(*  Title:      ZF/Tools/inductive_package.ML

    ID:         $Id$

    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 gen_assoc (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) = gen_assoc (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.$$$ "\\<subseteq>" || P.$$$ "<=") |-- 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;