(*  Title:      HOL/Nominal/nominal_inductive.ML

     ID:         $Id$

     Author:     Stefan Berghofer, TU Muenchen

 Infrastructure for proving equivariance and strong induction theorems

 for inductive predicates involving nominal datatypes.

 *)

 signature NOMINAL_INDUCTIVE =

 sig

   val prove_strong_ind: string -> (string * string list) list -> theory -> Proof.state

   val prove_eqvt: string -> string list -> theory -> theory

 end

 structure NominalInductive : NOMINAL_INDUCTIVE =

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

 val finite_Un = thm "finite_Un";

 val supp_prod = thm "supp_prod";

 val fresh_prod = thm "fresh_prod";

 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_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);

 val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE;

 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 NominalPackage.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 insert (op aconv o pairself fst)

                         (incr_boundvars (~i) u, T)) 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. P (pi_1 o ... o pi_n o t)                *)

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

 (* or    id (ALL 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 [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 prove_strong_ind s avoids thy =

   let

     val ctxt = ProofContext.init thy;

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

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

     val raw_induct = atomize_induct ctxt raw_induct;

     val monos = InductivePackage.get_monos ctxt;

     val eqvt_thms = NominalThmDecls.get_eqvt_thms ctxt;

     val _ = (case names \\ foldl (apfst prop_of #> add_term_consts) [] 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 raw_induct' = Logic.unvarify (prop_of raw_induct);

     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 _ = assert_all (null o duplicates op = o snd) avoids

       (fn (a, _) => error ("Duplicate variable names for case " ^ quote a));

     val _ = (case map fst avoids \\ induct_cases of

         [] => ()

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

     val avoids' = map (fn name =>

       (name, the_default [] (AList.lookup op = avoids name))) induct_cases;

     fun mk_avoids params (name, ps) =

       let val k = length params - 1

       in map (fn x => case find_index (equal x o fst) params of

           ~1 => error ("No such variable in case " ^ quote name ^

             " of inductive definition: " ^ quote x)

         | i => (Bound (k - i), snd (nth params i))) ps

       end;

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

       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,

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

            map (apfst (incr_boundvars 1)) (mk_avoids params avoid),

          prems, strip_comb (HOLogic.dest_Trueprop concl))

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

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

     val ind_sort = if null atomTs then HOLogic.typeS

       else Sign.certify_sort thy (map (fn T => Sign.intern_class thy

         ("fs_" ^ Sign.base_name (fst (dest_Type T)))) atomTs);

     val fs_ctxt_tyname = Name.variant (map fst (term_tfrees raw_induct')) "'n";

     val fs_ctxt_name = Name.variant (add_term_names (raw_induct', [])) "z";

     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_distinct [] = []

       | mk_distinct ((x, T) :: xs) = List.mapPartial (fn (y, U) =>

           if T = U then SOME (HOLogic.mk_Trueprop

             (HOLogic.mk_not (HOLogic.eq_const T $ x $ y)))

           else NONE) xs @ mk_distinct xs;

     fun mk_fresh (x, T) = HOLogic.mk_Trueprop

       (Const ("Nominal.fresh", T --> fsT --> HOLogic.boolT) $ x $ Bound 0);

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

       let

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

         val prem = Logic.list_implies

           (map mk_fresh bvars @ mk_distinct bvars @

            map (fn prem =>

              if null (term_frees prem inter ps) then prem

              else lift_prem prem) prems,

            HOLogic.mk_Trueprop (lift_pred p ts));

         val vs = map (Var o apfst (rpair 0)) (rename_wrt_term prem params')

       in

         (list_all (params', prem), (rev vs, subst_bounds (vs, 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 (NominalPackage.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 = map (fn (params, bvars, prems, (p, ts)) =>

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

           (List.mapPartial (fn prem =>

              if null (ps inter term_frees prem) then SOME prem

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

         (mk_distinct bvars @

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

            (Const ("Nominal.fresh", U --> T --> HOLogic.boolT) $ u $ t)) bvars)

