(*  Title:      HOL/Nominal/nominal_inductive2.ML

     Author:     Stefan Berghofer, TU Muenchen

 Infrastructure for proving equivariance and strong induction theorems

 for inductive predicates involving nominal datatypes.

 Experimental version that allows to avoid lists of atoms.

 *)

 signature NOMINAL_INDUCTIVE2 =

 sig

   val prove_strong_ind: string -> string option -> (string * string list) list ->

     local_theory -> Proof.state

 end

 structure NominalInductive2 : NOMINAL_INDUCTIVE2 =

 struct

 val inductive_forall_name = "HOL.induct_forall";

 val inductive_forall_def = @{thm induct_forall_def};

 val inductive_atomize = @{thms induct_atomize};

 val inductive_rulify = @{thms induct_rulify};

 fun rulify_term thy = Raw_Simplifier.rewrite_term thy inductive_rulify [];

 val atomize_conv =

   Raw_Simplifier.rewrite_cterm (true, false, false) (K (K NONE))

     (HOL_basic_ss addsimps inductive_atomize);

 val atomize_intr = Conv.fconv_rule (Conv.prems_conv ~1 atomize_conv);

 fun atomize_induct ctxt = Conv.fconv_rule (Conv.prems_conv ~1

   (Conv.params_conv ~1 (K (Conv.prems_conv ~1 atomize_conv)) ctxt));

 val fresh_postprocess =

   Simplifier.full_simplify (HOL_basic_ss addsimps

     [@{thm fresh_star_set_eq}, @{thm fresh_star_Un_elim},

      @{thm fresh_star_insert_elim}, @{thm fresh_star_empty_elim}]);

 fun preds_of ps t = inter (op = o apfst dest_Free) (Term.add_frees t []) ps;

 val perm_bool = mk_meta_eq @{thm perm_bool};

 val perm_boolI = @{thm perm_boolI};

 val (_, [perm_boolI_pi, _]) = Drule.strip_comb (snd (Thm.dest_comb

   (Drule.strip_imp_concl (cprop_of perm_boolI))));

 fun mk_perm_bool pi th = th RS Drule.cterm_instantiate

   [(perm_boolI_pi, pi)] perm_boolI;

 fun mk_perm_bool_simproc names = Simplifier.simproc_global_i

   (theory_of_thm perm_bool) "perm_bool" [@{term "perm pi x"}] (fn thy => fn ss =>

     fn Const ("Nominal.perm", _) $ _ $ t =>

          if member (op =) names (the_default "" (try (head_of #> dest_Const #> fst) t))

          then SOME perm_bool else NONE

      | _ => NONE);

 fun transp ([] :: _) = []

   | transp xs = map hd xs :: transp (map tl xs);

 fun add_binders thy i (t as (_ $ _)) bs = (case strip_comb t of

       (Const (s, T), ts) => (case strip_type T of

         (Ts, Type (tname, _)) =>

           (case NominalDatatype.get_nominal_datatype thy tname of

              NONE => fold (add_binders thy i) ts bs

            | SOME {descr, index, ...} => (case AList.lookup op =

                  (#3 (the (AList.lookup op = descr index))) s of

                NONE => fold (add_binders thy i) ts bs

              | SOME cargs => fst (fold (fn (xs, x) => fn (bs', cargs') =>

                  let val (cargs1, (u, _) :: cargs2) = chop (length xs) cargs'

                  in (add_binders thy i u

                    (fold (fn (u, T) =>

                       if exists (fn j => j < i) (loose_bnos u) then I

                       else AList.map_default op = (T, [])

                         (insert op aconv (incr_boundvars (~i) u)))

                           cargs1 bs'), cargs2)

                  end) cargs (bs, ts ~~ Ts))))

       | _ => fold (add_binders thy i) ts bs)

     | (u, ts) => add_binders thy i u (fold (add_binders thy i) ts bs))

   | add_binders thy i (Abs (_, _, t)) bs = add_binders thy (i + 1) t bs

   | add_binders thy i _ bs = bs;

 fun split_conj f names (Const (@{const_name HOL.conj}, _) $ p $ q) _ = (case head_of p of

       Const (name, _) =>

         if member (op =) names name then SOME (f p q) else NONE

     | _ => NONE)

   | split_conj _ _ _ _ = NONE;

 fun strip_all [] t = t

   | strip_all (_ :: xs) (Const (@{const_name All}, _) $ Abs (s, T, t)) = strip_all xs t;

