src/HOL/Tools/inductive_package.ML
 author haftmann Fri Dec 05 18:43:42 2008 +0100 (2008-12-05) changeset 29006 abe0f11cfa4e parent 28965 1de908189869 child 29064 70a61d58460e permissions -rw-r--r--
Name.name_of -> Binding.base_name
```     1 (*  Title:      HOL/Tools/inductive_package.ML
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
```
```     4
```
```     5 (Co)Inductive Definition module for HOL.
```
```     6
```
```     7 Features:
```
```     8   * least or greatest fixedpoints
```
```     9   * mutually recursive definitions
```
```    10   * definitions involving arbitrary monotone operators
```
```    11   * automatically proves introduction and elimination rules
```
```    12
```
```    13   Introduction rules have the form
```
```    14   [| M Pj ti, ..., Q x, ... |] ==> Pk t
```
```    15   where M is some monotone operator (usually the identity)
```
```    16   Q x is any side condition on the free variables
```
```    17   ti, t are any terms
```
```    18   Pj, Pk are two of the predicates being defined in mutual recursion
```
```    19 *)
```
```    20
```
```    21 signature BASIC_INDUCTIVE_PACKAGE =
```
```    22 sig
```
```    23   type inductive_result
```
```    24   val morph_result: morphism -> inductive_result -> inductive_result
```
```    25   type inductive_info
```
```    26   val the_inductive: Proof.context -> string -> inductive_info
```
```    27   val print_inductives: Proof.context -> unit
```
```    28   val mono_add: attribute
```
```    29   val mono_del: attribute
```
```    30   val get_monos: Proof.context -> thm list
```
```    31   val mk_cases: Proof.context -> term -> thm
```
```    32   val inductive_forall_name: string
```
```    33   val inductive_forall_def: thm
```
```    34   val rulify: thm -> thm
```
```    35   val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
```
```    36     thm list list * local_theory
```
```    37   val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
```
```    38     thm list list * local_theory
```
```    39   type inductive_flags
```
```    40   val add_inductive_i:
```
```    41     inductive_flags -> ((Binding.T * typ) * mixfix) list ->
```
```    42     (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
```
```    43     inductive_result * local_theory
```
```    44   val add_inductive: bool -> bool ->
```
```    45     (Binding.T * string option * mixfix) list ->
```
```    46     (Binding.T * string option * mixfix) list ->
```
```    47     (Attrib.binding * string) list ->
```
```    48     (Facts.ref * Attrib.src list) list ->
```
```    49     local_theory -> inductive_result * local_theory
```
```    50   val add_inductive_global: string -> inductive_flags ->
```
```    51     ((Binding.T * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
```
```    52     thm list -> theory -> inductive_result * theory
```
```    53   val arities_of: thm -> (string * int) list
```
```    54   val params_of: thm -> term list
```
```    55   val partition_rules: thm -> thm list -> (string * thm list) list
```
```    56   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
```
```    57   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
```
```    58   val infer_intro_vars: thm -> int -> thm list -> term list list
```
```    59   val setup: theory -> theory
```
```    60 end;
```
```    61
```
```    62 signature INDUCTIVE_PACKAGE =
```
```    63 sig
```
```    64   include BASIC_INDUCTIVE_PACKAGE
```
```    65   type add_ind_def
```
```    66   val declare_rules: string -> Binding.T -> bool -> bool -> string list ->
```
```    67     thm list -> Binding.T list -> Attrib.src list list -> (thm * string list) list ->
```
```    68     thm -> local_theory -> thm list * thm list * thm * local_theory
```
```    69   val add_ind_def: add_ind_def
```
```    70   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
```
```    71     ((Binding.T * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
```
```    72     thm list -> local_theory -> inductive_result * local_theory
```
```    73   val gen_add_inductive: add_ind_def -> bool -> bool ->
```
```    74     (Binding.T * string option * mixfix) list ->
```
```    75     (Binding.T * string option * mixfix) list ->
```
```    76     (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
```
```    77     local_theory -> inductive_result * local_theory
```
```    78   val gen_ind_decl: add_ind_def -> bool ->
```
```    79     OuterParse.token list -> (local_theory -> local_theory) * OuterParse.token list
```
```    80 end;
```
```    81
```
```    82 structure InductivePackage: INDUCTIVE_PACKAGE =
```
```    83 struct
```
```    84
```
```    85
```
```    86 (** theory context references **)
```
```    87
```
```    88 val inductive_forall_name = "HOL.induct_forall";
```
```    89 val inductive_forall_def = thm "induct_forall_def";
```
```    90 val inductive_conj_name = "HOL.induct_conj";
```
```    91 val inductive_conj_def = thm "induct_conj_def";
```
```    92 val inductive_conj = thms "induct_conj";
```
```    93 val inductive_atomize = thms "induct_atomize";
```
```    94 val inductive_rulify = thms "induct_rulify";
```
```    95 val inductive_rulify_fallback = thms "induct_rulify_fallback";
```
```    96
```
```    97 val notTrueE = TrueI RSN (2, notE);
```
```    98 val notFalseI = Seq.hd (atac 1 notI);
```
```    99 val simp_thms' = map (fn s => mk_meta_eq (the (find_first
```
```   100   (equal (OldGoals.read_prop HOL.thy s) o prop_of) simp_thms)))
```
```   101   ["(~True) = False", "(~False) = True",
```
```   102    "(True --> ?P) = ?P", "(False --> ?P) = True",
```
```   103    "(?P & True) = ?P", "(True & ?P) = ?P"];
```
```   104
```
```   105
```
```   106
```
```   107 (** context data **)
```
```   108
```
```   109 type inductive_result =
```
```   110   {preds: term list, elims: thm list, raw_induct: thm,
```
```   111    induct: thm, intrs: thm list};
```
```   112
```
```   113 fun morph_result phi {preds, elims, raw_induct: thm, induct, intrs} =
```
```   114   let
```
```   115     val term = Morphism.term phi;
```
```   116     val thm = Morphism.thm phi;
```
