src/HOL/Tools/inductive_package.ML
author wenzelm
Thu Nov 08 23:59:37 2001 +0100 (2001-11-08)
changeset 12109 bd6eb9194a5d
parent 11991 da6ee05d9f3d
child 12128 25565bbbd246
permissions -rw-r--r--
theory data: finish method;
     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     License:    GPL (GNU GENERAL PUBLIC LICENSE)
     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 the
    18 current theory!
    19 
    20   Introduction rules have the form
    21   [| ti:M(Sj), ..., P(x), ... |] ==> t: Sk
    22   where M is some monotone operator (usually the identity)
    23   P(x) is any side condition on the free variables
    24   ti, t are any terms
    25   Sj, Sk are two of the sets being defined in mutual recursion
    26 
    27 Sums are used only for mutual recursion.  Products are used only to
    28 derive "streamlined" induction rules for relations.
    29 *)
    30 
    31 signature INDUCTIVE_PACKAGE =
    32 sig
    33   val quiet_mode: bool ref
    34   val unify_consts: Sign.sg -> 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 print_inductives: theory -> unit
    40   val mono_add_global: theory attribute
    41   val mono_del_global: theory attribute
    42   val get_monos: theory -> thm list
    43   val inductive_forall_name: string
    44   val inductive_forall_def: thm
    45   val rulify: thm -> thm
    46   val inductive_cases: ((bstring * Args.src list) * string list) * Comment.text
    47     -> theory -> theory
    48   val inductive_cases_i: ((bstring * theory attribute list) * term list) * Comment.text
    49     -> theory -> theory
    50   val add_inductive_i: bool -> bool -> bstring -> bool -> bool -> bool -> term list ->
    51     ((bstring * term) * theory attribute list) list ->
    52       thm list -> thm list -> theory -> theory *
    53       {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
    54        intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
    55   val add_inductive: bool -> bool -> string list ->
    56     ((bstring * string) * Args.src list) list -> (xstring * Args.src list) list ->
    57       (xstring * Args.src list) list -> theory -> theory *
    58       {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
    59        intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
    60   val setup: (theory -> theory) list
    61 end;
    62 
    63 structure InductivePackage: INDUCTIVE_PACKAGE =
    64 struct
    65 
    66 
    67 (** theory context references **)
    68 
    69 val mono_name = "HOL.mono";
    70 val gfp_name = "Gfp.gfp";
    71 val lfp_name = "Lfp.lfp";
    72 val vimage_name = "Inverse_Image.vimage";
    73 val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (Thm.concl_of vimageD);
    74 
    75 val inductive_forall_name = "HOL.induct_forall";
    76 val inductive_forall_def = thm "induct_forall_def";
    77 val inductive_conj_name = "HOL.induct_conj";
    78 val inductive_conj_def = thm "induct_conj_def";
    79 val inductive_conj = thms "induct_conj";
    80 val inductive_atomize = thms "induct_atomize";
    81 val inductive_rulify1 = thms "induct_rulify1";
    82 val inductive_rulify2 = thms "induct_rulify2";
    83 
    84 
    85 
    86 (** theory data **)
    87 
    88 (* data kind 'HOL/inductive' *)
    89 
    90 type inductive_info =
    91   {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm,
    92     induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm};
    93 
    94 structure InductiveArgs =
    95 struct
    96   val name = "HOL/inductive";
    97   type T = inductive_info Symtab.table * thm list;
    98 
    99   val empty = (Symtab.empty, []);
   100   val copy = I;
   101   val finish = I;
   102   val prep_ext = I;
   103   fun merge ((tab1, monos1), (tab2, monos2)) =
   104     (Symtab.merge (K true) (tab1, tab2), Drule.merge_rules (monos1, monos2));
   105 
   106   fun print sg (tab, monos) =
   107     [Pretty.strs ("(co)inductives:" :: map #1 (Sign.cond_extern_table sg Sign.constK tab)),
   108      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_sg sg) monos)]
   109     |> Pretty.chunks |> Pretty.writeln;
   110 end;
   111 
   112 structure InductiveData = TheoryDataFun(InductiveArgs);
   113 val print_inductives = InductiveData.print;
   114 
   115 
   116 (* get and put data *)
   117 
   118 fun get_inductive thy name = Symtab.lookup (fst (InductiveData.get thy), name);
   119 
   120 fun the_inductive thy name =
   121   (case get_inductive thy name of
   122     None => error ("Unknown (co)inductive set " ^ quote name)
   123   | Some info => info);
   124 
   125 fun put_inductives names info thy =
   126   let
   127     fun upd ((tab, monos), name) = (Symtab.update_new ((name, info), tab), monos);
