src/HOL/Tools/inductive_package.ML
 author wenzelm Wed Dec 27 18:27:19 2000 +0100 (2000-12-27 ago) changeset 10743 8ea821d1e3a4 parent 10735 dfaf75f0076f child 10804 306aee77eadd permissions -rw-r--r--
Method.erule 0;
```     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     Copyright   1994  University of Cambridge
```
```     7                 1998  TU Muenchen
```
```     8
```
```     9 (Co)Inductive Definition module for HOL.
```
```    10
```
```    11 Features:
```
```    12   * least or greatest fixedpoints
```
```    13   * user-specified product and sum constructions
```
```    14   * mutually recursive definitions
```
```    15   * definitions involving arbitrary monotone operators
```
```    16   * automatically proves introduction and elimination rules
```
```    17
```
```    18 The recursive sets must *already* be declared as constants in the
```
```    19 current theory!
```
```    20
```
```    21   Introduction rules have the form
```
```    22   [| ti:M(Sj), ..., P(x), ... |] ==> t: Sk
```
```    23   where M is some monotone operator (usually the identity)
```
```    24   P(x) is any side condition on the free variables
```
```    25   ti, t are any terms
```
```    26   Sj, Sk are two of the sets being defined in mutual recursion
```
```    27
```
```    28 Sums are used only for mutual recursion.  Products are used only to
```
```    29 derive "streamlined" induction rules for relations.
```
```    30 *)
```
```    31
```
```    32 signature INDUCTIVE_PACKAGE =
```
```    33 sig
```
```    34   val quiet_mode: bool ref
```
```    35   val unify_consts: Sign.sg -> term list -> term list -> term list * term list
```
```    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_cases: ((bstring * Args.src list) * string list) * Comment.text
```
```    44     -> theory -> theory
```
```    45   val inductive_cases_i: ((bstring * theory attribute list) * term list) * Comment.text
```
```    46     -> theory -> theory
```
```    47   val add_inductive_i: bool -> bool -> bstring -> bool -> bool -> bool -> term list ->
```
```    48     theory attribute list -> ((bstring * term) * theory attribute list) list ->
```
```    49       thm list -> thm list -> theory -> theory *
```
```    50       {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
```
```    51        intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
```
```    52   val add_inductive: bool -> bool -> string list -> Args.src list ->
```
```    53     ((bstring * string) * Args.src list) list -> (xstring * Args.src list) list ->
```
```    54       (xstring * Args.src list) list -> theory -> theory *
```
```    55       {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
```
```    56        intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
```
```    57   val setup: (theory -> theory) list
```
```    58 end;
```
```    59
```
```    60 structure InductivePackage: INDUCTIVE_PACKAGE =
```
```    61 struct
```
```    62
```
```    63
```
```    64 (** theory context references **)
```
```    65
```
```    66 val mono_name = "Ord.mono";
```
```    67 val gfp_name = "Gfp.gfp";
```
```    68 val lfp_name = "Lfp.lfp";
```
```    69 val vimage_name = "Inverse_Image.vimage";
```
```    70 val Const _ \$ (vimage_f \$ _) \$ _ = HOLogic.dest_Trueprop (Thm.concl_of vimageD);
```
```    71
```
```    72 val inductive_conj_name = "Inductive.conj";
```
```    73 val inductive_conj_def = thm "conj_def";
```
```    74 val inductive_conj = thms "inductive_conj";
```
```    75 val inductive_atomize = thms "inductive_atomize";
```
```    76 val inductive_rulify1 = thms "inductive_rulify1";
```
```    77 val inductive_rulify2 = thms "inductive_rulify2";
```
```    78
```
```    79
```
```    80
```
```    81 (** theory data **)
```
```    82
```
```    83 (* data kind 'HOL/inductive' *)
```
```    84
```
```    85 type inductive_info =
```
```    86   {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm,
```
```    87     induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm};
```
```    88
```
```    89 structure InductiveArgs =
```
```    90 struct
```
```    91   val name = "HOL/inductive";
```
```    92   type T = inductive_info Symtab.table * thm list;
```
```    93
```
```    94   val empty = (Symtab.empty, []);
```
```    95   val copy = I;
```
```    96   val prep_ext = I;
```
```    97   fun merge ((tab1, monos1), (tab2, monos2)) = (Symtab.merge (K true) (tab1, tab2),
```
```    98     Library.generic_merge Thm.eq_thm I I monos1 monos2);
```
```    99
```
```   100   fun print sg (tab, monos) =
```
```   101     [Pretty.strs ("(co)inductives:" :: map #1 (Sign.cond_extern_table sg Sign.constK tab)),
```
```   102      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_sg sg) monos)]
```
```   103     |> Pretty.chunks |> Pretty.writeln;
```
```   104 end;
```
```   105
```
```   106 structure InductiveData = TheoryDataFun(InductiveArgs);
```
```   107 val print_inductives = InductiveData.print;
```
```   108
```
```   109
```
```   110 (* get and put data *)
```
```   111
```