```   117     val fact = Morphism.fact phi;
```
```   118   in
```
```   119    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
```
```   120     induct = thm induct, intrs = fact intrs}
```
```   121   end;
```
```   122
```
```   123 type inductive_info =
```
```   124   {names: string list, coind: bool} * inductive_result;
```
```   125
```
```   126 structure InductiveData = GenericDataFun
```
```   127 (
```
```   128   type T = inductive_info Symtab.table * thm list;
```
```   129   val empty = (Symtab.empty, []);
```
```   130   val extend = I;
```
```   131   fun merge _ ((tab1, monos1), (tab2, monos2)) =
```
```   132     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
```
```   133 );
```
```   134
```
```   135 val get_inductives = InductiveData.get o Context.Proof;
```
```   136
```
```   137 fun print_inductives ctxt =
```
```   138   let
```
```   139     val (tab, monos) = get_inductives ctxt;
```
```   140     val space = Consts.space_of (ProofContext.consts_of ctxt);
```
```   141   in
```
```   142     [Pretty.strs ("(co)inductives:" :: map #1 (NameSpace.extern_table (space, tab))),
```
```   143      Pretty.big_list "monotonicity rules:" (map (ProofContext.pretty_thm ctxt) monos)]
```
```   144     |> Pretty.chunks |> Pretty.writeln
```
```   145   end;
```
```   146
```
```   147
```
```   148 (* get and put data *)
```
```   149
```
```   150 fun the_inductive ctxt name =
```
```   151   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
```
```   152     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
```
```   153   | SOME info => info);
```
```   154
```
```   155 fun put_inductives names info = InductiveData.map
```
```   156   (apfst (fold (fn name => Symtab.update (name, info)) names));
```
```   157
```
```   158
```
```   159
```
```   160 (** monotonicity rules **)
```
```   161
```
```   162 val get_monos = #2 o get_inductives;
```
```   163 val map_monos = InductiveData.map o apsnd;
```
```   164
```
```   165 fun mk_mono thm =
```
```   166   let
```
```   167     val concl = concl_of thm;
```
```   168     fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @
```
```   169       (case concl of
```
```   170           (_ \$ (_ \$ (Const ("Not", _) \$ _) \$ _)) => []
```
```   171         | _ => [thm' RS (thm' RS eq_to_mono2)]);
```
```   172     fun dest_less_concl thm = dest_less_concl (thm RS le_funD)
```
```   173       handle THM _ => thm RS le_boolD
```
```   174   in
```
```   175     case concl of
```
```   176       Const ("==", _) \$ _ \$ _ => eq2mono (thm RS meta_eq_to_obj_eq)
```
```   177     | _ \$ (Const ("op =", _) \$ _ \$ _) => eq2mono thm
```
```   178     | _ \$ (Const ("HOL.ord_class.less_eq", _) \$ _ \$ _) =>
```
```   179       [dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
```
```   180          (resolve_tac [le_funI, le_boolI'])) thm))]
```
```   181     | _ => [thm]
```
```   182   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm thm);
```
```   183
```
```   184 val mono_add = Thm.declaration_attribute (map_monos o fold Thm.add_thm o mk_mono);
```
```   185 val mono_del = Thm.declaration_attribute (map_monos o fold Thm.del_thm o mk_mono);
```
```   186
```
```   187
```
```   188
```
```   189 (** misc utilities **)
```
```   190
```
```   191 fun message quiet_mode s = if quiet_mode then () else writeln s;
```
```   192 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
```
```   193
```
```   194 fun coind_prefix true = "co"
```
```   195   | coind_prefix false = "";
```
```   196
```
```   197 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
```
```   198
```
```   199 fun make_bool_args f g [] i = []
```
```   200   | make_bool_args f g (x :: xs) i =
```
```   201       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
```
```   202
```
```   203 fun make_bool_args' xs =
```
```   204   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
```
```   205
```
```   206 fun find_arg T x [] = sys_error "find_arg"
```
```   207   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
```
```   208       apsnd (cons p) (find_arg T x ps)
```
```   209   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
```
```   210       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
```
```   211       else apsnd (cons p) (find_arg T x ps);
```
```   212
```
```   213 fun make_args Ts xs =
```
```   214   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
```
```   215     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
```
```   216
```
```   217 fun make_args' Ts xs Us =
```
```   218   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
```
```   219
```
```   220 fun dest_predicate cs params t =
```
```   221   let
```
```   222     val k = length params;
```
```   223     val (c, ts) = strip_comb t;
```
```   224     val (xs, ys) = chop k ts;
```
```   225     val i = find_index_eq c cs;
```
```   226   in
```
```   227     if xs = params andalso i >= 0 then
```
```   228       SOME (c, i, ys, chop (length ys)
```
```   229         (List.drop (binder_types (fastype_of c), k)))
```
```   230     else NONE
```
```   231   end;
```
```   232
```
```   233 fun mk_names a 0 = []
```
```   234   | mk_names a 1 = [a]
```
```   235   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
```
```   236
```
```   237
```
```   238
```
```   239 (** process rules **)
```
```   240
```
```   241 local
```
```   242
```
```   243 fun err_in_rule ctxt name t msg =
```
```   244   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
```
```   245     Syntax.string_of_term ctxt t, msg]);
```
```   246
```
```   247 fun err_in_prem ctxt name t p msg =
```
```   248   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
```
```   249     "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
```
```   250
```
```   251 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
```
```   252
```
```   253 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
```
```   254
```
```   255 val bad_app = "Inductive predicate must be applied to parameter(s) ";
```
```   256
```
```   257 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
```
```   258
```
