = variant (foldr add_term_names [] intr_tms);

      val z' = mk_variant"z" and X' = mk_variant"X" and w' = mk_variant"w";

      fun dest_tprop (Const("Trueprop",_) $ P) = P
        | dest_tprop Q = error ("Ill-formed premise of introduction rule: " ^
                                Sign.string_of_term sign Q);

      (*Makes a disjunct from an introduction rule*)
      fun fp_part intr = (*quantify over rule's free vars except parameters*)
        let val prems = map dest_tprop (Logic.strip_imp_prems intr)
            val dummy = List.app (fn rec_hd => List.app (chk_prem rec_hd) prems) rec_hds
            val exfrees = term_frees intr \\ rec_params
            val zeq = FOLogic.mk_eq (Free(z',iT), #1 (rule_concl intr))
        in foldr FOLogic.mk_exists
                 (fold_bal FOLogic.mk_conj (zeq::prems)) exfrees
        end;

      (*The Part(A,h) terms -- compose injections to make h*)
      fun mk_Part (Bound 0) = Free(X',iT) (*no mutual rec, no Part needed*)
        | mk_Part h         = Part_const $ Free(X',iT) $ Abs(w',iT,h);

      (*Access to balanced disjoint sums via injections*)
      val parts =
          map mk_Part (accesses_bal (fn t => Su.inl $ t, fn t => Su.inr $ t, Bound 0)
                                    (length rec_tms));

      (*replace each set by the corresponding Part(A,h)*)
      val part_intrs = map (subst_free (rec_tms ~~ parts) o fp_part) intr_tms;

      val fp_abs = absfree(X', iT,
                       mk_Collect(z', dom_sum,
                                  fold_bal FOLogic.mk_disj part_intrs));

      val fp_rhs = Fp.oper $ dom_sum $ fp_abs

      val dummy = List.app (fn rec_hd => deny (Logic.occs (rec_hd, fp_rhs))
                                 "Illegal occurrence of recursion operator")
               rec_hds;

      (*** Make the new theory ***)

      (*A key definition:
        If no mutual recursion then it equals the one recursive set.
        If mutual recursion then it differs from all the recursive sets. *)
      val big_rec_base_name = space_implode "_" rec_base_names;
      val big_rec_name = Sign.intern_const sign big_rec_base_name;

      val dummy = conditional verbose (fn () =>
        writeln ((if coind then "Coind" else "Ind") ^ "uctive definition " ^ quote big_rec_name));

      (*Forbid the inductive definition structure from clashing with a theory
        name.  This restriction may become obsolete as ML is de-emphasized.*)
      val dummy = deny (big_rec_base_name mem (Context.names_of sign))
                   ("Definition " ^ big_rec_base_name ^
                    " would clash with the theory of the same name!");

      (*Big_rec... is the union of the mutually recursive sets*)
      val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);

      (*The individual sets must already be declared*)
      val axpairs = map Logic.mk_defpair
            ((big_rec_tm, fp_rhs) ::
             (case parts of
                 [_] => []                        (*no mutual recursion*)
               | _ => rec_tms ~~          (*define the sets as Parts*)
                      map (subst_atomic [(Free(X',iT),big_rec_tm)]) parts));

      (*tracing: print the fixedpoint definition*)
      val dummy = if !Ind_Syntax.trace then
                  List.app (writeln o Sign.string_of_term sign o #2) axpairs
              else ()

      (*add definitions of the inductive sets*)
      val thy1 = thy |> Theory.add_path big_rec_base_name
                     |> (#1 o PureThy.add_defs_i false (map Thm.no_attributes axpairs))

      (*fetch fp definitions from the theory*)
      val big_rec_def::part_rec_defs =
        map (get_def thy1)
            (case rec_names of [_] => rec_names
                             | _   => big_rec_base_name::rec_names);

      val sign1 = sign_of thy1;