```   112 fun get_inductive thy name = Symtab.lookup (fst (InductiveData.get thy), name);
```
```   113
```
```   114 fun the_inductive thy name =
```
```   115   (case get_inductive thy name of
```
```   116     None => error ("Unknown (co)inductive set " ^ quote name)
```
```   117   | Some info => info);
```
```   118
```
```   119 fun put_inductives names info thy =
```
```   120   let
```
```   121     fun upd ((tab, monos), name) = (Symtab.update_new ((name, info), tab), monos);
```
```   122     val tab_monos = foldl upd (InductiveData.get thy, names)
```
```   123       handle Symtab.DUP name => error ("Duplicate definition of (co)inductive set " ^ quote name);
```
```   124   in InductiveData.put tab_monos thy end;
```
```   125
```
```   126
```
```   127
```
```   128 (** monotonicity rules **)
```
```   129
```
```   130 val get_monos = #2 o InductiveData.get;
```
```   131 fun map_monos f = InductiveData.map (Library.apsnd f);
```
```   132
```
```   133 fun mk_mono thm =
```
```   134   let
```
```   135     fun eq2mono thm' = [standard (thm' RS (thm' RS eq_to_mono))] @
```
```   136       (case concl_of thm of
```
```   137           (_ \$ (_ \$ (Const ("Not", _) \$ _) \$ _)) => []
```
```   138         | _ => [standard (thm' RS (thm' RS eq_to_mono2))]);
```
```   139     val concl = concl_of thm
```
```   140   in
```
```   141     if Logic.is_equals concl then
```
```   142       eq2mono (thm RS meta_eq_to_obj_eq)
```
```   143     else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then
```
```   144       eq2mono thm
```
```   145     else [thm]
```
```   146   end;
```
```   147
```
```   148
```
```   149 (* attributes *)
```
```   150
```
```   151 fun mono_add_global (thy, thm) = (map_monos (Drule.add_rules (mk_mono thm)) thy, thm);
```
```   152 fun mono_del_global (thy, thm) = (map_monos (Drule.del_rules (mk_mono thm)) thy, thm);
```
```   153
```
```   154 val mono_attr =
```
```   155  (Attrib.add_del_args mono_add_global mono_del_global,
```
```   156   Attrib.add_del_args Attrib.undef_local_attribute Attrib.undef_local_attribute);
```
```   157
```
```   158
```
```   159
```
```   160 (** misc utilities **)
```
```   161
```
```   162 val quiet_mode = ref false;
```
```   163 fun message s = if ! quiet_mode then () else writeln s;
```
```   164 fun clean_message s = if ! quick_and_dirty then () else message s;
```
```   165
```
```   166 fun coind_prefix true = "co"
```
```   167   | coind_prefix false = "";
```
```   168
```
```   169
```
```   170 (*the following code ensures that each recursive set always has the
```
```   171   same type in all introduction rules*)
```
```   172 fun unify_consts sign cs intr_ts =
```
```   173   (let
```
```   174     val {tsig, ...} = Sign.rep_sg sign;
```
```   175     val add_term_consts_2 =
```
```   176       foldl_aterms (fn (cs, Const c) => c ins cs | (cs, _) => cs);
```
```   177     fun varify (t, (i, ts)) =
```
```   178       let val t' = map_term_types (incr_tvar (i + 1)) (Type.varify (t, []))
```
```   179       in (maxidx_of_term t', t'::ts) end;
```
```   180     val (i, cs') = foldr varify (cs, (~1, []));
```
```   181     val (i', intr_ts') = foldr varify (intr_ts, (i, []));
```
```   182     val rec_consts = foldl add_term_consts_2 ([], cs');
```
```   183     val intr_consts = foldl add_term_consts_2 ([], intr_ts');
```
```   184     fun unify (env, (cname, cT)) =
```
```   185       let val consts = map snd (filter (fn c => fst c = cname) intr_consts)
```
```   186       in foldl (fn ((env', j'), Tp) => (Type.unify tsig j' env' Tp))
```
```   187           (env, (replicate (length consts) cT) ~~ consts)
```
```   188       end;
```
```   189     val (env, _) = foldl unify ((Vartab.empty, i'), rec_consts);
```
```   190     fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars_Vartab env T
```
```   191       in if T = T' then T else typ_subst_TVars_2 env T' end;
```
```   192     val subst = fst o Type.freeze_thaw o
```
```   193       (map_term_types (typ_subst_TVars_2 env))
```
```   194
```
```   195   in (map subst cs', map subst intr_ts')
```
```   196   end) handle Type.TUNIFY =>
```
```   197     (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts));
```
```   198
```
```   199
```
```   200 (*make injections used in mutually recursive definitions*)
```
```   201 fun mk_inj cs sumT c x =
```
```   202   let
```
```   203     fun mk_inj' T n i =
```
```   204       if n = 1 then x else
```
```   205       let val n2 = n div 2;
```
```   206           val Type (_, [T1, T2]) = T
```
```   207       in
```
```   208         if i <= n2 then
```
```   209           Const ("Inl", T1 --> T) \$ (mk_inj' T1 n2 i)
```
```   210         else
```
```   211           Const ("Inr", T2 --> T) \$ (mk_inj' T2 (n - n2) (i - n2))
```
```   212       end
```
```   213   in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
```
```   214   end;
```
```   215
```
```   216 (*make "vimage" terms for selecting out components of mutually rec.def*)
```
```   217 fun mk_vimage cs sumT t c = if length cs < 2 then t else
```
