src/HOL/Tools/inductive_package.ML
 author wenzelm Tue Oct 09 00:20:13 2007 +0200 (2007-10-09) changeset 24920 2a45e400fdad parent 24915 fc90277c0dd7 child 24925 f38dd8d0a30d permissions -rw-r--r--
generic Syntax.pretty/string_of operations;
```     1 (*  Title:      HOL/Tools/inductive_package.ML
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
```
```     5
```
```     6 (Co)Inductive Definition module for HOL.
```
```     7
```
```     8 Features:
```
```     9   * least or greatest fixedpoints
```
```    10   * mutually recursive definitions
```
```    11   * definitions involving arbitrary monotone operators
```
```    12   * automatically proves introduction and elimination rules
```
```    13
```
```    14   Introduction rules have the form
```
```    15   [| M Pj ti, ..., Q x, ... |] ==> Pk t
```
```    16   where M is some monotone operator (usually the identity)
```
```    17   Q x is any side condition on the free variables
```
```    18   ti, t are any terms
```
```    19   Pj, Pk are two of the predicates being defined in mutual recursion
```
```    20 *)
```
```    21
```
```    22 signature BASIC_INDUCTIVE_PACKAGE =
```
```    23 sig
```
```    24   val quiet_mode: bool ref
```
```    25   type inductive_result
```
```    26   val morph_result: morphism -> inductive_result -> inductive_result
```
```    27   type inductive_info
```
```    28   val the_inductive: Proof.context -> string -> inductive_info
```
```    29   val print_inductives: Proof.context -> unit
```
```    30   val mono_add: attribute
```
```    31   val mono_del: attribute
```
```    32   val get_monos: Proof.context -> thm list
```
```    33   val mk_cases: Proof.context -> term -> thm
```
```    34   val inductive_forall_name: string
```
```    35   val inductive_forall_def: thm
```
```    36   val rulify: thm -> thm
```
```    37   val inductive_cases: ((bstring * Attrib.src list) * string list) list ->
```
```    38     Proof.context -> thm list list * local_theory
```
```    39   val inductive_cases_i: ((bstring * Attrib.src list) * term list) list ->
```
```    40     Proof.context -> thm list list * local_theory
```
```    41   val add_inductive_i:
```
```    42     {verbose: bool, kind: string, alt_name: bstring, coind: bool, no_elim: bool, no_ind: bool} ->
```
```    43     ((string * typ) * mixfix) list ->
```
```    44     (string * typ) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
```
```    45       local_theory -> inductive_result * local_theory
```
```    46   val add_inductive: bool -> bool -> (string * string option * mixfix) list ->
```
```    47     (string * string option * mixfix) list ->
```
```    48     ((bstring * Attrib.src list) * string) list -> (thmref * Attrib.src list) list ->
```
```    49     local_theory -> inductive_result * local_theory
```
```    50   val add_inductive_global:
```
```    51     {verbose: bool, kind: string, alt_name: bstring, coind: bool, no_elim: bool, no_ind: bool} ->
```
```    52     ((string * typ) * mixfix) list -> (string * typ) list ->
```
```    53     ((bstring * Attrib.src list) * term) list -> thm list -> theory -> inductive_result * theory
```
```    54   val arities_of: thm -> (string * int) list
```
```    55   val params_of: thm -> term list
```
```    56   val partition_rules: thm -> thm list -> (string * thm 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 -> bstring -> bool -> bool -> string list ->
```
```    67     thm list -> bstring 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 ->
```
```    71     {verbose: bool, kind: string, alt_name: bstring, coind: bool, no_elim: bool, no_ind: bool} ->
```
```    72     ((string * typ) * mixfix) list ->
```
```    73     (string * typ) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
```
```    74       local_theory -> inductive_result * local_theory
```
```    75   val gen_add_inductive: add_ind_def ->
```
```    76     bool -> bool -> (string * string option * mixfix) list ->
```
```    77     (string * string option * mixfix) list ->
```
```    78     ((bstring * Attrib.src list) * string) list -> (thmref * Attrib.src list) list ->
```
```    79     local_theory -> inductive_result * local_theory
```
```    80   val gen_ind_decl: add_ind_def ->
```
```    81     bool -> OuterParse.token list ->
```
```    82     (Toplevel.transition -> Toplevel.transition) * OuterParse.token list
```
```    83 end;
```
```    84
```
```    85 structure InductivePackage: INDUCTIVE_PACKAGE =
```
```    86 struct
```
```    87
```
```    88
```
```    89 (** theory context references **)
```
```    90
```
```    91 val inductive_forall_name = "HOL.induct_forall";
```
```    92 val inductive_forall_def = thm "induct_forall_def";
```
```    93 val inductive_conj_name = "HOL.induct_conj";
```
```    94 val inductive_conj_def = thm "induct_conj_def";
```
```    95 val inductive_conj = thms "induct_conj";
```
```    96 val inductive_atomize = thms "induct_atomize";
```
```    97 val inductive_rulify = thms "induct_rulify";
```
```    98 val inductive_rulify_fallback = thms "induct_rulify_fallback";
```
```    99
```
```   100 val notTrueE = TrueI RSN (2, notE);
```
```   101 val notFalseI = Seq.hd (atac 1 notI);
```
```   102 val simp_thms' = map (fn s => mk_meta_eq (the (find_first
```
```   103   (equal (Sign.read_prop HOL.thy s) o prop_of) simp_thms)))
```
```   104   ["(~True) = False", "(~False) = True",
```
```   105    "(True --> ?P) = ?P", "(False --> ?P) = True",
```
```   106    "(?P & True) = ?P", "(True & ?P) = ?P"];
```
```   107
```
```   108
```
```   109
```
```   110 (** context data **)
```
```   111
```
```   112 type inductive_result =
```
```   113   {preds: term list, elims: thm list, raw_induct: thm,
```
```   114    induct: thm, intrs: thm list};
```
```   115
```
```   116 fun morph_result phi {preds, elims, raw_induct: thm, induct, intrs} =
```
```   117   let
```
