# File inductive_package.ML

    (*  Title:      HOL/Tools/inductive_package.ML
        ID:         $Id: inductive_package.ML,v 1.139 2005/09/19 14:38:35 haftmann Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Author:     Stefan Berghofer, TU Muenchen
        Author:     Markus Wenzel, TU Muenchen

    (Co)Inductive Definition module for HOL.

    Features:
      * least or greatest fixedpoints
      * user-specified product and sum constructions
      * mutually recursive definitions
      * definitions involving arbitrary monotone operators
      * automatically proves introduction and elimination rules

    The recursive sets must *already* be declared as constants in the
    current theory!

      Introduction rules have the form
      [| ti:M(Sj), ..., P(x), ... |] ==> t: Sk
      where M is some monotone operator (usually the identity)
      P(x) is any side condition on the free variables
      ti, t are any terms
      Sj, Sk are two of the sets being defined in mutual recursion

    Sums are used only for mutual recursion.  Products are used only to
    derive "streamlined" induction rules for relations.
    *)

    signature INDUCTIVE_PACKAGE =
    sig
      val quiet_mode: bool ref
      val trace: bool ref
      val unify_consts: theory -> term list -> term list -> term list * term list
      val split_rule_vars: term list -> thm -> thm
      val get_inductive: theory -> string -> ({names: string list, coind: bool} *
        {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
         intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}) option
      val the_mk_cases: theory -> string -> string -> thm
      val print_inductives: theory -> unit
      val mono_add_global: theory attribute
      val mono_del_global: theory attribute
      val get_monos: theory -> thm list
      val inductive_forall_name: string
      val inductive_forall_def: thm
      val rulify: thm -> thm
      val inductive_cases: ((bstring * Attrib.src list) * string list) list -> theory -> theory
      val inductive_cases_i: ((bstring * theory attribute list) * term list) list -> theory -> theory
      val add_inductive_i: bool -> bool -> bstring -> bool -> bool -> bool -> term list ->
        ((bstring * term) * theory attribute list) list -> thm list -> theory -> theory *
          {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
           intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
      val add_inductive: bool -> bool -> string list ->
        ((bstring * string) * Attrib.src list) list -> (thmref * Attrib.src list) list ->
        theory -> theory *
          {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
           intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
      val setup: (theory -> theory) list
    end;

    structure InductivePackage: INDUCTIVE_PACKAGE =
    struct

    (** theory context references **)

    val mono_name = "Orderings.mono";
    val gfp_name = "FixedPoint.gfp";
    val lfp_name = "FixedPoint.lfp";
    val vimage_name = "Set.vimage";
    val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (Thm.concl_of vimageD);

    val inductive_forall_name = "HOL.induct_forall";
    val inductive_forall_def = thm "induct_forall_def";
    val inductive_conj_name = "HOL.induct_conj";
    val inductive_conj_def = thm "induct_conj_def";
    val inductive_conj = thms "induct_conj";
    val inductive_atomize = thms "induct_atomize";
    val inductive_rulify1 = thms "induct_rulify1";
    val inductive_rulify2 = thms "induct_rulify2";

    (** theory data **)

    (* data kind 'HOL/inductive' *)

    type inductive_info =
      {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm,
        induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm};

    structure InductiveData = TheoryDataFun
    (struct
      val name = "HOL/inductive";
      type T = inductive_info Symtab.table * thm list;

      val empty = (Symtab.empty, []);
      val copy = I;
      val extend = I;
      fun merge _ ((tab1, monos1), (tab2, monos2)) =
        (Symtab.merge (K true) (tab1, tab2), Drule.merge_rules (monos1, monos2));

      fun print thy (tab, monos) =
        [Pretty.strs ("(co)inductives:" ::
          map #1 (NameSpace.extern_table (Sign.const_space thy, tab))),
         Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_sg thy) monos)]
        |> Pretty.chunks |> Pretty.writeln;
    end);

    val print_inductives = InductiveData.print;