   128     val tab_monos = foldl upd (InductiveData.get thy, names)
   129       handle Symtab.DUP name => error ("Duplicate definition of (co)inductive set " ^ quote name);
   130   in InductiveData.put tab_monos thy end;
   131 
   132 
   133 
   134 (** monotonicity rules **)
   135 
   136 val get_monos = #2 o InductiveData.get;
   137 fun map_monos f = InductiveData.map (Library.apsnd f);
   138 
   139 fun mk_mono thm =
   140   let
   141     fun eq2mono thm' = [standard (thm' RS (thm' RS eq_to_mono))] @
   142       (case concl_of thm of
   143           (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
   144         | _ => [standard (thm' RS (thm' RS eq_to_mono2))]);
   145     val concl = concl_of thm
   146   in
   147     if Logic.is_equals concl then
   148       eq2mono (thm RS meta_eq_to_obj_eq)
   149     else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then
   150       eq2mono thm
   151     else [thm]
   152   end;
   153 
   154 
   155 (* attributes *)
   156 
   157 fun mono_add_global (thy, thm) = (map_monos (Drule.add_rules (mk_mono thm)) thy, thm);
   158 fun mono_del_global (thy, thm) = (map_monos (Drule.del_rules (mk_mono thm)) thy, thm);
   159 
   160 val mono_attr =
   161  (Attrib.add_del_args mono_add_global mono_del_global,
   162   Attrib.add_del_args Attrib.undef_local_attribute Attrib.undef_local_attribute);
   163 
   164 
   165 
   166 (** misc utilities **)
   167 
   168 val quiet_mode = ref false;
   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 sign cs intr_ts =
   179   (let
   180     val {tsig, ...} = Sign.rep_sg sign;
   181     val add_term_consts_2 =
   182       foldl_aterms (fn (cs, Const c) => c ins cs | (cs, _) => cs);
   183     fun varify (t, (i, ts)) =
   184       let val t' = map_term_types (incr_tvar (i + 1)) (Type.varify (t, []))
   185       in (maxidx_of_term t', t'::ts) end;
   186     val (i, cs') = foldr varify (cs, (~1, []));
   187     val (i', intr_ts') = foldr varify (intr_ts, (i, []));
   188     val rec_consts = foldl add_term_consts_2 ([], cs');
   189     val intr_consts = foldl add_term_consts_2 ([], intr_ts');
   190     fun unify (env, (cname, cT)) =
   191       let val consts = map snd (filter (fn c => fst c = cname) intr_consts)
   192       in foldl (fn ((env', j'), Tp) => (Type.unify tsig j' env' Tp))
   193           (env, (replicate (length consts) cT) ~~ consts)
   194       end;
   195     val (env, _) = foldl unify ((Vartab.empty, i'), rec_consts);
   196     fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars_Vartab env T
   197       in if T = T' then T else typ_subst_TVars_2 env T' end;
   198     val subst = fst o Type.freeze_thaw o
   199       (map_term_types (typ_subst_TVars_2 env))
   200 
   201   in (map subst cs', map subst intr_ts')
   202   end) handle Type.TUNIFY =>
   203     (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts));
   204 
   205 
   206 (*make injections used in mutually recursive definitions*)
   207 fun mk_inj cs sumT c x =
   208   let
   209     fun mk_inj' T n i =
   210       if n = 1 then x else
   211       let val n2 = n div 2;
   212           val Type (_, [T1, T2]) = T
   213       in
   214         if i <= n2 then
   215           Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   216         else
   217           Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
   218       end
   219   in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
   220   end;
   221 
   222 (*make "vimage" terms for selecting out components of mutually rec.def*)
   223 fun mk_vimage cs sumT t c = if length cs < 2 then t else
   224   let
   225     val cT = HOLogic.dest_setT (fastype_of c);
   226     val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
   227   in
   228     Const (vimage_name, vimageT) $
   229       Abs ("y", cT, mk_inj cs sumT c (Bound 0)) $ t
   230   end;
   231 
   232 (** proper splitting **)
   233 
   234 fun prod_factors p (Const ("Pair", _) $ t $ u) =
   235       p :: prod_factors (1::p) t @ prod_factors (2::p) u
   236   | prod_factors p _ = [];
   237 
   238 fun mg_prod_factors ts (fs, t $ u) = if t mem ts then
   239         let val f = prod_factors [] u
   240         in overwrite (fs, (t, f inter if_none (assoc (fs, t)) f)) end
   241       else mg_prod_factors ts (mg_prod_factors ts (fs, t), u)
   242   | mg_prod_factors ts (fs, Abs (_, _, t)) = mg_prod_factors ts (fs, t)
   243   | mg_prod_factors ts (fs, _) = fs;
   244 
   245 fun prodT_factors p ps (T as Type ("*", [T1, T2])) =
   246       if p mem ps then prodT_factors (1::p) ps T1 @ prodT_factors (2::p) ps T2
   247       else [T]