```   118     val term = Morphism.term phi;
```
```   119     val thm = Morphism.thm phi;
```
```   120     val fact = Morphism.fact phi;
```
```   121   in
```
```   122    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
```
```   123     induct = thm induct, intrs = fact intrs}
```
```   124   end;
```
```   125
```
```   126 type inductive_info =
```
```   127   {names: string list, coind: bool} * inductive_result;
```
```   128
```
```   129 structure InductiveData = GenericDataFun
```
```   130 (
```
```   131   type T = inductive_info Symtab.table * thm list;
```
```   132   val empty = (Symtab.empty, []);
```
```   133   val extend = I;
```
```   134   fun merge _ ((tab1, monos1), (tab2, monos2)) =
```
```   135     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
```
```   136 );
```
```   137
```
```   138 val get_inductives = InductiveData.get o Context.Proof;
```
```   139
```
```   140 fun print_inductives ctxt =
```
```   141   let
```
```   142     val (tab, monos) = get_inductives ctxt;
```
```   143     val space = Consts.space_of (ProofContext.consts_of ctxt);
```
```   144   in
```
```   145     [Pretty.strs ("(co)inductives:" :: map #1 (NameSpace.extern_table (space, tab))),
```
```   146      Pretty.big_list "monotonicity rules:" (map (ProofContext.pretty_thm ctxt) monos)]
```
```   147     |> Pretty.chunks |> Pretty.writeln
```
```   148   end;
```
```   149
```
```   150
```
```   151 (* get and put data *)
```
```   152
```
```   153 fun the_inductive ctxt name =
```
```   154   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
```
```   155     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
```
```   156   | SOME info => info);
```
```   157
```
```   158 fun put_inductives names info = InductiveData.map (apfst (fn tab =>
```
```   159   fold (fn name => Symtab.update_new (name, info)) names tab
```
```   160     handle Symtab.DUP d => error ("Duplicate definition of (co)inductive predicate " ^ quote d)));
```
```   161
```
```   162
```
```   163
```
```   164 (** monotonicity rules **)
```
```   165
```
```   166 val get_monos = #2 o get_inductives;
```
```   167 val map_monos = InductiveData.map o apsnd;
```
```   168
```
```   169 fun mk_mono thm =
```
```   170   let
```
```   171     val concl = concl_of thm;
```
```   172     fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @
```
```   173       (case concl of
```
```   174           (_ \$ (_ \$ (Const ("Not", _) \$ _) \$ _)) => []
```
```   175         | _ => [thm' RS (thm' RS eq_to_mono2)]);
```
```   176     fun dest_less_concl thm = dest_less_concl (thm RS le_funD)
```
```   177       handle THM _ => thm RS le_boolD
```
```   178   in
```
```   179     case concl of
```
```   180       Const ("==", _) \$ _ \$ _ => eq2mono (thm RS meta_eq_to_obj_eq)
```
```   181     | _ \$ (Const ("op =", _) \$ _ \$ _) => eq2mono thm
```
```   182     | _ \$ (Const ("HOL.ord_class.less_eq", _) \$ _ \$ _) =>
```
```   183       [dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
```
```   184          (resolve_tac [le_funI, le_boolI'])) thm))]
```
```   185     | _ => [thm]
```
```   186   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ string_of_thm thm);
```
```   187
```
```   188 val mono_add = Thm.declaration_attribute (map_monos o fold Thm.add_thm o mk_mono);
```
```   189 val mono_del = Thm.declaration_attribute (map_monos o fold Thm.del_thm o mk_mono);
```
```   190
```
```   191
```
```   192
```
```   193 (** misc utilities **)
```
```   194
```
```   195 val quiet_mode = ref false;
```
```   196 fun message s = if ! quiet_mode then () else writeln s;
```
```   197 fun clean_message s = if ! quick_and_dirty then () else message s;
```
```   198
```
```   199 fun coind_prefix true = "co"
```
```   200   | coind_prefix false = "";
```
```   201
```
```   202 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
```
```   203
```
```   204 fun make_bool_args f g [] i = []
```
```   205   | make_bool_args f g (x :: xs) i =
```
```   206       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
```
```   207
```
```   208 fun make_bool_args' xs =
```
```   209   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
```
```   210
```
```   211 fun find_arg T x [] = sys_error "find_arg"
```
```   212   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
```
```   213       apsnd (cons p) (find_arg T x ps)
```
```   214   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
```
```   215       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
```
```   216       else apsnd (cons p) (find_arg T x ps);
```
```   217
```
```   218 fun make_args Ts xs =
```
```   219   map (fn (T, (NONE, ())) => Const ("arbitrary", T) | (_, (SOME t, ())) => t)
```
```   220     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
```
```   221
```
```   222 fun make_args' Ts xs Us =
```
```   223   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
```
```   224
```
```   225 fun dest_predicate cs params t =
```
```   226   let
```
```   227     val k = length params;
```
```   228     val (c, ts) = strip_comb t;
```
```   229     val (xs, ys) = chop k ts;
```
```   230     val i = find_index_eq c cs;
```
```   231   in
```
```   232     if xs = params andalso i >= 0 then
```
```   233       SOME (c, i, ys, chop (length ys)
```
```   234         (List.drop (binder_types (fastype_of c), k)))
```
```   235     else NONE
```
```   236   end;
```
```   237
```
```   238 fun mk_names a 0 = []
```
```   239   | mk_names a 1 = [a]
```
```   240   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
```
```   241
```
```   242
```
```   243
```
```   244 (** process rules **)
```
```   245
```
```   246 local
```
```   247
```
```   248 fun err_in_rule ctxt name t msg =
```
```   249   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
```
```   250     Syntax.string_of_term ctxt t, msg]);
```
```   251
```
