src/HOL/Tools/inductive_package.ML
author wenzelm
Mon Aug 01 19:20:29 2005 +0200 (2005-08-01 ago)
changeset 16975 34ed8d5d4f16
parent 16934 9ef19e3c7fdd
child 17010 5abc26872268
permissions -rw-r--r--
Sign.read_term;
     1 (*  Title:      HOL/Tools/inductive_package.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Author:     Stefan Berghofer, TU Muenchen
     5     Author:     Markus Wenzel, TU Muenchen
     6 
     7 (Co)Inductive Definition module for HOL.
     8 
     9 Features:
    10   * least or greatest fixedpoints
    11   * user-specified product and sum constructions
    12   * mutually recursive definitions
    13   * definitions involving arbitrary monotone operators
    14   * automatically proves introduction and elimination rules
    15 
    16 The recursive sets must *already* be declared as constants in the
    17 current 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.  Products are used only to
    27 derive "streamlined" induction rules for relations.
    28 *)
    29 
    30 signature INDUCTIVE_PACKAGE =
    31 sig
    32   val quiet_mode: bool ref
    33   val trace: bool ref
    34   val unify_consts: theory -> term list -> term list -> term list * term list
    35   val split_rule_vars: term list -> thm -> thm
    36   val get_inductive: theory -> string -> ({names: string list, coind: bool} *
    37     {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
    38      intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}) option
    39   val the_mk_cases: theory -> string -> string -> thm
    40   val print_inductives: theory -> unit
    41   val mono_add_global: theory attribute
    42   val mono_del_global: theory attribute
    43   val get_monos: theory -> thm list
    44   val inductive_forall_name: string
    45   val inductive_forall_def: thm
    46   val rulify: thm -> thm
    47   val inductive_cases: ((bstring * Attrib.src list) * string list) list -> theory -> theory
    48   val inductive_cases_i: ((bstring * theory attribute list) * term list) list -> theory -> theory
    49   val add_inductive_i: bool -> bool -> bstring -> bool -> bool -> bool -> term list ->
    50     ((bstring * term) * theory attribute list) list -> thm list -> theory -> theory *
    51       {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
    52        intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
    53   val add_inductive: bool -> bool -> string list ->
    54     ((bstring * string) * Attrib.src list) list -> (thmref * Attrib.src list) list ->
    55     theory -> theory *
    56       {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
    57        intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
    58   val setup: (theory -> theory) list
    59 end;
    60 
    61 structure InductivePackage: INDUCTIVE_PACKAGE =
    62 struct
    63 
    64 
    65 (** theory context references **)
    66 
    67 val mono_name = "Orderings.mono";
    68 val gfp_name = "Gfp.gfp";
    69 val lfp_name = "Lfp.lfp";
    70 val vimage_name = "Set.vimage";
    71 val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (Thm.concl_of vimageD);
    72 
    73 val inductive_forall_name = "HOL.induct_forall";
    74 val inductive_forall_def = thm "induct_forall_def";
    75 val inductive_conj_name = "HOL.induct_conj";
    76 val inductive_conj_def = thm "induct_conj_def";
    77 val inductive_conj = thms "induct_conj";
    78 val inductive_atomize = thms "induct_atomize";
    79 val inductive_rulify1 = thms "induct_rulify1";
    80 val inductive_rulify2 = thms "induct_rulify2";
    81 
    82 
    83 
    84 (** theory data **)
    85 
    86 (* data kind 'HOL/inductive' *)
    87 
    88 type inductive_info =
    89   {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm,
    90     induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm};
    91 
    92 structure InductiveData = TheoryDataFun
    93 (struct
    94   val name = "HOL/inductive";
    95   type T = inductive_info Symtab.table * thm list;
    96 
    97   val empty = (Symtab.empty, []);
    98   val copy = I;
    99   val extend = I;
   100   fun merge _ ((tab1, monos1), (tab2, monos2)) =
   101     (Symtab.merge (K true) (tab1, tab2), Drule.merge_rules (monos1, monos2));
   102 
   103   fun print thy (tab, monos) =
   104     [Pretty.strs ("(co)inductives:" ::
   105       map #1 (NameSpace.extern_table (Sign.const_space thy, tab))),
   106      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_sg thy) monos)]
   107     |> Pretty.chunks |> Pretty.writeln;
   108 end);
   109 
   110 val print_inductives = InductiveData.print;
   111 
   112 
   113 (* get and put data *)
   114 
   115 fun get_inductive thy name = Symtab.lookup (fst (InductiveData.get thy), name);
   116 
   117 fun the_inductive thy name =
