src/HOL/Tools/inductive_package.ML
author berghofe
Fri Oct 16 18:55:34 1998 +0200 (1998-10-16)
changeset 5662 72a2e33d3b9e
parent 5553 ae42b36a50c2
child 5718 e5094d678285
permissions -rw-r--r--
Added quiet_mode flag.
     1 (*  Title:      HOL/Tools/inductive_package.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4                 Stefan Berghofer,   TU Muenchen
     5     Copyright   1994  University of Cambridge
     6                 1998  TU Muenchen     
     7 
     8 (Co)Inductive Definition module for HOL
     9 
    10 Features:
    11 * least or greatest fixedpoints
    12 * user-specified product and sum constructions
    13 * mutually recursive definitions
    14 * definitions involving arbitrary monotone operators
    15 * automatically proves introduction and elimination rules
    16 
    17 The recursive sets must *already* be declared as constants in parent theory!
    18 
    19   Introduction rules have the form
    20   [| ti:M(Sj), ..., P(x), ... |] ==> t: Sk |]
    21   where M is some monotone operator (usually the identity)
    22   P(x) is any side condition on the free variables
    23   ti, t are any terms
    24   Sj, Sk are two of the sets being defined in mutual recursion
    25 
    26 Sums are used only for mutual recursion;
    27 Products are used only to derive "streamlined" induction rules for relations
    28 *)
    29 
    30 signature INDUCTIVE_PACKAGE =
    31 sig
    32   val quiet_mode : bool ref
    33   val add_inductive_i : bool -> bool -> bstring -> bool -> bool -> bool -> term list ->
    34     term list -> thm list -> thm list -> theory -> theory *
    35       {defs:thm list, elims:thm list, raw_induct:thm, induct:thm,
    36        intrs:thm list,
    37        mk_cases:thm list -> string -> thm, mono:thm,
    38        unfold:thm}
    39   val add_inductive : bool -> bool -> string list -> string list
    40     -> thm list -> thm list -> theory -> theory *
    41       {defs:thm list, elims:thm list, raw_induct:thm, induct:thm,
    42        intrs:thm list,
    43        mk_cases:thm list -> string -> thm, mono:thm,
    44        unfold:thm}
    45 end;
    46 
    47 structure InductivePackage : INDUCTIVE_PACKAGE =
    48 struct
    49 
    50 val quiet_mode = ref false;
    51 fun message s = if !quiet_mode then () else writeln s;
    52 
    53 (*For proving monotonicity of recursion operator*)
    54 val basic_monos = [subset_refl, imp_refl, disj_mono, conj_mono, 
    55                    ex_mono, Collect_mono, in_mono, vimage_mono];
    56 
    57 val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (concl_of vimageD);
    58 
    59 (*Delete needless equality assumptions*)
    60 val refl_thin = prove_goal HOL.thy "!!P. [| a=a;  P |] ==> P"
    61      (fn _ => [assume_tac 1]);
    62 
    63 (*For simplifying the elimination rule*)
    64 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject];
    65 
    66 val vimage_name = Sign.intern_const (sign_of Vimage.thy) "op -``";
    67 val mono_name = Sign.intern_const (sign_of Ord.thy) "mono";
    68 
    69 (* make injections needed in mutually recursive definitions *)
    70 
    71 fun mk_inj cs sumT c x =
    72   let
    73     fun mk_inj' T n i =
    74       if n = 1 then x else
    75       let val n2 = n div 2;
    76           val Type (_, [T1, T2]) = T
    77       in
    78         if i <= n2 then
    79           Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
    80         else
    81           Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
    82       end
    83   in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
    84   end;
    85 
    86 (* make "vimage" terms for selecting out components of mutually rec.def. *)
    87 
    88 fun mk_vimage cs sumT t c = if length cs < 2 then t else
    89   let
    90     val cT = HOLogic.dest_setT (fastype_of c);
    91     val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
    92   in
    93     Const (vimage_name, vimageT) $
    94       Abs ("y", cT, mk_inj cs sumT c (Bound 0)) $ t
    95   end;
    96 
    97 (**************************** well-formedness checks **************************)
    98 
    99 fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