 (*********************************************************************)

 (* maps  R ... & (ALL pi_1 ... pi_n z. P z (pi_1 o ... o pi_n o t))  *)

 (* or    ALL pi_1 ... pi_n z. P z (pi_1 o ... o pi_n o t)            *)

 (* to    R ... & id (ALL z. P z (pi_1 o ... o pi_n o t))             *)

 (* or    id (ALL z. P z (pi_1 o ... o pi_n o t))                     *)

 (*                                                                   *)

 (* where "id" protects the subformula from simplification            *)

 (*********************************************************************)

 fun inst_conj_all names ps pis (Const (@{const_name HOL.conj}, _) $ p $ q) _ =

       (case head_of p of

          Const (name, _) =>

            if member (op =) names name then SOME (HOLogic.mk_conj (p,

              Const ("Fun.id", HOLogic.boolT --> HOLogic.boolT) $

                (subst_bounds (pis, strip_all pis q))))

            else NONE

        | _ => NONE)

   | inst_conj_all names ps pis t u =

       if member (op aconv) ps (head_of u) then

         SOME (Const ("Fun.id", HOLogic.boolT --> HOLogic.boolT) $

           (subst_bounds (pis, strip_all pis t)))

       else NONE

   | inst_conj_all _ _ _ _ _ = NONE;

 fun inst_conj_all_tac k = EVERY

   [TRY (EVERY [etac conjE 1, rtac conjI 1, atac 1]),

    REPEAT_DETERM_N k (etac allE 1),

    simp_tac (HOL_basic_ss addsimps [@{thm id_apply}]) 1];

 fun map_term f t u = (case f t u of

       NONE => map_term' f t u | x => x)

 and map_term' f (t $ u) (t' $ u') = (case (map_term f t t', map_term f u u') of

       (NONE, NONE) => NONE

     | (SOME t'', NONE) => SOME (t'' $ u)

     | (NONE, SOME u'') => SOME (t $ u'')

     | (SOME t'', SOME u'') => SOME (t'' $ u''))

   | map_term' f (Abs (s, T, t)) (Abs (s', T', t')) = (case map_term f t t' of

       NONE => NONE

     | SOME t'' => SOME (Abs (s, T, t'')))

   | map_term' _ _ _ = NONE;

 (*********************************************************************)

 (*         Prove  F[f t]  from  F[t],  where F is monotone           *)

 (*********************************************************************)

 fun map_thm ctxt f tac monos opt th =

   let

     val prop = prop_of th;

     fun prove t =

       Goal.prove ctxt [] [] t (fn _ =>

         EVERY [cut_facts_tac [th] 1, etac rev_mp 1,

           REPEAT_DETERM (FIRSTGOAL (resolve_tac monos)),

           REPEAT_DETERM (rtac impI 1 THEN (atac 1 ORELSE tac))])

   in Option.map prove (map_term f prop (the_default prop opt)) end;

 fun abs_params params t =

   let val vs =  map (Var o apfst (rpair 0)) (Term.rename_wrt_term t params)

   in (list_all (params, t), (rev vs, subst_bounds (vs, t))) end;

 fun inst_params thy (vs, p) th cts =

   let val env = Pattern.first_order_match thy (p, prop_of th)

     (Vartab.empty, Vartab.empty)

   in Thm.instantiate ([],

     map (Envir.subst_term env #> cterm_of thy) vs ~~ cts) th

   end;

 fun prove_strong_ind s alt_name avoids ctxt =

   let

     val thy = Proof_Context.theory_of ctxt;

     val ({names, ...}, {raw_induct, intrs, elims, ...}) =

       Inductive.the_inductive ctxt (Sign.intern_const thy s);

     val ind_params = Inductive.params_of raw_induct;

     val raw_induct = atomize_induct ctxt raw_induct;

     val elims = map (atomize_induct ctxt) elims;

     val monos = Inductive.get_monos ctxt;

     val eqvt_thms = NominalThmDecls.get_eqvt_thms ctxt;

     val _ = (case subtract (op =) (fold (Term.add_const_names o Thm.prop_of) eqvt_thms []) names of

