src/HOL/Tools/inductive_package.ML

New inductive definition package

(*  Title:      HOL/Tools/inductive_package.ML
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
                Stefan Berghofer,   TU Muenchen
    Copyright   1994  University of Cambridge
                1998  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 parent 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 add_inductive_i : bool -> bool -> bstring -> bool -> bool -> bool -> term list ->
    term list -> thm list -> thm list -> theory -> theory *
      {defs:thm list, elims:thm list, raw_induct:thm, induct:thm,
       intrs:thm list,
       mk_cases:thm list -> string -> thm, mono:thm,
       unfold:thm}
  val add_inductive : bool -> bool -> string list -> string list
    -> thm list -> thm list -> theory -> theory *
      {defs:thm list, elims:thm list, raw_induct:thm, induct:thm,
       intrs:thm list,
       mk_cases:thm list -> string -> thm, mono:thm,
       unfold:thm}
end;

structure InductivePackage : INDUCTIVE_PACKAGE =
struct

(*For proving monotonicity of recursion operator*)
val basic_monos = [subset_refl, imp_refl, disj_mono, conj_mono, 
                   ex_mono, Collect_mono, in_mono, vimage_mono];

val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (concl_of vimageD);

(*Delete needless equality assumptions*)
val refl_thin = prove_goal HOL.thy "!!P. [| a=a;  P |] ==> P"
     (fn _ => [assume_tac 1]);

(*For simplifying the elimination rule*)
val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, disjE, Pair_inject];

val vimage_name = Sign.intern_const (sign_of Vimage.thy) "op -``";
val mono_name = Sign.intern_const (sign_of Ord.thy) "mono";

(* make injections needed 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 ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
        else
          Const ("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;

(**************************** well-formedness checks **************************)

fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
  (Sign.string_of_term sign t) ^ "\n" ^ msg);

fun err_in_prem sign t p msg = error ("Ill-formed premise\n" ^
  (Sign.string_of_term sign p) ^ "\nin introduction rule\n" ^
  (Sign.string_of_term sign t) ^ "\n" ^ msg);

val msg1 = "Conclusion of introduction rule must have form\
          \ ' t : S_i '";
val msg2 = "Premises mentioning recursive sets must have form\
          \ ' t : M S_i '";
val msg3 = "Recursion term on left of member symbol";

fun check_rule sign cs r =
  let
    fun check_prem prem = if exists (Logic.occs o (rpair prem)) cs then
         (case prem of
           (Const ("op :", _) $ t $ u) =>
             if exists (Logic.occs o (rpair t)) cs then
               err_in_prem sign r prem msg3 else ()
         | _ => err_in_prem sign r prem msg2)
        else ()

  in (case (HOLogic.dest_Trueprop o Logic.strip_imp_concl) r of
        (Const ("op :", _) $ _ $ u) =>
          if u mem cs then map (check_prem o HOLogic.dest_Trueprop)
            (Logic.strip_imp_prems r)
          else err_in_rule sign r msg1
      | _ => err_in_rule sign r msg1)
  end;

fun try' f msg sign t = (f t) handle _ => error (msg ^ Sign.string_of_term sign t);

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

(***************************** elimination rules ******************************)

fun mk_elims cs cTs params intr_ts =
  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;

    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));
      in
        Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
          map mk_elim_prem (filter (equal c o #1) intrs), P)
      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);

        fun subst (prem as (Const ("op :", T) $ t $ u), prems) =
              let val n = find_index_eq u cs in
                if n >= 0 then prem :: (nth_elem (n, preds)) $ t :: prems else
                  (subst_free (map (fn (c, P) => (c, HOLogic.mk_binop "op Int"
                    (c, HOLogic.Collect_const (HOLogic.dest_setT
                      (fastype_of c)) $ P))) (cs ~~ preds)) prem)::prems
              end
          | subst (prem, prems) = prem::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 subst
             (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
               HOLogic.mk_Trueprop (nth_elem (find_index_eq u cs, preds) $ t)))
      end;

    val ind_prems = map mk_ind_prem intr_ts;

    (* make conclusions for induction rules *)

    fun mk_ind_concl ((c, P), (ts, x)) =
      let val T = HOLogic.dest_setT (fastype_of c);
          val Ts = HOLogic.prodT_factors T;
          val (frees, x') = foldr (fn (T', (fs, s)) =>
            ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
          val tuple = HOLogic.mk_tuple 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 (app HOLogic.conj)
        (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))

  in (preds, ind_prems, mutual_ind_concl)
  end;

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

(**************************** prove monotonicity ******************************)

fun prove_mono setT fp_fun monos thy =
  let
    val _ = writeln "  Proving monotonicity...";

    val mono = prove_goalw_cterm [] (cterm_of (sign_of thy) (HOLogic.mk_Trueprop
      (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
        (fn _ => [rtac monoI 1, REPEAT (ares_tac (basic_monos @ monos) 1)])

  in mono end;

(************************* prove introduction rules ***************************)

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

    val unfold = standard (mono RS (fp_def RS
      (if coind then def_gfp_Tarski else def_lfp_Tarski)));