   100   (Sign.string_of_term sign t) ^ "\n" ^ msg);
   101 
   102 fun err_in_prem sign t p msg = error ("Ill-formed premise\n" ^
   103   (Sign.string_of_term sign p) ^ "\nin introduction rule\n" ^
   104   (Sign.string_of_term sign t) ^ "\n" ^ msg);
   105 
   106 val msg1 = "Conclusion of introduction rule must have form\
   107           \ ' t : S_i '";
   108 val msg2 = "Premises mentioning recursive sets must have form\
   109           \ ' t : M S_i '";
   110 val msg3 = "Recursion term on left of member symbol";
   111 
   112 fun check_rule sign cs r =
   113   let
   114     fun check_prem prem = if exists (Logic.occs o (rpair prem)) cs then
   115          (case prem of
   116            (Const ("op :", _) $ t $ u) =>
   117              if exists (Logic.occs o (rpair t)) cs then
   118                err_in_prem sign r prem msg3 else ()
   119          | _ => err_in_prem sign r prem msg2)
   120         else ()
   121 
   122   in (case (HOLogic.dest_Trueprop o Logic.strip_imp_concl) r of
   123         (Const ("op :", _) $ _ $ u) =>
   124           if u mem cs then map (check_prem o HOLogic.dest_Trueprop)
   125             (Logic.strip_imp_prems r)
   126           else err_in_rule sign r msg1
   127       | _ => err_in_rule sign r msg1)
   128   end;
   129 
   130 fun try' f msg sign t = (f t) handle _ => error (msg ^ Sign.string_of_term sign t);
   131 
   132 (*********************** properties of (co)inductive sets *********************)
   133 
   134 (***************************** elimination rules ******************************)
   135 
   136 fun mk_elims cs cTs params intr_ts =
   137   let
   138     val used = foldr add_term_names (intr_ts, []);
   139     val [aname, pname] = variantlist (["a", "P"], used);
   140     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   141 
   142     fun dest_intr r =
   143       let val Const ("op :", _) $ t $ u =
   144         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   145       in (u, t, Logic.strip_imp_prems r) end;
   146 
   147     val intrs = map dest_intr intr_ts;
   148 
   149     fun mk_elim (c, T) =
   150       let
   151         val a = Free (aname, T);
   152 
   153         fun mk_elim_prem (_, t, ts) =
   154           list_all_free (map dest_Free ((foldr add_term_frees (t::ts, [])) \\ params),
   155             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
   156       in
   157         Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
   158           map mk_elim_prem (filter (equal c o #1) intrs), P)
   159       end
   160   in
   161     map mk_elim (cs ~~ cTs)
   162   end;
   163         
   164 (***************** premises and conclusions of induction rules ****************)
   165 
   166 fun mk_indrule cs cTs params intr_ts =
   167   let
   168     val used = foldr add_term_names (intr_ts, []);
   169 
   170     (* predicates for induction rule *)
   171 
   172     val preds = map Free (variantlist (if length cs < 2 then ["P"] else
   173       map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
   174         map (fn T => T --> HOLogic.boolT) cTs);
   175 
   176     (* transform an introduction rule into a premise for induction rule *)
   177 
   178     fun mk_ind_prem r =
   179       let
   180         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   181 
   182         fun subst (prem as (Const ("op :", T) $ t $ u), prems) =
   183               let val n = find_index_eq u cs in
   184                 if n >= 0 then prem :: (nth_elem (n, preds)) $ t :: prems else
   185                   (subst_free (map (fn (c, P) => (c, HOLogic.mk_binop "op Int"
   186                     (c, HOLogic.Collect_const (HOLogic.dest_setT
   187                       (fastype_of c)) $ P))) (cs ~~ preds)) prem)::prems
   188               end
   189           | subst (prem, prems) = prem::prems;
   190 
   191         val Const ("op :", _) $ t $ u =
   192           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   193 
   194       in list_all_free (frees,
   195            Logic.list_implies (map HOLogic.mk_Trueprop (foldr subst
   196              (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
