src/HOL/Tools/inductive_package.ML
 author wenzelm Sat May 17 13:54:30 2008 +0200 (2008-05-17) changeset 26928 ca87aff1ad2d parent 26736 e6091328718f child 26988 742e26213212 permissions -rw-r--r--
structure Display: less pervasive 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   type inductive_result
```
```    25   val morph_result: morphism -> inductive_result -> inductive_result
```
```    26   type inductive_info
```
```    27   val the_inductive: Proof.context -> string -> inductive_info
```
```    28   val print_inductives: Proof.context -> unit
```
```    29   val mono_add: attribute
```
```    30   val mono_del: attribute
```
```    31   val get_monos: Proof.context -> thm list
```
```    32   val mk_cases: Proof.context -> term -> thm
```
```    33   val inductive_forall_name: string
```
```    34   val inductive_forall_def: thm
```
```    35   val rulify: thm -> thm
```
```    36   val inductive_cases: ((bstring * Attrib.src list) * string list) list ->
```
```    37     Proof.context -> thm list list * local_theory
```
```    38   val inductive_cases_i: ((bstring * Attrib.src list) * term list) list ->
```
```    39     Proof.context -> thm list list * local_theory
```
```    40   type inductive_flags
```
```    41   val add_inductive_i:
```
```    42     inductive_flags -> ((string * typ) * mixfix) list ->
```
```    43     (string * typ) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
```
```    44       local_theory -> inductive_result * local_theory
```
```    45   val add_inductive: bool -> bool -> (string * string option * mixfix) list ->
```
```    46     (string * string option * mixfix) list ->
```
```    47     ((bstring * Attrib.src list) * string) list -> (Facts.ref * Attrib.src list) list ->
```
```    48     local_theory -> inductive_result * local_theory
```
```    49   val add_inductive_global: string -> inductive_flags ->
```
```    50     ((string * typ) * mixfix) list -> (string * typ) list ->
```
```    51     ((bstring * Attrib.src list) * term) list -> thm list -> theory -> inductive_result * theory
```
```    52   val arities_of: thm -> (string * int) list
```
```    53   val params_of: thm -> term list
```
```    54   val partition_rules: thm -> thm list -> (string * thm list) list
```
```    55   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
```
```    56   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
```
```    57   val infer_intro_vars: thm -> int -> thm list -> term list list
```
```    58   val setup: theory -> theory
```
```    59 end;
```
```    60
```
```    61 signature INDUCTIVE_PACKAGE =
```
```    62 sig
```
```    63   include BASIC_INDUCTIVE_PACKAGE
```
```    64   type add_ind_def
```
```    65   val declare_rules: string -> bstring -> bool -> bool -> string list ->
```
```    66     thm list -> bstring list -> Attrib.src list list -> (thm * string list) list ->
```
```    67     thm -> local_theory -> thm list * thm list * thm * local_theory
```
```    68   val add_ind_def: add_ind_def
```
```    69   val gen_add_inductive_i: add_ind_def ->
```
```    70     inductive_flags -> ((string * typ) * mixfix) list ->
```
```    71     (string * typ) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
```
```    72       local_theory -> inductive_result * local_theory
```
```    73   val gen_add_inductive: add_ind_def ->
```
```    74     bool -> bool -> (string * string option * mixfix) list ->
```
```    75     (string * string option * mixfix) list ->
```
```    76     ((bstring * Attrib.src list) * string) list -> (Facts.ref * Attrib.src list) list ->
```
```    77     local_theory -> inductive_result * local_theory
```
```    78   val gen_ind_decl: add_ind_def ->
```
```    79     bool -> OuterParse.token list ->
```
```    80     (Toplevel.transition -> Toplevel.transition) * OuterParse.token list
```
```    81 end;
```
```    82
```
```    83 structure InductivePackage: INDUCTIVE_PACKAGE =
```
```    84 struct
```
```    85
```
```    86
```
```    87 (** theory context references **)
```
```    88
```
```    89 val inductive_forall_name = "HOL.induct_forall";
```
```    90 val inductive_forall_def = thm "induct_forall_def";
```
```    91 val inductive_conj_name = "HOL.induct_conj";
```
```    92 val inductive_conj_def = thm "induct_conj_def";
```
```    93 val inductive_conj = thms "induct_conj";
```
```    94 val inductive_atomize = thms "induct_atomize";
```
```    95 val inductive_rulify = thms "induct_rulify";
```
```    96 val inductive_rulify_fallback = thms "induct_rulify_fallback";
```
```    97
```