   248   | prodT_factors _ _ T = [T];
   249 
   250 fun ap_split p ps (Type ("*", [T1, T2])) T3 u =
   251       if p mem ps then HOLogic.split_const (T1, T2, T3) $
   252         Abs ("v", T1, ap_split (2::p) ps T2 T3 (ap_split (1::p) ps T1
   253           (prodT_factors (2::p) ps T2 ---> T3) (incr_boundvars 1 u) $ Bound 0))
   254       else u
   255   | ap_split _ _ _ _ u =  u;
   256 
   257 fun mk_tuple p ps (Type ("*", [T1, T2])) (tms as t::_) =
   258       if p mem ps then HOLogic.mk_prod (mk_tuple (1::p) ps T1 tms, 
   259         mk_tuple (2::p) ps T2 (drop (length (prodT_factors (1::p) ps T1), tms)))
   260       else t
   261   | mk_tuple _ _ _ (t::_) = t;
   262 
   263 fun split_rule_var' ((t as Var (v, Type ("fun", [T1, T2])), ps), rl) =
   264       let val T' = prodT_factors [] ps T1 ---> T2
   265           val newt = ap_split [] ps T1 T2 (Var (v, T'))
   266           val cterm = Thm.cterm_of (#sign (rep_thm rl))
   267       in
   268           instantiate ([], [(cterm t, cterm newt)]) rl
   269       end
   270   | split_rule_var' (_, rl) = rl;
   271 
   272 val remove_split = rewrite_rule [split_conv RS eq_reflection];
   273 
   274 fun split_rule_vars vs rl = standard (remove_split (foldr split_rule_var'
   275   (mg_prod_factors vs ([], #prop (rep_thm rl)), rl)));
   276 
   277 fun split_rule vs rl = standard (remove_split (foldr split_rule_var'
   278   (mapfilter (fn (t as Var ((a, _), _)) =>
   279     apsome (pair t) (assoc (vs, a))) (term_vars (#prop (rep_thm rl))), rl)));
   280 
   281 
   282 (** process rules **)
   283 
   284 local
   285 
   286 fun err_in_rule sg name t msg =
   287   error (cat_lines ["Ill-formed introduction rule " ^ quote name, Sign.string_of_term sg t, msg]);
   288 
   289 fun err_in_prem sg name t p msg =
   290   error (cat_lines ["Ill-formed premise", Sign.string_of_term sg p,
   291     "in introduction rule " ^ quote name, Sign.string_of_term sg t, msg]);
   292 
   293 val bad_concl = "Conclusion of introduction rule must have form \"t : S_i\"";
   294 
   295 val all_not_allowed = 
   296     "Introduction rule must not have a leading \"!!\" quantifier";
   297 
   298 val atomize_cterm = Tactic.rewrite_cterm true inductive_atomize;
   299 
   300 in
   301 
   302 fun check_rule sg cs ((name, rule), att) =
   303   let
   304     val concl = Logic.strip_imp_concl rule;
   305     val prems = Logic.strip_imp_prems rule;
   306     val aprems = prems |> map (Thm.term_of o atomize_cterm o Thm.cterm_of sg);
   307     val arule = Logic.list_implies (aprems, concl);
   308 
   309     fun check_prem (prem, aprem) =
   310       if can HOLogic.dest_Trueprop aprem then ()
   311       else err_in_prem sg name rule prem "Non-atomic premise";
   312   in
   313     (case concl of
   314       Const ("Trueprop", _) $ (Const ("op :", _) $ t $ u) =>
   315         if u mem cs then
   316           if exists (Logic.occs o rpair t) cs then
   317             err_in_rule sg name rule "Recursion term on left of member symbol"
   318           else seq check_prem (prems ~~ aprems)
   319         else err_in_rule sg name rule bad_concl
   320       | Const ("all", _) $ _ => err_in_rule sg name rule all_not_allowed
   321       | _ => err_in_rule sg name rule bad_concl);
   322     ((name, arule), att)
   323   end;
   324 
   325 val rulify =
   326   standard o Tactic.norm_hhf o
   327   hol_simplify inductive_rulify2 o hol_simplify inductive_rulify1 o
   328   hol_simplify inductive_conj;
   329 
   330 end;
   331 
   332 
   333 
   334 (** properties of (co)inductive sets **)
   335 
   336 (* elimination rules *)
   337 
   338 fun mk_elims cs cTs params intr_ts intr_names =
   339   let
   340     val used = foldr add_term_names (intr_ts, []);
   341     val [aname, pname] = variantlist (["a", "P"], used);
   342     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   343 
   344     fun dest_intr r =
   345       let val Const ("op :", _) $ t $ u =
   346         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   347       in (u, t, Logic.strip_imp_prems r) end;
   348 
   349     val intrs = map dest_intr intr_ts ~~ intr_names;
   350 
   351     fun mk_elim (c, T) =
   352       let
   353         val a = Free (aname, T);
   354 
   355         fun mk_elim_prem (_, t, ts) =
   356           list_all_free (map dest_Free ((foldr add_term_frees (t::ts, [])) \\ params),
   357             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
   358         val c_intrs = (filter (equal c o #1 o #1) intrs);
   359       in
   360         (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
   361           map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs)
