(*  Title:      ZF/Tools/inductive_package.ML

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

 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 ->

     ((binding * term) * attribute list) list ->

     thm list * thm list * thm list * thm list -> theory -> theory * inductive_result

   val add_inductive: string list * string ->

     ((binding * string) * Attrib.src list) list ->

     (Facts.ref * Attrib.src list) list * (Facts.ref * Attrib.src list) list *

     (Facts.ref * Attrib.src list) list * (Facts.ref * 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)

   raw_intr_specs (monos, con_defs, type_intrs, type_elims) thy =

 let

   val _ = Theory.requires thy "Inductive_ZF" "(co)inductive definitions";

   val ctxt = ProofContext.init thy;

   val intr_specs = map (apfst (apfst Binding.name_of)) raw_intr_specs;

   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: " ^

                    Syntax.string_of_term ctxt 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 thy) 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: " ^

                 Syntax.string_of_term ctxt t);

   end;

   val dummy = assert_all is_Free rec_params

       (fn t => "Param in recursion term not a free variable: " ^

                Syntax.string_of_term ctxt t);

   (*** Construct the fixedpoint definition ***)

   val mk_variant = Name.variant (List.foldr OldTerm.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: " ^

                             Syntax.string_of_term ctxt 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 = OldTerm.term_frees intr \\ rec_params

         val zeq = FOLogic.mk_eq (Free(z',iT), #1 (rule_concl intr))

     in List.foldr FOLogic.mk_exists

              (BalancedTree.make 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         = @{const Part} $ Free(X',iT) $ Abs(w',iT,h);

   (*Access to balanced disjoint sums via injections*)

   val parts = map mk_Part

     (BalancedTree.accesses {left = fn t => Su.inl $ t, right = fn t => Su.inr $ t, init = 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,

                               BalancedTree.make FOLogic.mk_disj part_intrs));

   val fp_rhs = Fp.oper $ dom_sum $ fp_abs

   val dummy = List.app (fn rec_hd => (Logic.occs (rec_hd, fp_rhs) andalso

                              error "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 thy big_rec_base_name;

   val _ =

     if verbose then

       writeln ((if coind then "Coind" else "Ind") ^ "uctive definition " ^ quote big_rec_name)

     else ();

   (*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 PrimitiveDefs.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

               writeln (cat_lines (map (Syntax.string_of_term ctxt o #2) axpairs))

           else ()

   (*add definitions of the inductive sets*)

   val (_, thy1) =

     thy

     |> Sign.add_path big_rec_base_name

     |> PureThy.add_defs false (map (Thm.no_attributes o apfst Binding.name) axpairs);

   val ctxt1 = ProofContext.init thy1;

   (*fetch fp definitions from the theory*)

   val big_rec_def::part_rec_defs =

     map (Thm.get_def thy1)

         (case rec_names of [_] => rec_names

                          | _   => big_rec_base_name::rec_names);

   (********)

   val dummy = writeln "  Proving monotonicity...";

   val bnd_mono =

     Goal.prove_global thy1 [] [] (FOLogic.mk_Trueprop (Fp.bnd_mono $ dom_sum $ fp_abs))

       (fn _ => EVERY

         [rtac (@{thm Collect_subset} RS @{thm bnd_monoI}) 1,

          REPEAT (ares_tac (@{thms 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 (@{thm Part_subset} RS @{thm 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 [@{thm subsetD}];

   (*Type-checking is hardest aspect of proof;

     disjIn selects the correct disjunct after unfolding*)

   fun intro_tacsf disjIn =

     [DETERM (stac unfold 1),

      REPEAT (resolve_tac [@{thm Part_eqI}, @{thm 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 @{thm 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::@{thm PartE}::@{thm 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 (@{thm SigmaI}::@{thm subsetI}::type_intrs) 1)];