    (* get and put data *)

    val get_inductive = Symtab.lookup o #1 o InductiveData.get;

    fun the_inductive thy name =
      (case get_inductive thy name of
        NONE => error ("Unknown (co)inductive set " ^ quote name)
      | SOME info => info);

    val the_mk_cases = (#mk_cases o #2) oo the_inductive;

    fun put_inductives names info thy =
      let
        fun upd ((tab, monos), name) = (Symtab.update_new (name, info) tab, monos);
        val tab_monos = Library.foldl upd (InductiveData.get thy, names)
          handle Symtab.DUP name => error ("Duplicate definition of (co)inductive set " ^ quote name);
      in InductiveData.put tab_monos thy end;

    (** monotonicity rules **)

    val get_monos = #2 o InductiveData.get;
    fun map_monos f = InductiveData.map (Library.apsnd f);

    fun mk_mono thm =
      let
        fun eq2mono thm' = [standard (thm' RS (thm' RS eq_to_mono))] @
          (case concl_of thm of
              (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
            | _ => [standard (thm' RS (thm' RS eq_to_mono2))]);
        val concl = concl_of thm
      in
        if Logic.is_equals concl then
          eq2mono (thm RS meta_eq_to_obj_eq)
        else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then
          eq2mono thm
        else [thm]
      end;

    (* attributes *)

    fun mono_add_global (thy, thm) = (map_monos (Drule.add_rules (mk_mono thm)) thy, thm);
    fun mono_del_global (thy, thm) = (map_monos (Drule.del_rules (mk_mono thm)) thy, thm);

    val mono_attr =
     (Attrib.add_del_args mono_add_global mono_del_global,
      Attrib.add_del_args Attrib.undef_local_attribute Attrib.undef_local_attribute);

    (** misc utilities **)

    val quiet_mode = ref false;
    val trace = ref false;  (*for debugging*)
    fun message s = if ! quiet_mode then () else writeln s;
    fun clean_message s = if ! quick_and_dirty then () else message s;

    fun coind_prefix true = "co"
      | coind_prefix false = "";

    (*the following code ensures that each recursive set always has the
      same type in all introduction rules*)
    fun unify_consts thy cs intr_ts =
      (let
        val add_term_consts_2 = fold_aterms (fn Const c => insert (op =) c | _ => I);
        fun varify (t, (i, ts)) =
          let val t' = map_term_types (Logic.incr_tvar (i + 1)) (#1 (Type.varify (t, [])))
          in (maxidx_of_term t', t'::ts) end;
        val (i, cs') = foldr varify (~1, []) cs;
        val (i', intr_ts') = foldr varify (i, []) intr_ts;
        val rec_consts = fold add_term_consts_2 cs' [];
        val intr_consts = fold add_term_consts_2 intr_ts' [];
        fun unify (cname, cT) =
          let val consts = map snd (List.filter (fn c => fst c = cname) intr_consts)
          in fold (Sign.typ_unify thy) ((replicate (length consts) cT) ~~ consts) end;
        val (env, _) = fold unify rec_consts (Vartab.empty, i');
        val subst = Type.freeze o map_term_types (Envir.norm_type env)

      in (map subst cs', map subst intr_ts')
      end) handle Type.TUNIFY =>
        (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts));

    (*make injections used in mutually recursive definitions*)
    fun mk_inj cs sumT c x =
      let
        fun mk_inj' T n i =
          if n = 1 then x else
          let val n2 = n div 2;
              val Type (_, [T1, T2]) = T
          in
            if i <= n2 then
              Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
            else
              Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
          end
      in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
      end;

    (*make "vimage" terms for selecting out components of mutually rec.def*)
    fun mk_vimage cs sumT t c = if length cs < 2 then t else
      let
        val cT = HOLogic.dest_setT (fastype_of c);
        val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
      in
        Const (vimage_name, vimageT) $
          Abs ("y", cT, mk_inj cs sumT c (Bound 0)) $ t
      end;

