src/HOL/Nominal/nominal_inductive2.ML

renamed simpset_of to global_simpset_of, and local_simpset_of to simpset_of -- same for claset and clasimpset;

(*  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 * 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 = MetaSimplifier.rewrite_term thy inductive_rulify [];

val atomize_conv =
  MetaSimplifier.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 = gen_inter (op = o apfst dest_Free) (ps, Term.add_frees t []);

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_i
  (theory_of_thm perm_bool) "perm_bool" [@{term "perm pi x"}] (fn thy => fn ss =>
    fn Const ("Nominal.perm", _) $ _ $ t =>
         if the_default "" (try (head_of #> dest_Const #> fst) t) mem names
         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 ("op &", _) $ p $ q) _ = (case head_of p of
      Const (name, _) =>
        if name mem names then SOME (f p q) else NONE
    | _ => NONE)
  | split_conj _ _ _ _ = NONE;

fun strip_all [] t = t
  | strip_all (_ :: xs) (Const ("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 ("op &", _) $ p $ q) _ =
      (case head_of p of
         Const (name, _) =>
           if name mem names 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 avoids ctxt =
  let
    val thy = ProofContext.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 names \\ fold (Term.add_const_names o Thm.prop_of) eqvt_thms [] of
        [] => ()
      | xs => error ("Missing equivariance theorem for predicate(s): " ^
          commas_quote xs));
    val induct_cases = map fst (fst (RuleCases.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 map fst avoids \\ induct_cases of
        [] => ()
      | xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs));
    fun mk_avoids params name sets =
      let
        val (_, ctxt') = ProofContext.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 union}, 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.certify_sort thy (map (fn a => Sign.intern_class thy
        ("fs_" ^ Long_Name.base_name a)) atoms);
    val ([fs_ctxt_tyname], _) = Name.variants ["'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 p mem ps 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 =
      (DatatypeProp.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
          (List.mapPartial (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 = PureThy.get_thms thy "perm_pi_simp";
    val pt2_atoms = map (fn a => PureThy.get_thm thy
      ("pt_" ^ Long_Name.base_name a ^ "2")) atoms;
    val eqvt_ss = Simplifier.theory_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 = PureThy.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) (atoms \ a)) 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 = ProofContext.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 = PureThy.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) 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 = List.mapPartial (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' = ProofContext.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 (ProofContext.export ctxt' ctxt);

  in
    ctxt'' |>
    Proof.theorem_i 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 = RuleCases.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, RuleCases.consumes 0])
          else (strong_raw_induct RSN (2, rev_mp),
            [ind_case_names, RuleCases.consumes 1]);
        val ((_, [strong_induct']), ctxt') = LocalTheory.note Thm.generatedK
          ((rec_qualified (Binding.name "strong_induct"),
            map (Attrib.internal o K) (#2 strong_induct)), [#1 strong_induct])
          ctxt;
        val strong_inducts =
          ProjectRule.projects ctxt' (1 upto length names) strong_induct'
      in
        ctxt' |>
        LocalTheory.note Thm.generatedK
          ((rec_qualified (Binding.name "strong_inducts"),
            [Attrib.internal (K ind_case_names),
             Attrib.internal (K (RuleCases.consumes 1))]),
           strong_inducts) |> snd
      end)
      (map (map (rulify_term thy #> rpair [])) vc_compat)
  end;


(* outer syntax *)

local structure P = OuterParse and K = OuterKeyword in

val _ =
  OuterSyntax.local_theory_to_proof "nominal_inductive2"
    "prove strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal
    (P.xname -- Scan.optional (P.$$$ "avoids" |-- P.enum1 "|" (P.name --
      (P.$$$ ":" |-- P.and_list1 P.term))) [] >> (fn (name, avoids) =>
        prove_strong_ind name avoids));

end;

end