   (*combines disjI1 and disjI2 to get the corresponding nested disjunct...*)

   val mk_disj_rls = BalancedTree.accesses

     {left = fn rl => rl RS @{thm disjI1},

      right = fn rl => rl RS @{thm disjI2},

      init = @{thm asm_rl}};

   val intrs =

     (intr_tms, map intro_tacsf (mk_disj_rls (length intr_tms)))

     |> ListPair.map (fn (t, tacs) =>

       Goal.prove_global thy1 [] [] t

         (fn _ => EVERY (rewrite_goals_tac part_rec_defs :: tacs)))

     handle MetaSimplifier.SIMPLIFIER (msg, thm) => (Display.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 (PRIMITIVE (fold_rule 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 ())

     let val thy = Thm.theory_of_thm elim in cterm_of thy (Syntax.read_prop_global thy a) end;

   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 (@{const_name Trueprop}, _) $

                       (Const (@{const_name mem}, _) $ 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, @{const Collect} $ 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 (OldTerm.term_frees intr \\ rec_params)

            val iprems = List.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 Syntax.string_of_term ctxt1) ind_prems;

                   writeln "raw_induct = "; Display.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 = Simplifier.theory_context thy 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 =

        Goal.prove_global thy1 [] ind_prems

          (FOLogic.mk_Trueprop (Ind_Syntax.mk_all_imp (big_rec_tm, pred)))

          (fn {prems, ...} => EVERY

            [rewrite_goals_tac part_rec_defs,

             rtac (@{thm impI} RS @{thm allI}) 1,

             DETERM (etac raw_induct 1),

             (*Push Part inside Collect*)

             full_simp_tac (min_ss addsimps [@{thm Part_Collect}]) 1,

             (*This CollectE and disjE separates out the introduction rules*)

             REPEAT (FIRSTGOAL (eresolve_tac [@{thm CollectE}, @{thm disjE}])),

             (*Now break down the individual cases.  No disjE here in case

               some premise involves disjunction.*)

             REPEAT (FIRSTGOAL (eresolve_tac [@{thm CollectE}, @{thm exE}, @{thm conjE}]

                                ORELSE' bound_hyp_subst_tac)),

             ind_tac (rev (map (rewrite_rule part_rec_defs) prems)) (length prems)]);

      val dummy = if !Ind_Syntax.trace then

                  (writeln "quant_induct = "; Display.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 =

              List.foldr FOLogic.mk_all

                (FOLogic.imp $

                 (@{const mem} $ 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 $ (@{const mem} $ 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,

                  BalancedTree.make FOLogic.mk_conj

                  (ListPair.map mk_rec_imp (rec_tms, preds)))))

      and mutual_induct_concl =

       FOLogic.mk_Trueprop(BalancedTree.make FOLogic.mk_conj qconcls);

      val dummy = if !Ind_Syntax.trace then

                  (writeln ("induct_concl = " ^

                            Syntax.string_of_term ctxt1 induct_concl);

                   writeln ("mutual_induct_concl = " ^

                            Syntax.string_of_term ctxt1 mutual_induct_concl))

              else ();

      val lemma_tac = FIRST' [eresolve_tac [@{thm asm_rl}, @{thm conjE}, @{thm PartE}, @{thm mp}],

                              resolve_tac [@{thm allI}, @{thm impI}, @{thm conjI}, @{thm Part_eqI}],

                              dresolve_tac [@{thm spec}, @{thm 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...";

            Goal.prove_global thy1 [] []

              (Logic.mk_implies (induct_concl, mutual_induct_concl))

              (fn _ => EVERY

                [rewrite_goals_tac part_rec_defs,

                 REPEAT (rewrite_goals_tac [Pr.split_eq] THEN lemma_tac 1)]))

        else (writeln "  [ No mutual induction rule needed ]"; @{thm TrueI});

      val dummy = if !Ind_Syntax.trace then

                  (writeln "lemma = "; Display.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 = [@{thm subset_refl} RS @{thm Collect_mono}] RL monos