   118   (case get_inductive thy name of
   119     NONE => error ("Unknown (co)inductive set " ^ quote name)
   120   | SOME info => info);
   121 
   122 val the_mk_cases = (#mk_cases o #2) oo the_inductive;
   123 
   124 fun put_inductives names info thy =
   125   let
   126     fun upd ((tab, monos), name) = (Symtab.update_new ((name, info), tab), monos);
   127     val tab_monos = Library.foldl upd (InductiveData.get thy, names)
   128       handle Symtab.DUP name => error ("Duplicate definition of (co)inductive set " ^ quote name);
   129   in InductiveData.put tab_monos thy end;
   130 
   131 
   132 
   133 (** monotonicity rules **)
   134 
   135 val get_monos = #2 o InductiveData.get;
   136 fun map_monos f = InductiveData.map (Library.apsnd f);
   137 
   138 fun mk_mono thm =
   139   let
   140     fun eq2mono thm' = [standard (thm' RS (thm' RS eq_to_mono))] @
   141       (case concl_of thm of
   142           (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
   143         | _ => [standard (thm' RS (thm' RS eq_to_mono2))]);
   144     val concl = concl_of thm
   145   in
   146     if Logic.is_equals concl then
   147       eq2mono (thm RS meta_eq_to_obj_eq)
   148     else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then
   149       eq2mono thm
   150     else [thm]
   151   end;
   152 
   153 
   154 (* attributes *)
   155 
   156 fun mono_add_global (thy, thm) = (map_monos (Drule.add_rules (mk_mono thm)) thy, thm);
   157 fun mono_del_global (thy, thm) = (map_monos (Drule.del_rules (mk_mono thm)) thy, thm);
   158 
   159 val mono_attr =
   160  (Attrib.add_del_args mono_add_global mono_del_global,
   161   Attrib.add_del_args Attrib.undef_local_attribute Attrib.undef_local_attribute);
   162 
   163 
   164 
   165 (** misc utilities **)
   166 
   167 val quiet_mode = ref false;
   168 val trace = ref false;  (*for debugging*)
   169 fun message s = if ! quiet_mode then () else writeln s;
   170 fun clean_message s = if ! quick_and_dirty then () else message s;
   171 
   172 fun coind_prefix true = "co"
   173   | coind_prefix false = "";
   174 
   175 
   176 (*the following code ensures that each recursive set always has the
   177   same type in all introduction rules*)
   178 fun unify_consts thy cs intr_ts =
   179   (let
   180     val add_term_consts_2 = fold_aterms (fn Const c => insert (op =) c | _ => I);
   181     fun varify (t, (i, ts)) =
   182       let val t' = map_term_types (Logic.incr_tvar (i + 1)) (#1 (Type.varify (t, [])))
   183       in (maxidx_of_term t', t'::ts) end;
   184     val (i, cs') = foldr varify (~1, []) cs;
   185     val (i', intr_ts') = foldr varify (i, []) intr_ts;
   186     val rec_consts = fold add_term_consts_2 cs' [];
   187     val intr_consts = fold add_term_consts_2 intr_ts' [];
   188     fun unify (cname, cT) =
   189       let val consts = map snd (List.filter (fn c => fst c = cname) intr_consts)
   190       in fold (Sign.typ_unify thy) ((replicate (length consts) cT) ~~ consts) end;
   191     val (env, _) = fold unify rec_consts (Vartab.empty, i');
   192     val subst = Type.freeze o map_term_types (Envir.norm_type env)
   193 
   194   in (map subst cs', map subst intr_ts')
   195   end) handle Type.TUNIFY =>
   196     (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts));
   197 
   198 
   199 (*make injections used in mutually recursive definitions*)
   200 fun mk_inj cs sumT c x =
   201   let
   202     fun mk_inj' T n i =
   203       if n = 1 then x else
   204       let val n2 = n div 2;
   205           val Type (_, [T1, T2]) = T
   206       in
   207         if i <= n2 then
   208           Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   209         else
   210           Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
   211       end
   212   in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
   213   end;
   214 
   215 (*make "vimage" terms for selecting out components of mutually rec.def*)
   216 fun mk_vimage cs sumT t c = if length cs < 2 then t else
   217   let
   218     val cT = HOLogic.dest_setT (fastype_of c);
   219     val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
   220   in
   221     Const (vimage_name, vimageT) $
   222       Abs ("y", cT, mk_inj cs sumT c (Bound 0)) $ t
   223   end;
   224 
   225 (** proper splitting **)
   226 
   227 fun prod_factors p (Const ("Pair", _) $ t $ u) =
   228       p :: prod_factors (1::p) t @ prod_factors (2::p) u
   229   | prod_factors p _ = [];
   230 
   231 fun mg_prod_factors ts (fs, t $ u) = if t mem ts then
   232         let val f = prod_factors [] u
   233         in overwrite (fs, (t, f inter (curry getOpt) (assoc (fs, t)) f)) end
   234       else mg_prod_factors ts (mg_prod_factors ts (fs, t), u)