      (********)
      val dummy = writeln "  Proving monotonicity...";

      val bnd_mono =
          prove_goalw_cterm []
            (cterm_of sign1
                      (FOLogic.mk_Trueprop (Fp.bnd_mono $ dom_sum $ fp_abs)))
            (fn _ =>
             [rtac (Collect_subset RS bnd_monoI) 1,
              REPEAT (ares_tac (basic_monos @ monos) 1)]);

      val dom_subset = standard (big_rec_def RS Fp.subs);

      val unfold = standard ([big_rec_def, bnd_mono] MRS Fp.Tarski);

      (********)
      val dummy = writeln "  Proving the introduction rules...";

      (*Mutual recursion?  Helps to derive subset rules for the
        individual sets.*)
      val Part_trans =
          case rec_names of
               [_] => asm_rl
             | _   => standard (Part_subset RS subset_trans);

      (*To type-check recursive occurrences of the inductive sets, possibly
        enclosed in some monotonic operator M.*)
      val rec_typechecks =
         [dom_subset] RL (asm_rl :: ([Part_trans] RL monos))
         RL [subsetD];

      (*Type-checking is hardest aspect of proof;
        disjIn selects the correct disjunct after unfolding*)
      fun intro_tacsf disjIn prems =
        [(*insert prems and underlying sets*)
         cut_facts_tac prems 1,
         DETERM (stac unfold 1),
         REPEAT (resolve_tac [Part_eqI,CollectI] 1),
         (*Now 2-3 subgoals: typechecking, the disjunction, perhaps equality.*)
         rtac disjIn 2,
         (*Not ares_tac, since refl must be tried before equality assumptions;
           backtracking may occur if the premises have extra variables!*)
         DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 2 APPEND assume_tac 2),
         (*Now solve the equations like Tcons(a,f) = Inl(?b4)*)
         rewrite_goals_tac con_defs,
         REPEAT (rtac refl 2),
         (*Typechecking; this can fail*)
         if !Ind_Syntax.trace then print_tac "The type-checking subgoal:"
         else all_tac,
         REPEAT (FIRSTGOAL (        dresolve_tac rec_typechecks
                            ORELSE' eresolve_tac (asm_rl::PartE::SigmaE2::
                                                  type_elims)
                            ORELSE' hyp_subst_tac)),
         if !Ind_Syntax.trace then print_tac "The subgoal after monos, type_elims:"
         else all_tac,
         DEPTH_SOLVE (swap_res_tac (SigmaI::subsetI::type_intrs) 1)];

      (*combines disjI1 and disjI2 to get the corresponding nested disjunct...*)
      val mk_disj_rls =
          let fun f rl = rl RS disjI1
              and g rl = rl RS disjI2
          in  accesses_bal(f, g, asm_rl)  end;

      fun prove_intr (ct, tacsf) = prove_goalw_cterm part_rec_defs ct tacsf;

      val intrs = ListPair.map prove_intr
                    (map (cterm_of sign1) intr_tms,
                     map intro_tacsf (mk_disj_rls(length intr_tms)))
                   handle MetaSimplifier.SIMPLIFIER (msg,thm) => (print_thm thm; error msg);

      (********)
      val dummy = writeln "  Proving the elimination rule...";

      (*Breaks down logical connectives in the monotonic function*)
      val basic_elim_tac =
          REPEAT (SOMEGOAL (eresolve_tac (Ind_Syntax.elim_rls @ Su.free_SEs)
                    ORELSE' bound_hyp_subst_tac))
          THEN prune_params_tac
              (*Mutual recursion: collapse references to Part(D,h)*)
          THEN fold_tac part_rec_defs;

      (*Elimination*)
      val elim = rule_by_tactic basic_elim_tac
                     (unfold RS Ind_Syntax.equals_CollectD)