   197                HOLogic.mk_Trueprop (nth_elem (find_index_eq u cs, preds) $ t)))
   198       end;
   199 
   200     val ind_prems = map mk_ind_prem intr_ts;
   201 
   202     (* make conclusions for induction rules *)
   203 
   204     fun mk_ind_concl ((c, P), (ts, x)) =
   205       let val T = HOLogic.dest_setT (fastype_of c);
   206           val Ts = HOLogic.prodT_factors T;
   207           val (frees, x') = foldr (fn (T', (fs, s)) =>
   208             ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
   209           val tuple = HOLogic.mk_tuple T frees;
   210       in ((HOLogic.mk_binop "op -->"
   211         (HOLogic.mk_mem (tuple, c), P $ tuple))::ts, x')
   212       end;
   213 
   214     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 (app HOLogic.conj)
   215         (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
   216 
   217   in (preds, ind_prems, mutual_ind_concl)
   218   end;
   219 
   220 (********************** proofs for (co)inductive sets *************************)
   221 
   222 (**************************** prove monotonicity ******************************)
   223 
   224 fun prove_mono setT fp_fun monos thy =
   225   let
   226     val _ = message "  Proving monotonicity...";
   227 
   228     val mono = prove_goalw_cterm [] (cterm_of (sign_of thy) (HOLogic.mk_Trueprop
   229       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
   230         (fn _ => [rtac monoI 1, REPEAT (ares_tac (basic_monos @ monos) 1)])
   231 
   232   in mono end;
   233 
   234 (************************* prove introduction rules ***************************)
   235 
   236 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
   237   let
   238     val _ = message "  Proving the introduction rules...";
   239 
   240     val unfold = standard (mono RS (fp_def RS
   241       (if coind then def_gfp_Tarski else def_lfp_Tarski)));
   242 
   243     fun select_disj 1 1 = []
   244       | select_disj _ 1 = [rtac disjI1]
   245       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   246 
   247     val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
   248       (cterm_of (sign_of thy) intr) (fn prems =>
   249        [(*insert prems and underlying sets*)
   250        cut_facts_tac prems 1,
   251        stac unfold 1,
   252        REPEAT (resolve_tac [vimageI2, CollectI] 1),
   253        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
   254        EVERY1 (select_disj (length intr_ts) i),
   255        (*Not ares_tac, since refl must be tried before any equality assumptions;
   256          backtracking may occur if the premises have extra variables!*)
   257        DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
   258        (*Now solve the equations like Inl 0 = Inl ?b2*)
   259        rewrite_goals_tac con_defs,
   260        REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
   261 
   262   in (intrs, unfold) end;
   263 
   264 (*************************** prove elimination rules **************************)
   265 
   266 fun prove_elims cs cTs params intr_ts unfold rec_sets_defs thy =
   267   let
   268     val _ = message "  Proving the elimination rules...";
   269 
   270     val rules1 = [CollectE, disjE, make_elim vimageD];
   271     val rules2 = [exE, conjE, Inl_neq_Inr, Inr_neq_Inl] @
   272       map make_elim [Inl_inject, Inr_inject];
   273 
   274     val elims = map (fn t => prove_goalw_cterm rec_sets_defs
   275       (cterm_of (sign_of thy) t) (fn prems =>
   276         [cut_facts_tac [hd prems] 1,
   277          dtac (unfold RS subst) 1,
   278          REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   279          REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   280          EVERY (map (fn prem =>
   281            DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))]))
   282       (mk_elims cs cTs params intr_ts)
   283 
   284   in elims end;
   285 
   286 (** derivation of simplified elimination rules **)
   287 
   288 (*Applies freeness of the given constructors, which *must* be unfolded by
   289   the given defs.  Cannot simply use the local con_defs because con_defs=[] 
   290   for inference systems.