    (** proper splitting **)

    fun prod_factors p (Const ("Pair", _) $ t $ u) =
          p :: prod_factors (1::p) t @ prod_factors (2::p) u
      | prod_factors p _ = [];

    fun mg_prod_factors ts (t $ u) fs = if t mem ts then
            let val f = prod_factors [] u
            in AList.update (op =) (t, f inter (AList.lookup (op =) fs t) |> the_default f) fs end
          else mg_prod_factors ts u (mg_prod_factors ts t fs)
      | mg_prod_factors ts (Abs (_, _, t)) fs = mg_prod_factors ts t fs
      | mg_prod_factors ts _ fs = fs;

    fun prodT_factors p ps (T as Type ("*", [T1, T2])) =
          if p mem ps then prodT_factors (1::p) ps T1 @ prodT_factors (2::p) ps T2
          else [T]
      | prodT_factors _ _ T = [T];

    fun ap_split p ps (Type ("*", [T1, T2])) T3 u =
          if p mem ps then HOLogic.split_const (T1, T2, T3) $
            Abs ("v", T1, ap_split (2::p) ps T2 T3 (ap_split (1::p) ps T1
              (prodT_factors (2::p) ps T2 ---> T3) (incr_boundvars 1 u) $ Bound 0))
          else u
      | ap_split _ _ _ _ u =  u;

    fun mk_tuple p ps (Type ("*", [T1, T2])) (tms as t::_) =
          if p mem ps then HOLogic.mk_prod (mk_tuple (1::p) ps T1 tms,
            mk_tuple (2::p) ps T2 (Library.drop (length (prodT_factors (1::p) ps T1), tms)))
          else t
      | mk_tuple _ _ _ (t::_) = t;

    fun split_rule_var' ((t as Var (v, Type ("fun", [T1, T2])), ps), rl) =
          let val T' = prodT_factors [] ps T1 ---> T2
              val newt = ap_split [] ps T1 T2 (Var (v, T'))
              val cterm = Thm.cterm_of (Thm.theory_of_thm rl)
          in
              instantiate ([], [(cterm t, cterm newt)]) rl
          end
      | split_rule_var' (_, rl) = rl;

    val remove_split = rewrite_rule [split_conv RS eq_reflection];

    fun split_rule_vars vs rl = standard (remove_split (foldr split_rule_var'
      rl (mg_prod_factors vs (Thm.prop_of rl) [])));

    fun split_rule vs rl = standard (remove_split (foldr split_rule_var'
      rl (List.mapPartial (fn (t as Var ((a, _), _)) =>
          Option.map (pair t) (AList.lookup (op =) vs a)) (term_vars (Thm.prop_of rl)))));

    (** process rules **)

    local

    fun err_in_rule thy name t msg =
      error (cat_lines ["Ill-formed introduction rule " ^ quote name,
        Sign.string_of_term thy t, msg]);

    fun err_in_prem thy name t p msg =
      error (cat_lines ["Ill-formed premise", Sign.string_of_term thy p,
        "in introduction rule " ^ quote name, Sign.string_of_term thy t, msg]);

    val bad_concl = "Conclusion of introduction rule must have form \"t : S_i\"";

    val all_not_allowed =
        "Introduction rule must not have a leading \"!!\" quantifier";

    fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];

    in

    fun check_rule thy cs ((name, rule), att) =
      let
        val concl = Logic.strip_imp_concl rule;
        val prems = Logic.strip_imp_prems rule;
        val aprems = map (atomize_term thy) prems;
        val arule = Logic.list_implies (aprems, concl);

        fun check_prem (prem, aprem) =
          if can HOLogic.dest_Trueprop aprem then ()
          else err_in_prem thy name rule prem "Non-atomic premise";
      in
        (case concl of
          Const ("Trueprop", _) $ (Const ("op :", _) $ t $ u) =>
            if u mem cs then
              if exists (Logic.occs o rpair t) cs then
                err_in_rule thy name rule "Recursion term on left of member symbol"
              else List.app check_prem (prems ~~ aprems)
            else err_in_rule thy name rule bad_concl
          | Const ("all", _) $ _ => err_in_rule thy name rule all_not_allowed
          | _ => err_in_rule thy name rule bad_concl);
        ((name, arule), att)
      end;

    val rulify =
      standard o
      hol_simplify inductive_rulify2 o hol_simplify inductive_rulify1 o
      hol_simplify inductive_conj;

    end;