```   218   let
```
```   219     val cT = HOLogic.dest_setT (fastype_of c);
```
```   220     val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
```
```   221   in
```
```   222     Const (vimage_name, vimageT) \$
```
```   223       Abs ("y", cT, mk_inj cs sumT c (Bound 0)) \$ t
```
```   224   end;
```
```   225
```
```   226
```
```   227
```
```   228 (** process rules **)
```
```   229
```
```   230 local
```
```   231
```
```   232 fun err_in_rule sg name t msg =
```
```   233   error (cat_lines ["Ill-formed introduction rule " ^ quote name, Sign.string_of_term sg t, msg]);
```
```   234
```
```   235 fun err_in_prem sg name t p msg =
```
```   236   error (cat_lines ["Ill-formed premise", Sign.string_of_term sg p,
```
```   237     "in introduction rule " ^ quote name, Sign.string_of_term sg t, msg]);
```
```   238
```
```   239 val bad_concl = "Conclusion of introduction rule must have form \"t : S_i\"";
```
```   240
```
```   241 val atomize_cterm = InductMethod.rewrite_cterm inductive_atomize;
```
```   242 fun full_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
```
```   243
```
```   244 in
```
```   245
```
```   246 fun check_rule sg cs ((name, rule), att) =
```
```   247   let
```
```   248     val concl = Logic.strip_imp_concl rule;
```
```   249     val prems = Logic.strip_imp_prems rule;
```
```   250     val aprems = prems |> map (Thm.term_of o atomize_cterm o Thm.cterm_of sg);
```
```   251     val arule = Logic.list_implies (aprems, concl);
```
```   252
```
```   253     fun check_prem (prem, aprem) =
```
```   254       if can HOLogic.dest_Trueprop aprem then ()
```
```   255       else err_in_prem sg name rule prem "Non-atomic premise";
```
```   256   in
```
```   257     (case HOLogic.dest_Trueprop concl of
```
```   258       (Const ("op :", _) \$ t \$ u) =>
```
```   259         if u mem cs then
```
```   260           if exists (Logic.occs o rpair t) cs then
```
```   261             err_in_rule sg name rule "Recursion term on left of member symbol"
```
```   262           else seq check_prem (prems ~~ aprems)
```
```   263         else err_in_rule sg name rule bad_concl
```
```   264       | _ => err_in_rule sg name rule bad_concl);
```
```   265     ((name, arule), att)
```
```   266   end;
```
```   267
```
```   268 val rulify =
```
```   269   standard o full_simplify [Drule.norm_hhf_eq] o
```
```   270   full_simplify inductive_rulify2 o full_simplify inductive_rulify1 o
```
```   271   full_simplify inductive_conj;
```
```   272
```
```   273 end;
```
```   274
```
```   275
```
```   276
```
```   277 (** properties of (co)inductive sets **)
```
```   278
```
```   279 (* elimination rules *)
```
```   280
```
```   281 fun mk_elims cs cTs params intr_ts intr_names =
```
```   282   let
```
```   283     val used = foldr add_term_names (intr_ts, []);
```
```   284     val [aname, pname] = variantlist (["a", "P"], used);
```
```   285     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
```
```   286
```
```   287     fun dest_intr r =
```
```   288       let val Const ("op :", _) \$ t \$ u =
```
```   289         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
```
```   290       in (u, t, Logic.strip_imp_prems r) end;
```
```   291
```
```   292     val intrs = map dest_intr intr_ts ~~ intr_names;
```
```   293
```
```   294     fun mk_elim (c, T) =
```
```   295       let
```
```   296         val a = Free (aname, T);
```
```   297
```
```   298         fun mk_elim_prem (_, t, ts) =
```
```   299           list_all_free (map dest_Free ((foldr add_term_frees (t::ts, [])) \\ params),
```
```   300             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
```
```   301         val c_intrs = (filter (equal c o #1 o #1) intrs);
```
```   302       in
```
```   303         (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
```
```   304           map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs)
```
```   305       end
```
```   306   in
```
```   307     map mk_elim (cs ~~ cTs)
```
```   308   end;
```
```   309
```
```   310
```
```   311 (* premises and conclusions of induction rules *)
```
```   312
```
```   313 fun mk_indrule cs cTs params intr_ts =
```
```   314   let
```
```   315     val used = foldr add_term_names (intr_ts, []);
```
```   316
```
```   317     (* predicates for induction rule *)
```
```   318
```
```   319     val preds = map Free (variantlist (if length cs < 2 then ["P"] else
```
```   320       map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
```
```   321         map (fn T => T --> HOLogic.boolT) cTs);
```
```   322
```
```   323     (* transform an introduction rule into a premise for induction rule *)
```
```   324
```
```   325     fun mk_ind_prem r =
```
```   326       let
```
```   327         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
```
```   328
```
```   329         val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
```
```   330
```
```   331         fun subst (s as ((m as Const ("op :", T)) \$ t \$ u)) =
```
```   332               (case pred_of u of