```   259 in
```
```   260
```
```   261 fun check_rule ctxt cs params ((binding, att), rule) =
```
```   262   let
```
```   263     val err_name = Binding.display binding;
```
```   264     val params' = Term.variant_frees rule (Logic.strip_params rule);
```
```   265     val frees = rev (map Free params');
```
```   266     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
```
```   267     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
```
```   268     val rule' = Logic.list_implies (prems, concl);
```
```   269     val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
```
```   270     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
```
```   271
```
```   272     fun check_ind err t = case dest_predicate cs params t of
```
```   273         NONE => err (bad_app ^
```
```   274           commas (map (Syntax.string_of_term ctxt) params))
```
```   275       | SOME (_, _, ys, _) =>
```
```   276           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
```
```   277           then err bad_ind_occ else ();
```
```   278
```
```   279     fun check_prem' prem t =
```
```   280       if head_of t mem cs then
```
```   281         check_ind (err_in_prem ctxt err_name rule prem) t
```
```   282       else (case t of
```
```   283           Abs (_, _, t) => check_prem' prem t
```
```   284         | t \$ u => (check_prem' prem t; check_prem' prem u)
```
```   285         | _ => ());
```
```   286
```
```   287     fun check_prem (prem, aprem) =
```
```   288       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
```
```   289       else err_in_prem ctxt err_name rule prem "Non-atomic premise";
```
```   290   in
```
```   291     (case concl of
```
```   292        Const ("Trueprop", _) \$ t =>
```
```   293          if head_of t mem cs then
```
```   294            (check_ind (err_in_rule ctxt err_name rule') t;
```
```   295             List.app check_prem (prems ~~ aprems))
```
```   296          else err_in_rule ctxt err_name rule' bad_concl
```
```   297      | _ => err_in_rule ctxt err_name rule' bad_concl);
```
```   298     ((binding, att), arule)
```
```   299   end;
```
```   300
```
```   301 val rulify =
```
```   302   hol_simplify inductive_conj
```
```   303   #> hol_simplify inductive_rulify
```
```   304   #> hol_simplify inductive_rulify_fallback
```
```   305   #> MetaSimplifier.norm_hhf;
```
```   306
```
```   307 end;
```
```   308
```
```   309
```
```   310
```
```   311 (** proofs for (co)inductive predicates **)
```
```   312
```
```   313 (* prove monotonicity *)
```
```   314
```
```   315 fun prove_mono quiet_mode skip_mono predT fp_fun monos ctxt =
```
```   316  (message (quiet_mode orelse skip_mono andalso !quick_and_dirty)
```
```   317     "  Proving monotonicity ...";
```
```   318   (if skip_mono then SkipProof.prove else Goal.prove) ctxt [] []
```
```   319     (HOLogic.mk_Trueprop
```
```   320       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) \$ fp_fun))
```
```   321     (fn _ => EVERY [rtac @{thm monoI} 1,
```
```   322       REPEAT (resolve_tac [le_funI, le_boolI'] 1),
```
```   323       REPEAT (FIRST
```
```   324         [atac 1,
```
```   325          resolve_tac (List.concat (map mk_mono monos) @ get_monos ctxt) 1,
```
```   326          etac le_funE 1, dtac le_boolD 1])]));
```
```   327
```
```   328
```
```   329 (* prove introduction rules *)
```
```   330
```
```   331 fun prove_intrs quiet_mode coind mono fp_def k params intr_ts rec_preds_defs ctxt =
```
```   332   let
```
```   333     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
```
```   334
```
```   335     val unfold = funpow k (fn th => th RS fun_cong)
```
```   336       (mono RS (fp_def RS
```
```   337         (if coind then def_gfp_unfold else def_lfp_unfold)));
```
```   338
```
```   339     fun select_disj 1 1 = []
```
```   340       | select_disj _ 1 = [rtac disjI1]
```
```   341       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
```
```   342
```
```   343     val rules = [refl, TrueI, notFalseI, exI, conjI];
```
```   344
```
```   345     val intrs = map_index (fn (i, intr) => rulify
```
```   346       (SkipProof.prove ctxt (map (fst o dest_Free) params) [] intr (fn _ => EVERY
```
```   347        [rewrite_goals_tac rec_preds_defs,
```
```   348         rtac (unfold RS iffD2) 1,
```
```   349         EVERY1 (select_disj (length intr_ts) (i + 1)),
```
```   350         (*Not ares_tac, since refl must be tried before any equality assumptions;
```
```   351           backtracking may occur if the premises have extra variables!*)
```
```   352         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts
```
```   353
```
```   354   in (intrs, unfold) end;
```
```   355
```
```   356
```
```   357 (* prove elimination rules *)
```
```   358
```
```   359 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt =
```
```   360   let
```
```   361     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
```
```   362
```
```   363     val ([pname], ctxt') = ctxt |>
```
```   364       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
```
```   365       Variable.variant_fixes ["P"];
```
```   366     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
```
```   367
```
```   368     fun dest_intr r =
```
```   369       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
```
```   370        Logic.strip_assums_hyp r, Logic.strip_params r);
```
```   371
```
```   372     val intrs = map dest_intr intr_ts ~~ intr_names;
```
```   373
```
```   374     val rules1 = [disjE, exE, FalseE];
```
```   375     val rules2 = [conjE, FalseE, notTrueE];
```
```   376
```
```   377     fun prove_elim c =
```
```   378       let
```
```   379         val Ts = List.drop (binder_types (fastype_of c), length params);
```
```   380         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
```
```   381         val frees = map Free (anames ~~ Ts);
```
```   382
```
```   383         fun mk_elim_prem ((_, _, us, _), ts, params') =
```
```   384           list_all (params',
```
```   385             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
```