   362       end
   363   in
   364     map mk_elim (cs ~~ cTs)
   365   end;
   366 
   367 
   368 (* premises and conclusions of induction rules *)
   369 
   370 fun mk_indrule cs cTs params intr_ts =
   371   let
   372     val used = foldr add_term_names (intr_ts, []);
   373 
   374     (* predicates for induction rule *)
   375 
   376     val preds = map Free (variantlist (if length cs < 2 then ["P"] else
   377       map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
   378         map (fn T => T --> HOLogic.boolT) cTs);
   379 
   380     (* transform an introduction rule into a premise for induction rule *)
   381 
   382     fun mk_ind_prem r =
   383       let
   384         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   385 
   386         val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
   387 
   388         fun subst (s as ((m as Const ("op :", T)) $ t $ u)) =
   389               (case pred_of u of
   390                   None => (m $ fst (subst t) $ fst (subst u), None)
   391                 | Some P => (HOLogic.mk_binop inductive_conj_name (s, P $ t), Some (s, P $ t)))
   392           | subst s =
   393               (case pred_of s of
   394                   Some P => (HOLogic.mk_binop "op Int"
   395                     (s, HOLogic.Collect_const (HOLogic.dest_setT
   396                       (fastype_of s)) $ P), None)
   397                 | None => (case s of
   398                      (t $ u) => (fst (subst t) $ fst (subst u), None)
   399                    | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), None)
   400                    | _ => (s, None)));
   401 
   402         fun mk_prem (s, prems) = (case subst s of
   403               (_, Some (t, u)) => t :: u :: prems
   404             | (t, _) => t :: prems);
   405 
   406         val Const ("op :", _) $ t $ u =
   407           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   408 
   409       in list_all_free (frees,
   410            Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
   411              (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
   412                HOLogic.mk_Trueprop (the (pred_of u) $ t)))
   413       end;
   414 
   415     val ind_prems = map mk_ind_prem intr_ts;
   416     val factors = foldl (mg_prod_factors preds) ([], ind_prems);
   417 
   418     (* make conclusions for induction rules *)
   419 
   420     fun mk_ind_concl ((c, P), (ts, x)) =
   421       let val T = HOLogic.dest_setT (fastype_of c);
   422           val ps = if_none (assoc (factors, P)) [];
   423           val Ts = prodT_factors [] ps T;
   424           val (frees, x') = foldr (fn (T', (fs, s)) =>
   425             ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
   426           val tuple = mk_tuple [] ps T frees;
   427       in ((HOLogic.mk_binop "op -->"
   428         (HOLogic.mk_mem (tuple, c), P $ tuple))::ts, x')
   429       end;
   430 
   431     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   432         (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
   433 
   434   in (preds, ind_prems, mutual_ind_concl,
   435     map (apfst (fst o dest_Free)) factors)
   436   end;
   437 
   438 
   439 (* prepare cases and induct rules *)
   440 
   441 (*
   442   transform mutual rule:
   443     HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
   444   into i-th projection:
   445     xi:Ai ==> HH ==> Pi xi
   446 *)
   447 
   448 fun project_rules [name] rule = [(name, rule)]
   449   | project_rules names mutual_rule =
   450       let
   451         val n = length names;
   452         fun proj i =
   453           (if i < n then (fn th => th RS conjunct1) else I)
   454             (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
   455             RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
   456       in names ~~ map proj (1 upto n) end;
   457 
   458 fun add_cases_induct no_elim no_ind names elims induct =
   459   let
   460     fun cases_spec (name, elim) thy =
   461       thy
   462       |> Theory.add_path (Sign.base_name name)
   463       |> (#1 o PureThy.add_thms [(("cases", elim), [InductAttrib.cases_set_global name])])
   464       |> Theory.parent_path;
   465     val cases_specs = if no_elim then [] else map2 cases_spec (names, elims);
   466 
   467     fun induct_spec (name, th) = #1 o PureThy.add_thms
   468       [(("", RuleCases.save induct th), [InductAttrib.induct_set_global name])];
   469     val induct_specs = if no_ind then [] else map induct_spec (project_rules names induct);
   470   in Library.apply (cases_specs @ induct_specs) end;
   471 
   472 
   473 
   474 (** proofs for (co)inductive sets **)