   235   | mg_prod_factors ts (fs, Abs (_, _, t)) = mg_prod_factors ts (fs, t)
   236   | mg_prod_factors ts (fs, _) = fs;
   237 
   238 fun prodT_factors p ps (T as Type ("*", [T1, T2])) =
   239       if p mem ps then prodT_factors (1::p) ps T1 @ prodT_factors (2::p) ps T2
   240       else [T]
   241   | prodT_factors _ _ T = [T];
   242 
   243 fun ap_split p ps (Type ("*", [T1, T2])) T3 u =
   244       if p mem ps then HOLogic.split_const (T1, T2, T3) $
   245         Abs ("v", T1, ap_split (2::p) ps T2 T3 (ap_split (1::p) ps T1
   246           (prodT_factors (2::p) ps T2 ---> T3) (incr_boundvars 1 u) $ Bound 0))
   247       else u
   248   | ap_split _ _ _ _ u =  u;
   249 
   250 fun mk_tuple p ps (Type ("*", [T1, T2])) (tms as t::_) =
   251       if p mem ps then HOLogic.mk_prod (mk_tuple (1::p) ps T1 tms, 
   252         mk_tuple (2::p) ps T2 (Library.drop (length (prodT_factors (1::p) ps T1), tms)))
   253       else t
   254   | mk_tuple _ _ _ (t::_) = t;
   255 
   256 fun split_rule_var' ((t as Var (v, Type ("fun", [T1, T2])), ps), rl) =
   257       let val T' = prodT_factors [] ps T1 ---> T2
   258           val newt = ap_split [] ps T1 T2 (Var (v, T'))
   259           val cterm = Thm.cterm_of (Thm.theory_of_thm rl)
   260       in
   261           instantiate ([], [(cterm t, cterm newt)]) rl
   262       end
   263   | split_rule_var' (_, rl) = rl;
   264 
   265 val remove_split = rewrite_rule [split_conv RS eq_reflection];
   266 
   267 fun split_rule_vars vs rl = standard (remove_split (foldr split_rule_var'
   268   rl (mg_prod_factors vs ([], Thm.prop_of rl))));
   269 
   270 fun split_rule vs rl = standard (remove_split (foldr split_rule_var'
   271   rl (List.mapPartial (fn (t as Var ((a, _), _)) =>
   272       Option.map (pair t) (assoc (vs, a))) (term_vars (Thm.prop_of rl)))));
   273 
   274 
   275 (** process rules **)
   276 
   277 local
   278 
   279 fun err_in_rule thy name t msg =
   280   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
   281     Sign.string_of_term thy t, msg]);
   282 
   283 fun err_in_prem thy name t p msg =
   284   error (cat_lines ["Ill-formed premise", Sign.string_of_term thy p,
   285     "in introduction rule " ^ quote name, Sign.string_of_term thy t, msg]);
   286 
   287 val bad_concl = "Conclusion of introduction rule must have form \"t : S_i\"";
   288 
   289 val all_not_allowed = 
   290     "Introduction rule must not have a leading \"!!\" quantifier";
   291 
   292 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
   293 
   294 in
   295 
   296 fun check_rule thy cs ((name, rule), att) =
   297   let
   298     val concl = Logic.strip_imp_concl rule;
   299     val prems = Logic.strip_imp_prems rule;
   300     val aprems = map (atomize_term thy) prems;
   301     val arule = Logic.list_implies (aprems, concl);
   302 
   303     fun check_prem (prem, aprem) =
   304       if can HOLogic.dest_Trueprop aprem then ()
   305       else err_in_prem thy name rule prem "Non-atomic premise";
   306   in
   307     (case concl of
   308       Const ("Trueprop", _) $ (Const ("op :", _) $ t $ u) =>
   309         if u mem cs then
   310           if exists (Logic.occs o rpair t) cs then
   311             err_in_rule thy name rule "Recursion term on left of member symbol"
   312           else List.app check_prem (prems ~~ aprems)
   313         else err_in_rule thy name rule bad_concl
   314       | Const ("all", _) $ _ => err_in_rule thy name rule all_not_allowed
   315       | _ => err_in_rule thy name rule bad_concl);
   316     ((name, arule), att)
   317   end;
   318 
   319 val rulify =
   320   standard o
   321   hol_simplify inductive_rulify2 o hol_simplify inductive_rulify1 o
   322   hol_simplify inductive_conj;
   323 
   324 end;
   325 
   326 
   327 
   328 (** properties of (co)inductive sets **)
   329 
   330 (* elimination rules *)
   331 
   332 fun mk_elims cs cTs params intr_ts intr_names =
   333   let
   334     val used = foldr add_term_names [] intr_ts;
   335     val [aname, pname] = variantlist (["a", "P"], used);
   336     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   337 
   338     fun dest_intr r =
   339       let val Const ("op :", _) $ t $ u =
   340         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   341       in (u, t, Logic.strip_imp_prems r) end;
   342 
   343     val intrs = map dest_intr intr_ts ~~ intr_names;
   344 
   345     fun mk_elim (c, T) =
   346       let
   347         val a = Free (aname, T);
   348 
   349         fun mk_elim_prem (_, t, ts) =
   350           list_all_free (map dest_Free ((foldr add_term_frees [] (t::ts)) \\ params),
   351             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
   352         val c_intrs = (List.filter (equal c o #1 o #1) intrs);