```   252 fun err_in_prem ctxt name t p msg =
```
```   253   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
```
```   254     "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
```
```   255
```
```   256 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
```
```   257
```
```   258 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
```
```   259
```
```   260 val bad_app = "Inductive predicate must be applied to parameter(s) ";
```
```   261
```
```   262 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
```
```   263
```
```   264 in
```
```   265
```
```   266 fun check_rule ctxt cs params ((name, att), rule) =
```
```   267   let
```
```   268     val params' = Term.variant_frees rule (Logic.strip_params rule);
```
```   269     val frees = rev (map Free params');
```
```   270     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
```
```   271     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
```
```   272     val rule' = Logic.list_implies (prems, concl);
```
```   273     val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
```
```   274     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
```
```   275
```
```   276     fun check_ind err t = case dest_predicate cs params t of
```
```   277         NONE => err (bad_app ^
```
```   278           commas (map (Syntax.string_of_term ctxt) params))
```
```   279       | SOME (_, _, ys, _) =>
```
```   280           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
```
```   281           then err bad_ind_occ else ();
```
```   282
```
```   283     fun check_prem' prem t =
```
```   284       if head_of t mem cs then
```
```   285         check_ind (err_in_prem ctxt name rule prem) t
```
```   286       else (case t of
```
```   287           Abs (_, _, t) => check_prem' prem t
```
```   288         | t \$ u => (check_prem' prem t; check_prem' prem u)
```
```   289         | _ => ());
```
```   290
```
```   291     fun check_prem (prem, aprem) =
```
```   292       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
```
```   293       else err_in_prem ctxt name rule prem "Non-atomic premise";
```
```   294   in
```
```   295     (case concl of
```
```   296        Const ("Trueprop", _) \$ t =>
```
```   297          if head_of t mem cs then
```
```   298            (check_ind (err_in_rule ctxt name rule') t;
```
```   299             List.app check_prem (prems ~~ aprems))
```
```   300          else err_in_rule ctxt name rule' bad_concl
```
```   301      | _ => err_in_rule ctxt name rule' bad_concl);
```
```   302     ((name, att), arule)
```
```   303   end;
```
```   304
```
```   305 val rulify =
```
```   306   hol_simplify inductive_conj
```
```   307   #> hol_simplify inductive_rulify
```
```   308   #> hol_simplify inductive_rulify_fallback
```
```   309   #> MetaSimplifier.norm_hhf;
```
```   310
```
```   311 end;
```
```   312
```
```   313
```
```   314
```
```   315 (** proofs for (co)inductive predicates **)
```
```   316
```
```   317 (* prove monotonicity -- NOT subject to quick_and_dirty! *)
```
```   318
```
```   319 fun prove_mono predT fp_fun monos ctxt =
```
```   320  (message "  Proving monotonicity ...";
```
```   321   Goal.prove ctxt [] []   (*NO quick_and_dirty here!*)
```
```   322     (HOLogic.mk_Trueprop
```
```   323       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) \$ fp_fun))
```
```   324     (fn _ => EVERY [rtac monoI 1,
```
```   325       REPEAT (resolve_tac [le_funI, le_boolI'] 1),
```
```   326       REPEAT (FIRST
```
```   327         [atac 1,
```
```   328          resolve_tac (List.concat (map mk_mono monos) @ get_monos ctxt) 1,
```
```   329          etac le_funE 1, dtac le_boolD 1])]));
```
```   330
```
```   331
```
```   332 (* prove introduction rules *)
```
```   333
```
```   334 fun prove_intrs coind mono fp_def k params intr_ts rec_preds_defs ctxt =
```
```   335   let
```
```   336     val _ = clean_message "  Proving the introduction rules ...";
```
```   337
```
```   338     val unfold = funpow k (fn th => th RS fun_cong)
```
```   339       (mono RS (fp_def RS
```
```   340         (if coind then def_gfp_unfold else def_lfp_unfold)));
```
```   341
```
```   342     fun select_disj 1 1 = []
```
```   343       | select_disj _ 1 = [rtac disjI1]
```
```   344       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
```
```   345
```
```   346     val rules = [refl, TrueI, notFalseI, exI, conjI];
```
```   347
```
```   348     val intrs = map_index (fn (i, intr) => rulify
```
```   349       (SkipProof.prove ctxt (map (fst o dest_Free) params) [] intr (fn _ => EVERY
```
```   350        [rewrite_goals_tac rec_preds_defs,
```
```   351         rtac (unfold RS iffD2) 1,
```
```   352         EVERY1 (select_disj (length intr_ts) (i + 1)),
```
```   353         (*Not ares_tac, since refl must be tried before any equality assumptions;
```
```   354           backtracking may occur if the premises have extra variables!*)
```
```   355         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts
```
```   356
```
```   357   in (intrs, unfold) end;
```
```   358
```
```   359
```
```   360 (* prove elimination rules *)
```
```   361
```
```   362 fun prove_elims cs params intr_ts intr_names unfold rec_preds_defs ctxt =
```
```   363   let
```
```   364     val _ = clean_message "  Proving the elimination rules ...";
```
```   365
```
```   366     val ([pname], ctxt') = ctxt |>
```
```   367       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
```
```   368       Variable.variant_fixes ["P"];
```
```   369     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
```
```   370
```
```   371     fun dest_intr r =
```
```   372       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
```
```   373        Logic.strip_assums_hyp r, Logic.strip_params r);
```
```   374
```
```   375     val intrs = map dest_intr intr_ts ~~ intr_names;