    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) => prove_goalw_cterm rec_sets_defs
      (cterm_of (sign_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*)
       rewrite_goals_tac con_defs,
       REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)

  in (intrs, unfold) end;

(*************************** prove elimination rules **************************)

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

    val rules1 = [CollectE, disjE, make_elim vimageD];
    val rules2 = [exE, conjE, Inl_neq_Inr, Inr_neq_Inl] @
      map make_elim [Inl_inject, Inr_inject];

    val elims = map (fn t => prove_goalw_cterm rec_sets_defs
      (cterm_of (sign_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))]))
      (mk_elims cs cTs params intr_ts)

  in elims end;

(** derivation of simplified elimination rules **)

(*Applies freeness of the given constructors, which *must* be unfolded by
  the given defs.  Cannot simply use the local con_defs because con_defs=[] 
  for inference systems.
  FIXME: proper handling of conjunctive / disjunctive side conditions?!
 *)
fun con_elim_tac simps =
  let val elim_tac = REPEAT o (eresolve_tac elim_rls)
  in ALLGOALS(EVERY'[elim_tac,
                 asm_full_simp_tac (simpset_of Nat.thy addsimps simps),
                 elim_tac,
                 REPEAT o bound_hyp_subst_tac])
     THEN prune_params_tac
  end;

(*String s should have the form t:Si where Si is an inductive set*)
fun mk_cases elims defs s =
  let val prem = assume (read_cterm (sign_of_thm (hd elims)) (s, propT));
      val elims' = map (try (fn r =>
        rule_by_tactic (con_elim_tac defs) (prem RS r) |> standard)) elims
  in case find_first is_some elims' of
       Some (Some r) => r
     | None => error ("mk_cases: string '" ^ s ^ "' not of form 't : S_i'")
  end;

(**************************** prove induction rule ****************************)

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

    val sign = sign_of thy;

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

    (* 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 ("sum_case",
           [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
             mk_ind_pred T1 (take (n, Ps)) $ mk_ind_pred T2 (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 => prove_goalw_cterm [] (cterm_of sign
        (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)) $
             nth_elem (find_index_eq c cs, preds)))))
        (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac
           (map mk_meta_eq [sum_case_Inl, sum_case_Inr]),
          rtac refl 1])) cs;

    val induct = prove_goalw_cterm [] (cterm_of sign
      (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
        [rtac (impI RS allI) 1,
         DETERM (etac (mono RS (fp_def RS def_induct)) 1),
         rewrite_goals_tac (map mk_meta_eq (vimage_Int::vimage_simps)),
         fold_goals_tac rec_sets_defs,
         (*This CollectE and disjE separates out the introduction rules*)
         REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE])),
         (*Now break down the individual cases.  No disjE here in case
           some premise involves disjunction.*)
         REPEAT (FIRSTGOAL (eresolve_tac [IntE, CollectE, exE, conjE] 
                     ORELSE' hyp_subst_tac)),
         rewrite_goals_tac (map mk_meta_eq [sum_case_Inl, sum_case_Inr]),
         EVERY (map (fn prem =>
           DEPTH_SOLVE_1 (ares_tac (prem::[conjI, refl]) 1)) prems)]);

    val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
      (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 (map mk_meta_eq [sum_case_Inl, sum_case_Inr]),
            atac 1])])

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

(*************** definitional introduction of (co)inductive sets **************)

fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
    intr_ts monos con_defs thy params paramTs cTs cnames =
  let
    val _ = if verbose then writeln ("Proofs for " ^
      (if coind then "co" else "") ^ "inductive set(s) " ^ commas cnames) else ();

    val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
    val setT = HOLogic.mk_setT sumT;

    val fp_name = if coind then Sign.intern_const (sign_of Gfp.thy) "gfp"
      else Sign.intern_const (sign_of Lfp.thy) "lfp";

    (* 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 idx = length frees;
        val subst = subst_free (cs ~~ (map (mk_vimage cs sumT (Bound (idx + 1))) 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))
        (frees, foldr1 (app HOLogic.conj)
          (((HOLogic.eq_const sumT) $ Bound idx $ (mk_inj cs sumT u t))::
            (map (subst o HOLogic.dest_Trueprop)
              (Logic.strip_imp_prems r))))
      end

    (* make a disjunction of all introduction rules *)

    val fp_fun = Abs ("S", setT, (HOLogic.Collect_const sumT) $
      Abs ("x", sumT, foldr1 (app HOLogic.disj) (map transform_rule intr_ts)));

    (* add definiton of recursive sets to theory *)

    val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
    val full_rec_name = Sign.full_name (sign_of 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' = thy |>
      (if declare_consts then
        Theory.add_consts_i (map (fn (c, n) =>
          (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
       else I) |>
      (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 [(("defs", def_terms), [])];

    (* get definitions from theory *)

    val fp_def::rec_sets_defs = get_thms thy' "defs";