      (*Applies freeness of the given constructors, which *must* be unfolded by
          the given defs.  Cannot simply use the local con_defs because
          con_defs=[] for inference systems.
        Proposition A should have the form t:Si where Si is an inductive set*)
      fun make_cases ss A =
        rule_by_tactic
          (basic_elim_tac THEN ALLGOALS (asm_full_simp_tac ss) THEN basic_elim_tac)
          (Thm.assume A RS elim)
          |> Drule.standard';
      fun mk_cases a = make_cases (*delayed evaluation of body!*)
        (simpset ()) (read_cterm (Thm.sign_of_thm elim) (a, propT));

      fun induction_rules raw_induct thy =
       let
         val dummy = writeln "  Proving the induction rule...";

         (*** Prove the main induction rule ***)

         val pred_name = "P";            (*name for predicate variables*)

         (*Used to make induction rules;
            ind_alist = [(rec_tm1,pred1),...] associates predicates with rec ops
            prem is a premise of an intr rule*)
         fun add_induct_prem ind_alist (prem as Const("Trueprop",_) $
                          (Const("op :",_)$t$X), iprems) =
              (case AList.lookup (op aconv) ind_alist X of
                   SOME pred => prem :: FOLogic.mk_Trueprop (pred $ t) :: iprems
                 | NONE => (*possibly membership in M(rec_tm), for M monotone*)
                     let fun mk_sb (rec_tm,pred) =
                                 (rec_tm, Ind_Syntax.Collect_const$rec_tm$pred)
                     in  subst_free (map mk_sb ind_alist) prem :: iprems  end)
           | add_induct_prem ind_alist (prem,iprems) = prem :: iprems;

         (*Make a premise of the induction rule.*)
         fun induct_prem ind_alist intr =
           let val quantfrees = map dest_Free (term_frees intr \\ rec_params)
               val iprems = foldr (add_induct_prem ind_alist) []
                                  (Logic.strip_imp_prems intr)
               val (t,X) = Ind_Syntax.rule_concl intr
               val (SOME pred) = AList.lookup (op aconv) ind_alist X
               val concl = FOLogic.mk_Trueprop (pred $ t)
           in list_all_free (quantfrees, Logic.list_implies (iprems,concl)) end
           handle Bind => error"Recursion term not found in conclusion";

         (*Minimizes backtracking by delivering the correct premise to each goal.
           Intro rules with extra Vars in premises still cause some backtracking *)
         fun ind_tac [] 0 = all_tac
           | ind_tac(prem::prems) i =
                 DEPTH_SOLVE_1 (ares_tac [prem, refl] i) THEN ind_tac prems (i-1);

         val pred = Free(pred_name, Ind_Syntax.iT --> FOLogic.oT);

         val ind_prems = map (induct_prem (map (rpair pred) rec_tms))
                             intr_tms;

         val dummy = if !Ind_Syntax.trace then
                     (writeln "ind_prems = ";
                      List.app (writeln o Sign.string_of_term sign1) ind_prems;
                      writeln "raw_induct = "; print_thm raw_induct)
                 else ();

         (*We use a MINIMAL simpset. Even FOL_ss contains too many simpules.
           If the premises get simplified, then the proofs could fail.*)
         val min_ss = empty_ss
               setmksimps (map mk_eq o ZF_atomize o gen_all)
               setSolver (mk_solver "minimal"
                          (fn prems => resolve_tac (triv_rls@prems)
                                       ORELSE' assume_tac
                                       ORELSE' etac FalseE));

         val quant_induct =
             prove_goalw_cterm part_rec_defs
               (cterm_of sign1
                (Logic.list_implies
                 (ind_prems,
                  FOLogic.mk_Trueprop (Ind_Syntax.mk_all_imp(big_rec_tm,pred)))))
               (fn prems =>
                [rtac (impI RS allI) 1,
                 DETERM (etac raw_induct 1),
                 (*Push Part inside Collect*)
                 full_simp_tac (min_ss addsimps [Part_Collect]) 1,
                 (*This CollectE and disjE separates out the introduction rules*)
                 REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE])),
                 (*Now break down the individual cases.  No disjE here in case
                   some premise involves disjunction.*)
                 REPEAT (FIRSTGOAL (eresolve_tac [CollectE, exE, conjE]
    	                        ORELSE' bound_hyp_subst_tac)),
                 ind_tac (rev prems) (length prems) ]);

         val dummy = if !Ind_Syntax.trace then
                     (writeln "quant_induct = "; print_thm quant_induct)
                 else ();