```
```   376
```
```   377     val rules1 = [disjE, exE, FalseE];
```
```   378     val rules2 = [conjE, FalseE, notTrueE];
```
```   379
```
```   380     fun prove_elim c =
```
```   381       let
```
```   382         val Ts = List.drop (binder_types (fastype_of c), length params);
```
```   383         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
```
```   384         val frees = map Free (anames ~~ Ts);
```
```   385
```
```   386         fun mk_elim_prem ((_, _, us, _), ts, params') =
```
```   387           list_all (params',
```
```   388             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
```
```   389               (frees ~~ us) @ ts, P));
```
```   390         val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs);
```
```   391         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
```
```   392            map mk_elim_prem (map #1 c_intrs)
```
```   393       in
```
```   394         (SkipProof.prove ctxt'' [] prems P
```
```   395           (fn {prems, ...} => EVERY
```
```   396             [cut_facts_tac [hd prems] 1,
```
```   397              rewrite_goals_tac rec_preds_defs,
```
```   398              dtac (unfold RS iffD1) 1,
```
```   399              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
```
```   400              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
```
```   401              EVERY (map (fn prem =>
```
```   402                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
```
```   403           |> rulify
```
```   404           |> singleton (ProofContext.export ctxt'' ctxt),
```
```   405          map #2 c_intrs)
```
```   406       end
```
```   407
```
```   408    in map prove_elim cs end;
```
```   409
```
```   410
```
```   411 (* derivation of simplified elimination rules *)
```
```   412
```
```   413 local
```
```   414
```
```   415 (*delete needless equality assumptions*)
```
```   416 val refl_thin = Goal.prove_global HOL.thy [] []
```
```   417   (Sign.read_prop HOL.thy "!!P. a = a ==> P ==> P")
```
```   418   (fn _ => assume_tac 1);
```
```   419 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
```
```   420 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
```
```   421
```
```   422 fun simp_case_tac ss i =
```
```   423   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
```
```   424
```
```   425 in
```
```   426
```
```   427 fun mk_cases ctxt prop =
```
```   428   let
```
```   429     val thy = ProofContext.theory_of ctxt;
```
```   430     val ss = Simplifier.local_simpset_of ctxt;
```
```   431
```
```   432     fun err msg =
```
```   433       error (Pretty.string_of (Pretty.block
```
```   434         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
```
```   435
```
```   436     val elims = Induct.find_casesP ctxt prop;
```
```   437
```
```   438     val cprop = Thm.cterm_of thy prop;
```
```   439     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
```
```   440     fun mk_elim rl =
```
```   441       Thm.implies_intr cprop (Tactic.rule_by_tactic tac (Thm.assume cprop RS rl))
```
```   442       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
```
```   443   in
```
```   444     (case get_first (try mk_elim) elims of
```
```   445       SOME r => r
```
```   446     | NONE => err "Proposition not an inductive predicate:")
```
```   447   end;
```
```   448
```
```   449 end;
```
```   450
```
```   451
```
```   452 (* inductive_cases *)
```
```   453
```
```   454 fun gen_inductive_cases prep_att prep_prop args lthy =
```
```   455   let
```
```   456     val thy = ProofContext.theory_of lthy;
```
```   457     val facts = args |> map (fn ((a, atts), props) =>
```
```   458       ((a, map (prep_att thy) atts),
```
```   459         map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
```
```   460   in lthy |> LocalTheory.notes Thm.theoremK facts |>> map snd end;
```
```   461
```
```   462 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
```
```   463 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
```
```   464
```
```   465
```
```   466 fun ind_cases src = Method.syntax (Scan.lift (Scan.repeat1 Args.name --
```
```   467     Scan.optional (Args.\$\$\$ "for" |-- Scan.repeat1 Args.name) [])) src
```
```   468   #> (fn ((raw_props, fixes), ctxt) =>
```
```   469     let
```
```   470       val (_, ctxt') = Variable.add_fixes fixes ctxt;
```
```   471       val props = Syntax.read_props ctxt' raw_props;
```
```   472       val ctxt'' = fold Variable.declare_term props ctxt';
```
```   473       val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props)
```
```   474     in Method.erule 0 rules end);
```
```   475
```
```   476
```
```   477
```
```   478 (* prove induction rule *)
```
```   479
```
```   480 fun prove_indrule cs argTs bs xs rec_const params intr_ts mono
```
```   481     fp_def rec_preds_defs ctxt =
```
```   482   let
```
```   483     val _ = clean_message "  Proving the induction rule ...";
```
```   484     val thy = ProofContext.theory_of ctxt;
```
```   485
```
```   486     (* predicates for induction rule *)
```
```   487
```
```   488     val (pnames, ctxt') = ctxt |>
```
```   489       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
```
```   490       Variable.variant_fixes (mk_names "P" (length cs));
```
```   491     val preds = map Free (pnames ~~
```
```   492       map (fn c => List.drop (binder_types (fastype_of c), length params) --->
```
```   493         HOLogic.boolT) cs);
```
```   494
```
```   495     (* transform an introduction rule into a premise for induction rule *)
```
```   496
```
```   497     fun mk_ind_prem r =
```
```   498       let
```
```   499         fun subst s = (case dest_predicate cs params s of
```
```   500             SOME (_, i, ys, (_, Ts)) =>
```
```   501               let
```
```   502                 val k = length Ts;
```