   291  *)
   292 fun con_elim_tac simps =
   293   let val elim_tac = REPEAT o (eresolve_tac elim_rls)
   294   in ALLGOALS(EVERY'[elim_tac,
   295                  asm_full_simp_tac (simpset_of NatDef.thy addsimps simps),
   296                  elim_tac,
   297                  REPEAT o bound_hyp_subst_tac])
   298      THEN prune_params_tac
   299   end;
   300 
   301 (*String s should have the form t:Si where Si is an inductive set*)
   302 fun mk_cases elims simps s =
   303   let val prem = assume (read_cterm (sign_of_thm (hd elims)) (s, propT));
   304       val elims' = map (try (fn r =>
   305         rule_by_tactic (con_elim_tac simps) (prem RS r) |> standard)) elims
   306   in case find_first is_some elims' of
   307        Some (Some r) => r
   308      | None => error ("mk_cases: string '" ^ s ^ "' not of form 't : S_i'")
   309   end;
   310 
   311 (**************************** prove induction rule ****************************)
   312 
   313 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
   314     fp_def rec_sets_defs thy =
   315   let
   316     val _ = message "  Proving the induction rule...";
   317 
   318     val sign = sign_of thy;
   319 
   320     val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
   321 
   322     (* make predicate for instantiation of abstract induction rule *)
   323 
   324     fun mk_ind_pred _ [P] = P
   325       | mk_ind_pred T Ps =
   326          let val n = (length Ps) div 2;
   327              val Type (_, [T1, T2]) = T
   328          in Const ("sum_case",
   329            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
   330              mk_ind_pred T1 (take (n, Ps)) $ mk_ind_pred T2 (drop (n, Ps))
   331          end;
   332 
   333     val ind_pred = mk_ind_pred sumT preds;
   334 
   335     val ind_concl = HOLogic.mk_Trueprop
   336       (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->"
   337         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0)));
   338 
   339     (* simplification rules for vimage and Collect *)
   340 
   341     val vimage_simps = if length cs < 2 then [] else
   342       map (fn c => prove_goalw_cterm [] (cterm_of sign
   343         (HOLogic.mk_Trueprop (HOLogic.mk_eq
   344           (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c,
   345            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $
   346              nth_elem (find_index_eq c cs, preds)))))
   347         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac
   348            (map mk_meta_eq [sum_case_Inl, sum_case_Inr]),
   349           rtac refl 1])) cs;
   350 
   351     val induct = prove_goalw_cterm [] (cterm_of sign
   352       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
   353         [rtac (impI RS allI) 1,
   354          DETERM (etac (mono RS (fp_def RS def_induct)) 1),
   355          rewrite_goals_tac (map mk_meta_eq (vimage_Int::vimage_simps)),
   356          fold_goals_tac rec_sets_defs,
   357          (*This CollectE and disjE separates out the introduction rules*)
   358          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE])),
   359          (*Now break down the individual cases.  No disjE here in case
   360            some premise involves disjunction.*)
   361          REPEAT (FIRSTGOAL (eresolve_tac [IntE, CollectE, exE, conjE] 
   362                      ORELSE' hyp_subst_tac)),
   363          rewrite_goals_tac (map mk_meta_eq [sum_case_Inl, sum_case_Inr]),
   364          EVERY (map (fn prem =>
   365            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
   366 
   367     val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
   368       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
   369         [cut_facts_tac prems 1,
   370          REPEAT (EVERY
   371            [REPEAT (resolve_tac [conjI, impI] 1),
   372             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
   373             rewrite_goals_tac (map mk_meta_eq [sum_case_Inl, sum_case_Inr]),
   374             atac 1])])
   375 
   376   in standard (split_rule (induct RS lemma))
   377   end;
   378 
   379 (*************** definitional introduction of (co)inductive sets **************)
   380 
   381 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
   382     intr_ts monos con_defs thy params paramTs cTs cnames =
   383   let
   384     val _ = if verbose then message ("Proofs for " ^
   385       (if coind then "co" else "") ^ "inductive set(s) " ^ commas cnames) else ();
   386 
   387     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
   388     val setT = HOLogic.mk_setT sumT;
   389 
   390     val fp_name = if coind then Sign.intern_const (sign_of Gfp.thy) "gfp"