         (*** Prove the simultaneous induction rule ***)

         (*Make distinct predicates for each inductive set*)

         (*The components of the element type, several if it is a product*)
         val elem_type = CP.pseudo_type dom_sum;
         val elem_factors = CP.factors elem_type;
         val elem_frees = mk_frees "za" elem_factors;
         val elem_tuple = CP.mk_tuple Pr.pair elem_type elem_frees;

         (*Given a recursive set and its domain, return the "fsplit" predicate
           and a conclusion for the simultaneous induction rule.
           NOTE.  This will not work for mutually recursive predicates.  Previously
           a summand 'domt' was also an argument, but this required the domain of
           mutual recursion to invariably be a disjoint sum.*)
         fun mk_predpair rec_tm =
           let val rec_name = (#1 o dest_Const o head_of) rec_tm
               val pfree = Free(pred_name ^ "_" ^ Sign.base_name rec_name,
                                elem_factors ---> FOLogic.oT)
               val qconcl =
                 foldr FOLogic.mk_all
                   (FOLogic.imp $
                    (Ind_Syntax.mem_const $ elem_tuple $ rec_tm)
                          $ (list_comb (pfree, elem_frees))) elem_frees
           in  (CP.ap_split elem_type FOLogic.oT pfree,
                qconcl)
           end;

         val (preds,qconcls) = split_list (map mk_predpair rec_tms);

         (*Used to form simultaneous induction lemma*)
         fun mk_rec_imp (rec_tm,pred) =
             FOLogic.imp $ (Ind_Syntax.mem_const $ Bound 0 $ rec_tm) $
                              (pred $ Bound 0);

         (*To instantiate the main induction rule*)
         val induct_concl =
             FOLogic.mk_Trueprop
               (Ind_Syntax.mk_all_imp
                (big_rec_tm,
                 Abs("z", Ind_Syntax.iT,
                     fold_bal FOLogic.mk_conj
                     (ListPair.map mk_rec_imp (rec_tms, preds)))))
         and mutual_induct_concl =
          FOLogic.mk_Trueprop(fold_bal FOLogic.mk_conj qconcls);

         val dummy = if !Ind_Syntax.trace then
                     (writeln ("induct_concl = " ^
                               Sign.string_of_term sign1 induct_concl);
                      writeln ("mutual_induct_concl = " ^
                               Sign.string_of_term sign1 mutual_induct_concl))
                 else ();

         val lemma_tac = FIRST' [eresolve_tac [asm_rl, conjE, PartE, mp],
                                 resolve_tac [allI, impI, conjI, Part_eqI],
                                 dresolve_tac [spec, mp, Pr.fsplitD]];

         val need_mutual = length rec_names > 1;

         val lemma = (*makes the link between the two induction rules*)
           if need_mutual then
              (writeln "  Proving the mutual induction rule...";
               prove_goalw_cterm part_rec_defs
                     (cterm_of sign1 (Logic.mk_implies (induct_concl,
                                                       mutual_induct_concl)))
                     (fn prems =>
                      [cut_facts_tac prems 1,
                       REPEAT (rewrite_goals_tac [Pr.split_eq] THEN
                               lemma_tac 1)]))
           else (writeln "  [ No mutual induction rule needed ]";
                 TrueI);

         val dummy = if !Ind_Syntax.trace then
                     (writeln "lemma = "; print_thm lemma)
                 else ();

         (*Mutual induction follows by freeness of Inl/Inr.*)