```   503                 val bs = map Bound (k - 1 downto 0);
```
```   504                 val P = list_comb (List.nth (preds, i),
```
```   505                   map (incr_boundvars k) ys @ bs);
```
```   506                 val Q = list_abs (mk_names "x" k ~~ Ts,
```
```   507                   HOLogic.mk_binop inductive_conj_name
```
```   508                     (list_comb (incr_boundvars k s, bs), P))
```
```   509               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
```
```   510           | NONE => (case s of
```
```   511               (t \$ u) => (fst (subst t) \$ fst (subst u), NONE)
```
```   512             | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
```
```   513             | _ => (s, NONE)));
```
```   514
```
```   515         fun mk_prem (s, prems) = (case subst s of
```
```   516               (_, SOME (t, u)) => t :: u :: prems
```
```   517             | (t, _) => t :: prems);
```
```   518
```
```   519         val SOME (_, i, ys, _) = dest_predicate cs params
```
```   520           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
```
```   521
```
```   522       in list_all_free (Logic.strip_params r,
```
```   523         Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
```
```   524           [] (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r))),
```
```   525             HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
```
```   526       end;
```
```   527
```
```   528     val ind_prems = map mk_ind_prem intr_ts;
```
```   529
```
```   530
```
```   531     (* make conclusions for induction rules *)
```
```   532
```
```   533     val Tss = map (binder_types o fastype_of) preds;
```
```   534     val (xnames, ctxt'') =
```
```   535       Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
```
```   536     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
```
```   537         (map (fn (((xnames, Ts), c), P) =>
```
```   538            let val frees = map Free (xnames ~~ Ts)
```
```   539            in HOLogic.mk_imp
```
```   540              (list_comb (c, params @ frees), list_comb (P, frees))
```
```   541            end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
```
```   542
```
```   543
```
```   544     (* make predicate for instantiation of abstract induction rule *)
```
```   545
```
```   546     val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
```
```   547       (map_index (fn (i, P) => foldr HOLogic.mk_imp
```
```   548          (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))
```
```   549          (make_bool_args HOLogic.mk_not I bs i)) preds));
```
```   550
```
```   551     val ind_concl = HOLogic.mk_Trueprop
```
```   552       (HOLogic.mk_binrel "HOL.ord_class.less_eq" (rec_const, ind_pred));
```
```   553
```
```   554     val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));
```
```   555
```
```   556     val induct = SkipProof.prove ctxt'' [] ind_prems ind_concl
```
```   557       (fn {prems, ...} => EVERY
```
```   558         [rewrite_goals_tac [inductive_conj_def],
```
```   559          DETERM (rtac raw_fp_induct 1),
```
```   560          REPEAT (resolve_tac [le_funI, le_boolI] 1),
```
```   561          rewrite_goals_tac (inf_fun_eq :: inf_bool_eq :: simp_thms'),
```
```   562          (*This disjE separates out the introduction rules*)
```
```   563          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
```
```   564          (*Now break down the individual cases.  No disjE here in case
```
```   565            some premise involves disjunction.*)
```
```   566          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
```
```   567          REPEAT (FIRSTGOAL
```
```   568            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
```
```   569          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
```
```   570              (inductive_conj_def :: rec_preds_defs @ simp_thms') prem,
```
```   571            conjI, refl] 1)) prems)]);
```
```   572
```
```   573     val lemma = SkipProof.prove ctxt'' [] []
```
```   574       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
```
```   575         [rewrite_goals_tac rec_preds_defs,
```
```   576          REPEAT (EVERY
```
```   577            [REPEAT (resolve_tac [conjI, impI] 1),
```
```   578             REPEAT (eresolve_tac [le_funE, le_boolE] 1),
```
```   579             atac 1,
```
```   580             rewrite_goals_tac simp_thms',
```
```   581             atac 1])])
```
```   582
```
```   583   in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end;
```
```   584
```
```   585
```
```   586
```
```   587 (** specification of (co)inductive predicates **)
```
```   588
```
```   589 fun mk_ind_def alt_name coind cs intr_ts monos
```
```   590       params cnames_syn ctxt =
```
```   591   let
```
```   592     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
```
```   593
```
```   594     val argTs = fold (fn c => fn Ts => Ts @
```
```   595       (List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs [];
```
```   596     val k = log 2 1 (length cs);
```
```   597     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
```
```   598     val p :: xs = map Free (Variable.variant_frees ctxt intr_ts
```
```   599       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
```
```   600     val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts)
```
```   601       (map (rpair HOLogic.boolT) (mk_names "b" k)));
```
```   602
```
```   603     fun subst t = (case dest_predicate cs params t of
```
```   604         SOME (_, i, ts, (Ts, Us)) =>
```
```   605           let
```
```   606             val l = length Us;
```
```   607             val zs = map Bound (l - 1 downto 0)
```
```   608           in
```
```   609             list_abs (map (pair "z") Us, list_comb (p,
```
```   610               make_bool_args' bs i @ make_args argTs
```