```   386               (frees ~~ us) @ ts, P));
```
```   387         val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs);
```
```   388         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
```
```   389            map mk_elim_prem (map #1 c_intrs)
```
```   390       in
```
```   391         (SkipProof.prove ctxt'' [] prems P
```
```   392           (fn {prems, ...} => EVERY
```
```   393             [cut_facts_tac [hd prems] 1,
```
```   394              rewrite_goals_tac rec_preds_defs,
```
```   395              dtac (unfold RS iffD1) 1,
```
```   396              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
```
```   397              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
```
```   398              EVERY (map (fn prem =>
```
```   399                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
```
```   400           |> rulify
```
```   401           |> singleton (ProofContext.export ctxt'' ctxt),
```
```   402          map #2 c_intrs)
```
```   403       end
```
```   404
```
```   405    in map prove_elim cs end;
```
```   406
```
```   407
```
```   408 (* derivation of simplified elimination rules *)
```
```   409
```
```   410 local
```
```   411
```
```   412 (*delete needless equality assumptions*)
```
```   413 val refl_thin = Goal.prove_global HOL.thy [] [] @{prop "!!P. a = a ==> P ==> P"}
```
```   414   (fn _ => assume_tac 1);
```
```   415 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
```
```   416 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
```
```   417
```
```   418 fun simp_case_tac ss i =
```
```   419   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
```
```   420
```
```   421 in
```
```   422
```
```   423 fun mk_cases ctxt prop =
```
```   424   let
```
```   425     val thy = ProofContext.theory_of ctxt;
```
```   426     val ss = Simplifier.local_simpset_of ctxt;
```
```   427
```
```   428     fun err msg =
```
```   429       error (Pretty.string_of (Pretty.block
```
```   430         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
```
```   431
```
```   432     val elims = Induct.find_casesP ctxt prop;
```
```   433
```
```   434     val cprop = Thm.cterm_of thy prop;
```
```   435     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
```
```   436     fun mk_elim rl =
```
```   437       Thm.implies_intr cprop (Tactic.rule_by_tactic tac (Thm.assume cprop RS rl))
```
```   438       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
```
```   439   in
```
```   440     (case get_first (try mk_elim) elims of
```
```   441       SOME r => r
```
```   442     | NONE => err "Proposition not an inductive predicate:")
```
```   443   end;
```
```   444
```
```   445 end;
```
```   446
```
```   447
```
```   448 (* inductive_cases *)
```
```   449
```
```   450 fun gen_inductive_cases prep_att prep_prop args lthy =
```
```   451   let
```
```   452     val thy = ProofContext.theory_of lthy;
```
```   453     val facts = args |> map (fn ((a, atts), props) =>
```
```   454       ((a, map (prep_att thy) atts),
```
```   455         map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
```
```   456   in lthy |> LocalTheory.notes Thm.theoremK facts |>> map snd end;
```
```   457
```
```   458 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
```
```   459 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
```
```   460
```
```   461
```
```   462 fun ind_cases src = Method.syntax (Scan.lift (Scan.repeat1 Args.name_source --
```
```   463     Scan.optional (Args.\$\$\$ "for" |-- Scan.repeat1 Args.name) [])) src
```
```   464   #> (fn ((raw_props, fixes), ctxt) =>
```
```   465     let
```
```   466       val (_, ctxt') = Variable.add_fixes fixes ctxt;
```
```   467       val props = Syntax.read_props ctxt' raw_props;
```
```   468       val ctxt'' = fold Variable.declare_term props ctxt';
```
```   469       val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props)
```
```   470     in Method.erule 0 rules end);
```
```   471
```
```   472
```
```   473
```
```   474 (* prove induction rule *)
```
```   475
```
```   476 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
```
```   477     fp_def rec_preds_defs ctxt =
```
```   478   let
```
```   479     val _ = clean_message quiet_mode "  Proving the induction rule ...";
```
```   480     val thy = ProofContext.theory_of ctxt;
```
```   481
```
```   482     (* predicates for induction rule *)
```
```   483
```
```   484     val (pnames, ctxt') = ctxt |>
```
```   485       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
```
```   486       Variable.variant_fixes (mk_names "P" (length cs));
```
```   487     val preds = map Free (pnames ~~
```
```   488       map (fn c => List.drop (binder_types (fastype_of c), length params) --->
```
```   489         HOLogic.boolT) cs);
```
```   490
```
```   491     (* transform an introduction rule into a premise for induction rule *)
```
```   492
```
```   493     fun mk_ind_prem r =
```
```   494       let
```
```   495         fun subst s = (case dest_predicate cs params s of
```
```   496             SOME (_, i, ys, (_, Ts)) =>
```
```   497               let
```
```   498                 val k = length Ts;
```
```   499                 val bs = map Bound (k - 1 downto 0);
```
```   500                 val P = list_comb (List.nth (preds, i),
```
```   501                   map (incr_boundvars k) ys @ bs);
```
```   502                 val Q = list_abs (mk_names "x" k ~~ Ts,
```
```   503                   HOLogic.mk_binop inductive_conj_name
```
```   504                     (list_comb (incr_boundvars k s, bs), P))
```
```   505               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
```
```   506           | NONE => (case s of
```
```   507               (t \$ u) => (fst (subst t) \$ fst (subst u), NONE)
```
```   508             | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
```
```   509             | _ => (s, NONE)));
```
```   510
```
```   511         fun mk_prem (s, prems) = (case subst s of
```
```   512               (_, SOME (t, u)) => t :: u :: prems
```
```   513             | (t, _) => t :: prems);
```