   391       else Sign.intern_const (sign_of Lfp.thy) "lfp";
   392 
   393     val used = foldr add_term_names (intr_ts, []);
   394     val [sname, xname] = variantlist (["S", "x"], used);
   395 
   396     (* transform an introduction rule into a conjunction  *)
   397     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
   398     (* is transformed into                                *)
   399     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
   400 
   401     fun transform_rule r =
   402       let
   403         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   404         val subst = subst_free
   405           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
   406         val Const ("op :", _) $ t $ u =
   407           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   408 
   409       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
   410         (frees, foldr1 (app HOLogic.conj)
   411           (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t))::
   412             (map (subst o HOLogic.dest_Trueprop)
   413               (Logic.strip_imp_prems r))))
   414       end
   415 
   416     (* make a disjunction of all introduction rules *)
   417 
   418     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
   419       absfree (xname, sumT, foldr1 (app HOLogic.disj) (map transform_rule intr_ts)));
   420 
   421     (* add definiton of recursive sets to theory *)
   422 
   423     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
   424     val full_rec_name = Sign.full_name (sign_of thy) rec_name;
   425 
   426     val rec_const = list_comb
   427       (Const (full_rec_name, paramTs ---> setT), params);
   428 
   429     val fp_def_term = Logic.mk_equals (rec_const,
   430       Const (fp_name, (setT --> setT) --> setT) $ fp_fun)
   431 
   432     val def_terms = fp_def_term :: (if length cs < 2 then [] else
   433       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
   434 
   435     val thy' = thy |>
   436       (if declare_consts then
   437         Theory.add_consts_i (map (fn (c, n) =>
   438           (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
   439        else I) |>
   440       (if length cs < 2 then I else
   441        Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)]) |>
   442       Theory.add_path rec_name |>
   443       PureThy.add_defss_i [(("defs", def_terms), [])];
   444 
   445     (* get definitions from theory *)
   446 
   447     val fp_def::rec_sets_defs = get_thms thy' "defs";
   448 
   449     (* prove and store theorems *)
   450 
   451     val mono = prove_mono setT fp_fun monos thy';
   452     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
   453       rec_sets_defs thy';
   454     val elims = if no_elim then [] else
   455       prove_elims cs cTs params intr_ts unfold rec_sets_defs thy';
   456     val raw_induct = if no_ind then TrueI else
   457       if coind then standard (rule_by_tactic
   458         (rewrite_tac [mk_meta_eq vimage_Un] THEN
   459           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
   460       else
   461         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
   462           rec_sets_defs thy';
   463     val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
   464       else standard (raw_induct RSN (2, rev_mp));
   465 
   466     val thy'' = thy' |>
   467       PureThy.add_tthmss [(("intrs", map Attribute.tthm_of intrs), [])] |>
   468       (if no_elim then I else PureThy.add_tthmss
   469         [(("elims", map Attribute.tthm_of elims), [])]) |>
   470       (if no_ind then I else PureThy.add_tthms
   471         [(((if coind then "co" else "") ^ "induct",
   472           Attribute.tthm_of induct), [])]) |>
   473       Theory.parent_path;
   474 
   475   in (thy'',
   476     {defs = fp_def::rec_sets_defs,
   477      mono = mono,
   478      unfold = unfold,
   479      intrs = intrs,
   480      elims = elims,
   481      mk_cases = mk_cases elims,
   482      raw_induct = raw_induct,
   483      induct = induct})
   484   end;
   485 
   486 (***************** axiomatic introduction of (co)inductive sets ***************)
   487 
   488 fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
   489     intr_ts monos con_defs thy params paramTs cTs cnames =
   490   let
   491     val _ = if verbose then message ("Adding axioms for " ^
   492       (if coind then "co" else "") ^ "inductive set(s) " ^ commas cnames) else ();
   493 
   494     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
   495 
   496     val elim_ts = mk_elims cs cTs params intr_ts;
   497 
   498     val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
   499     val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