    (** properties of (co)inductive sets **)

    (* elimination rules *)

    fun mk_elims cs cTs params intr_ts intr_names =
      let
        val used = foldr add_term_names [] intr_ts;
        val [aname, pname] = variantlist (["a", "P"], used);
        val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));

        fun dest_intr r =
          let val Const ("op :", _) $ t $ u =
            HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
          in (u, t, Logic.strip_imp_prems r) end;

        val intrs = map dest_intr intr_ts ~~ intr_names;

        fun mk_elim (c, T) =
          let
            val a = Free (aname, T);

            fun mk_elim_prem (_, t, ts) =
              list_all_free (map dest_Free ((foldr add_term_frees [] (t::ts)) \\ params),
                Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
            val c_intrs = (List.filter (equal c o #1 o #1) intrs);
          in
            (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
              map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs)
          end
      in
        map mk_elim (cs ~~ cTs)
      end;

    (* premises and conclusions of induction rules *)

    fun mk_indrule cs cTs params intr_ts =
      let
        val used = foldr add_term_names [] intr_ts;

        (* predicates for induction rule *)

        val preds = map Free (variantlist (if length cs < 2 then ["P"] else
          map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
            map (fn T => T --> HOLogic.boolT) cTs);

        (* transform an introduction rule into a premise for induction rule *)

        fun mk_ind_prem r =
          let
            val frees = map dest_Free ((add_term_frees (r, [])) \\ params);

            val pred_of = AList.lookup (op aconv) (cs ~~ preds);

            fun subst (s as ((m as Const ("op :", T)) $ t $ u)) =
                  (case pred_of u of
                      NONE => (m $ fst (subst t) $ fst (subst u), NONE)
                    | SOME P => (HOLogic.mk_binop inductive_conj_name (s, P $ t), SOME (s, P $ t)))
              | subst s =
                  (case pred_of s of
                      SOME P => (HOLogic.mk_binop "op Int"
                        (s, HOLogic.Collect_const (HOLogic.dest_setT
                          (fastype_of s)) $ P), NONE)
                    | NONE => (case s of
                         (t $ u) => (fst (subst t) $ fst (subst u), NONE)
                       | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
                       | _ => (s, NONE)));

            fun mk_prem (s, prems) = (case subst s of
                  (_, SOME (t, u)) => t :: u :: prems
                | (t, _) => t :: prems);

            val Const ("op :", _) $ t $ u =
              HOLogic.dest_Trueprop (Logic.strip_imp_concl r)

          in list_all_free (frees,
               Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
                 [] (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r))),
                   HOLogic.mk_Trueprop (valOf (pred_of u) $ t)))
          end;

        val ind_prems = map mk_ind_prem intr_ts;

        val factors = Library.fold (mg_prod_factors preds) ind_prems [];

        (* make conclusions for induction rules *)

        fun mk_ind_concl ((c, P), (ts, x)) =
          let val T = HOLogic.dest_setT (fastype_of c);
              val ps = AList.lookup (op =) factors P |> the_default [];
              val Ts = prodT_factors [] ps T;
              val (frees, x') = foldr (fn (T', (fs, s)) =>
                ((Free (s, T'))::fs, Symbol.bump_string s)) ([], x) Ts;
              val tuple = mk_tuple [] ps T frees;
          in ((HOLogic.mk_binop "op -->"
            (HOLogic.mk_mem (tuple, c), P $ tuple))::ts, x')
          end;

        val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
            (fst (foldr mk_ind_concl ([], "xa") (cs ~~ preds))))

      in (preds, ind_prems, mutual_ind_concl,
        map (apfst (fst o dest_Free)) factors)
      end;

    (* prepare cases and induct rules *)