```    98 val notTrueE = TrueI RSN (2, notE);
```
```    99 val notFalseI = Seq.hd (atac 1 notI);
```
```   100 val simp_thms' = map (fn s => mk_meta_eq (the (find_first
```
```   101   (equal (Sign.read_prop HOL.thy s) o prop_of) simp_thms)))
```
```   102   ["(~True) = False", "(~False) = True",
```
```   103    "(True --> ?P) = ?P", "(False --> ?P) = True",
```
```   104    "(?P & True) = ?P", "(True & ?P) = ?P"];
```
```   105
```
```   106
```
```   107
```
```   108 (** context data **)
```
```   109
```
```   110 type inductive_result =
```
```   111   {preds: term list, elims: thm list, raw_induct: thm,
```
```   112    induct: thm, intrs: thm list};
```
```   113
```
```   114 fun morph_result phi {preds, elims, raw_induct: thm, induct, intrs} =
```
```   115   let
```
```   116     val term = Morphism.term phi;
```
```   117     val thm = Morphism.thm phi;
```
```   118     val fact = Morphism.fact phi;
```
```   119   in
```
```   120    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
```
```   121     induct = thm induct, intrs = fact intrs}
```
```   122   end;
```
```   123
```
```   124 type inductive_info =
```
```   125   {names: string list, coind: bool} * inductive_result;
```
```   126
```
```   127 structure InductiveData = GenericDataFun
```
```   128 (
```
```   129   type T = inductive_info Symtab.table * thm list;
```
```   130   val empty = (Symtab.empty, []);
```
```   131   val extend = I;
```
```   132   fun merge _ ((tab1, monos1), (tab2, monos2)) =
```
```   133     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
```
```   134 );
```
```   135
```
```   136 val get_inductives = InductiveData.get o Context.Proof;
```
```   137
```
```   138 fun print_inductives ctxt =
```
```   139   let
```
```   140     val (tab, monos) = get_inductives ctxt;
```
```   141     val space = Consts.space_of (ProofContext.consts_of ctxt);
```
```   142   in
```
```   143     [Pretty.strs ("(co)inductives:" :: map #1 (NameSpace.extern_table (space, tab))),
```
```   144      Pretty.big_list "monotonicity rules:" (map (ProofContext.pretty_thm ctxt) monos)]
```
```   145     |> Pretty.chunks |> Pretty.writeln
```
```   146   end;
```
```   147
```
```   148
```
```   149 (* get and put data *)
```
```   150
```
```   151 fun the_inductive ctxt name =
```
```   152   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
```
```   153     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
```
```   154   | SOME info => info);
```
```   155
```
```   156 fun put_inductives names info = InductiveData.map
```
```   157   (apfst (fold (fn name => Symtab.update (name, info)) names));
```
```   158
```
```   159
```
```   160
```
```   161 (** monotonicity rules **)
```
```   162
```
```   163 val get_monos = #2 o get_inductives;
```
```   164 val map_monos = InductiveData.map o apsnd;
```
```   165
```
```   166 fun mk_mono thm =
```
```   167   let
```
```   168     val concl = concl_of thm;
```
```   169     fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @
```
```   170       (case concl of
```
```   171           (_ \$ (_ \$ (Const ("Not", _) \$ _) \$ _)) => []
```
```   172         | _ => [thm' RS (thm' RS eq_to_mono2)]);
```
```   173     fun dest_less_concl thm = dest_less_concl (thm RS le_funD)
```
```   174       handle THM _ => thm RS le_boolD
```
```   175   in
```
```   176     case concl of
```
```   177       Const ("==", _) \$ _ \$ _ => eq2mono (thm RS meta_eq_to_obj_eq)
```
```   178     | _ \$ (Const ("op =", _) \$ _ \$ _) => eq2mono thm
```
```   179     | _ \$ (Const ("HOL.ord_class.less_eq", _) \$ _ \$ _) =>
```
```   180       [dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
```
```   181          (resolve_tac [le_funI, le_boolI'])) thm))]
```
```   182     | _ => [thm]
```
```   183   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm thm);
```
```   184
```
```   185 val mono_add = Thm.declaration_attribute (map_monos o fold Thm.add_thm o mk_mono);
```
```   186 val mono_del = Thm.declaration_attribute (map_monos o fold Thm.del_thm o mk_mono);
```
```   187
```
```   188
```
```   189
```
```   190 (** misc utilities **)
```
```   191
```
```   192 fun message quiet_mode s = if quiet_mode then () else writeln s;
```
```   193 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
```
```   194
```
```   195 fun coind_prefix true = "co"
```
```   196   | coind_prefix false = "";
```
```   197
```
```   198 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
```
```   199
```
```   200 fun make_bool_args f g [] i = []
```
```   201   | make_bool_args f g (x :: xs) i =