```   611                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
```
```   612           end
```
```   613       | NONE => (case t of
```
```   614           t1 \$ t2 => subst t1 \$ subst t2
```
```   615         | Abs (x, T, u) => Abs (x, T, subst u)
```
```   616         | _ => t));
```
```   617
```
```   618     (* transform an introduction rule into a conjunction  *)
```
```   619     (*   [| p_i t; ... |] ==> p_j u                       *)
```
```   620     (* is transformed into                                *)
```
```   621     (*   b_j & x_j = u & p b_j t & ...                    *)
```
```   622
```
```   623     fun transform_rule r =
```
```   624       let
```
```   625         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
```
```   626           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
```
```   627         val ps = make_bool_args HOLogic.mk_not I bs i @
```
```   628           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
```
```   629           map (subst o HOLogic.dest_Trueprop)
```
```   630             (Logic.strip_assums_hyp r)
```
```   631       in foldr (fn ((x, T), P) => HOLogic.exists_const T \$ (Abs (x, T, P)))
```
```   632         (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
```
```   633         (Logic.strip_params r)
```
```   634       end
```
```   635
```
```   636     (* make a disjunction of all introduction rules *)
```
```   637
```
```   638     val fp_fun = fold_rev lambda (p :: bs @ xs)
```
```   639       (if null intr_ts then HOLogic.false_const
```
```   640        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
```
```   641
```
```   642     (* add definiton of recursive predicates to theory *)
```
```   643
```
```   644     val rec_name = if alt_name = "" then
```
```   645       space_implode "_" (map fst cnames_syn) else alt_name;
```
```   646
```
```   647     val ((rec_const, (_, fp_def)), ctxt') = ctxt |>
```
```   648       LocalTheory.def Thm.internalK
```
```   649         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
```
```   650          (("", []), fold_rev lambda params
```
```   651            (Const (fp_name, (predT --> predT) --> predT) \$ fp_fun)));
```
```   652     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
```
```   653       (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params)));
```
```   654     val specs = if length cs < 2 then [] else
```
```   655       map_index (fn (i, (name_mx, c)) =>
```
```   656         let
```
```   657           val Ts = List.drop (binder_types (fastype_of c), length params);
```
```   658           val xs = map Free (Variable.variant_frees ctxt intr_ts
```
```   659             (mk_names "x" (length Ts) ~~ Ts))
```
```   660         in
```
```   661           (name_mx, (("", []), fold_rev lambda (params @ xs)
```
```   662             (list_comb (rec_const, params @ make_bool_args' bs i @
```
```   663               make_args argTs (xs ~~ Ts)))))
```
```   664         end) (cnames_syn ~~ cs);
```
```   665     val (consts_defs, ctxt'') = LocalTheory.defs Thm.internalK specs ctxt';
```
```   666     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
```
```   667
```
```   668     val mono = prove_mono predT fp_fun monos ctxt''
```
```   669
```
```   670   in (ctxt'', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
```
```   671     list_comb (rec_const, params), preds, argTs, bs, xs)
```
```   672   end;
```
```   673
```
```   674 fun declare_rules kind rec_name coind no_ind cnames intrs intr_names intr_atts
```
```   675       elims raw_induct ctxt =
```
```   676   let
```
```   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 (NameSpace.qualified rec_name) intr_names ~~
```
```   691          intr_atts ~~ map (fn th => [([th],
```
```   692            [Attrib.internal (K (ContextRules.intro_query NONE))])]) intrs) |>>
```
```   693       map (hd o snd);
```
```   694     val (((_, elims'), (_, [induct'])), ctxt2) =
```
```   695       ctxt1 |>
```
```   696       LocalTheory.note kind ((NameSpace.qualified rec_name "intros", []), intrs') ||>>
```
```   697       fold_map (fn (name, (elim, cases)) =>
```
```   698         LocalTheory.note kind ((NameSpace.qualified (Sign.base_name name) "cases",
```
```   699           [Attrib.internal (K (RuleCases.case_names cases)),
```
```   700            Attrib.internal (K (RuleCases.consumes 1)),
```
```   701            Attrib.internal (K (Induct.cases_pred name)),
```
```   702            Attrib.internal (K (ContextRules.elim_query NONE))]), [elim]) #>
```
```   703         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
```
```   704       LocalTheory.note kind ((NameSpace.qualified rec_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 [((NameSpace.qualified rec_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 add_ind_def =
```
```   720   {verbose: bool, kind: string, alt_name: bstring, coind: bool, no_elim: bool, no_ind: bool} ->
```
```   721   term list -> ((string * Attrib.src list) * term) list -> thm list ->
```
```   722   term list -> (string * mixfix) list ->
```
```   723   local_theory -> inductive_result * local_theory
```
```   724
```
```   725 fun add_ind_def {verbose, kind, alt_name, coind, no_elim, no_ind}
```
```   726     cs intros monos params cnames_syn ctxt =
```
```   727   let
```
```   728     val _ =
```
```   729       if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^
```
```   730         commas_quote (map fst cnames_syn)) else ();
```
```   731
```
```   732     val cnames = map (Sign.full_name (ProofContext.theory_of ctxt) o #1) cnames_syn;  (* FIXME *)
```
```   733     val ((intr_names, intr_atts), intr_ts) =
```
```   734       apfst split_list (split_list (map (check_rule ctxt cs params) intros));
```
```   735
```
```   736     val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
```