         [] => ()

       | xs => error ("Missing equivariance theorem for predicate(s): " ^

           commas_quote xs));

     val induct_cases = map fst (fst (Rule_Cases.get (the

       (Induct.lookup_inductP ctxt (hd names)))));

     val induct_cases' = if null induct_cases then replicate (length intrs) ""

       else induct_cases;

     val ([raw_induct'], ctxt') = Variable.import_terms false [prop_of raw_induct] ctxt;

     val concls = raw_induct' |> Logic.strip_imp_concl |> HOLogic.dest_Trueprop |>

       HOLogic.dest_conj |> map (HOLogic.dest_imp ##> strip_comb);

     val ps = map (fst o snd) concls;

     val _ = (case duplicates (op = o pairself fst) avoids of

         [] => ()

       | xs => error ("Duplicate case names: " ^ commas_quote (map fst xs)));

     val _ = (case subtract (op =) induct_cases (map fst avoids) of

         [] => ()

       | xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs));

     fun mk_avoids params name sets =

       let

         val (_, ctxt') = Proof_Context.add_fixes

           (map (fn (s, T) => (Binding.name s, SOME T, NoSyn)) params) ctxt;

         fun mk s =

           let

             val t = Syntax.read_term ctxt' s;

             val t' = list_abs_free (params, t) |>

               funpow (length params) (fn Abs (_, _, t) => t)

           in (t', HOLogic.dest_setT (fastype_of t)) end

           handle TERM _ =>

             error ("Expression " ^ quote s ^ " to be avoided in case " ^

               quote name ^ " is not a set type");

         fun add_set p [] = [p]

           | add_set (t, T) ((u, U) :: ps) =

               if T = U then

                 let val S = HOLogic.mk_setT T

                 in (Const (@{const_name sup}, S --> S --> S) $ u $ t, T) :: ps

                 end

               else (u, U) :: add_set (t, T) ps

       in

         fold (mk #> add_set) sets []

       end;

     val prems = map (fn (prem, name) =>

       let

         val prems = map (incr_boundvars 1) (Logic.strip_assums_hyp prem);

         val concl = incr_boundvars 1 (Logic.strip_assums_concl prem);

         val params = Logic.strip_params prem

       in

         (params,

          if null avoids then

            map (fn (T, ts) => (HOLogic.mk_set T ts, T))

              (fold (add_binders thy 0) (prems @ [concl]) [])

          else case AList.lookup op = avoids name of

            NONE => []

          | SOME sets =>

              map (apfst (incr_boundvars 1)) (mk_avoids params name sets),

          prems, strip_comb (HOLogic.dest_Trueprop concl))

       end) (Logic.strip_imp_prems raw_induct' ~~ induct_cases');

     val atomTs = distinct op = (maps (map snd o #2) prems);

     val atoms = map (fst o dest_Type) atomTs;

     val ind_sort = if null atomTs then HOLogic.typeS

       else Sign.minimize_sort thy (Sign.certify_sort thy (map (fn a => Sign.intern_class thy

         ("fs_" ^ Long_Name.base_name a)) atoms));

     val (fs_ctxt_tyname, _) = Name.variant "'n" (Variable.names_of ctxt');

     val ([fs_ctxt_name], ctxt'') = Variable.variant_fixes ["z"] ctxt';

     val fsT = TFree (fs_ctxt_tyname, ind_sort);

     val inductive_forall_def' = Drule.instantiate'

       [SOME (ctyp_of thy fsT)] [] inductive_forall_def;

     fun lift_pred' t (Free (s, T)) ts =

       list_comb (Free (s, fsT --> T), t :: ts);

     val lift_pred = lift_pred' (Bound 0);

     fun lift_prem (t as (f $ u)) =

           let val (p, ts) = strip_comb t

           in

             if member (op =) ps p then

               Const (inductive_forall_name,

                 (fsT --> HOLogic.boolT) --> HOLogic.boolT) $

                   Abs ("z", fsT, lift_pred p (map (incr_boundvars 1) ts))

             else lift_prem f $ lift_prem u

           end

       | lift_prem (Abs (s, T, t)) = Abs (s, T, lift_prem t)

       | lift_prem t = t;

     fun mk_fresh (x, T) = HOLogic.mk_Trueprop

       (NominalDatatype.fresh_star_const T fsT $ x $ Bound 0);

     val (prems', prems'') = split_list (map (fn (params, sets, prems, (p, ts)) =>

       let

         val params' = params @ [("y", fsT)];

         val prem = Logic.list_implies

           (map mk_fresh sets @

            map (fn prem =>

              if null (preds_of ps prem) then prem

              else lift_prem prem) prems,

            HOLogic.mk_Trueprop (lift_pred p ts));

       in abs_params params' prem end) prems);

     val ind_vars =

       (Datatype_Prop.indexify_names (replicate (length atomTs) "pi") ~~

        map NominalAtoms.mk_permT atomTs) @ [("z", fsT)];

     val ind_Ts = rev (map snd ind_vars);

     val concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

       (map (fn (prem, (p, ts)) => HOLogic.mk_imp (prem,

         HOLogic.list_all (ind_vars, lift_pred p

           (map (fold_rev (NominalDatatype.mk_perm ind_Ts)