    (*
      transform mutual rule:
        HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
      into i-th projection:
        xi:Ai ==> HH ==> Pi xi
    *)

    fun project_rules [name] rule = [(name, rule)]
      | project_rules names mutual_rule =
          let
            val n = length names;
            fun proj i =
              (if i < n then (fn th => th RS conjunct1) else I)
                (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
                RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
          in names ~~ map proj (1 upto n) end;

    fun add_cases_induct no_elim no_induct names elims induct =
      let
        fun cases_spec (name, elim) thy =
          thy
          |> Theory.add_path (Sign.base_name name)
          |> (#1 o PureThy.add_thms [(("cases", elim), [InductAttrib.cases_set_global name])])
          |> Theory.parent_path;
        val cases_specs = if no_elim then [] else map2 cases_spec (names, elims);

        fun induct_spec (name, th) = #1 o PureThy.add_thms
          [(("", RuleCases.save induct th), [InductAttrib.induct_set_global name])];
        val induct_specs = if no_induct then [] else map induct_spec (project_rules names induct);
      in Library.apply (cases_specs @ induct_specs) end;

    (** proofs for (co)inductive sets **)

    (* prove monotonicity -- NOT subject to quick_and_dirty! *)

    fun prove_mono setT fp_fun monos thy =
     (message "  Proving monotonicity ...";
      Goals.prove_goalw_cterm []      (*NO quick_and_dirty_prove_goalw_cterm here!*)
        (Thm.cterm_of thy (HOLogic.mk_Trueprop
          (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
        (fn _ => [rtac monoI 1, REPEAT (ares_tac (List.concat (map mk_mono monos) @ get_monos thy) 1)]));

    (* prove introduction rules *)

    fun prove_intrs coind mono fp_def intr_ts rec_sets_defs thy =
      let
        val _ = clean_message "  Proving the introduction rules ...";

        val unfold = standard' (mono RS (fp_def RS
          (if coind then def_gfp_unfold else def_lfp_unfold)));

        fun select_disj 1 1 = []
          | select_disj _ 1 = [rtac disjI1]
          | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));

        val intrs = map (fn (i, intr) => quick_and_dirty_prove_goalw_cterm thy rec_sets_defs
          (Thm.cterm_of thy intr) (fn prems =>
           [(*insert prems and underlying sets*)
           cut_facts_tac prems 1,
           stac unfold 1,
           REPEAT (resolve_tac [vimageI2, CollectI] 1),
           (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
           EVERY1 (select_disj (length intr_ts) i),
           (*Not ares_tac, since refl must be tried before any equality assumptions;
             backtracking may occur if the premises have extra variables!*)
           DEPTH_SOLVE_1 (resolve_tac [refl, exI, conjI] 1 APPEND assume_tac 1),
           (*Now solve the equations like Inl 0 = Inl ?b2*)
           REPEAT (rtac refl 1)])
          |> rulify) (1 upto (length intr_ts) ~~ intr_ts)

      in (intrs, unfold) end;

    (* prove elimination rules *)

    fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
      let
        val _ = clean_message "  Proving the elimination rules ...";

        val rules1 = [CollectE, disjE, make_elim vimageD, exE];
        val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @ map make_elim [Inl_inject, Inr_inject];
      in
        mk_elims cs cTs params intr_ts intr_names |> map (fn (t, cases) =>
          quick_and_dirty_prove_goalw_cterm thy rec_sets_defs
            (Thm.cterm_of thy t) (fn prems =>
              [cut_facts_tac [hd prems] 1,
               dtac (unfold RS subst) 1,
               REPEAT (FIRSTGOAL (eresolve_tac rules1)),
               REPEAT (FIRSTGOAL (eresolve_tac rules2)),
               EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))])
            |> rulify
            |> RuleCases.name cases)
      end;

    (* derivation of simplified elimination rules *)

    local

    (*cprop should have the form t:Si where Si is an inductive set*)
    val mk_cases_err = "mk_cases: proposition not of form \"t : S_i\"";