   475 
   476 (* prove monotonicity -- NOT subject to quick_and_dirty! *)
   477 
   478 fun prove_mono setT fp_fun monos thy =
   479  (message "  Proving monotonicity ...";
   480   Goals.prove_goalw_cterm []      (*NO quick_and_dirty_prove_goalw_cterm here!*)
   481     (Thm.cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
   482       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
   483     (fn _ => [rtac monoI 1, REPEAT (ares_tac (flat (map mk_mono monos) @ get_monos thy) 1)]));
   484 
   485 
   486 (* prove introduction rules *)
   487 
   488 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
   489   let
   490     val _ = clean_message "  Proving the introduction rules ...";
   491 
   492     val unfold = standard (mono RS (fp_def RS
   493       (if coind then def_gfp_unfold else def_lfp_unfold)));
   494 
   495     fun select_disj 1 1 = []
   496       | select_disj _ 1 = [rtac disjI1]
   497       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   498 
   499     val intrs = map (fn (i, intr) => quick_and_dirty_prove_goalw_cterm thy rec_sets_defs
   500       (Thm.cterm_of (Theory.sign_of thy) intr) (fn prems =>
   501        [(*insert prems and underlying sets*)
   502        cut_facts_tac prems 1,
   503        stac unfold 1,
   504        REPEAT (resolve_tac [vimageI2, CollectI] 1),
   505        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
   506        EVERY1 (select_disj (length intr_ts) i),
   507        (*Not ares_tac, since refl must be tried before any equality assumptions;
   508          backtracking may occur if the premises have extra variables!*)
   509        DEPTH_SOLVE_1 (resolve_tac [refl, exI, conjI] 1 APPEND assume_tac 1),
   510        (*Now solve the equations like Inl 0 = Inl ?b2*)
   511        rewrite_goals_tac con_defs,
   512        REPEAT (rtac refl 1)])
   513       |> rulify) (1 upto (length intr_ts) ~~ intr_ts)
   514 
   515   in (intrs, unfold) end;
   516 
   517 
   518 (* prove elimination rules *)
   519 
   520 fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
   521   let
   522     val _ = clean_message "  Proving the elimination rules ...";
   523 
   524     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
   525     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @ map make_elim [Inl_inject, Inr_inject];
   526   in
   527     mk_elims cs cTs params intr_ts intr_names |> map (fn (t, cases) =>
   528       quick_and_dirty_prove_goalw_cterm thy rec_sets_defs
   529         (Thm.cterm_of (Theory.sign_of thy) t) (fn prems =>
   530           [cut_facts_tac [hd prems] 1,
   531            dtac (unfold RS subst) 1,
   532            REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   533            REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   534            EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))])
   535         |> rulify
   536         |> RuleCases.name cases)
   537   end;
   538 
   539 
   540 (* derivation of simplified elimination rules *)
   541 
   542 (*Applies freeness of the given constructors, which *must* be unfolded by
   543   the given defs.  Cannot simply use the local con_defs because con_defs=[]
   544   for inference systems. (??) *)
   545 
   546 local
   547 
   548 (*cprop should have the form t:Si where Si is an inductive set*)
   549 val mk_cases_err = "mk_cases: proposition not of form \"t : S_i\"";
   550 
   551 (*delete needless equality assumptions*)
   552 val refl_thin = prove_goal HOL.thy "!!P. a = a ==> P ==> P" (fn _ => [assume_tac 1]);
   553 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject];
   554 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   555 
   556 fun simp_case_tac solved ss i =
   557   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i
   558   THEN_MAYBE (if solved then no_tac else all_tac);
   559 
   560 in
   561 
   562 fun mk_cases_i elims ss cprop =
   563   let
   564     val prem = Thm.assume cprop;
   565     val tac = ALLGOALS (simp_case_tac false ss) THEN prune_params_tac;
   566     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic tac (prem RS rl));
   567   in
   568     (case get_first (try mk_elim) elims of
   569       Some r => r
   570     | None => error (Pretty.string_of (Pretty.block
   571         [Pretty.str mk_cases_err, Pretty.fbrk, Display.pretty_cterm cprop])))
   572   end;
   573 
   574 fun mk_cases elims s =
   575   mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
   576 
   577 fun smart_mk_cases thy ss cprop =
   578   let
   579     val c = #1 (Term.dest_Const (Term.head_of (#2 (HOLogic.dest_mem (HOLogic.dest_Trueprop