             (map Bound (length atomTs downto 1))) ts)))) concls));

     val concl' = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

       (map (fn (prem, (p, ts)) => HOLogic.mk_imp (prem,

         lift_pred' (Free (fs_ctxt_name, fsT)) p ts)) concls));

     val (vc_compat, vc_compat') = map (fn (params, sets, prems, (p, ts)) =>

       map (fn q => abs_params params (incr_boundvars ~1 (Logic.list_implies

           (map_filter (fn prem =>

              if null (preds_of ps prem) then SOME prem

              else map_term (split_conj (K o I) names) prem prem) prems, q))))

         (maps (fn (t, T) => map (fn (u, U) => HOLogic.mk_Trueprop

            (NominalDatatype.fresh_star_const U T $ u $ t)) sets)

              (ts ~~ binder_types (fastype_of p)) @

          map (fn (u, U) => HOLogic.mk_Trueprop (Const (@{const_name finite},

            HOLogic.mk_setT U --> HOLogic.boolT) $ u)) sets) |>

       split_list) prems |> split_list;

     val perm_pi_simp = Global_Theory.get_thms thy "perm_pi_simp";

     val pt2_atoms = map (fn a => Global_Theory.get_thm thy

       ("pt_" ^ Long_Name.base_name a ^ "2")) atoms;

     val eqvt_ss = Simplifier.global_context thy HOL_basic_ss

       addsimps (eqvt_thms @ perm_pi_simp @ pt2_atoms)

       addsimprocs [mk_perm_bool_simproc ["Fun.id"],

         NominalPermeq.perm_simproc_app, NominalPermeq.perm_simproc_fun];

     val fresh_star_bij = Global_Theory.get_thms thy "fresh_star_bij";

     val pt_insts = map (NominalAtoms.pt_inst_of thy) atoms;

     val at_insts = map (NominalAtoms.at_inst_of thy) atoms;

     val dj_thms = maps (fn a =>

       map (NominalAtoms.dj_thm_of thy a) (remove (op =) a atoms)) atoms;

     val finite_ineq = map2 (fn th => fn th' => th' RS (th RS

       @{thm pt_set_finite_ineq})) pt_insts at_insts;

     val perm_set_forget =

       map (fn th => th RS @{thm dj_perm_set_forget}) dj_thms;

     val perm_freshs_freshs = atomTs ~~ map2 (fn th => fn th' => th' RS (th RS

       @{thm pt_freshs_freshs})) pt_insts at_insts;

     fun obtain_fresh_name ts sets (T, fin) (freshs, ths1, ths2, ths3, ctxt) =

       let

         val thy = Proof_Context.theory_of ctxt;

         (** protect terms to avoid that fresh_star_prod_set interferes with  **)

         (** pairs used in introduction rules of inductive predicate          **)

         fun protect t =

           let val T = fastype_of t in Const ("Fun.id", T --> T) $ t end;

         val p = foldr1 HOLogic.mk_prod (map protect ts);

         val atom = fst (dest_Type T);

         val {at_inst, ...} = NominalAtoms.the_atom_info thy atom;

         val fs_atom = Global_Theory.get_thm thy

           ("fs_" ^ Long_Name.base_name atom ^ "1");

         val avoid_th = Drule.instantiate'

           [SOME (ctyp_of thy (fastype_of p))] [SOME (cterm_of thy p)]

           ([at_inst, fin, fs_atom] MRS @{thm at_set_avoiding});

         val (([(_, cx)], th1 :: th2 :: ths), ctxt') = Obtain.result

           (fn _ => EVERY

             [rtac avoid_th 1,

              full_simp_tac (HOL_ss addsimps [@{thm fresh_star_prod_set}]) 1,

              full_simp_tac (HOL_basic_ss addsimps [@{thm id_apply}]) 1,

              rotate_tac 1 1,

              REPEAT (etac conjE 1)])

           [] ctxt;

         val (Ts1, _ :: Ts2) = take_prefix (not o equal T) (map snd sets);

         val pTs = map NominalAtoms.mk_permT (Ts1 @ Ts2);

         val (pis1, pis2) = chop (length Ts1)