    (*delete needless equality assumptions*)
    val refl_thin = prove_goal HOL.thy "!!P. a = a ==> P ==> P" (fn _ => [assume_tac 1]);
    val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject];
    val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;

    fun simp_case_tac solved ss i =
      EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i
      THEN_MAYBE (if solved then no_tac else all_tac);

    in

    fun mk_cases_i elims ss cprop =
      let
        val prem = Thm.assume cprop;
        val tac = ALLGOALS (simp_case_tac false ss) THEN prune_params_tac;
        fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic tac (prem RS rl));
      in
        (case get_first (try mk_elim) elims of
          SOME r => r
        | NONE => error (Pretty.string_of (Pretty.block
            [Pretty.str mk_cases_err, Pretty.fbrk, Display.pretty_cterm cprop])))
      end;

    fun mk_cases elims s =
      mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.theory_of_thm (hd elims)) (s, propT));

    fun smart_mk_cases thy ss cprop =
      let
        val c = #1 (Term.dest_Const (Term.head_of (#2 (HOLogic.dest_mem (HOLogic.dest_Trueprop
          (Logic.strip_imp_concl (Thm.term_of cprop))))))) handle TERM _ => error mk_cases_err;
        val (_, {elims, ...}) = the_inductive thy c;
      in mk_cases_i elims ss cprop end;

    end;

    (* inductive_cases(_i) *)

    fun gen_inductive_cases prep_att prep_prop args thy =
      let
        val cert_prop = Thm.cterm_of thy o prep_prop (ProofContext.init thy);
        val mk_cases = smart_mk_cases thy (Simplifier.simpset_of thy) o cert_prop;

        val facts = args |> map (fn ((a, atts), props) =>
         ((a, map (prep_att thy) atts), map (Thm.no_attributes o single o mk_cases) props));
      in thy |> IsarThy.theorems_i Drule.lemmaK facts |> #1 end;

    val inductive_cases = gen_inductive_cases Attrib.global_attribute ProofContext.read_prop;
    val inductive_cases_i = gen_inductive_cases (K I) ProofContext.cert_prop;

    (* mk_cases_meth *)

    fun mk_cases_meth (ctxt, raw_props) =
      let
        val thy = ProofContext.theory_of ctxt;
        val ss = local_simpset_of ctxt;
        val cprops = map (Thm.cterm_of thy o ProofContext.read_prop ctxt) raw_props;
      in Method.erule 0 (map (smart_mk_cases thy ss) cprops) end;

    val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name));

    (* prove induction rule *)

    fun prove_indrule cs cTs sumT rec_const params intr_ts mono
        fp_def rec_sets_defs thy =
      let
        val _ = clean_message "  Proving the induction rule ...";

        val sum_case_rewrites =
          (if Context.theory_name thy = "Datatype" then
            PureThy.get_thms thy (Name "sum.cases")
          else
            (case ThyInfo.lookup_theory "Datatype" of
              NONE => []
            | SOME thy' =>
                if Theory.subthy (thy', thy) then
                  PureThy.get_thms thy' (Name "sum.cases")
                else []))
          |> map mk_meta_eq;

        val (preds, ind_prems, mutual_ind_concl, factors) =
          mk_indrule cs cTs params intr_ts;

        val dummy = if !trace then
    		(writeln "ind_prems = ";
    		 List.app (writeln o Sign.string_of_term thy) ind_prems)
    	    else ();

        (* make predicate for instantiation of abstract induction rule *)

        fun mk_ind_pred _ [P] = P
          | mk_ind_pred T Ps =
             let val n = (length Ps) div 2;
                 val Type (_, [T1, T2]) = T
             in Const ("Datatype.sum.sum_case",
               [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
                 mk_ind_pred T1 (Library.take (n, Ps)) $ mk_ind_pred T2 (Library.drop (n, Ps))
             end;

        val ind_pred = mk_ind_pred sumT preds;

        val ind_concl = HOLogic.mk_Trueprop
          (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->"
            (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0)));

        (* simplification rules for vimage and Collect *)

        val vimage_simps = if length cs < 2 then [] else
          map (fn c => quick_and_dirty_prove_goalw_cterm thy [] (Thm.cterm_of thy
            (HOLogic.mk_Trueprop (HOLogic.mk_eq
              (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c,
               HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $
                 List.nth (preds, find_index_eq c cs)))))
            (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites, rtac refl 1])) cs;

        val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));

        val dummy = if !trace then
    		(writeln "raw_fp_induct = "; print_thm raw_fp_induct)
    	    else ();