```   514
```
```   515         val SOME (_, i, ys, _) = dest_predicate cs params
```
```   516           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
```
```   517
```
```   518       in list_all_free (Logic.strip_params r,
```
```   519         Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
```
```   520           [] (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r))),
```
```   521             HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
```
```   522       end;
```
```   523
```
```   524     val ind_prems = map mk_ind_prem intr_ts;
```
```   525
```
```   526
```
```   527     (* make conclusions for induction rules *)
```
```   528
```
```   529     val Tss = map (binder_types o fastype_of) preds;
```
```   530     val (xnames, ctxt'') =
```
```   531       Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
```
```   532     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
```
```   533         (map (fn (((xnames, Ts), c), P) =>
```
```   534            let val frees = map Free (xnames ~~ Ts)
```
```   535            in HOLogic.mk_imp
```
```   536              (list_comb (c, params @ frees), list_comb (P, frees))
```
```   537            end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
```
```   538
```
```   539
```
```   540     (* make predicate for instantiation of abstract induction rule *)
```
```   541
```
```   542     val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
```
```   543       (map_index (fn (i, P) => foldr HOLogic.mk_imp
```
```   544          (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))
```
```   545          (make_bool_args HOLogic.mk_not I bs i)) preds));
```
```   546
```
```   547     val ind_concl = HOLogic.mk_Trueprop
```
```   548       (HOLogic.mk_binrel "HOL.ord_class.less_eq" (rec_const, ind_pred));
```
```   549
```
```   550     val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));
```
```   551
```
```   552     val induct = SkipProof.prove ctxt'' [] ind_prems ind_concl
```
```   553       (fn {prems, ...} => EVERY
```
```   554         [rewrite_goals_tac [inductive_conj_def],
```
```   555          DETERM (rtac raw_fp_induct 1),
```
```   556          REPEAT (resolve_tac [le_funI, le_boolI] 1),
```
```   557          rewrite_goals_tac (inf_fun_eq :: inf_bool_eq :: simp_thms'),
```
```   558          (*This disjE separates out the introduction rules*)
```
```   559          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
```
```   560          (*Now break down the individual cases.  No disjE here in case
```
```   561            some premise involves disjunction.*)
```
```   562          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
```
```   563          REPEAT (FIRSTGOAL
```
```   564            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
```
```   565          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
```
```   566              (inductive_conj_def :: rec_preds_defs @ simp_thms') prem,
```
```   567            conjI, refl] 1)) prems)]);
```
```   568
```
```   569     val lemma = SkipProof.prove ctxt'' [] []
```
```   570       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
```
```   571         [rewrite_goals_tac rec_preds_defs,
```
```   572          REPEAT (EVERY
```
```   573            [REPEAT (resolve_tac [conjI, impI] 1),
```
```   574             REPEAT (eresolve_tac [le_funE, le_boolE] 1),
```
```   575             atac 1,
```
```   576             rewrite_goals_tac simp_thms',
```
```   577             atac 1])])
```
```   578
```
```   579   in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end;
```
```   580
```
```   581
```
```   582
```
```   583 (** specification of (co)inductive predicates **)
```
```   584
```
```   585 fun mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts monos params cnames_syn ctxt =
```
```   586   let
```
```   587     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
```
```   588
```
```   589     val argTs = fold (fn c => fn Ts => Ts @
```
```   590       (List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs [];
```
```   591     val k = log 2 1 (length cs);
```
```   592     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
```
```   593     val p :: xs = map Free (Variable.variant_frees ctxt intr_ts
```
```   594       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
```
```   595     val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts)
```
```   596       (map (rpair HOLogic.boolT) (mk_names "b" k)));
```
```   597
```
```   598     fun subst t = (case dest_predicate cs params t of
```
```   599         SOME (_, i, ts, (Ts, Us)) =>
```
```   600           let
```
```   601             val l = length Us;
```
```   602             val zs = map Bound (l - 1 downto 0)
```
```   603           in
```
```   604             list_abs (map (pair "z") Us, list_comb (p,
```
```   605               make_bool_args' bs i @ make_args argTs
```
```   606                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
```
```   607           end
```
```   608       | NONE => (case t of
```
```   609           t1 \$ t2 => subst t1 \$ subst t2
```
```   610         | Abs (x, T, u) => Abs (x, T, subst u)
```
```   611         | _ => t));
```
```   612
```
```   613     (* transform an introduction rule into a conjunction  *)
```
```   614     (*   [| p_i t; ... |] ==> p_j u                       *)
```
```   615     (* is transformed into                                *)
```
```   616     (*   b_j & x_j = u & p b_j t & ...                    *)
```
```   617
```
```   618     fun transform_rule r =
```
```   619       let
```
```   620         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
```
```   621           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
```
```   622         val ps = make_bool_args HOLogic.mk_not I bs i @
```
```   623           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
```
```   624           map (subst o HOLogic.dest_Trueprop)
```
```   625             (Logic.strip_assums_hyp r)
```
```   626       in foldr (fn ((x, T), P) => HOLogic.exists_const T \$ (Abs (x, T, P)))