         (*Simplification largely reduces the mutual induction rule to the
           standard rule*)
         val mut_ss =
             min_ss addsimps [Su.distinct, Su.distinct', Su.inl_iff, Su.inr_iff];

         val all_defs = con_defs @ part_rec_defs;

         (*Removes Collects caused by M-operators in the intro rules.  It is very
           hard to simplify
             list({v: tf. (v : t --> P_t(v)) & (v : f --> P_f(v))})
           where t==Part(tf,Inl) and f==Part(tf,Inr) to  list({v: tf. P_t(v)}).
           Instead the following rules extract the relevant conjunct.
         *)
         val cmonos = [subset_refl RS Collect_mono] RL monos
                       RLN (2,[rev_subsetD]);

         (*Minimizes backtracking by delivering the correct premise to each goal*)
         fun mutual_ind_tac [] 0 = all_tac
           | mutual_ind_tac(prem::prems) i =
               DETERM
                (SELECT_GOAL
                   (
                    (*Simplify the assumptions and goal by unfolding Part and
                      using freeness of the Sum constructors; proves all but one
                      conjunct by contradiction*)
                    rewrite_goals_tac all_defs  THEN
                    simp_tac (mut_ss addsimps [Part_iff]) 1  THEN
                    IF_UNSOLVED (*simp_tac may have finished it off!*)
                      ((*simplify assumptions*)
                       (*some risk of excessive simplification here -- might have
                         to identify the bare minimum set of rewrites*)
                       full_simp_tac
                          (mut_ss addsimps conj_simps @ imp_simps @ quant_simps) 1
                       THEN
                       (*unpackage and use "prem" in the corresponding place*)
                       REPEAT (rtac impI 1)  THEN
                       rtac (rewrite_rule all_defs prem) 1  THEN
                       (*prem must not be REPEATed below: could loop!*)
                       DEPTH_SOLVE (FIRSTGOAL (ares_tac [impI] ORELSE'
                                               eresolve_tac (conjE::mp::cmonos))))
                   ) i)
                THEN mutual_ind_tac prems (i-1);

         val mutual_induct_fsplit =
           if need_mutual then
             prove_goalw_cterm []
                   (cterm_of sign1
                    (Logic.list_implies
                       (map (induct_prem (rec_tms~~preds)) intr_tms,
                        mutual_induct_concl)))
                   (fn prems =>
                    [rtac (quant_induct RS lemma) 1,
                     mutual_ind_tac (rev prems) (length prems)])
           else TrueI;

         (** Uncurrying the predicate in the ordinary induction rule **)

         (*instantiate the variable to a tuple, if it is non-trivial, in order to
           allow the predicate to be "opened up".
           The name "x.1" comes from the "RS spec" !*)
         val inst =
             case elem_frees of [_] => I
                | _ => instantiate ([], [(cterm_of sign1 (Var(("x",1), Ind_Syntax.iT)),
                                          cterm_of sign1 elem_tuple)]);

         (*strip quantifier and the implication*)
         val induct0 = inst (quant_induct RS spec RSN (2,rev_mp));

         val Const ("Trueprop", _) $ (pred_var $ _) = concl_of induct0

         val induct = CP.split_rule_var(pred_var, elem_type-->FOLogic.oT, induct0)
                      |> standard
         and mutual_induct = CP.remove_split mutual_induct_fsplit

         val (thy', [induct', mutual_induct']) = thy |> PureThy.add_thms
          [((co_prefix ^ "induct", induct), [case_names, InductAttrib.induct_set_global big_rec_name]),
           (("mutual_induct", mutual_induct), [case_names])];
        in ((thy', induct'), mutual_induct')
        end;  (*of induction_rules*)

      val raw_induct = standard ([big_rec_def, bnd_mono] MRS Fp.induct)

      val ((thy2, induct), mutual_induct) =
        if not coind then induction_rules raw_induct thy1
        else (thy1 |> PureThy.add_thms [((co_prefix ^ "induct", raw_induct), [])] |> apsnd hd, TrueI)
      and defs = big_rec_def :: part_rec_defs