```
```   333                   None => (m \$ fst (subst t) \$ fst (subst u), None)
```
```   334                 | Some P => (HOLogic.mk_binop inductive_conj_name (s, P \$ t), Some (s, P \$ t)))
```
```   335           | subst s =
```
```   336               (case pred_of s of
```
```   337                   Some P => (HOLogic.mk_binop "op Int"
```
```   338                     (s, HOLogic.Collect_const (HOLogic.dest_setT
```
```   339                       (fastype_of s)) \$ P), None)
```
```   340                 | None => (case s of
```
```   341                      (t \$ u) => (fst (subst t) \$ fst (subst u), None)
```
```   342                    | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), None)
```
```   343                    | _ => (s, None)));
```
```   344
```
```   345         fun mk_prem (s, prems) = (case subst s of
```
```   346               (_, Some (t, u)) => t :: u :: prems
```
```   347             | (t, _) => t :: prems);
```
```   348
```
```   349         val Const ("op :", _) \$ t \$ u =
```
```   350           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
```
```   351
```
```   352       in list_all_free (frees,
```
```   353            Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
```
```   354              (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
```
```   355                HOLogic.mk_Trueprop (the (pred_of u) \$ t)))
```
```   356       end;
```
```   357
```
```   358     val ind_prems = map mk_ind_prem intr_ts;
```
```   359
```
```   360     (* make conclusions for induction rules *)
```
```   361
```
```   362     fun mk_ind_concl ((c, P), (ts, x)) =
```
```   363       let val T = HOLogic.dest_setT (fastype_of c);
```
```   364           val Ts = HOLogic.prodT_factors T;
```
```   365           val (frees, x') = foldr (fn (T', (fs, s)) =>
```
```   366             ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
```
```   367           val tuple = HOLogic.mk_tuple T frees;
```
```   368       in ((HOLogic.mk_binop "op -->"
```
```   369         (HOLogic.mk_mem (tuple, c), P \$ tuple))::ts, x')
```
```   370       end;
```
```   371
```
```   372     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
```
```   373         (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
```
```   374
```
```   375   in (preds, ind_prems, mutual_ind_concl)
```
```   376   end;
```
```   377
```
```   378
```
```   379 (* prepare cases and induct rules *)
```
```   380
```
```   381 (*
```
```   382   transform mutual rule:
```
```   383     HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
```
```   384   into i-th projection:
```
```   385     xi:Ai ==> HH ==> Pi xi
```
```   386 *)
```
```   387
```
```   388 fun project_rules [name] rule = [(name, rule)]
```
```   389   | project_rules names mutual_rule =
```
```   390       let
```
```   391         val n = length names;
```
```   392         fun proj i =
```
```   393           (if i < n then (fn th => th RS conjunct1) else I)
```
```   394             (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
```
```   395             RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
```
```   396       in names ~~ map proj (1 upto n) end;
```
```   397
```
```   398 fun add_cases_induct no_elim no_ind names elims induct induct_cases =
```
```   399   let
```
```   400     fun cases_spec (name, elim) thy =
```
```   401       thy
```
```   402       |> Theory.add_path (Sign.base_name name)
```
```   403       |> (#1 o PureThy.add_thms [(("cases", elim), [InductAttrib.cases_set_global name])])
```
```   404       |> Theory.parent_path;
```
```   405     val cases_specs = if no_elim then [] else map2 cases_spec (names, elims);
```
```   406
```
```   407     fun induct_spec (name, th) = (#1 o PureThy.add_thms
```
```   408       [(("", th), [RuleCases.case_names induct_cases, InductAttrib.induct_set_global name])]);
```
```   409     val induct_specs = if no_ind then [] else map induct_spec (project_rules names induct);
```
```   410   in Library.apply (cases_specs @ induct_specs) end;
```
```   411
```
```   412
```
```   413
```
```   414 (** proofs for (co)inductive sets **)
```
```   415
```
```   416 (* prove monotonicity -- NOT subject to quick_and_dirty! *)
```
```   417
```
```   418 fun prove_mono setT fp_fun monos thy =
```
```   419  (message "  Proving monotonicity ...";
```
```   420   Goals.prove_goalw_cterm []      (*NO SkipProof.prove_goalw_cterm here!*)
```
```   421     (Thm.cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
```
```   422       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) \$ fp_fun)))
```
```   423     (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)]));
```
```   424
```
```   425
```
```   426 (* prove introduction rules *)
```
```   427
```
```   428 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
```
```   429   let
```
```   430     val _ = clean_message "  Proving the introduction rules ...";
```
```   431
```
```   432     val unfold = standard (mono RS (fp_def RS
```