   353       in
   354         (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
   355           map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs)
   356       end
   357   in
   358     map mk_elim (cs ~~ cTs)
   359   end;
   360 
   361 
   362 (* premises and conclusions of induction rules *)
   363 
   364 fun mk_indrule cs cTs params intr_ts =
   365   let
   366     val used = foldr add_term_names [] intr_ts;
   367 
   368     (* predicates for induction rule *)
   369 
   370     val preds = map Free (variantlist (if length cs < 2 then ["P"] else
   371       map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
   372         map (fn T => T --> HOLogic.boolT) cTs);
   373 
   374     (* transform an introduction rule into a premise for induction rule *)
   375 
   376     fun mk_ind_prem r =
   377       let
   378         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   379 
   380         val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
   381 
   382         fun subst (s as ((m as Const ("op :", T)) $ t $ u)) =
   383               (case pred_of u of
   384                   NONE => (m $ fst (subst t) $ fst (subst u), NONE)
   385                 | SOME P => (HOLogic.mk_binop inductive_conj_name (s, P $ t), SOME (s, P $ t)))
   386           | subst s =
   387               (case pred_of s of
   388                   SOME P => (HOLogic.mk_binop "op Int"
   389                     (s, HOLogic.Collect_const (HOLogic.dest_setT
   390                       (fastype_of s)) $ P), NONE)
   391                 | NONE => (case s of
   392                      (t $ u) => (fst (subst t) $ fst (subst u), NONE)
   393                    | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
   394                    | _ => (s, NONE)));
   395 
   396         fun mk_prem (s, prems) = (case subst s of
   397               (_, SOME (t, u)) => t :: u :: prems
   398             | (t, _) => t :: prems);
   399 
   400         val Const ("op :", _) $ t $ u =
   401           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   402 
   403       in list_all_free (frees,
   404            Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
   405              [] (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r))),
   406                HOLogic.mk_Trueprop (valOf (pred_of u) $ t)))
   407       end;
   408 
   409     val ind_prems = map mk_ind_prem intr_ts;
   410 
   411     val factors = Library.foldl (mg_prod_factors preds) ([], ind_prems);
   412 
   413     (* make conclusions for induction rules *)
   414 
   415     fun mk_ind_concl ((c, P), (ts, x)) =
   416       let val T = HOLogic.dest_setT (fastype_of c);
   417           val ps = getOpt (assoc (factors, P), []);
   418           val Ts = prodT_factors [] ps T;
   419           val (frees, x') = foldr (fn (T', (fs, s)) =>
   420             ((Free (s, T'))::fs, Symbol.bump_string s)) ([], x) Ts;
   421           val tuple = mk_tuple [] ps T frees;
   422       in ((HOLogic.mk_binop "op -->"
   423         (HOLogic.mk_mem (tuple, c), P $ tuple))::ts, x')
   424       end;
   425 
   426     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   427         (fst (foldr mk_ind_concl ([], "xa") (cs ~~ preds))))
   428 
   429   in (preds, ind_prems, mutual_ind_concl,
   430     map (apfst (fst o dest_Free)) factors)
   431   end;
   432 
   433 
   434 (* prepare cases and induct rules *)
   435 
   436 (*
   437   transform mutual rule:
   438     HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
   439   into i-th projection:
   440     xi:Ai ==> HH ==> Pi xi
   441 *)
   442 
   443 fun project_rules [name] rule = [(name, rule)]
   444   | project_rules names mutual_rule =
   445       let
   446         val n = length names;
   447         fun proj i =
   448           (if i < n then (fn th => th RS conjunct1) else I)
   449             (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
   450             RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
   451       in names ~~ map proj (1 upto n) end;
   452 
   453 fun add_cases_induct no_elim no_induct names elims induct =
   454   let
   455     fun cases_spec (name, elim) thy =
   456       thy
   457       |> Theory.add_path (Sign.base_name name)
   458       |> (#1 o PureThy.add_thms [(("cases", elim), [InductAttrib.cases_set_global name])])
   459       |> Theory.parent_path;
   460     val cases_specs = if no_elim then [] else map2 cases_spec (names, elims);
   461 
   462     fun induct_spec (name, th) = #1 o PureThy.add_thms
   463       [(("", RuleCases.save induct th), [InductAttrib.induct_set_global name])];
   464     val induct_specs = if no_induct then [] else map induct_spec (project_rules names induct);