```
```   627         (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
```
```   628         (Logic.strip_params r)
```
```   629       end
```
```   630
```
```   631     (* make a disjunction of all introduction rules *)
```
```   632
```
```   633     val fp_fun = fold_rev lambda (p :: bs @ xs)
```
```   634       (if null intr_ts then HOLogic.false_const
```
```   635        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
```
```   636
```
```   637     (* add definiton of recursive predicates to theory *)
```
```   638
```
```   639     val rec_name =
```
```   640       if Binding.is_empty alt_name then
```
```   641         Binding.name (space_implode "_" (map (Binding.base_name o fst) cnames_syn))
```
```   642       else alt_name;
```
```   643
```
```   644     val ((rec_const, (_, fp_def)), ctxt') = ctxt |>
```
```   645       LocalTheory.define Thm.internalK
```
```   646         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
```
```   647          (Attrib.empty_binding, fold_rev lambda params
```
```   648            (Const (fp_name, (predT --> predT) --> predT) \$ fp_fun)));
```
```   649     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
```
```   650       (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params)));
```
```   651     val specs = if length cs < 2 then [] else
```
```   652       map_index (fn (i, (name_mx, c)) =>
```
```   653         let
```
```   654           val Ts = List.drop (binder_types (fastype_of c), length params);
```
```   655           val xs = map Free (Variable.variant_frees ctxt intr_ts
```
```   656             (mk_names "x" (length Ts) ~~ Ts))
```
```   657         in
```
```   658           (name_mx, (Attrib.empty_binding, fold_rev lambda (params @ xs)
```
```   659             (list_comb (rec_const, params @ make_bool_args' bs i @
```
```   660               make_args argTs (xs ~~ Ts)))))
```
```   661         end) (cnames_syn ~~ cs);
```
```   662     val (consts_defs, ctxt'') = fold_map (LocalTheory.define Thm.internalK) specs ctxt';
```
```   663     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
```
```   664
```
```   665     val mono = prove_mono quiet_mode skip_mono predT fp_fun monos ctxt''
```
```   666
```
```   667   in (ctxt'', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
```
```   668     list_comb (rec_const, params), preds, argTs, bs, xs)
```
```   669   end;
```
```   670
```
```   671 fun declare_rules kind rec_binding coind no_ind cnames intrs intr_bindings intr_atts
```
```   672       elims raw_induct ctxt =
```
```   673   let
```
```   674     val rec_name = Binding.base_name rec_binding;
```
```   675     val rec_qualified = Binding.qualify rec_name;
```
```   676     val intr_names = map Binding.base_name intr_bindings;
```
```   677     val ind_case_names = RuleCases.case_names intr_names;
```
```   678     val induct =
```
```   679       if coind then
```
```   680         (raw_induct, [RuleCases.case_names [rec_name],
```
```   681           RuleCases.case_conclusion (rec_name, intr_names),
```
```   682           RuleCases.consumes 1, Induct.coinduct_pred (hd cnames)])
```
```   683       else if no_ind orelse length cnames > 1 then
```
```   684         (raw_induct, [ind_case_names, RuleCases.consumes 0])
```
```   685       else (raw_induct RSN (2, rev_mp), [ind_case_names, RuleCases.consumes 1]);
```
```   686
```
```   687     val (intrs', ctxt1) =
```
```   688       ctxt |>
```
```   689       LocalTheory.notes kind
```
```   690         (map rec_qualified intr_bindings ~~ intr_atts ~~ map (fn th => [([th],
```
```   691            [Attrib.internal (K (ContextRules.intro_query NONE))])]) intrs) |>>
```
```   692       map (hd o snd);
```
```   693     val (((_, elims'), (_, [induct'])), ctxt2) =
```
```   694       ctxt1 |>
```
```   695       LocalTheory.note kind ((rec_qualified (Binding.name "intros"), []), intrs') ||>>
```
```   696       fold_map (fn (name, (elim, cases)) =>
```
```   697         LocalTheory.note kind ((Binding.name (NameSpace.qualified (Sign.base_name name) "cases"),
```
```   698           [Attrib.internal (K (RuleCases.case_names cases)),
```
```   699            Attrib.internal (K (RuleCases.consumes 1)),
```
```   700            Attrib.internal (K (Induct.cases_pred name)),
```
```   701            Attrib.internal (K (ContextRules.elim_query NONE))]), [elim]) #>
```
```   702         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
```
```   703       LocalTheory.note kind
```
```   704         ((rec_qualified (Binding.name (coind_prefix coind ^ "induct")),
```
```   705           map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
```
```   706
```
```   707     val ctxt3 = if no_ind orelse coind then ctxt2 else
```
```   708       let val inducts = cnames ~~ ProjectRule.projects ctxt2 (1 upto length cnames) induct'
```
```   709       in
```
```   710         ctxt2 |>
```
```   711         LocalTheory.notes kind [((rec_qualified (Binding.name "inducts"), []),
```
```   712           inducts |> map (fn (name, th) => ([th],
```
```   713             [Attrib.internal (K ind_case_names),
```
```   714              Attrib.internal (K (RuleCases.consumes 1)),
```
```   715              Attrib.internal (K (Induct.induct_pred name))])))] |> snd
```
```   716       end
```
```   717   in (intrs', elims', induct', ctxt3) end;
```
```   718
```
```   719 type inductive_flags =
```
```   720   {quiet_mode: bool, verbose: bool, kind: string, alt_name: Binding.T,
```
```   721    coind: bool, no_elim: bool, no_ind: bool, skip_mono: bool}
```
```   722
```
```   723 type add_ind_def =
```
```   724   inductive_flags ->
```
```   725   term list -> (Attrib.binding * term) list -> thm list ->
```
```   726   term list -> (Binding.T * mixfix) list ->
```
```   727   local_theory -> inductive_result * local_theory
```
```   728
```
```   729 fun add_ind_def {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono}
```
```   730     cs intros monos params cnames_syn ctxt =
```
```   731   let
```
```   732     val _ = null cnames_syn andalso error "No inductive predicates given";
```