      val (thy3, ([bnd_mono', dom_subset', elim'], [defs', intrs'])) =
        thy2
        |> IndCases.declare big_rec_name make_cases
        |> PureThy.add_thms
          [(("bnd_mono", bnd_mono), []),
           (("dom_subset", dom_subset), []),
           (("cases", elim), [case_names, InductAttrib.cases_set_global big_rec_name])]
        |>>> (PureThy.add_thmss o map Thm.no_attributes)
            [("defs", defs),
             ("intros", intrs)];
      val (thy4, intrs'') =
        thy3 |> PureThy.add_thms ((intr_names ~~ intrs') ~~ map #2 intr_specs)
        |>> Theory.parent_path;
      in
        (thy4,
          {defs = defs',
           bnd_mono = bnd_mono',
           dom_subset = dom_subset',
           intrs = intrs'',
           elim = elim',
           mk_cases = mk_cases,
           induct = induct,
           mutual_induct = mutual_induct})
      end;

    (*external version, accepting strings*)
    fun add_inductive_x (srec_tms, sdom_sum) sintrs (monos, con_defs, type_intrs, type_elims) thy =
      let
        val read = Sign.simple_read_term (Theory.sign_of thy);
        val rec_tms = map (read Ind_Syntax.iT) srec_tms;
        val dom_sum = read Ind_Syntax.iT sdom_sum;
        val intr_tms = map (read propT o snd o fst) sintrs;
        val intr_specs = (map (fst o fst) sintrs ~~ intr_tms) ~~ map snd sintrs;
      in
        add_inductive_i true (rec_tms, dom_sum) intr_specs
          (monos, con_defs, type_intrs, type_elims) thy
      end

    (*source version*)
    fun add_inductive (srec_tms, sdom_sum) intr_srcs
        (raw_monos, raw_con_defs, raw_type_intrs, raw_type_elims) thy =
      let
        val intr_atts = map (map (Attrib.global_attribute thy) o snd) intr_srcs;
        val (thy', (((monos, con_defs), type_intrs), type_elims)) = thy
          |> IsarThy.apply_theorems raw_monos
          |>>> IsarThy.apply_theorems raw_con_defs
          |>>> IsarThy.apply_theorems raw_type_intrs
          |>>> IsarThy.apply_theorems raw_type_elims;
      in
        add_inductive_x (srec_tms, sdom_sum) (map fst intr_srcs ~~ intr_atts)
          (monos, con_defs, type_intrs, type_elims) thy'
      end;

    (* outer syntax *)

    local structure P = OuterParse and K = OuterKeyword in

    fun mk_ind (((((doms, intrs), monos), con_defs), type_intrs), type_elims) =
      #1 o add_inductive doms (map P.triple_swap intrs) (monos, con_defs, type_intrs, type_elims);

    val ind_decl =
      (P.$$$ "domains" |-- P.!!! (P.enum1 "+" P.term --
          ((P.$$$ "⊆" || P.$$$ "<=") |-- P.term))) --
      (P.$$$ "intros" |--
        P.!!! (Scan.repeat1 (P.opt_thm_name ":" -- P.prop))) --
      Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1) [] --
      Scan.optional (P.$$$ "con_defs" |-- P.!!! P.xthms1) [] --
      Scan.optional (P.$$$ "type_intros" |-- P.!!! P.xthms1) [] --
      Scan.optional (P.$$$ "type_elims" |-- P.!!! P.xthms1) []
      >> (Toplevel.theory o mk_ind);

    val inductiveP = OuterSyntax.command (co_prefix ^ "inductive")
      ("define " ^ co_prefix ^ "inductive sets") K.thy_decl ind_decl;

    val _ = OuterSyntax.add_keywords
      ["domains", "intros", "monos", "con_defs", "type_intros", "type_elims"];
    val _ = OuterSyntax.add_parsers [inductiveP];

    end;

    end;