   465   in Library.apply (cases_specs @ induct_specs) end;
   466 
   467 
   468 
   469 (** proofs for (co)inductive sets **)
   470 
   471 (* prove monotonicity -- NOT subject to quick_and_dirty! *)
   472 
   473 fun prove_mono setT fp_fun monos thy =
   474  (message "  Proving monotonicity ...";
   475   Goals.prove_goalw_cterm []      (*NO quick_and_dirty_prove_goalw_cterm here!*)
   476     (Thm.cterm_of thy (HOLogic.mk_Trueprop
   477       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
   478     (fn _ => [rtac monoI 1, REPEAT (ares_tac (List.concat (map mk_mono monos) @ get_monos thy) 1)]));
   479 
   480 
   481 (* prove introduction rules *)
   482 
   483 fun prove_intrs coind mono fp_def intr_ts rec_sets_defs thy =
   484   let
   485     val _ = clean_message "  Proving the introduction rules ...";
   486 
   487     val unfold = standard' (mono RS (fp_def RS
   488       (if coind then def_gfp_unfold else def_lfp_unfold)));
   489 
   490     fun select_disj 1 1 = []
   491       | select_disj _ 1 = [rtac disjI1]
   492       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   493 
   494     val intrs = map (fn (i, intr) => quick_and_dirty_prove_goalw_cterm thy rec_sets_defs
   495       (Thm.cterm_of thy intr) (fn prems =>
   496        [(*insert prems and underlying sets*)
   497        cut_facts_tac prems 1,
   498        stac unfold 1,
   499        REPEAT (resolve_tac [vimageI2, CollectI] 1),
   500        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
   501        EVERY1 (select_disj (length intr_ts) i),
   502        (*Not ares_tac, since refl must be tried before any equality assumptions;
   503          backtracking may occur if the premises have extra variables!*)
   504        DEPTH_SOLVE_1 (resolve_tac [refl, exI, conjI] 1 APPEND assume_tac 1),
   505        (*Now solve the equations like Inl 0 = Inl ?b2*)
   506        REPEAT (rtac refl 1)])
   507       |> rulify) (1 upto (length intr_ts) ~~ intr_ts)
   508 
   509   in (intrs, unfold) end;
   510 
   511 
   512 (* prove elimination rules *)
   513 
   514 fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
   515   let
   516     val _ = clean_message "  Proving the elimination rules ...";
   517 
   518     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
   519     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @ map make_elim [Inl_inject, Inr_inject];
   520   in
   521     mk_elims cs cTs params intr_ts intr_names |> map (fn (t, cases) =>
   522       quick_and_dirty_prove_goalw_cterm thy rec_sets_defs
   523         (Thm.cterm_of thy t) (fn prems =>
   524           [cut_facts_tac [hd prems] 1,
   525            dtac (unfold RS subst) 1,
   526            REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   527            REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   528            EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))])
   529         |> rulify
   530         |> RuleCases.name cases)
   531   end;
   532 
   533 
   534 (* derivation of simplified elimination rules *)
   535 
   536 local
   537 
   538 (*cprop should have the form t:Si where Si is an inductive set*)
   539 val mk_cases_err = "mk_cases: proposition not of form \"t : S_i\"";
   540 
   541 (*delete needless equality assumptions*)
   542 val refl_thin = prove_goal HOL.thy "!!P. a = a ==> P ==> P" (fn _ => [assume_tac 1]);
   543 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject];
   544 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   545 
   546 fun simp_case_tac solved ss i =
   547   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i
   548   THEN_MAYBE (if solved then no_tac else all_tac);
   549 
   550 in
   551 
   552 fun mk_cases_i elims ss cprop =
   553   let
   554     val prem = Thm.assume cprop;
   555     val tac = ALLGOALS (simp_case_tac false ss) THEN prune_params_tac;
   556     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic tac (prem RS rl));
   557   in
   558     (case get_first (try mk_elim) elims of
   559       SOME r => r
   560     | NONE => error (Pretty.string_of (Pretty.block
   561         [Pretty.str mk_cases_err, Pretty.fbrk, Display.pretty_cterm cprop])))
   562   end;
   563 
   564 fun mk_cases elims s =
   565   mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.theory_of_thm (hd elims)) (s, propT));
   566 
   567 fun smart_mk_cases thy ss cprop =
   568   let
   569     val c = #1 (Term.dest_Const (Term.head_of (#2 (HOLogic.dest_mem (HOLogic.dest_Trueprop
   570       (Logic.strip_imp_concl (Thm.term_of cprop))))))) handle TERM _ => error mk_cases_err;
   571     val (_, {elims, ...}) = the_inductive thy c;
   572   in mk_cases_i elims ss cprop end;
   573 
   574 end;
   575 
   576 
   577 (* inductive_cases(_i) *)
   578 