   580       (Logic.strip_imp_concl (Thm.term_of cprop))))))) handle TERM _ => error mk_cases_err;
   581     val (_, {elims, ...}) = the_inductive thy c;
   582   in mk_cases_i elims ss cprop end;
   583 
   584 end;
   585 
   586 
   587 (* inductive_cases(_i) *)
   588 
   589 fun gen_inductive_cases prep_att prep_const prep_prop
   590     (((name, raw_atts), raw_props), comment) thy =
   591   let
   592     val ss = Simplifier.simpset_of thy;
   593     val sign = Theory.sign_of thy;
   594     val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
   595     val atts = map (prep_att thy) raw_atts;
   596     val thms = map (smart_mk_cases thy ss) cprops;
   597   in
   598     thy |>
   599     IsarThy.have_theorems_i Drule.lemmaK [(((name, atts), map Thm.no_attributes thms), comment)]
   600   end;
   601 
   602 val inductive_cases =
   603   gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
   604 
   605 val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
   606 
   607 
   608 (* mk_cases_meth *)
   609 
   610 fun mk_cases_meth (ctxt, raw_props) =
   611   let
   612     val thy = ProofContext.theory_of ctxt;
   613     val ss = Simplifier.get_local_simpset ctxt;
   614     val cprops = map (Thm.cterm_of (Theory.sign_of thy) o ProofContext.read_prop ctxt) raw_props;
   615   in Method.erule 0 (map (smart_mk_cases thy ss) cprops) end;
   616 
   617 val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name));
   618 
   619 
   620 (* prove induction rule *)
   621 
   622 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
   623     fp_def rec_sets_defs thy =
   624   let
   625     val _ = clean_message "  Proving the induction rule ...";
   626 
   627     val sign = Theory.sign_of thy;
   628 
   629     val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
   630         None => []
   631       | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
   632 
   633     val (preds, ind_prems, mutual_ind_concl, factors) =
   634       mk_indrule cs cTs params intr_ts;
   635 
   636     (* make predicate for instantiation of abstract induction rule *)
   637 
   638     fun mk_ind_pred _ [P] = P
   639       | mk_ind_pred T Ps =
   640          let val n = (length Ps) div 2;
   641              val Type (_, [T1, T2]) = T
   642          in Const ("Datatype.sum.sum_case",
   643            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
   644              mk_ind_pred T1 (take (n, Ps)) $ mk_ind_pred T2 (drop (n, Ps))
   645          end;
   646 
   647     val ind_pred = mk_ind_pred sumT preds;
   648 
   649     val ind_concl = HOLogic.mk_Trueprop
   650       (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->"
   651         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0)));
   652 
   653     (* simplification rules for vimage and Collect *)
   654 
   655     val vimage_simps = if length cs < 2 then [] else
   656       map (fn c => quick_and_dirty_prove_goalw_cterm thy [] (Thm.cterm_of sign
   657         (HOLogic.mk_Trueprop (HOLogic.mk_eq
   658           (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c,
   659            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $
   660              nth_elem (find_index_eq c cs, preds)))))
   661         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites, rtac refl 1])) cs;
   662 
   663     val induct = quick_and_dirty_prove_goalw_cterm thy [inductive_conj_def] (Thm.cterm_of sign
   664       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
   665         [rtac (impI RS allI) 1,
   666          DETERM (etac (mono RS (fp_def RS def_lfp_induct)) 1),
   667          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
   668          fold_goals_tac rec_sets_defs,
   669          (*This CollectE and disjE separates out the introduction rules*)
   670          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
   671          (*Now break down the individual cases.  No disjE here in case
   672            some premise involves disjunction.*)
   673          REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
   674          rewrite_goals_tac sum_case_rewrites,
   675          EVERY (map (fn prem =>
   676            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
   677 
   678     val lemma = quick_and_dirty_prove_goalw_cterm thy rec_sets_defs (Thm.cterm_of sign
   679       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
   680         [cut_facts_tac prems 1,
   681          REPEAT (EVERY
   682            [REPEAT (resolve_tac [conjI, impI] 1),
   683             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
   684             rewrite_goals_tac sum_case_rewrites,
   685             atac 1])])
   686 