```   733     val names = map (Binding.base_name o fst) cnames_syn;
```
```   734     val _ = message (quiet_mode andalso not verbose)
```
```   735       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
```
```   736
```
```   737     val cnames = map (Sign.full_name (ProofContext.theory_of ctxt) o #1) cnames_syn;  (* FIXME *)
```
```   738     val ((intr_names, intr_atts), intr_ts) =
```
```   739       apfst split_list (split_list (map (check_rule ctxt cs params) intros));
```
```   740
```
```   741     val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
```
```   742       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts
```
```   743         monos params cnames_syn ctxt;
```
```   744
```
```   745     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
```
```   746       params intr_ts rec_preds_defs ctxt1;
```
```   747     val elims = if no_elim then [] else
```
```   748       prove_elims quiet_mode cs params intr_ts (map Binding.base_name intr_names)
```
```   749         unfold rec_preds_defs ctxt1;
```
```   750     val raw_induct = zero_var_indexes
```
```   751       (if no_ind then Drule.asm_rl else
```
```   752        if coind then
```
```   753          singleton (ProofContext.export
```
```   754            (snd (Variable.add_fixes (map (fst o dest_Free) params) ctxt1)) ctxt1)
```
```   755            (rotate_prems ~1 (ObjectLogic.rulify
```
```   756              (fold_rule rec_preds_defs
```
```   757                (rewrite_rule [le_fun_def, le_bool_def, sup_fun_eq, sup_bool_eq]
```
```   758                 (mono RS (fp_def RS def_coinduct))))))
```
```   759        else
```
```   760          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
```
```   761            rec_preds_defs ctxt1);
```
```   762
```
```   763     val (intrs', elims', induct, ctxt2) = declare_rules kind rec_name coind no_ind
```
```   764       cnames intrs intr_names intr_atts elims raw_induct ctxt1;
```
```   765
```
```   766     val result =
```
```   767       {preds = preds,
```
```   768        intrs = intrs',
```
```   769        elims = elims',
```
```   770        raw_induct = rulify raw_induct,
```
```   771        induct = induct};
```
```   772
```
```   773     val ctxt3 = ctxt2
```
```   774       |> LocalTheory.declaration (fn phi =>
```
```   775         let val result' = morph_result phi result;
```
```   776         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
```
```   777   in (result, ctxt3) end;
```
```   778
```
```   779
```
```   780 (* external interfaces *)
```
```   781
```
```   782 fun gen_add_inductive_i mk_def
```
```   783     (flags as {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono})
```
```   784     cnames_syn pnames spec monos lthy =
```
```   785   let
```
```   786     val thy = ProofContext.theory_of lthy;
```
```   787     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
```
```   788
```
```   789
```
```   790     (* abbrevs *)
```
```   791
```
```   792     val (_, ctxt1) = Variable.add_fixes (map (Binding.base_name o fst o fst) cnames_syn) lthy;
```
```   793
```
```   794     fun get_abbrev ((name, atts), t) =
```
```   795       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
```
```   796         let
```
```   797           val _ = Binding.is_empty name andalso null atts orelse
```
```   798             error "Abbreviations may not have names or attributes";
```
```   799           val ((x, T), rhs) = LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt1 t));
```
```   800           val var =
```
```   801             (case find_first (fn ((c, _), _) => Binding.base_name c = x) cnames_syn of
```
```   802               NONE => error ("Undeclared head of abbreviation " ^ quote x)
```
```   803             | SOME ((b, T'), mx) =>
```
```   804                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
```
```   805                 else (b, mx));
```
```   806         in SOME (var, rhs) end
```
```   807       else NONE;
```
```   808
```
```   809     val abbrevs = map_filter get_abbrev spec;
```
```   810     val bs = map (Binding.base_name o fst o fst) abbrevs;
```
```   811
```
```   812
```
```   813     (* predicates *)
```
```   814
```
```   815     val pre_intros = filter_out (is_some o get_abbrev) spec;
```
```   816     val cnames_syn' = filter_out (member (op =) bs o Binding.base_name o fst o fst) cnames_syn;
```
```   817     val cs = map (Free o apfst Binding.base_name o fst) cnames_syn';
```
```   818     val ps = map Free pnames;
```
```   819
```
```   820     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.base_name o fst o fst) cnames_syn');
```
```   821     val _ = map (fn abbr => LocalDefs.fixed_abbrev abbr ctxt2) abbrevs;
```
```   822     val ctxt3 = ctxt2 |> fold (snd oo LocalDefs.fixed_abbrev) abbrevs;
```
```   823     val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy;
```
```   824
```
```   825     fun close_rule r = list_all_free (rev (fold_aterms
```
```   826       (fn t as Free (v as (s, _)) =>
```
```   827           if Variable.is_fixed ctxt1 s orelse
```
```   828             member (op =) ps t then I else insert (op =) v
```
```   829         | _ => I) r []), r);
```
```   830
```
```   831     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
```
```   832     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
```
```   833   in
```
```   834     lthy
```
```   835     |> mk_def flags cs intros monos ps preds
```
```   836     ||> fold (snd oo LocalTheory.abbrev Syntax.mode_default) abbrevs
```
```   837   end;
```
```   838
```
```   839 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
```
```   840   let
```
```   841     val ((vars, specs), _) = lthy |> ProofContext.set_mode ProofContext.mode_abbrev
```
```   842       |> Specification.read_specification
```
```   843           (cnames_syn @ pnames_syn) (map (fn (a, s) => [(a, [s])]) intro_srcs);
```
```   844     val (cs, ps) = chop (length cnames_syn) vars;
```
```   845     val intrs = map (apsnd the_single) specs;
```
```   846     val monos = Attrib.eval_thms lthy raw_monos;
```