                    RLN (2,[@{thm 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 [@{thm 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 @{thms conj_simps} @ @{thms imp_simps} @ @{thms 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

          Goal.prove_global thy1 [] (map (induct_prem (rec_tms~~preds)) intr_tms)

            mutual_induct_concl

            (fn {prems, ...} => EVERY

              [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 thy1 (Var(("x",1), Ind_Syntax.iT)),

                                       cterm_of thy1 elem_tuple)]);

      (*strip quantifier and the implication*)

      val induct0 = inst (quant_induct RS spec RSN (2, @{thm rev_mp}));

      val Const (@{const_name 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 ([induct', mutual_induct'], thy') =

        thy

        |> PureThy.add_thms [((Binding.name (co_prefix ^ "induct"), induct),

              [case_names, Induct.induct_pred big_rec_name]),

            ((Binding.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 [((Binding.name (co_prefix ^ "induct"), raw_induct), [])]

       |> apfst hd |> Library.swap, TrueI)

   and defs = big_rec_def :: part_rec_defs

   val (([bnd_mono', dom_subset', elim'], [defs', intrs']), thy3) =

     thy2

     |> IndCases.declare big_rec_name make_cases

     |> PureThy.add_thms

       [((Binding.name "bnd_mono", bnd_mono), []),

        ((Binding.name "dom_subset", dom_subset), []),

        ((Binding.name "cases", elim), [case_names, Induct.cases_pred big_rec_name])]

     ||>> (PureThy.add_thmss o map Thm.no_attributes)

         [(Binding.name "defs", defs),

          (Binding.name "intros", intrs)];

   val (intrs'', thy4) =

     thy3

     |> PureThy.add_thms ((map Binding.name intr_names ~~ intrs') ~~ map #2 intr_specs)

     ||> Sign.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;

 (*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 ctxt = ProofContext.init thy;

     val read_terms = map (Syntax.parse_term ctxt #> TypeInfer.constrain Ind_Syntax.iT)

       #> Syntax.check_terms ctxt;

     val intr_atts = map (map (Attrib.attribute thy) o snd) intr_srcs;

     val sintrs = map fst intr_srcs ~~ intr_atts;

     val rec_tms = read_terms srec_tms;

     val dom_sum = singleton read_terms sdom_sum;

     val intr_tms = Syntax.read_props ctxt (map (snd o fst) sintrs);

     val intr_specs = (map (fst o fst) sintrs ~~ intr_tms) ~~ map snd sintrs;

     val monos = Attrib.eval_thms ctxt raw_monos;

     val con_defs = Attrib.eval_thms ctxt raw_con_defs;

     val type_intrs = Attrib.eval_thms ctxt raw_type_intrs;

     val type_elims = Attrib.eval_thms ctxt raw_type_elims;

   in

     thy

     |> add_inductive_i true (rec_tms, dom_sum) intr_specs (monos, con_defs, type_intrs, type_elims)

   end;

 (* outer syntax *)

 local structure P = OuterParse and K = OuterKeyword in

 val _ = List.app OuterKeyword.keyword

   ["domains", "intros", "monos", "con_defs", "type_intros", "type_elims"];

 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 (SpecParse.opt_thm_name ":" -- P.prop))) --

   Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.xthms1) [] --

   Scan.optional (P.$$$ "con_defs" |-- P.!!! SpecParse.xthms1) [] --

   Scan.optional (P.$$$ "type_intros" |-- P.!!! SpecParse.xthms1) [] --

   Scan.optional (P.$$$ "type_elims" |-- P.!!! SpecParse.xthms1) []

   >> (Toplevel.theory o mk_ind);

 val _ = OuterSyntax.command (co_prefix ^ "inductive")

   ("define " ^ co_prefix ^ "inductive sets") K.thy_decl ind_decl;

 end;

 end;