              (ts ~~ binder_types (fastype_of p)))) prems;

     val perm_pi_simp = PureThy.get_thms thy (Name "perm_pi_simp");

     val pt2_atoms = map (fn aT => PureThy.get_thm thy

       (Name ("pt_" ^ Sign.base_name (fst (dest_Type aT)) ^ "2"))) atomTs;

     val eqvt_ss = HOL_basic_ss addsimps (eqvt_thms @ perm_pi_simp @ pt2_atoms)

       addsimprocs [mk_perm_bool_simproc ["Fun.id"]];

     val fresh_bij = PureThy.get_thms thy (Name "fresh_bij");

     val perm_bij = PureThy.get_thms thy (Name "perm_bij");

     val fs_atoms = map (fn aT => PureThy.get_thm thy

       (Name ("fs_" ^ Sign.base_name (fst (dest_Type aT)) ^ "1"))) atomTs;

     val exists_fresh' = PureThy.get_thms thy (Name "exists_fresh'");

     val fresh_atm = PureThy.get_thms thy (Name "fresh_atm");

     val calc_atm = PureThy.get_thms thy (Name "calc_atm");

     val perm_fresh_fresh = PureThy.get_thms thy (Name "perm_fresh_fresh");

     fun obtain_fresh_name ts T (freshs1, freshs2, ctxt) =

       let

         (** protect terms to avoid that supp_prod 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 @ freshs1);

         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop

             (HOLogic.exists_const T $ Abs ("x", T,

               Const ("Nominal.fresh", T --> fastype_of p --> HOLogic.boolT) $

                 Bound 0 $ p)))

           (fn _ => EVERY

             [resolve_tac exists_fresh' 1,

              simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);

         val (([cx], ths), ctxt') = Obtain.result

           (fn _ => EVERY

             [etac exE 1,

              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,

              full_simp_tac (HOL_basic_ss addsimps [id_apply]) 1,

              REPEAT (etac conjE 1)])

           [ex] ctxt

       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;

     fun mk_proof thy thss =

       let val ctxt = ProofContext.init thy

       in Goal.prove_global thy [] prems' concl' (fn ihyps =>

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

           rtac raw_induct 1 THEN

           EVERY (maps (fn ((((_, bvars, oprems, _), vc_compat_ths), ihyp), (vs, ihypt)) =>

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

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

                let

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

                    chop (length params - length atomTs - 1) (map term_of params) ||>

                    split_last;

                  val bvars' = map

                    (fn (Bound i, T) => (nth params' (length params' - i), T)

                      | (t, T) => (t, T)) bvars;

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

                    fold_rev (NominalPackage.mk_perm []) pis t) bvars';

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

                  val (freshs1, freshs2, ctxt'') = fold

                    (obtain_fresh_name (ts @ pi_bvars))

                    (map snd bvars') ([], [], ctxt');

                  val freshs2' = NominalPackage.mk_not_sym freshs2;

                  val pis' = map NominalPackage.perm_of_pair (pi_bvars ~~ freshs1);

                  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 (concat_perm #> map) pis' pis;

                  val env = Pattern.first_order_match thy (ihypt, prop_of ihyp)

                    (Vartab.empty, Vartab.empty);

                  val ihyp' = Thm.instantiate ([], map (pairself (cterm_of thy))

                    (map (Envir.subst_vars env) vs ~~

                     map (fold_rev (NominalPackage.mk_perm [])

                       (rev pis' @ pis)) params' @ [z])) ihyp;

                  fun mk_pi th =

                    Simplifier.simplify (HOL_basic_ss addsimps [id_apply]

                        addsimprocs [NominalPackage.perm_simproc])

                      (Simplifier.simplify eqvt_ss

                        (fold_rev (fn pi => fn th' => th' RS Drule.cterm_instantiate

                          [(perm_boolI_pi, cterm_of thy pi)] perm_boolI)