    (* prove and store theorems *)

    val mono = prove_mono setT fp_fun monos thy';
    val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
      rec_sets_defs thy';
    val elims = if no_elim then [] else
      prove_elims cs cTs params intr_ts unfold rec_sets_defs thy';
    val raw_induct = if no_ind then TrueI 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 thy';
    val induct = if coind orelse length cs > 1 then raw_induct
      else standard (raw_induct RSN (2, rev_mp));

    val thy'' = thy' |>
      PureThy.add_tthmss [(("intrs", map Attribute.tthm_of intrs), [])] |>
      (if no_elim then I else PureThy.add_tthmss
        [(("elims", map Attribute.tthm_of elims), [])]) |>
      (if no_ind then I else PureThy.add_tthms
        [(((if coind then "co" else "") ^ "induct",
          Attribute.tthm_of induct), [])]) |>
      Theory.parent_path;

  in (thy'',
    {defs = fp_def::rec_sets_defs,
     mono = mono,
     unfold = unfold,
     intrs = intrs,
     elims = elims,
     mk_cases = mk_cases elims,
     raw_induct = raw_induct,
     induct = induct})
  end;

(***************** axiomatic introduction of (co)inductive sets ***************)

fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
    intr_ts monos con_defs thy params paramTs cTs cnames =
  let
    val _ = if verbose then writeln ("Adding axioms for " ^
      (if coind then "co" else "") ^ "inductive set(s) " ^ commas cnames) else ();

    val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;

    val elim_ts = mk_elims cs cTs params intr_ts;

    val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
    val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
    
    val thy' = thy |>
      (if declare_consts then
        Theory.add_consts_i (map (fn (c, n) =>
          (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
       else I) |>
      Theory.add_path rec_name |>
      PureThy.add_axiomss_i [(("intrs", intr_ts), []), (("elims", elim_ts), [])] |>
      (if coind then I
       else PureThy.add_axioms_i [(("internal_induct", ind_t), [])]);

    val intrs = get_thms thy' "intrs";
    val elims = get_thms thy' "elims";
    val raw_induct = if coind then TrueI else
      standard (split_rule (get_thm thy' "internal_induct"));
    val induct = if coind orelse length cs > 1 then raw_induct
      else standard (raw_induct RSN (2, rev_mp));

    val thy'' = thy' |>
      (if coind then I
       else PureThy.add_tthms [(("induct", Attribute.tthm_of induct), [])]) |>
      Theory.parent_path

  in (thy'',
    {defs = [],
     mono = TrueI,
     unfold = TrueI,
     intrs = intrs,
     elims = elims,
     mk_cases = mk_cases elims,
     raw_induct = raw_induct,
     induct = induct})
  end;

(********************** introduction of (co)inductive sets ********************)

fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
    intr_ts monos con_defs thy =
  let
    val _ = Theory.requires thy "Inductive"
      ((if coind then "co" else "") ^ "inductive definitions");

    val sign = sign_of thy;

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

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

    val cnames = map (try' (Sign.base_name o fst o dest_Const o head_of)
      "Recursive set not previously declared as constant: " sign) cs;

    val _ = assert_all Syntax.is_identifier cnames
       (fn a => "Base name of recursive set not an identifier: " ^ a);

    val _ = map (check_rule sign cs) intr_ts;

  in
    (if !quick_and_dirty then add_ind_axm else add_ind_def)
      verbose declare_consts alt_name coind no_elim no_ind cs intr_ts monos
        con_defs thy params paramTs cTs cnames
  end;

(***************************** external interface *****************************)

fun add_inductive verbose coind c_strings intr_strings monos con_defs thy =
  let
    val sign = sign_of thy;
    val cs = map (readtm (sign_of thy) HOLogic.termTVar) c_strings;
    val intr_ts = map (readtm (sign_of thy) propT) intr_strings;

    (* the following code ensures that each recursive set *)
    (* always has the same type in all introduction rules *)

    val {tsig, ...} = Sign.rep_sg sign;
    val add_term_consts_2 =
      foldl_aterms (fn (cs, Const c) => c ins cs | (cs, _) => cs);
    fun varify (t, (i, ts)) =
      let val t' = map_term_types (incr_tvar (i + 1)) (Type.varify (t, []))
      in (maxidx_of_term t', t'::ts) end;
    val (i, cs') = foldr varify (cs, (~1, []));
    val (i', intr_ts') = foldr varify (intr_ts, (i, []));
    val rec_consts = foldl add_term_consts_2 ([], cs');
    val intr_consts = foldl add_term_consts_2 ([], intr_ts');
    fun unify (env, (cname, cT)) =
      let val consts = map snd (filter (fn c => fst c = cname) intr_consts)
      in (foldl (fn ((env', j'), Tp) => Type.unify tsig j' env' Tp)
        (env, (replicate (length consts) cT) ~~ consts)) handle _ =>
          error ("Occurrences of constant '" ^ cname ^
            "' have incompatible types")
      end;
    val (env, _) = foldl unify (([], i'), rec_consts);
    fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars env T
      in if T = T' then T else typ_subst_TVars_2 env T' end;
    val subst = fst o Type.freeze_thaw o
      (map_term_types (typ_subst_TVars_2 env));
    val cs'' = map subst cs';
    val intr_ts'' = map subst intr_ts';

  in add_inductive_i verbose false "" coind false false cs'' intr_ts''
    monos con_defs thy
  end;

end;