   687   in standard (split_rule factors (induct RS lemma)) end;
   688 
   689 
   690 
   691 (** specification of (co)inductive sets **)
   692 
   693 fun cond_declare_consts declare_consts cs paramTs cnames =
   694   if declare_consts then
   695     Theory.add_consts_i (map (fn (c, n) => (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
   696   else I;
   697 
   698 fun mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
   699       params paramTs cTs cnames =
   700   let
   701     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
   702     val setT = HOLogic.mk_setT sumT;
   703 
   704     val fp_name = if coind then gfp_name else lfp_name;
   705 
   706     val used = foldr add_term_names (intr_ts, []);
   707     val [sname, xname] = variantlist (["S", "x"], used);
   708 
   709     (* transform an introduction rule into a conjunction  *)
   710     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
   711     (* is transformed into                                *)
   712     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
   713 
   714     fun transform_rule r =
   715       let
   716         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   717         val subst = subst_free
   718           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
   719         val Const ("op :", _) $ t $ u =
   720           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   721 
   722       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
   723         (frees, foldr1 HOLogic.mk_conj
   724           (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t))::
   725             (map (subst o HOLogic.dest_Trueprop)
   726               (Logic.strip_imp_prems r))))
   727       end
   728 
   729     (* make a disjunction of all introduction rules *)
   730 
   731     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
   732       absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
   733 
   734     (* add definiton of recursive sets to theory *)
   735 
   736     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
   737     val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
   738 
   739     val rec_const = list_comb
   740       (Const (full_rec_name, paramTs ---> setT), params);
   741 
   742     val fp_def_term = Logic.mk_equals (rec_const,
   743       Const (fp_name, (setT --> setT) --> setT) $ fp_fun);
   744 
   745     val def_terms = fp_def_term :: (if length cs < 2 then [] else
   746       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
   747 
   748     val (thy', [fp_def :: rec_sets_defs]) =
   749       thy
   750       |> cond_declare_consts declare_consts cs paramTs cnames
   751       |> (if length cs < 2 then I
   752           else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)])
   753       |> Theory.add_path rec_name
   754       |> PureThy.add_defss_i false [(("defs", def_terms), [])];
   755 
   756     val mono = prove_mono setT fp_fun monos thy'
   757 
   758   in (thy', mono, fp_def, rec_sets_defs, rec_const, sumT) end;
   759 
   760 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
   761     intros monos con_defs thy params paramTs cTs cnames induct_cases =
   762   let
   763     val _ =
   764       if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
   765         commas_quote cnames) else ();
   766 
   767     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
   768 
   769     val (thy1, mono, fp_def, rec_sets_defs, rec_const, sumT) =
   770       mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
   771         params paramTs cTs cnames;
   772 
   773     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
   774       rec_sets_defs thy1;
   775     val elims = if no_elim then [] else
   776       prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy1;
   777     val raw_induct = if no_ind then Drule.asm_rl else
   778       if coind then standard (rule_by_tactic
   779         (rewrite_tac [mk_meta_eq vimage_Un] THEN
   780           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
   781       else
   782         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
   783           rec_sets_defs thy1;
   784     val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
   785       else standard (raw_induct RSN (2, rev_mp));
   786 
   787     val (thy2, intrs') =
   788       thy1 |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts);
   789     val (thy3, ([intrs'', elims'], [induct'])) =
   790       thy2
   791       |> PureThy.add_thmss
   792         [(("intros", intrs'), []),
   793           (("elims", elims), [RuleCases.consumes 1])]
   794       |>>> PureThy.add_thms