                            (rev pis' @ pis) th));

                  val (gprems1, gprems2) = split_list

                    (map (fn (th, t) =>

                       if null (term_frees t inter ps) then (SOME th, mk_pi th)

                       else

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

                            (etac conjunct1 1) monos NONE th,

                          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)) |>> List.mapPartial I;

                  val vc_compat_ths' = map (fn th =>

                    let

                      val th' = gprems1 MRS

                        Thm.instantiate (Thm.first_order_match

                          (Conjunction.mk_conjunction_balanced (cprems_of th),

                           Conjunction.mk_conjunction_balanced (map cprop_of gprems1))) th;

                      val (bop, lhs, rhs) = (case concl_of th' of

                          _ $ (fresh $ lhs $ rhs) =>

                            (fn t => fn u => fresh $ t $ u, lhs, rhs)

                        | _ $ (_ $ (_ $ lhs $ rhs)) =>

                            (curry (HOLogic.mk_not o HOLogic.mk_eq), lhs, rhs));

                      val th'' = Goal.prove ctxt'' [] [] (HOLogic.mk_Trueprop

                          (bop (fold_rev (NominalPackage.mk_perm []) pis lhs)

                             (fold_rev (NominalPackage.mk_perm []) pis rhs)))

                        (fn _ => simp_tac (HOL_basic_ss addsimps

                           (fresh_bij @ perm_bij)) 1 THEN rtac th' 1)

                    in Simplifier.simplify (eqvt_ss addsimps fresh_atm) th'' end)

                      vc_compat_ths;

                  val vc_compat_ths'' = NominalPackage.mk_not_sym vc_compat_ths';

                  (** Since calc_atm simplifies (pi :: 'a prm) o (x :: 'b) to x **)

                  (** we have to pre-simplify the rewrite rules                 **)

                  val calc_atm_ss = HOL_ss addsimps calc_atm @

                     map (Simplifier.simplify (HOL_ss addsimps calc_atm))

                       (vc_compat_ths'' @ freshs2');

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

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

                      map (fold (NominalPackage.mk_perm []) pis') (tl ts))))

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

                      REPEAT_DETERM_N (nprems_of ihyp - length gprems)

                        (simp_tac calc_atm_ss 1),

                      REPEAT_DETERM_N (length gprems)

                        (simp_tac (HOL_ss

                           addsimps inductive_forall_def' :: gprems2

                           addsimprocs [NominalPackage.perm_simproc]) 1)]));

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

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

                      addsimps vc_compat_ths'' @ freshs2' @

                        perm_fresh_fresh @ fresh_atm) 1);

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

                in resolve_tac final' 1 end) context 1])

                  (prems ~~ thss ~~ 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 allE_Nil 1) THEN

             asm_full_simp_tac (simpset_of thy) 1)

         end)

       end;

   in

     thy |>

     ProofContext.init |>

     Proof.theorem_i NONE (fn thss => ProofContext.theory (fn thy =>

       let

         val ctxt = ProofContext.init thy;

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

         val ind_case_names = RuleCases.case_names induct_cases;

         val strong_raw_induct =

           mk_proof thy (map (map atomize_intr) thss) |>

           InductivePackage.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'], thy') = thy |>

           Sign.add_path rec_name |>

           PureThy.add_thms [(("strong_induct", #1 strong_induct), #2 strong_induct)];

         val strong_inducts =

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

       in

         thy' |>

         PureThy.add_thmss [(("strong_inducts", strong_inducts),

           [ind_case_names, RuleCases.consumes 1])] |> snd |>

         Sign.parent_path

       end))

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

   end;

 fun prove_eqvt s xatoms thy =

   let

     val ctxt = ProofContext.init thy;

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

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

     val raw_induct = atomize_induct ctxt raw_induct;

     val elims = map (atomize_induct ctxt) elims;

     val intrs = map atomize_intr intrs;

     val monos = InductivePackage.get_monos ctxt;

     val intrs' = InductivePackage.unpartition_rules intrs

       (map (fn (((s, ths), (_, k)), th) =>

            (s, ths ~~ InductivePackage.infer_intro_vars th k ths))