        val induct = quick_and_dirty_prove_goalw_cterm thy [inductive_conj_def] (Thm.cterm_of thy
          (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
            [rtac (impI RS allI) 1,
             DETERM (etac raw_fp_induct 1),
             rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
             fold_goals_tac rec_sets_defs,
             (*This CollectE and disjE separates out the introduction rules*)
             REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
             (*Now break down the individual cases.  No disjE here in case
               some premise involves disjunction.*)
             REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
             ALLGOALS (simp_tac (HOL_basic_ss addsimps sum_case_rewrites)),
             EVERY (map (fn prem =>
       	             DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);

        val lemma = quick_and_dirty_prove_goalw_cterm thy rec_sets_defs (Thm.cterm_of thy
          (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
            [cut_facts_tac prems 1,
             REPEAT (EVERY
               [REPEAT (resolve_tac [conjI, impI] 1),
                TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
                rewrite_goals_tac sum_case_rewrites,
                atac 1])])

      in standard (split_rule factors (induct RS lemma)) end;

    (** specification of (co)inductive sets **)

    fun cond_declare_consts declare_consts cs paramTs cnames =
      if declare_consts then
        Theory.add_consts_i (map (fn (c, n) => (Sign.base_name n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
      else I;

    fun mk_ind_def declare_consts alt_name coind cs intr_ts monos thy
          params paramTs cTs cnames =
      let
        val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
        val setT = HOLogic.mk_setT sumT;

        val fp_name = if coind then gfp_name else lfp_name;

        val used = foldr add_term_names [] intr_ts;
        val [sname, xname] = variantlist (["S", "x"], used);

        (* transform an introduction rule into a conjunction  *)
        (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
        (* is transformed into                                *)
        (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)

        fun transform_rule r =
          let
            val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
            val subst = subst_free
              (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
            val Const ("op :", _) $ t $ u =
              HOLogic.dest_Trueprop (Logic.strip_imp_concl r)

          in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
            (foldr1 HOLogic.mk_conj
              (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t))::
                (map (subst o HOLogic.dest_Trueprop)
                  (Logic.strip_imp_prems r)))) frees
          end

        (* make a disjunction of all introduction rules *)

        val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
          absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));

        (* add definiton of recursive sets to theory *)

        val rec_name = if alt_name = "" then
          space_implode "_" (map Sign.base_name cnames) else alt_name;
        val full_rec_name = if length cs < 2 then hd cnames
          else Sign.full_name thy rec_name;

        val rec_const = list_comb
          (Const (full_rec_name, paramTs ---> setT), params);

        val fp_def_term = Logic.mk_equals (rec_const,
          Const (fp_name, (setT --> setT) --> setT) $ fp_fun);

        val def_terms = fp_def_term :: (if length cs < 2 then [] else
          map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);

        val (thy', [fp_def :: rec_sets_defs]) =
          thy
          |> cond_declare_consts declare_consts cs paramTs cnames
          |> (if length cs < 2 then I
              else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)])
          |> Theory.add_path rec_name
          |> PureThy.add_defss_i false [(("defs", def_terms), [])];

        val mono = prove_mono setT fp_fun monos thy'

      in (thy', mono, fp_def, rec_sets_defs, rec_const, sumT) end;

    fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
        intros monos thy params paramTs cTs cnames induct_cases =
      let
        val _ =
          if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
            commas_quote (map Sign.base_name cnames)) else ();

        val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);

        val (thy1, mono, fp_def, rec_sets_defs, rec_const, sumT) =
          mk_ind_def declare_consts alt_name coind cs intr_ts monos thy
            params paramTs cTs cnames;

        val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts rec_sets_defs thy1;
        val elims = if no_elim then [] else
          prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy1;
        val raw_induct = if no_ind then Drule.asm_rl else
          if coind then standard (rule_by_tactic
            (rewrite_tac [mk_meta_eq vimage_Un] THEN
              fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
          else
            prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
              rec_sets_defs thy1;
        val induct =
          if coind orelse no_ind orelse length cs > 1 then (raw_induct, [RuleCases.consumes 0])
          else (raw_induct RSN (2, rev_mp), [RuleCases.consumes 1]);