```   433       (if coind then def_gfp_unfold else def_lfp_unfold)));
```
```   434
```
```   435     fun select_disj 1 1 = []
```
```   436       | select_disj _ 1 = [rtac disjI1]
```
```   437       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
```
```   438
```
```   439     val intrs = map (fn (i, intr) => SkipProof.prove_goalw_cterm thy rec_sets_defs
```
```   440       (Thm.cterm_of (Theory.sign_of thy) intr) (fn prems =>
```
```   441        [(*insert prems and underlying sets*)
```
```   442        cut_facts_tac prems 1,
```
```   443        stac unfold 1,
```
```   444        REPEAT (resolve_tac [vimageI2, CollectI] 1),
```
```   445        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
```
```   446        EVERY1 (select_disj (length intr_ts) i),
```
```   447        (*Not ares_tac, since refl must be tried before any equality assumptions;
```
```   448          backtracking may occur if the premises have extra variables!*)
```
```   449        DEPTH_SOLVE_1 (resolve_tac [refl, exI, conjI] 1 APPEND assume_tac 1),
```
```   450        (*Now solve the equations like Inl 0 = Inl ?b2*)
```
```   451        rewrite_goals_tac con_defs,
```
```   452        REPEAT (rtac refl 1)])
```
```   453       |> rulify) (1 upto (length intr_ts) ~~ intr_ts)
```
```   454
```
```   455   in (intrs, unfold) end;
```
```   456
```
```   457
```
```   458 (* prove elimination rules *)
```
```   459
```
```   460 fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
```
```   461   let
```
```   462     val _ = clean_message "  Proving the elimination rules ...";
```
```   463
```
```   464     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
```
```   465     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @ map make_elim [Inl_inject, Inr_inject];
```
```   466   in
```
```   467     map (fn (t, cases) => SkipProof.prove_goalw_cterm thy rec_sets_defs
```
```   468       (Thm.cterm_of (Theory.sign_of thy) t) (fn prems =>
```
```   469         [cut_facts_tac [hd prems] 1,
```
```   470          dtac (unfold RS subst) 1,
```
```   471          REPEAT (FIRSTGOAL (eresolve_tac rules1)),
```
```   472          REPEAT (FIRSTGOAL (eresolve_tac rules2)),
```
```   473          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))])
```
```   474       |> rulify
```
```   475       |> RuleCases.name cases)
```
```   476       (mk_elims cs cTs params intr_ts intr_names)
```
```   477   end;
```
```   478
```
```   479
```
```   480 (* derivation of simplified elimination rules *)
```
```   481
```
```   482 (*Applies freeness of the given constructors, which *must* be unfolded by
```
```   483   the given defs.  Cannot simply use the local con_defs because con_defs=[]
```
```   484   for inference systems. (??) *)
```
```   485
```
```   486 (*cprop should have the form t:Si where Si is an inductive set*)
```
```   487
```
```   488 val mk_cases_err = "mk_cases: proposition not of form \"t : S_i\"";
```
```   489
```
```   490 fun mk_cases_i elims ss cprop =
```
```   491   let
```
```   492     val prem = Thm.assume cprop;
```
```   493     val tac = ALLGOALS (InductMethod.simp_case_tac false ss) THEN prune_params_tac;
```
```   494     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic tac (prem RS rl));
```
```   495   in
```
```   496     (case get_first (try mk_elim) elims of
```
```   497       Some r => r
```
```   498     | None => error (Pretty.string_of (Pretty.block
```
```   499         [Pretty.str mk_cases_err, Pretty.fbrk, Display.pretty_cterm cprop])))
```
```   500   end;
```
```   501
```
```   502 fun mk_cases elims s =
```
```   503   mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
```
```   504
```
```   505 fun smart_mk_cases thy ss cprop =
```
```   506   let
```
```   507     val c = #1 (Term.dest_Const (Term.head_of (#2 (HOLogic.dest_mem (HOLogic.dest_Trueprop
```
```   508       (Logic.strip_imp_concl (Thm.term_of cprop))))))) handle TERM _ => error mk_cases_err;
```
```   509     val (_, {elims, ...}) = the_inductive thy c;
```
```   510   in mk_cases_i elims ss cprop end;
```
```   511
```
```   512
```
```   513 (* inductive_cases(_i) *)
```
```   514
```
```   515 fun gen_inductive_cases prep_att prep_const prep_prop
```
```   516     (((name, raw_atts), raw_props), comment) thy =
```
```   517   let
```
```   518     val ss = Simplifier.simpset_of thy;
```
```   519     val sign = Theory.sign_of thy;
```
```   520     val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
```
```   521     val atts = map (prep_att thy) raw_atts;
```
```   522     val thms = map (smart_mk_cases thy ss) cprops;
```
```   523   in thy |> IsarThy.have_theorems_i [(((name, atts), map Thm.no_attributes thms), comment)] end;
```
```   524
```
```   525 val inductive_cases =
```
```   526   gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
```
```   527
```
```   528 val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
```
```   529
```
```   530
```
```   531 (* mk_cases_meth *)
```
```   532
```
```   533 fun mk_cases_meth (ctxt, raw_props) =
```
```   534   let
```