          (InductivePackage.partition_rules raw_induct intrs ~~

           InductivePackage.arities_of raw_induct ~~ elims));

     val atoms' = NominalAtoms.atoms_of thy;

     val atoms =

       if null xatoms then atoms' else

       let val atoms = map (Sign.intern_type thy) xatoms

       in

         (case duplicates op = atoms of

              [] => ()

            | xs => error ("Duplicate atoms: " ^ commas xs);

          case atoms \\ atoms' of

              [] => ()

            | xs => error ("No such atoms: " ^ commas xs);

          atoms)

       end;

     val perm_pi_simp = PureThy.get_thms thy (Name "perm_pi_simp");

     val eqvt_ss = HOL_basic_ss addsimps

       (NominalThmDecls.get_eqvt_thms ctxt @ perm_pi_simp) addsimprocs

       [mk_perm_bool_simproc names];

     val t = Logic.unvarify (concl_of raw_induct);

     val pi = Name.variant (add_term_names (t, [])) "pi";

     val ps = map (fst o HOLogic.dest_imp)

       (HOLogic.dest_conj (HOLogic.dest_Trueprop t));

     fun eqvt_tac th pi (intr, vs) st =

       let

         fun eqvt_err s = error

           ("Could not prove equivariance for introduction rule\n" ^

            Sign.string_of_term (theory_of_thm intr)

              (Logic.unvarify (prop_of intr)) ^ "\n" ^ s);

         val res = SUBPROOF (fn {prems, params, ...} =>

           let

             val prems' = map (fn th => the_default th (map_thm ctxt

               (split_conj (K I) names) (etac conjunct2 1) monos NONE th)) prems;

             val prems'' = map (fn th' =>

               Simplifier.simplify eqvt_ss (th' RS th)) prems';

             val intr' = Drule.cterm_instantiate (map (cterm_of thy) vs ~~

                map (cterm_of thy o NominalPackage.mk_perm [] pi o term_of) params)

                intr

           in (rtac intr' THEN_ALL_NEW (TRY o resolve_tac prems'')) 1

           end) ctxt 1 st

       in

         case (Seq.pull res handle THM (s, _, _) => eqvt_err s) of

           NONE => eqvt_err ("Rule does not match goal\n" ^

             Sign.string_of_term (theory_of_thm st) (hd (prems_of st)))

         | SOME (th, _) => Seq.single th

       end;

     val thss = map (fn atom =>

       let

         val pi' = Free (pi, NominalAtoms.mk_permT (Type (atom, [])));

         val perm_boolI' = Drule.cterm_instantiate

           [(perm_boolI_pi, cterm_of thy pi')] perm_boolI

       in map (fn th => zero_var_indexes (th RS mp))

         (DatatypeAux.split_conj_thm (Goal.prove_global thy [] []

           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn p =>

             HOLogic.mk_imp (p, list_comb

              (apsnd (map (NominalPackage.mk_perm [] pi')) (strip_comb p)))) ps)))

           (fn _ => EVERY (rtac raw_induct 1 :: map (fn intr_vs =>

               full_simp_tac eqvt_ss 1 THEN

               eqvt_tac perm_boolI' pi' intr_vs) intrs'))))

       end) atoms

   in

     fold (fn (name, ths) =>

       Sign.add_path (Sign.base_name name) #>

       PureThy.add_thmss [(("eqvt", ths), [NominalThmDecls.eqvt_add])] #> snd #>

       Sign.parent_path) (names ~~ transp thss) thy

   end;

 (* outer syntax *)

 local structure P = OuterParse and K = OuterKeyword in

 val _ = OuterSyntax.keywords ["avoids"];

 val _ =

   OuterSyntax.command "nominal_inductive"

     "prove equivariance and strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal

     (P.name -- Scan.optional (P.$$$ "avoids" |-- P.and_list1 (P.name --

       (P.$$$ ":" |-- Scan.repeat1 P.name))) [] >> (fn (name, avoids) =>

         Toplevel.print o Toplevel.theory_to_proof (prove_strong_ind name avoids)));

 val _ =

   OuterSyntax.command "equivariance"

     "prove equivariance for inductive predicate involving nominal datatypes" K.thy_decl

     (P.name -- Scan.optional (P.$$$ "[" |-- P.list1 P.name --| P.$$$ "]") [] >>

       (fn (name, atoms) => Toplevel.theory (prove_eqvt name atoms)));

 end;

 end