           (map Bound (length pTs - 1 downto 0));

         val _ $ (f $ (_ $ pi $ l) $ r) = prop_of th2

         val th2' =

           Goal.prove ctxt [] []

             (list_all (map (pair "pi") pTs, HOLogic.mk_Trueprop

                (f $ fold_rev (NominalDatatype.mk_perm (rev pTs))

                   (pis1 @ pi :: pis2) l $ r)))

             (fn _ => cut_facts_tac [th2] 1 THEN

                full_simp_tac (HOL_basic_ss addsimps perm_set_forget) 1) |>

           Simplifier.simplify eqvt_ss

       in

         (freshs @ [term_of cx],

          ths1 @ ths, ths2 @ [th1], ths3 @ [th2'], ctxt')

       end;

     fun mk_ind_proof ctxt' thss =

       Goal.prove ctxt' [] prems' concl' (fn {prems = ihyps, context = ctxt} =>

         let val th = Goal.prove ctxt [] [] concl (fn {context, ...} =>

           rtac raw_induct 1 THEN

           EVERY (maps (fn (((((_, sets, oprems, _),

               vc_compat_ths), vc_compat_vs), ihyp), vs_ihypt) =>

             [REPEAT (rtac allI 1), simp_tac eqvt_ss 1,

              SUBPROOF (fn {prems = gprems, params, concl, context = ctxt', ...} =>

                let

                  val (cparams', (pis, z)) =

                    chop (length params - length atomTs - 1) (map #2 params) ||>

                    (map term_of #> split_last);

                  val params' = map term_of cparams'

                  val sets' = map (apfst (curry subst_bounds (rev params'))) sets;

                  val pi_sets = map (fn (t, _) =>

                    fold_rev (NominalDatatype.mk_perm []) pis t) sets';

                  val (P, ts) = strip_comb (HOLogic.dest_Trueprop (term_of concl));

                  val gprems1 = map_filter (fn (th, t) =>

                    if null (preds_of ps t) then SOME th

                    else

                      map_thm ctxt' (split_conj (K o I) names)

                        (etac conjunct1 1) monos NONE th)

                    (gprems ~~ oprems);

                  val vc_compat_ths' = map2 (fn th => fn p =>

                    let

                      val th' = gprems1 MRS inst_params thy p th cparams';

                      val (h, ts) =

                        strip_comb (HOLogic.dest_Trueprop (concl_of th'))

                    in

                      Goal.prove ctxt' [] []

                        (HOLogic.mk_Trueprop (list_comb (h,

                           map (fold_rev (NominalDatatype.mk_perm []) pis) ts)))

                        (fn _ => simp_tac (HOL_basic_ss addsimps

                           (fresh_star_bij @ finite_ineq)) 1 THEN rtac th' 1)

                    end) vc_compat_ths vc_compat_vs;

                  val (vc_compat_ths1, vc_compat_ths2) =

                    chop (length vc_compat_ths - length sets) vc_compat_ths';

                  val vc_compat_ths1' = map

                    (Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv

                       (Simplifier.rewrite eqvt_ss)))) vc_compat_ths1;

                  val (pis', fresh_ths1, fresh_ths2, fresh_ths3, ctxt'') = fold

                    (obtain_fresh_name ts sets)

                    (map snd sets' ~~ vc_compat_ths2) ([], [], [], [], ctxt');

                  fun concat_perm pi1 pi2 =

                    let val T = fastype_of pi1

                    in if T = fastype_of pi2 then

                        Const ("List.append", T --> T --> T) $ pi1 $ pi2

                      else pi2

                    end;

                  val pis'' = fold_rev (concat_perm #> map) pis' pis;

                  val ihyp' = inst_params thy vs_ihypt ihyp

                    (map (fold_rev (NominalDatatype.mk_perm [])

                       (pis' @ pis) #> cterm_of thy) params' @ [cterm_of thy z]);

                  fun mk_pi th =

                    Simplifier.simplify (HOL_basic_ss addsimps [@{thm id_apply}]

                        addsimprocs [NominalDatatype.perm_simproc])

                      (Simplifier.simplify eqvt_ss

                        (fold_rev (mk_perm_bool o cterm_of thy)

                          (pis' @ pis) th));

                  val gprems2 = map (fn (th, t) =>

                    if null (preds_of ps t) then mk_pi th

                    else

                      mk_pi (the (map_thm ctxt (inst_conj_all names ps (rev pis''))