```   535     val thy = ProofContext.theory_of ctxt;
```
```   536     val ss = Simplifier.get_local_simpset ctxt;
```
```   537     val cprops = map (Thm.cterm_of (Theory.sign_of thy) o ProofContext.read_prop ctxt) raw_props;
```
```   538   in Method.erule 0 (map (smart_mk_cases thy ss) cprops) end;
```
```   539
```
```   540 val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name));
```
```   541
```
```   542
```
```   543 (* prove induction rule *)
```
```   544
```
```   545 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
```
```   546     fp_def rec_sets_defs thy =
```
```   547   let
```
```   548     val _ = clean_message "  Proving the induction rule ...";
```
```   549
```
```   550     val sign = Theory.sign_of thy;
```
```   551
```
```   552     val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
```
```   553         None => []
```
```   554       | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
```
```   555
```
```   556     val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
```
```   557
```
```   558     (* make predicate for instantiation of abstract induction rule *)
```
```   559
```
```   560     fun mk_ind_pred _ [P] = P
```
```   561       | mk_ind_pred T Ps =
```
```   562          let val n = (length Ps) div 2;
```
```   563              val Type (_, [T1, T2]) = T
```
```   564          in Const ("Datatype.sum.sum_case",
```
```   565            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) \$
```
```   566              mk_ind_pred T1 (take (n, Ps)) \$ mk_ind_pred T2 (drop (n, Ps))
```
```   567          end;
```
```   568
```
```   569     val ind_pred = mk_ind_pred sumT preds;
```
```   570
```
```   571     val ind_concl = HOLogic.mk_Trueprop
```
```   572       (HOLogic.all_const sumT \$ Abs ("x", sumT, HOLogic.mk_binop "op -->"
```
```   573         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred \$ Bound 0)));
```
```   574
```
```   575     (* simplification rules for vimage and Collect *)
```
```   576
```
```   577     val vimage_simps = if length cs < 2 then [] else
```
```   578       map (fn c => SkipProof.prove_goalw_cterm thy [] (Thm.cterm_of sign
```
```   579         (HOLogic.mk_Trueprop (HOLogic.mk_eq
```
```   580           (mk_vimage cs sumT (HOLogic.Collect_const sumT \$ ind_pred) c,
```
```   581            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) \$
```
```   582              nth_elem (find_index_eq c cs, preds)))))
```
```   583         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites, rtac refl 1])) cs;
```
```   584
```
```   585     val induct = SkipProof.prove_goalw_cterm thy [inductive_conj_def] (Thm.cterm_of sign
```
```   586       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
```
```   587         [rtac (impI RS allI) 1,
```
```   588          DETERM (etac (mono RS (fp_def RS def_lfp_induct)) 1),
```
```   589          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
```
```   590          fold_goals_tac rec_sets_defs,
```
```   591          (*This CollectE and disjE separates out the introduction rules*)
```
```   592          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
```
```   593          (*Now break down the individual cases.  No disjE here in case
```
```   594            some premise involves disjunction.*)
```
```   595          REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
```
```   596          rewrite_goals_tac sum_case_rewrites,
```
```   597          EVERY (map (fn prem =>
```
```   598            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
```
```   599
```
```   600     val lemma = SkipProof.prove_goalw_cterm thy rec_sets_defs (Thm.cterm_of sign
```
```   601       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
```
```   602         [cut_facts_tac prems 1,
```
```   603          REPEAT (EVERY
```
```   604            [REPEAT (resolve_tac [conjI, impI] 1),
```
```   605             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
```
```   606             rewrite_goals_tac sum_case_rewrites,
```
```   607             atac 1])])
```
```   608
```
```   609   in standard (split_rule (induct RS lemma)) end;
```
```   610
```
```   611
```
```   612
```
```   613 (** specification of (co)inductive sets **)
```
```   614
```
```   615 fun cond_declare_consts declare_consts cs paramTs cnames =
```
```   616   if declare_consts then
```
```   617     Theory.add_consts_i (map (fn (c, n) => (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
```
```   618   else I;
```
```   619
```
```   620 fun mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
```
```   621       params paramTs cTs cnames =
```
```   622   let
```
```   623     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
```
```   624     val setT = HOLogic.mk_setT sumT;
```
```   625
```
```   626     val fp_name = if coind then gfp_name else lfp_name;
```
```   627
```
```   628     val used = foldr add_term_names (intr_ts, []);
```
```   629     val [sname, xname] = variantlist (["S", "x"], used);
```
```   630
```