   579 fun gen_inductive_cases prep_att prep_prop args thy =
   580   let
   581     val cert_prop = Thm.cterm_of thy o prep_prop (ProofContext.init thy);
   582     val mk_cases = smart_mk_cases thy (Simplifier.simpset_of thy) o cert_prop;
   583 
   584     val facts = args |> map (fn ((a, atts), props) =>
   585      ((a, map (prep_att thy) atts), map (Thm.no_attributes o single o mk_cases) props));
   586   in thy |> IsarThy.theorems_i Drule.lemmaK facts |> #1 end;
   587 
   588 val inductive_cases = gen_inductive_cases Attrib.global_attribute ProofContext.read_prop;
   589 val inductive_cases_i = gen_inductive_cases (K I) ProofContext.cert_prop;
   590 
   591 
   592 (* mk_cases_meth *)
   593 
   594 fun mk_cases_meth (ctxt, raw_props) =
   595   let
   596     val thy = ProofContext.theory_of ctxt;
   597     val ss = local_simpset_of ctxt;
   598     val cprops = map (Thm.cterm_of thy o ProofContext.read_prop ctxt) raw_props;
   599   in Method.erule 0 (map (smart_mk_cases thy ss) cprops) end;
   600 
   601 val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name));
   602 
   603 
   604 (* prove induction rule *)
   605 
   606 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
   607     fp_def rec_sets_defs thy =
   608   let
   609     val _ = clean_message "  Proving the induction rule ...";
   610 
   611     val sum_case_rewrites =
   612       (if Context.theory_name thy = "Datatype" then
   613         PureThy.get_thms thy (Name "sum.cases")
   614       else
   615         (case ThyInfo.lookup_theory "Datatype" of
   616           NONE => []
   617         | SOME thy' =>
   618             if Theory.subthy (thy', thy) then
   619               PureThy.get_thms thy' (Name "sum.cases")
   620             else []))
   621       |> map mk_meta_eq;
   622 
   623     val (preds, ind_prems, mutual_ind_concl, factors) =
   624       mk_indrule cs cTs params intr_ts;
   625 
   626     val dummy = if !trace then
   627 		(writeln "ind_prems = ";
   628 		 List.app (writeln o Sign.string_of_term thy) ind_prems)
   629 	    else ();
   630 
   631     (* make predicate for instantiation of abstract induction rule *)
   632 
   633     fun mk_ind_pred _ [P] = P
   634       | mk_ind_pred T Ps =
   635          let val n = (length Ps) div 2;
   636              val Type (_, [T1, T2]) = T
   637          in Const ("Datatype.sum.sum_case",
   638            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
   639              mk_ind_pred T1 (Library.take (n, Ps)) $ mk_ind_pred T2 (Library.drop (n, Ps))
   640          end;
   641 
   642     val ind_pred = mk_ind_pred sumT preds;
   643 
   644     val ind_concl = HOLogic.mk_Trueprop
   645       (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->"
   646         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0)));
   647 
   648     (* simplification rules for vimage and Collect *)
   649 
   650     val vimage_simps = if length cs < 2 then [] else
   651       map (fn c => quick_and_dirty_prove_goalw_cterm thy [] (Thm.cterm_of thy
   652         (HOLogic.mk_Trueprop (HOLogic.mk_eq
   653           (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c,
   654            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $
   655              List.nth (preds, find_index_eq c cs)))))
   656         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites, rtac refl 1])) cs;
   657 
   658     val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));
   659 
   660     val dummy = if !trace then
   661 		(writeln "raw_fp_induct = "; print_thm raw_fp_induct)
   662 	    else ();
   663 
   664     val induct = quick_and_dirty_prove_goalw_cterm thy [inductive_conj_def] (Thm.cterm_of thy
   665       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
   666         [rtac (impI RS allI) 1,
   667          DETERM (etac raw_fp_induct 1),
   668          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
   669          fold_goals_tac rec_sets_defs,
   670          (*This CollectE and disjE separates out the introduction rules*)
   671          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
   672          (*Now break down the individual cases.  No disjE here in case
   673            some premise involves disjunction.*)
   674          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
   675          ALLGOALS (simp_tac (HOL_basic_ss addsimps sum_case_rewrites)),
   676          EVERY (map (fn prem =>
   677    	             DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
   678 
   679     val lemma = quick_and_dirty_prove_goalw_cterm thy rec_sets_defs (Thm.cterm_of thy
   680       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
   681         [cut_facts_tac prems 1,
   682          REPEAT (EVERY
   683            [REPEAT (resolve_tac [conjI, impI] 1),