   795         [((coind_prefix coind ^ "induct", rulify induct),
   796          [RuleCases.case_names induct_cases,
   797           RuleCases.consumes 1])]
   798       |>> Theory.parent_path;
   799   in (thy3,
   800     {defs = fp_def :: rec_sets_defs,
   801      mono = mono,
   802      unfold = unfold,
   803      intrs = intrs'',
   804      elims = elims',
   805      mk_cases = mk_cases elims',
   806      raw_induct = rulify raw_induct,
   807      induct = induct'})
   808   end;
   809 
   810 
   811 (* external interfaces *)
   812 
   813 fun try_term f msg sign t =
   814   (case Library.try f t of
   815     Some x => x
   816   | None => error (msg ^ Sign.string_of_term sign t));
   817 
   818 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
   819     pre_intros monos con_defs thy =
   820   let
   821     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   822     val sign = Theory.sign_of thy;
   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: " sign) params;
   828 
   829     val cTs = map (try_term (HOLogic.dest_setT o fastype_of)
   830       "Recursive component not of type set: " sign) cs;
   831 
   832     val full_cnames = map (try_term (fst o dest_Const o head_of)
   833       "Recursive set not previously declared as constant: " sign) cs;
   834     val cnames = map Sign.base_name full_cnames;
   835 
   836     val save_sign =
   837       thy |> Theory.copy |> cond_declare_consts declare_consts cs paramTs cnames |> Theory.sign_of;
   838     val intros = map (check_rule save_sign cs) pre_intros;
   839     val induct_cases = map (#1 o #1) intros;
   840 
   841     val (thy1, result as {elims, induct, ...}) =
   842       add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs intros monos
   843         con_defs thy params paramTs cTs cnames induct_cases;
   844     val thy2 = thy1
   845       |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
   846       |> add_cases_induct no_elim (no_ind orelse coind) full_cnames elims induct;
   847   in (thy2, result) end;
   848 
   849 fun add_inductive verbose coind c_strings intro_srcs raw_monos raw_con_defs thy =
   850   let
   851     val sign = Theory.sign_of thy;
   852     val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
   853 
   854     val intr_names = map (fst o fst) intro_srcs;
   855     fun read_rule s = Thm.read_cterm sign (s, propT)
   856       handle ERROR => error ("The error(s) above occurred for " ^ s);
   857     val intr_ts = map (Thm.term_of o read_rule o snd o fst) intro_srcs;
   858     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
   859     val (cs', intr_ts') = unify_consts sign cs intr_ts;
   860 
   861     val ((thy', con_defs), monos) = thy
   862       |> IsarThy.apply_theorems raw_monos
   863       |> apfst (IsarThy.apply_theorems raw_con_defs);
   864   in
   865     add_inductive_i verbose false "" coind false false cs'
   866       ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
   867   end;
   868 
   869 
   870 
   871 (** package setup **)
   872 
   873 (* setup theory *)
   874 
   875 val setup =
   876  [InductiveData.init,
   877   Method.add_methods [("ind_cases", mk_cases_meth oo mk_cases_args,
   878     "dynamic case analysis on sets")],
   879   Attrib.add_attributes [("mono", mono_attr, "declaration of monotonicity rule")]];
   880 
   881 
   882 (* outer syntax *)
   883 
   884 local structure P = OuterParse and K = OuterSyntax.Keyword in
   885 
   886 fun mk_ind coind (((sets, intrs), monos), con_defs) =
   887   #1 o add_inductive true coind sets (map P.triple_swap intrs) monos con_defs;
   888 
   889 fun ind_decl coind =
   890   (Scan.repeat1 P.term --| P.marg_comment) --
   891   (P.$$$ "intros" |--
   892     P.!!! (Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
   893   Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
   894   Scan.optional (P.$$$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
   895   >> (Toplevel.theory o mk_ind coind);
   896 
   897 val inductiveP =
   898   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
   899 
   900 val coinductiveP =
   901   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
   902 
   903 
   904 val ind_cases =
   905   P.opt_thm_name ":" -- Scan.repeat1 P.prop -- P.marg_comment
   906   >> (Toplevel.theory o inductive_cases);
   907 
   908 val inductive_casesP =
   909   OuterSyntax.command "inductive_cases"
   910     "create simplified instances of elimination rules (improper)" K.thy_script ind_cases;
   911 
   912 val _ = OuterSyntax.add_keywords ["intros", "monos", "con_defs"];
   913 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
   914 
   915 end;
   916 
   917 end;