```   737       argTs, bs, xs) = mk_ind_def alt_name coind cs intr_ts
```
```   738         monos params cnames_syn ctxt;
```
```   739
```
```   740     val (intrs, unfold) = prove_intrs coind mono fp_def (length bs + length xs)
```
```   741       params intr_ts rec_preds_defs ctxt1;
```
```   742     val elims = if no_elim then [] else
```
```   743       prove_elims cs params intr_ts intr_names unfold rec_preds_defs ctxt1;
```
```   744     val raw_induct = zero_var_indexes
```
```   745       (if no_ind then Drule.asm_rl else
```
```   746        if coind then
```
```   747          singleton (ProofContext.export
```
```   748            (snd (Variable.add_fixes (map (fst o dest_Free) params) ctxt1)) ctxt1)
```
```   749            (rotate_prems ~1 (ObjectLogic.rulify (rule_by_tactic
```
```   750              (rewrite_tac [le_fun_def, le_bool_def, @{thm sup_fun_eq}, @{thm sup_bool_eq}] THEN
```
```   751                fold_tac rec_preds_defs) (mono RS (fp_def RS def_coinduct)))))
```
```   752        else
```
```   753          prove_indrule cs argTs bs xs rec_const params intr_ts mono fp_def
```
```   754            rec_preds_defs ctxt1);
```
```   755
```
```   756     val (intrs', elims', induct, ctxt2) = declare_rules kind rec_name coind no_ind
```
```   757       cnames intrs intr_names intr_atts elims raw_induct ctxt1;
```
```   758
```
```   759     val names = map #1 cnames_syn;
```
```   760     val result =
```
```   761       {preds = preds,
```
```   762        intrs = intrs',
```
```   763        elims = elims',
```
```   764        raw_induct = rulify raw_induct,
```
```   765        induct = induct};
```
```   766
```
```   767     val ctxt3 = ctxt2
```
```   768       |> Context.proof_map (put_inductives names ({names = names, coind = coind}, result))
```
```   769       |> LocalTheory.declaration (fn phi =>
```
```   770         let
```
```   771           val names' = map (LocalTheory.target_name ctxt2 o Morphism.name phi) names;
```
```   772           val result' = morph_result phi result;
```
```   773         in put_inductives names' ({names = names', coind = coind}, result') end);
```
```   774   in (result, ctxt3) end;
```
```   775
```
```   776
```
```   777 (* external interfaces *)
```
```   778
```
```   779 fun gen_add_inductive_i mk_def (flags as {verbose, kind, alt_name, coind, no_elim, no_ind})
```
```   780     cnames_syn pnames pre_intros monos ctxt =
```
```   781   let
```
```   782     val thy = ProofContext.theory_of ctxt;
```
```   783     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
```
```   784
```
```   785     fun is_abbrev ((name, atts), t) =
```
```   786       can (Logic.strip_assums_concl #> Logic.dest_equals) t andalso
```
```   787       (name = "" andalso null atts orelse
```
```   788        error "Abbreviations may not have names or attributes");
```
```   789
```
```   790     fun expand_atom tab (t as Free xT) =
```
```   791           the_default t (AList.lookup op = tab xT)
```
```   792       | expand_atom tab t = t;
```
```   793     fun expand [] r = r
```
```   794       | expand tab r = Envir.beta_norm (Term.map_aterms (expand_atom tab) r);
```
```   795
```
```   796     val (_, ctxt') = Variable.add_fixes (map (fst o fst) cnames_syn) ctxt;
```
```   797
```
```   798     fun prep_abbrevs [] abbrevs' abbrevs'' = (rev abbrevs', rev abbrevs'')
```
```   799       | prep_abbrevs ((_, abbrev) :: abbrevs) abbrevs' abbrevs'' =
```
```   800           let val ((s, T), t) =
```
```   801             LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt' abbrev))
```
```   802           in case find_first (equal s o fst o fst) cnames_syn of
```
```   803               NONE => error ("Head of abbreviation " ^ quote s ^ " undeclared")
```
```   804             | SOME (_, mx) => prep_abbrevs abbrevs
```
```   805                 (((s, T), expand abbrevs' t) :: abbrevs')
```
```   806                 (((s, mx), expand abbrevs' t) :: abbrevs'') (* FIXME: do not expand *)
```
```   807           end;
```
```   808
```
```   809     val (abbrevs, pre_intros') = List.partition is_abbrev pre_intros;
```
```   810     val (abbrevs', abbrevs'') = prep_abbrevs abbrevs [] [];
```
```   811     val _ = (case gen_inter (op = o apsnd fst)
```
```   812       (fold (Term.add_frees o snd) abbrevs' [], abbrevs') of
```
```   813         [] => ()
```
```   814       | xs => error ("Bad abbreviation(s): " ^ commas (map fst xs)));
```
```   815
```
```   816     val params = map Free pnames;
```
```   817     val cnames_syn' = filter_out (fn ((s, _), _) =>
```
```   818       exists (equal s o fst o fst) abbrevs') cnames_syn;
```
```   819     val cs = map (Free o fst) cnames_syn';
```
```   820     val cnames_syn'' = map (fn ((s, _), mx) => (s, mx)) cnames_syn';
```
```   821
```
```   822     fun close_rule (x, r) = (x, list_all_free (rev (fold_aterms
```
```   823       (fn t as Free (v as (s, _)) =>
```
```   824             if Variable.is_fixed ctxt' s orelse
```
```   825               member op = params t then I else insert op = v
```
```   826         | _ => I) r []), r));
```
```   827
```
```   828     val intros = map (close_rule ##> expand abbrevs') pre_intros';
```
```   829   in
```
```   830     ctxt
```
```   831     |> mk_def flags cs intros monos params cnames_syn''
```
```   832     ||> fold (snd oo LocalTheory.abbrev Syntax.default_mode) abbrevs''
```
```   833   end;
```
```   834
```
```   835 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
```
```   836   let
```
```   837     val ((vars, specs), _) = lthy |> Specification.read_specification
```
```   838       (cnames_syn @ pnames_syn) (map (fn (a, s) => [(a, [s])]) intro_srcs);
```
```   839     val (cs, ps) = chop (length cnames_syn) vars;
```
```   840     val intrs = map (apsnd the_single) specs;
```
```   841     val monos = Attrib.eval_thms lthy raw_monos;
```