        val (thy2, intrs') =
          thy1 |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts);
        val (thy3, ([intrs'', elims'], [induct'])) =
          thy2
          |> PureThy.add_thmss
            [(("intros", intrs'), []),
              (("elims", elims), [RuleCases.consumes 1])]
          |>>> PureThy.add_thms
            [((coind_prefix coind ^ "induct", rulify (#1 induct)),
             (RuleCases.case_names induct_cases :: #2 induct))]
          |>> Theory.parent_path;
      in (thy3,
        {defs = fp_def :: rec_sets_defs,
         mono = mono,
         unfold = unfold,
         intrs = intrs',
         elims = elims',
         mk_cases = mk_cases elims',
         raw_induct = rulify raw_induct,
         induct = induct'})
      end;

    (* external interfaces *)

    fun try_term f msg thy t =
      (case Library.try f t of
        SOME x => x
      | NONE => error (msg ^ Sign.string_of_term thy t));

    fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs pre_intros monos thy =
      let
        val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");

        (*parameters should agree for all mutually recursive components*)
        val (_, params) = strip_comb (hd cs);
        val paramTs = map (try_term (snd o dest_Free) "Parameter in recursive\
          \ component is not a free variable: " thy) params;

        val cTs = map (try_term (HOLogic.dest_setT o fastype_of)
          "Recursive component not of type set: " thy) cs;

        val cnames = map (try_term (fst o dest_Const o head_of)
          "Recursive set not previously declared as constant: " thy) cs;

        val save_thy = thy
          |> Theory.copy |> cond_declare_consts declare_consts cs paramTs cnames;
        val intros = map (check_rule save_thy cs) pre_intros;
        val induct_cases = map (#1 o #1) intros;

        val (thy1, result as {elims, induct, ...}) =
          add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs intros monos
            thy params paramTs cTs cnames induct_cases;
        val thy2 = thy1
          |> put_inductives cnames ({names = cnames, coind = coind}, result)
          |> add_cases_induct no_elim (no_ind orelse coind orelse length cs > 1)
              cnames elims induct;
      in (thy2, result) end;

    fun add_inductive verbose coind c_strings intro_srcs raw_monos thy =
      let
        val cs = map (Sign.read_term thy) c_strings;

        val intr_names = map (fst o fst) intro_srcs;
        fun read_rule s = Thm.read_cterm thy (s, propT)
          handle ERROR => error ("The error(s) above occurred for " ^ s);
        val intr_ts = map (Thm.term_of o read_rule o snd o fst) intro_srcs;
        val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
        val (cs', intr_ts') = unify_consts thy cs intr_ts;

        val (thy', monos) = thy |> IsarThy.apply_theorems raw_monos;
      in
        add_inductive_i verbose false "" coind false false cs'
          ((intr_names ~~ intr_ts') ~~ intr_atts) monos thy'
      end;

    (** package setup **)

    (* setup theory *)

    val setup =
     [InductiveData.init,
      Method.add_methods [("ind_cases", mk_cases_meth oo mk_cases_args,
        "dynamic case analysis on sets")],
      Attrib.add_attributes [("mono", mono_attr, "declaration of monotonicity rule")]];

    (* outer syntax *)

    local structure P = OuterParse and K = OuterKeyword in

    fun mk_ind coind ((sets, intrs), monos) =
      #1 o add_inductive true coind sets (map P.triple_swap intrs) monos;

    fun ind_decl coind =
      Scan.repeat1 P.term --
      (P.$$$ "intros" |--
        P.!!! (Scan.repeat1 (P.opt_thm_name ":" -- P.prop))) --
      Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1) []
      >> (Toplevel.theory o mk_ind coind);

    val inductiveP =
      OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);

    val coinductiveP =
      OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);

    val ind_cases =
      P.and_list1 (P.opt_thm_name ":" -- Scan.repeat1 P.prop)
      >> (Toplevel.theory o inductive_cases);

    val inductive_casesP =
      OuterSyntax.command "inductive_cases"
        "create simplified instances of elimination rules (improper)" K.thy_script ind_cases;

    val _ = OuterSyntax.add_keywords ["intros", "monos"];
    val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];

    end;

    end;