```   847     val flags = {quiet_mode = false, verbose = verbose, kind = Thm.theoremK,
```
```   848       alt_name = Binding.empty, coind = coind, no_elim = false, no_ind = false, skip_mono = false};
```
```   849   in
```
```   850     lthy
```
```   851     |> LocalTheory.set_group (serial_string ())
```
```   852     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.base_name o fst) ps) intrs monos
```
```   853   end;
```
```   854
```
```   855 val add_inductive_i = gen_add_inductive_i add_ind_def;
```
```   856 val add_inductive = gen_add_inductive add_ind_def;
```
```   857
```
```   858 fun add_inductive_global group flags cnames_syn pnames pre_intros monos thy =
```
```   859   let
```
```   860     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
```
```   861     val ctxt' = thy
```
```   862       |> TheoryTarget.init NONE
```
```   863       |> LocalTheory.set_group group
```
```   864       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
```
```   865       |> LocalTheory.exit;
```
```   866     val info = #2 (the_inductive ctxt' name);
```
```   867   in (info, ProofContext.theory_of ctxt') end;
```
```   868
```
```   869
```
```   870 (* read off arities of inductive predicates from raw induction rule *)
```
```   871 fun arities_of induct =
```
```   872   map (fn (_ \$ t \$ u) =>
```
```   873       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
```
```   874     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
```
```   875
```
```   876 (* read off parameters of inductive predicate from raw induction rule *)
```
```   877 fun params_of induct =
```
```   878   let
```
```   879     val (_ \$ t \$ u :: _) =
```
```   880       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
```
```   881     val (_, ts) = strip_comb t;
```
```   882     val (_, us) = strip_comb u
```
```   883   in
```
```   884     List.take (ts, length ts - length us)
```
```   885   end;
```
```   886
```
```   887 val pname_of_intr =
```
```   888   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
```
```   889
```
```   890 (* partition introduction rules according to predicate name *)
```
```   891 fun gen_partition_rules f induct intros =
```
```   892   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
```
```   893     (map (rpair [] o fst) (arities_of induct));
```
```   894
```
```   895 val partition_rules = gen_partition_rules I;
```
```   896 fun partition_rules' induct = gen_partition_rules fst induct;
```
```   897
```
```   898 fun unpartition_rules intros xs =
```
```   899   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
```
```   900     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
```
```   901
```
```   902 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
```
```   903 fun infer_intro_vars elim arity intros =
```
```   904   let
```
```   905     val thy = theory_of_thm elim;
```
```   906     val _ :: cases = prems_of elim;
```
```   907     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
```
```   908     fun mtch (t, u) =
```
```   909       let
```
```   910         val params = Logic.strip_params t;
```
```   911         val vars = map (Var o apfst (rpair 0))
```
```   912           (Name.variant_list used (map fst params) ~~ map snd params);
```
```   913         val ts = map (curry subst_bounds (rev vars))
```
```   914           (List.drop (Logic.strip_assums_hyp t, arity));
```
```   915         val us = Logic.strip_imp_prems u;
```
```   916         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
```
```   917           (Vartab.empty, Vartab.empty);
```
```   918       in
```
```   919         map (Envir.subst_vars tab) vars
```
```   920       end
```
```   921   in
```
```   922     map (mtch o apsnd prop_of) (cases ~~ intros)
```
```   923   end;
```
```   924
```
```   925
```
```   926
```
```   927 (** package setup **)
```
```   928
```
```   929 (* setup theory *)
```
```   930
```
```   931 val setup =
```
```   932   Method.add_methods [("ind_cases", ind_cases,
```
```   933     "dynamic case analysis on predicates")] #>
```
```   934   Attrib.add_attributes [("mono", Attrib.add_del_args mono_add mono_del,
```
```   935     "declaration of monotonicity rule")];
```
```   936
```
```   937
```
```   938 (* outer syntax *)
```
```   939
```
```   940 local structure P = OuterParse and K = OuterKeyword in
```
```   941
```
```   942 val _ = OuterKeyword.keyword "monos";
```
```   943
```
```   944 (* FIXME eliminate *)
```
```   945 fun flatten_specification specs = specs |> maps
```
```   946   (fn (a, (concl, [])) => concl |> map
```
```   947         (fn ((b, atts), [B]) =>
```
```   948               if Binding.is_empty a then ((b, atts), B)
```
```   949               else if Binding.is_empty b then ((a, atts), B)
```
```   950               else error "Illegal nested case names"
```
```   951           | ((b, _), _) => error "Illegal simultaneous specification")
```
```   952     | (a, _) => error ("Illegal local specification parameters for " ^ quote (Binding.base_name a)));
```
```   953
```
```   954 fun gen_ind_decl mk_def coind =
```
```   955   P.fixes -- P.for_fixes --
```
```   956   Scan.optional (P.\$\$\$ "where" |-- P.!!! SpecParse.specification) [] --
```
```   957   Scan.optional (P.\$\$\$ "monos" |-- P.!!! SpecParse.xthms1) []
```
```   958   >> (fn (((preds, params), specs), monos) =>
```
```   959       (snd o gen_add_inductive mk_def true coind preds params (flatten_specification specs) monos));
```
```   960
```
```   961 val ind_decl = gen_ind_decl add_ind_def;
```
```   962
```
```   963 val _ = OuterSyntax.local_theory "inductive" "define inductive predicates" K.thy_decl (ind_decl false);
```
```   964 val _ = OuterSyntax.local_theory "coinductive" "define coinductive predicates" K.thy_decl (ind_decl true);
```
```   965
```
```   966 val _ =
```
```   967   OuterSyntax.local_theory "inductive_cases"
```
```   968     "create simplified instances of elimination rules (improper)" K.thy_script
```
```   969     (P.and_list1 SpecParse.spec >> (fn specs => snd o inductive_cases specs));
```
```   970
```
```   971 end;
```
```   972
```
```   973 end;
```