```   842     val flags = {verbose = verbose, kind = Thm.theoremK, alt_name = "",
```
```   843       coind = coind, no_elim = false, no_ind = false};
```
```   844   in gen_add_inductive_i mk_def flags cs (map fst ps) intrs monos lthy end;
```
```   845
```
```   846 val add_inductive_i = gen_add_inductive_i add_ind_def;
```
```   847 val add_inductive = gen_add_inductive add_ind_def;
```
```   848
```
```   849 fun add_inductive_global flags cnames_syn pnames pre_intros monos =
```
```   850   TheoryTarget.init NONE #>
```
```   851   add_inductive_i flags cnames_syn pnames pre_intros monos #>
```
```   852   (fn (_, lthy) =>
```
```   853     (#2 (the_inductive (LocalTheory.target_of lthy)
```
```   854       (LocalTheory.target_name lthy (fst (fst (hd cnames_syn))))),
```
```   855     ProofContext.theory_of (LocalTheory.exit lthy)));
```
```   856
```
```   857
```
```   858 (* read off arities of inductive predicates from raw induction rule *)
```
```   859 fun arities_of induct =
```
```   860   map (fn (_ \$ t \$ u) =>
```
```   861       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
```
```   862     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
```
```   863
```
```   864 (* read off parameters of inductive predicate from raw induction rule *)
```
```   865 fun params_of induct =
```
```   866   let
```
```   867     val (_ \$ t \$ u :: _) =
```
```   868       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
```
```   869     val (_, ts) = strip_comb t;
```
```   870     val (_, us) = strip_comb u
```
```   871   in
```
```   872     List.take (ts, length ts - length us)
```
```   873   end;
```
```   874
```
```   875 val pname_of_intr =
```
```   876   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
```
```   877
```
```   878 (* partition introduction rules according to predicate name *)
```
```   879 fun partition_rules induct intros =
```
```   880   fold_rev (fn r => AList.map_entry op = (pname_of_intr r) (cons r)) intros
```
```   881     (map (rpair [] o fst) (arities_of induct));
```
```   882
```
```   883 fun unpartition_rules intros xs =
```
```   884   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
```
```   885     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
```
```   886
```
```   887 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
```
```   888 fun infer_intro_vars elim arity intros =
```
```   889   let
```
```   890     val thy = theory_of_thm elim;
```
```   891     val _ :: cases = prems_of elim;
```
```   892     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
```
```   893     fun mtch (t, u) =
```
```   894       let
```
```   895         val params = Logic.strip_params t;
```
```   896         val vars = map (Var o apfst (rpair 0))
```
```   897           (Name.variant_list used (map fst params) ~~ map snd params);
```
```   898         val ts = map (curry subst_bounds (rev vars))
```
```   899           (List.drop (Logic.strip_assums_hyp t, arity));
```
```   900         val us = Logic.strip_imp_prems u;
```
```   901         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
```
```   902           (Vartab.empty, Vartab.empty);
```
```   903       in
```
```   904         map (Envir.subst_vars tab) vars
```
```   905       end
```
```   906   in
```
```   907     map (mtch o apsnd prop_of) (cases ~~ intros)
```
```   908   end;
```
```   909
```
```   910
```
```   911 (** package setup **)
```
```   912
```
```   913 (* setup theory *)
```
```   914
```
```   915 val setup =
```
```   916   Method.add_methods [("ind_cases", ind_cases,
```
```   917     "dynamic case analysis on predicates")] #>
```
```   918   Attrib.add_attributes [("mono", Attrib.add_del_args mono_add mono_del,
```
```   919     "declaration of monotonicity rule")];
```
```   920
```
```   921
```
```   922 (* outer syntax *)
```
```   923
```
```   924 local structure P = OuterParse and K = OuterKeyword in
```
```   925
```
```   926 val _ = OuterSyntax.keywords ["monos"];
```
```   927
```
```   928 (* FIXME tmp *)
```
```   929 fun flatten_specification specs = specs |> maps
```
```   930   (fn (a, (concl, [])) => concl |> map
```
```   931         (fn ((b, atts), [B]) =>
```
```   932               if a = "" then ((b, atts), B)
```
```   933               else if b = "" then ((a, atts), B)
```
```   934               else error ("Illegal nested case names " ^ quote (NameSpace.append a b))
```
```   935           | ((b, _), _) => error ("Illegal simultaneous specification " ^ quote b))
```
```   936     | (a, _) => error ("Illegal local specification parameters for " ^ quote a));
```
```   937
```
```   938 fun gen_ind_decl mk_def coind =
```
```   939   P.opt_target --
```
```   940   P.fixes -- P.for_fixes --
```
```   941   Scan.optional (P.\$\$\$ "where" |-- P.!!! SpecParse.specification) [] --
```
```   942   Scan.optional (P.\$\$\$ "monos" |-- P.!!! SpecParse.xthms1) []
```
```   943   >> (fn ((((loc, preds), params), specs), monos) =>
```
```   944     Toplevel.local_theory loc
```
```   945       (fn lthy => lthy |> gen_add_inductive mk_def true coind preds params
```
```   946          (flatten_specification specs) monos |> snd));
```
```   947
```
```   948 val ind_decl = gen_ind_decl add_ind_def;
```
```   949
```
```   950 val _ = OuterSyntax.command "inductive" "define inductive predicates" K.thy_decl (ind_decl false);
```
```   951 val _ = OuterSyntax.command "coinductive" "define coinductive predicates" K.thy_decl (ind_decl true);
```
```   952
```
```   953 val _ =
```
```   954   OuterSyntax.command "inductive_cases"
```
```   955     "create simplified instances of elimination rules (improper)" K.thy_script
```
```   956     (P.opt_target -- P.and_list1 SpecParse.spec
```
```   957       >> (fn (loc, specs) => Toplevel.local_theory loc (snd o inductive_cases specs)));
```
```   958
```
```   959 end;
```
```   960
```
```   961 end;
```