   684             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
   685             rewrite_goals_tac sum_case_rewrites,
   686             atac 1])])
   687 
   688   in standard (split_rule factors (induct RS lemma)) end;
   689 
   690 
   691 
   692 (** specification of (co)inductive sets **)
   693 
   694 fun cond_declare_consts declare_consts cs paramTs cnames =
   695   if declare_consts then
   696     Theory.add_consts_i (map (fn (c, n) => (Sign.base_name n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
   697   else I;
   698 
   699 fun mk_ind_def declare_consts alt_name coind cs intr_ts monos thy
   700       params paramTs cTs cnames =
   701   let
   702     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
   703     val setT = HOLogic.mk_setT sumT;
   704 
   705     val fp_name = if coind then gfp_name else lfp_name;
   706 
   707     val used = foldr add_term_names [] intr_ts;
   708     val [sname, xname] = variantlist (["S", "x"], used);
   709 
   710     (* transform an introduction rule into a conjunction  *)
   711     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
   712     (* is transformed into                                *)
   713     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
   714 
   715     fun transform_rule r =
   716       let
   717         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   718         val subst = subst_free
   719           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
   720         val Const ("op :", _) $ t $ u =
   721           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   722 
   723       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
   724         (foldr1 HOLogic.mk_conj
   725           (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t))::
   726             (map (subst o HOLogic.dest_Trueprop)
   727               (Logic.strip_imp_prems r)))) frees
   728       end
   729 
   730     (* make a disjunction of all introduction rules *)
   731 
   732     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
   733       absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
   734 
   735     (* add definiton of recursive sets to theory *)
   736 
   737     val rec_name = if alt_name = "" then
   738       space_implode "_" (map Sign.base_name cnames) else alt_name;
   739     val full_rec_name = if length cs < 2 then hd cnames
   740       else Sign.full_name thy rec_name;
   741 
   742     val rec_const = list_comb
   743       (Const (full_rec_name, paramTs ---> setT), params);
   744 
   745     val fp_def_term = Logic.mk_equals (rec_const,
   746       Const (fp_name, (setT --> setT) --> setT) $ fp_fun);
   747 
   748     val def_terms = fp_def_term :: (if length cs < 2 then [] else
   749       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
   750 
   751     val (thy', [fp_def :: rec_sets_defs]) =
   752       thy
   753       |> cond_declare_consts declare_consts cs paramTs cnames
   754       |> (if length cs < 2 then I
   755           else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)])
   756       |> Theory.add_path rec_name
   757       |> PureThy.add_defss_i false [(("defs", def_terms), [])];
   758 
   759     val mono = prove_mono setT fp_fun monos thy'
   760 
   761   in (thy', mono, fp_def, rec_sets_defs, rec_const, sumT) end;
   762 
   763 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
   764     intros monos thy params paramTs cTs cnames induct_cases =
   765   let
   766     val _ =
   767       if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
   768         commas_quote (map Sign.base_name cnames)) else ();
   769 
   770     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
   771 
   772     val (thy1, mono, fp_def, rec_sets_defs, rec_const, sumT) =
   773       mk_ind_def declare_consts alt_name coind cs intr_ts monos thy
   774         params paramTs cTs cnames;
   775 
   776     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts rec_sets_defs thy1;
   777     val elims = if no_elim then [] else
   778       prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy1;
   779     val raw_induct = if no_ind then Drule.asm_rl else
   780       if coind then standard (rule_by_tactic
   781         (rewrite_tac [mk_meta_eq vimage_Un] THEN
   782           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
   783       else
   784         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
   785           rec_sets_defs thy1;
   786     val induct =
   787       if coind orelse no_ind orelse length cs > 1 then (raw_induct, [RuleCases.consumes 0])
   788       else (raw_induct RSN (2, rev_mp), [RuleCases.consumes 1]);
   789 
   790     val (thy2, intrs') =