```   631     (* transform an introduction rule into a conjunction  *)
```
```   632     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
```
```   633     (* is transformed into                                *)
```
```   634     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
```
```   635
```
```   636     fun transform_rule r =
```
```   637       let
```
```   638         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
```
```   639         val subst = subst_free
```
```   640           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
```
```   641         val Const ("op :", _) \$ t \$ u =
```
```   642           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
```
```   643
```
```   644       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
```
```   645         (frees, foldr1 HOLogic.mk_conj
```
```   646           (((HOLogic.eq_const sumT) \$ Free (xname, sumT) \$ (mk_inj cs sumT u t))::
```
```   647             (map (subst o HOLogic.dest_Trueprop)
```
```   648               (Logic.strip_imp_prems r))))
```
```   649       end
```
```   650
```
```   651     (* make a disjunction of all introduction rules *)
```
```   652
```
```   653     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) \$
```
```   654       absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
```
```   655
```
```   656     (* add definiton of recursive sets to theory *)
```
```   657
```
```   658     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
```
```   659     val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
```
```   660
```
```   661     val rec_const = list_comb
```
```   662       (Const (full_rec_name, paramTs ---> setT), params);
```
```   663
```
```   664     val fp_def_term = Logic.mk_equals (rec_const,
```
```   665       Const (fp_name, (setT --> setT) --> setT) \$ fp_fun);
```
```   666
```
```   667     val def_terms = fp_def_term :: (if length cs < 2 then [] else
```
```   668       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
```
```   669
```
```   670     val (thy', [fp_def :: rec_sets_defs]) =
```
```   671       thy
```
```   672       |> cond_declare_consts declare_consts cs paramTs cnames
```
```   673       |> (if length cs < 2 then I
```
```   674           else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)])
```
```   675       |> Theory.add_path rec_name
```
```   676       |> PureThy.add_defss_i false [(("defs", def_terms), [])];
```
```   677
```
```   678     val mono = prove_mono setT fp_fun monos thy'
```
```   679
```
```   680   in (thy', mono, fp_def, rec_sets_defs, rec_const, sumT) end;
```
```   681
```
```   682 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
```
```   683     atts intros monos con_defs thy params paramTs cTs cnames induct_cases =
```
```   684   let
```
```   685     val _ =
```
```   686       if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
```
```   687         commas_quote cnames) else ();
```
```   688
```
```   689     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
```
```   690
```
```   691     val (thy1, mono, fp_def, rec_sets_defs, rec_const, sumT) =
```
```   692       mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
```
```   693         params paramTs cTs cnames;
```
```   694
```
```   695     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
```
```   696       rec_sets_defs thy1;
```
```   697     val elims = if no_elim then [] else
```
```   698       prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy1;
```
```   699     val raw_induct = if no_ind then Drule.asm_rl else
```
```   700       if coind then standard (rule_by_tactic
```
```   701         (rewrite_tac [mk_meta_eq vimage_Un] THEN
```
```   702           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
```
```   703       else
```
```   704         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
```
```   705           rec_sets_defs thy1;
```
```   706     val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
```
```   707       else standard (raw_induct RSN (2, rev_mp));
```
```   708
```
```   709     val (thy2, intrs') =
```
```   710       thy1 |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts);
```
```   711     val (thy3, ([intrs'', elims'], [induct'])) =
```
```   712       thy2
```
```   713       |> PureThy.add_thmss [(("intros", intrs'), atts), (("elims", elims), [])]
```
```   714       |>>> PureThy.add_thms
```
```   715         [((coind_prefix coind ^ "induct", rulify induct), [RuleCases.case_names induct_cases])]
```
```   716       |>> Theory.parent_path;
```
```   717   in (thy3,
```
```   718     {defs = fp_def :: rec_sets_defs,
```
```   719      mono = mono,
```
```   720      unfold = unfold,
```
```   721      intrs = intrs'',
```
```   722      elims = elims',
```
```   723      mk_cases = mk_cases elims',
```
```   724      raw_induct = rulify raw_induct,
```
```   725      induct = induct'})
```
```   726   end;
```
```   727
```
```   728
```
```   729 (* external interfaces *)
```
```   730
```
```   731 fun try_term f msg sign t =
```