```
```   202       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
```
```   203
```
```   204 fun make_bool_args' xs =
```
```   205   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
```
```   206
```
```   207 fun find_arg T x [] = sys_error "find_arg"
```
```   208   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
```
```   209       apsnd (cons p) (find_arg T x ps)
```
```   210   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
```
```   211       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
```
```   212       else apsnd (cons p) (find_arg T x ps);
```
```   213
```
```   214 fun make_args Ts xs =
```
```   215   map (fn (T, (NONE, ())) => Const ("arbitrary", T) | (_, (SOME t, ())) => t)
```
```   216     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
```
```   217
```
```   218 fun make_args' Ts xs Us =
```
```   219   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
```
```   220
```
```   221 fun dest_predicate cs params t =
```
```   222   let
```
```   223     val k = length params;
```
```   224     val (c, ts) = strip_comb t;
```
```   225     val (xs, ys) = chop k ts;
```
```   226     val i = find_index_eq c cs;
```
```   227   in
```
```   228     if xs = params andalso i >= 0 then
```
```   229       SOME (c, i, ys, chop (length ys)
```
```   230         (List.drop (binder_types (fastype_of c), k)))
```
```   231     else NONE
```
```   232   end;
```
```   233
```
```   234 fun mk_names a 0 = []
```
```   235   | mk_names a 1 = [a]
```
```   236   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
```
```   237
```
```   238
```
```   239
```
```   240 (** process rules **)
```
```   241
```
```   242 local
```
```   243
```
```   244 fun err_in_rule ctxt name t msg =
```
```   245   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
```
```   246     Syntax.string_of_term ctxt t, msg]);
```
```   247
```
```   248 fun err_in_prem ctxt name t p msg =
```
```   249   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
```
```   250     "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
```
```   251
```
```   252 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
```
```   253
```
```   254 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
```
```   255
```
```   256 val bad_app = "Inductive predicate must be applied to parameter(s) ";
```
```   257
```
```   258 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
```
```   259
```
```   260 in
```
```   261
```
```   262 fun check_rule ctxt cs params ((name, att), rule) =
```
```   263   let
```
```   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 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 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 name rule') t;
```
```   295             List.app check_prem (prems ~~ aprems))
```
```   296          else err_in_rule ctxt name rule' bad_concl
```
```   297      | _ => err_in_rule ctxt name rule' bad_concl);
```
```   298     ((name, 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 --
```
```   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 = if alt_name = "" then
```
```   640       space_implode "_" (map fst cnames_syn) else alt_name;
```
```   641
```
```   642     val ((rec_const, (_, fp_def)), ctxt') = ctxt |>
```
```   643       LocalTheory.define Thm.internalK
```
```   644         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
```
```   645          (("", []), fold_rev lambda params
```
```   646            (Const (fp_name, (predT --> predT) --> predT) \$ fp_fun)));
```
```   647     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
```
```   648       (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params)));
```
```   649     val specs = if length cs < 2 then [] else
```
```   650       map_index (fn (i, (name_mx, c)) =>
```
```   651         let
```
```   652           val Ts = List.drop (binder_types (fastype_of c), length params);
```
```   653           val xs = map Free (Variable.variant_frees ctxt intr_ts
```
```   654             (mk_names "x" (length Ts) ~~ Ts))
```
```   655         in
```
```   656           (name_mx, (("", []), fold_rev lambda (params @ xs)
```
```   657             (list_comb (rec_const, params @ make_bool_args' bs i @
```
```   658               make_args argTs (xs ~~ Ts)))))
```
```   659         end) (cnames_syn ~~ cs);
```
```   660     val (consts_defs, ctxt'') = fold_map (LocalTheory.define Thm.internalK) specs ctxt';
```
```   661     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
```
```   662
```