   791       thy1 |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts);
   792     val (thy3, ([intrs'', elims'], [induct'])) =
   793       thy2
   794       |> PureThy.add_thmss
   795         [(("intros", intrs'), []),
   796           (("elims", elims), [RuleCases.consumes 1])]
   797       |>>> PureThy.add_thms
   798         [((coind_prefix coind ^ "induct", rulify (#1 induct)),
   799          (RuleCases.case_names induct_cases :: #2 induct))]
   800       |>> Theory.parent_path;
   801   in (thy3,
   802     {defs = fp_def :: rec_sets_defs,
   803      mono = mono,
   804      unfold = unfold,
   805      intrs = intrs',
   806      elims = elims',
   807      mk_cases = mk_cases elims',
   808      raw_induct = rulify raw_induct,
   809      induct = induct'})
   810   end;
   811 
   812 
   813 (* external interfaces *)
   814 
   815 fun try_term f msg thy t =
   816   (case Library.try f t of
   817     SOME x => x
   818   | NONE => error (msg ^ Sign.string_of_term thy t));
   819 
   820 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs pre_intros monos thy =
   821   let
   822     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   823 
   824     (*parameters should agree for all mutually recursive components*)
   825     val (_, params) = strip_comb (hd cs);
   826     val paramTs = map (try_term (snd o dest_Free) "Parameter in recursive\
   827       \ component is not a free variable: " thy) params;
   828 
   829     val cTs = map (try_term (HOLogic.dest_setT o fastype_of)
   830       "Recursive component not of type set: " thy) cs;
   831 
   832     val cnames = map (try_term (fst o dest_Const o head_of)
   833       "Recursive set not previously declared as constant: " thy) cs;
   834 
   835     val save_thy = thy
   836       |> Theory.copy |> cond_declare_consts declare_consts cs paramTs cnames;
   837     val intros = map (check_rule save_thy cs) pre_intros;
   838     val induct_cases = map (#1 o #1) intros;
   839 
   840     val (thy1, result as {elims, induct, ...}) =
   841       add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs intros monos
   842         thy params paramTs cTs cnames induct_cases;
   843     val thy2 = thy1
   844       |> put_inductives cnames ({names = cnames, coind = coind}, result)
   845       |> add_cases_induct no_elim (no_ind orelse coind orelse length cs > 1)
   846           cnames elims induct;
   847   in (thy2, result) end;
   848 
   849 fun add_inductive verbose coind c_strings intro_srcs raw_monos thy =
   850   let
   851     val cs = map (Sign.read_term thy) c_strings;
   852 
   853     val intr_names = map (fst o fst) intro_srcs;
   854     fun read_rule s = Thm.read_cterm thy (s, propT)
   855       handle ERROR => error ("The error(s) above occurred for " ^ s);
   856     val intr_ts = map (Thm.term_of o read_rule o snd o fst) intro_srcs;
   857     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
   858     val (cs', intr_ts') = unify_consts thy cs intr_ts;
   859 
   860     val (thy', monos) = thy |> IsarThy.apply_theorems raw_monos;
   861   in
   862     add_inductive_i verbose false "" coind false false cs'
   863       ((intr_names ~~ intr_ts') ~~ intr_atts) monos thy'
   864   end;
   865 
   866 
   867 
   868 (** package setup **)
   869 
   870 (* setup theory *)
   871 
   872 val setup =
   873  [InductiveData.init,
   874   Method.add_methods [("ind_cases", mk_cases_meth oo mk_cases_args,
   875     "dynamic case analysis on sets")],
   876   Attrib.add_attributes [("mono", mono_attr, "declaration of monotonicity rule")]];
   877 
   878 
   879 (* outer syntax *)
   880 
   881 local structure P = OuterParse and K = OuterSyntax.Keyword in
   882 
   883 fun mk_ind coind ((sets, intrs), monos) =
   884   #1 o add_inductive true coind sets (map P.triple_swap intrs) monos;
   885 
   886 fun ind_decl coind =
   887   Scan.repeat1 P.term --
   888   (P.$$$ "intros" |--
   889     P.!!! (Scan.repeat1 (P.opt_thm_name ":" -- P.prop))) --
   890   Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1) []
   891   >> (Toplevel.theory o mk_ind coind);
   892 
   893 val inductiveP =
   894   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
   895 
   896 val coinductiveP =
   897   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
   898 
   899 
   900 val ind_cases =
   901   P.and_list1 (P.opt_thm_name ":" -- Scan.repeat1 P.prop)
   902   >> (Toplevel.theory o inductive_cases);
   903 
   904 val inductive_casesP =
   905   OuterSyntax.command "inductive_cases"
   906     "create simplified instances of elimination rules (improper)" K.thy_script ind_cases;
   907 
   908 val _ = OuterSyntax.add_keywords ["intros", "monos"];
   909 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
   910 
   911 end;
   912 
   913 end;
   914