```   732   (case Library.try f t of
```
```   733     Some x => x
```
```   734   | None => error (msg ^ Sign.string_of_term sign t));
```
```   735
```
```   736 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
```
```   737     atts pre_intros monos con_defs thy =
```
```   738   let
```
```   739     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
```
```   740     val sign = Theory.sign_of thy;
```
```   741
```
```   742     (*parameters should agree for all mutually recursive components*)
```
```   743     val (_, params) = strip_comb (hd cs);
```
```   744     val paramTs = map (try_term (snd o dest_Free) "Parameter in recursive\
```
```   745       \ component is not a free variable: " sign) params;
```
```   746
```
```   747     val cTs = map (try_term (HOLogic.dest_setT o fastype_of)
```
```   748       "Recursive component not of type set: " sign) cs;
```
```   749
```
```   750     val full_cnames = map (try_term (fst o dest_Const o head_of)
```
```   751       "Recursive set not previously declared as constant: " sign) cs;
```
```   752     val cnames = map Sign.base_name full_cnames;
```
```   753
```
```   754     val save_sign =
```
```   755       thy |> Theory.copy |> cond_declare_consts declare_consts cs paramTs cnames |> Theory.sign_of;
```
```   756     val intros = map (check_rule save_sign cs) pre_intros;
```
```   757     val induct_cases = map (#1 o #1) intros;
```
```   758
```
```   759     val (thy1, result as {elims, induct, ...}) =
```
```   760       add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
```
```   761         con_defs thy params paramTs cTs cnames induct_cases;
```
```   762     val thy2 = thy1
```
```   763       |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
```
```   764       |> add_cases_induct no_elim (no_ind orelse coind) full_cnames elims induct induct_cases;
```
```   765   in (thy2, result) end;
```
```   766
```
```   767 fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
```
```   768   let
```
```   769     val sign = Theory.sign_of thy;
```
```   770     val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
```
```   771
```
```   772     val atts = map (Attrib.global_attribute thy) srcs;
```
```   773     val intr_names = map (fst o fst) intro_srcs;
```
```   774     fun read_rule s = Thm.read_cterm sign (s, propT)
```
```   775       handle ERROR => error ("The error(s) above occurred for " ^ s);
```
```   776     val intr_ts = map (Thm.term_of o read_rule o snd o fst) intro_srcs;
```
```   777     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
```
```   778     val (cs', intr_ts') = unify_consts sign cs intr_ts;
```
```   779
```
```   780     val ((thy', con_defs), monos) = thy
```
```   781       |> IsarThy.apply_theorems raw_monos
```
```   782       |> apfst (IsarThy.apply_theorems raw_con_defs);
```
```   783   in
```
```   784     add_inductive_i verbose false "" coind false false cs'
```
```   785       atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
```
```   786   end;
```
```   787
```
```   788
```
```   789
```
```   790 (** package setup **)
```
```   791
```
```   792 (* setup theory *)
```
```   793
```
```   794 val setup =
```
```   795  [InductiveData.init,
```
```   796   Method.add_methods [("ind_cases", mk_cases_meth oo mk_cases_args,
```
```   797     "dynamic case analysis on sets")],
```
```   798   Attrib.add_attributes [("mono", mono_attr, "declaration of monotonicity rule")]];
```
```   799
```
```   800
```
```   801 (* outer syntax *)
```
```   802
```
```   803 local structure P = OuterParse and K = OuterSyntax.Keyword in
```
```   804
```
```   805 fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
```
```   806   #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
```
```   807
```
```   808 fun ind_decl coind =
```
```   809   (Scan.repeat1 P.term --| P.marg_comment) --
```
```   810   (P.\$\$\$ "intros" |--
```
```   811     P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
```
```   812   Scan.optional (P.\$\$\$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
```
```   813   Scan.optional (P.\$\$\$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
```
```   814   >> (Toplevel.theory o mk_ind coind);
```
```   815
```
```   816 val inductiveP =
```
```   817   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
```
```   818
```
```   819 val coinductiveP =
```
```   820   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
```
```   821
```
```   822
```
```   823 val ind_cases =
```
```   824   P.opt_thm_name ":" -- Scan.repeat1 P.prop -- P.marg_comment
```
```   825   >> (Toplevel.theory o inductive_cases);
```
```   826
```
```   827 val inductive_casesP =
```
```   828   OuterSyntax.command "inductive_cases"
```
```   829     "create simplified instances of elimination rules (improper)" K.thy_script ind_cases;
```
```   830
```
```   831 val _ = OuterSyntax.add_keywords ["intros", "monos", "con_defs"];
```
```   832 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
```
```   833
```
```   834 end;
```
```   835
```
```   836 end;
```