                        (inst_conj_all_tac (length pis'')) monos (SOME t) th)))

                    (gprems ~~ oprems);

                  val perm_freshs_freshs' = map (fn (th, (_, T)) =>

                    th RS the (AList.lookup op = perm_freshs_freshs T))

                      (fresh_ths2 ~~ sets);

                  val th = Goal.prove ctxt'' [] []

                    (HOLogic.mk_Trueprop (list_comb (P $ hd ts,

                      map (fold_rev (NominalDatatype.mk_perm []) pis') (tl ts))))

                    (fn _ => EVERY ([simp_tac eqvt_ss 1, rtac ihyp' 1] @

                      map (fn th => rtac th 1) fresh_ths3 @

                      [REPEAT_DETERM_N (length gprems)

                        (simp_tac (HOL_basic_ss

                           addsimps [inductive_forall_def']

                           addsimprocs [NominalDatatype.perm_simproc]) 1 THEN

                         resolve_tac gprems2 1)]));

                  val final = Goal.prove ctxt'' [] [] (term_of concl)

                    (fn _ => cut_facts_tac [th] 1 THEN full_simp_tac (HOL_ss

                      addsimps vc_compat_ths1' @ fresh_ths1 @

                        perm_freshs_freshs') 1);

                  val final' = Proof_Context.export ctxt'' ctxt' [final];

                in resolve_tac final' 1 end) context 1])

                  (prems ~~ thss ~~ vc_compat' ~~ ihyps ~~ prems'')))

         in

           cut_facts_tac [th] 1 THEN REPEAT (etac conjE 1) THEN

           REPEAT (REPEAT (resolve_tac [conjI, impI] 1) THEN

             etac impE 1 THEN atac 1 THEN REPEAT (etac @{thm allE_Nil} 1) THEN

             asm_full_simp_tac (simpset_of ctxt) 1)

         end) |>

         fresh_postprocess |>

         singleton (Proof_Context.export ctxt' ctxt);

   in

     ctxt'' |>

     Proof.theorem NONE (fn thss => fn ctxt =>

       let

         val rec_name = space_implode "_" (map Long_Name.base_name names);

         val rec_qualified = Binding.qualify false rec_name;

         val ind_case_names = Rule_Cases.case_names induct_cases;

         val induct_cases' = Inductive.partition_rules' raw_induct

           (intrs ~~ induct_cases); 

         val thss' = map (map atomize_intr) thss;

         val thsss = Inductive.partition_rules' raw_induct (intrs ~~ thss');

         val strong_raw_induct =

           mk_ind_proof ctxt thss' |> Inductive.rulify;

         val strong_induct =

           if length names > 1 then

             (strong_raw_induct, [ind_case_names, Rule_Cases.consumes 0])

           else (strong_raw_induct RSN (2, rev_mp),

             [ind_case_names, Rule_Cases.consumes 1]);

         val (induct_name, inducts_name) =

           case alt_name of

             NONE => (rec_qualified (Binding.name "strong_induct"),

                      rec_qualified (Binding.name "strong_inducts"))

           | SOME s => (Binding.name s, Binding.name (s ^ "s"));

         val ((_, [strong_induct']), ctxt') = ctxt |> Local_Theory.note

           ((induct_name,

             map (Attrib.internal o K) (#2 strong_induct)), [#1 strong_induct]);

         val strong_inducts =

           Project_Rule.projects ctxt' (1 upto length names) strong_induct'

       in

         ctxt' |>

         Local_Theory.note

           ((inducts_name,

             [Attrib.internal (K ind_case_names),

              Attrib.internal (K (Rule_Cases.consumes 1))]),

            strong_inducts) |> snd

       end)

       (map (map (rulify_term thy #> rpair [])) vc_compat)

   end;

 (* outer syntax *)

 val _ =

   Outer_Syntax.local_theory_to_proof "nominal_inductive2"

     "prove strong induction theorem for inductive predicate involving nominal datatypes"

     Keyword.thy_goal

     (Parse.xname -- 

      Scan.option (Parse.$$$ "(" |-- Parse.!!! (Parse.name --| Parse.$$$ ")")) --

      (Scan.optional (Parse.$$$ "avoids" |-- Parse.enum1 "|" (Parse.name --

       (Parse.$$$ ":" |-- Parse.and_list1 Parse.term))) []) >> (fn ((name, rule_name), avoids) =>

         prove_strong_ind name rule_name avoids));

 end