   500     
   501     val thy' = thy |>
   502       (if declare_consts then
   503         Theory.add_consts_i (map (fn (c, n) =>
   504           (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
   505        else I) |>
   506       Theory.add_path rec_name |>
   507       PureThy.add_axiomss_i [(("intrs", intr_ts), []), (("elims", elim_ts), [])] |>
   508       (if coind then I
   509        else PureThy.add_axioms_i [(("internal_induct", ind_t), [])]);
   510 
   511     val intrs = get_thms thy' "intrs";
   512     val elims = get_thms thy' "elims";
   513     val raw_induct = if coind then TrueI else
   514       standard (split_rule (get_thm thy' "internal_induct"));
   515     val induct = if coind orelse length cs > 1 then raw_induct
   516       else standard (raw_induct RSN (2, rev_mp));
   517 
   518     val thy'' = thy' |>
   519       (if coind then I
   520        else PureThy.add_tthms [(("induct", Attribute.tthm_of induct), [])]) |>
   521       Theory.parent_path
   522 
   523   in (thy'',
   524     {defs = [],
   525      mono = TrueI,
   526      unfold = TrueI,
   527      intrs = intrs,
   528      elims = elims,
   529      mk_cases = mk_cases elims,
   530      raw_induct = raw_induct,
   531      induct = induct})
   532   end;
   533 
   534 (********************** introduction of (co)inductive sets ********************)
   535 
   536 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
   537     intr_ts monos con_defs thy =
   538   let
   539     val _ = Theory.requires thy "Inductive"
   540       ((if coind then "co" else "") ^ "inductive definitions");
   541 
   542     val sign = sign_of thy;
   543 
   544     (*parameters should agree for all mutually recursive components*)
   545     val (_, params) = strip_comb (hd cs);
   546     val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
   547       \ component is not a free variable: " sign) params;
   548 
   549     val cTs = map (try' (HOLogic.dest_setT o fastype_of)
   550       "Recursive component not of type set: " sign) cs;
   551 
   552     val cnames = map (try' (Sign.base_name o fst o dest_Const o head_of)
   553       "Recursive set not previously declared as constant: " sign) cs;
   554 
   555     val _ = assert_all Syntax.is_identifier cnames
   556        (fn a => "Base name of recursive set not an identifier: " ^ a);
   557 
   558     val _ = map (check_rule sign cs) intr_ts;
   559 
   560   in
   561     (if !quick_and_dirty then add_ind_axm else add_ind_def)
   562       verbose declare_consts alt_name coind no_elim no_ind cs intr_ts monos
   563         con_defs thy params paramTs cTs cnames
   564   end;
   565 
   566 (***************************** external interface *****************************)
   567 
   568 fun add_inductive verbose coind c_strings intr_strings monos con_defs thy =
   569   let
   570     val sign = sign_of thy;
   571     val cs = map (readtm (sign_of thy) HOLogic.termTVar) c_strings;
   572     val intr_ts = map (readtm (sign_of thy) propT) intr_strings;
   573 
   574     (* the following code ensures that each recursive set *)
   575     (* always has the same type in all introduction rules *)
   576 
   577     val {tsig, ...} = Sign.rep_sg sign;
   578     val add_term_consts_2 =
   579       foldl_aterms (fn (cs, Const c) => c ins cs | (cs, _) => cs);
   580     fun varify (t, (i, ts)) =
   581       let val t' = map_term_types (incr_tvar (i + 1)) (Type.varify (t, []))
   582       in (maxidx_of_term t', t'::ts) end;
   583     val (i, cs') = foldr varify (cs, (~1, []));
   584     val (i', intr_ts') = foldr varify (intr_ts, (i, []));
   585     val rec_consts = foldl add_term_consts_2 ([], cs');
   586     val intr_consts = foldl add_term_consts_2 ([], intr_ts');
   587     fun unify (env, (cname, cT)) =
   588       let val consts = map snd (filter (fn c => fst c = cname) intr_consts)
   589       in (foldl (fn ((env', j'), Tp) => Type.unify tsig j' env' Tp)
   590         (env, (replicate (length consts) cT) ~~ consts)) handle _ =>
   591           error ("Occurrences of constant '" ^ cname ^
   592             "' have incompatible types")
   593       end;
   594     val (env, _) = foldl unify (([], i'), rec_consts);
   595     fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars env T
   596       in if T = T' then T else typ_subst_TVars_2 env T' end;
   597     val subst = fst o Type.freeze_thaw o
   598       (map_term_types (typ_subst_TVars_2 env));
   599     val cs'' = map subst cs';
   600     val intr_ts'' = map subst intr_ts';
   601 
   602   in add_inductive_i verbose false "" coind false false cs'' intr_ts''
   603     monos con_defs thy
   604   end;
   605 
   606 end;