```   663     val mono = prove_mono quiet_mode skip_mono predT fp_fun monos ctxt''
```
```   664
```
```   665   in (ctxt'', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
```
```   666     list_comb (rec_const, params), preds, argTs, bs, xs)
```
```   667   end;
```
```   668
```
```   669 fun declare_rules kind rec_name coind no_ind cnames intrs intr_names intr_atts
```
```   670       elims raw_induct ctxt =
```
```   671   let
```
```   672     val ind_case_names = RuleCases.case_names intr_names;
```
```   673     val induct =
```
```   674       if coind then
```
```   675         (raw_induct, [RuleCases.case_names [rec_name],
```
```   676           RuleCases.case_conclusion (rec_name, intr_names),
```
```   677           RuleCases.consumes 1, Induct.coinduct_pred (hd cnames)])
```
```   678       else if no_ind orelse length cnames > 1 then
```
```   679         (raw_induct, [ind_case_names, RuleCases.consumes 0])
```
```   680       else (raw_induct RSN (2, rev_mp), [ind_case_names, RuleCases.consumes 1]);
```
```   681
```
```   682     val (intrs', ctxt1) =
```
```   683       ctxt |>
```
```   684       LocalTheory.notes kind
```
```   685         (map (NameSpace.qualified rec_name) intr_names ~~
```
```   686          intr_atts ~~ map (fn th => [([th],
```
```   687            [Attrib.internal (K (ContextRules.intro_query NONE))])]) intrs) |>>
```
```   688       map (hd o snd);
```
```   689     val (((_, elims'), (_, [induct'])), ctxt2) =
```
```   690       ctxt1 |>
```
```   691       LocalTheory.note kind ((NameSpace.qualified rec_name "intros", []), intrs') ||>>
```
```   692       fold_map (fn (name, (elim, cases)) =>
```
```   693         LocalTheory.note kind ((NameSpace.qualified (Sign.base_name name) "cases",
```
```   694           [Attrib.internal (K (RuleCases.case_names cases)),
```
```   695            Attrib.internal (K (RuleCases.consumes 1)),
```
```   696            Attrib.internal (K (Induct.cases_pred name)),
```
```   697            Attrib.internal (K (ContextRules.elim_query NONE))]), [elim]) #>
```
```   698         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
```
```   699       LocalTheory.note kind ((NameSpace.qualified rec_name (coind_prefix coind ^ "induct"),
```
```   700         map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
```
```   701
```
```   702     val ctxt3 = if no_ind orelse coind then ctxt2 else
```
```   703       let val inducts = cnames ~~ ProjectRule.projects ctxt2 (1 upto length cnames) induct'
```
```   704       in
```
```   705         ctxt2 |>
```
```   706         LocalTheory.notes kind [((NameSpace.qualified rec_name "inducts", []),
```
```   707           inducts |> map (fn (name, th) => ([th],
```
```   708             [Attrib.internal (K ind_case_names),
```
```   709              Attrib.internal (K (RuleCases.consumes 1)),
```
```   710              Attrib.internal (K (Induct.induct_pred name))])))] |> snd
```
```   711       end
```
```   712   in (intrs', elims', induct', ctxt3) end;
```
```   713
```
```   714 type inductive_flags =
```
```   715   {quiet_mode: bool, verbose: bool, kind: string, alt_name: bstring,
```
```   716    coind: bool, no_elim: bool, no_ind: bool, skip_mono: bool}
```
```   717
```
```   718 type add_ind_def =
```
```   719   inductive_flags ->
```
```   720   term list -> ((string * Attrib.src list) * term) list -> thm list ->
```
```   721   term list -> (string * mixfix) list ->
```
```   722   local_theory -> inductive_result * local_theory
```
```   723
```
```   724 fun add_ind_def {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono}
```
```   725     cs intros monos params cnames_syn ctxt =
```
```   726   let
```
```   727     val _ = null cnames_syn andalso error "No inductive predicates given";
```
```   728     val _ = message (quiet_mode andalso not verbose)
```
```   729       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^
```
```   730         commas_quote (map fst cnames_syn));
```
```   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 quiet_mode skip_mono alt_name coind cs intr_ts
```
```   738         monos params cnames_syn ctxt;
```
```   739
```
```   740     val (intrs, unfold) = prove_intrs quiet_mode 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 quiet_mode 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, sup_fun_eq, sup_bool_eq] THEN
```
```   751                fold_tac rec_preds_defs) (mono RS (fp_def RS def_coinduct)))))
```
```   752        else
```
```   753          prove_indrule quiet_mode 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       |> LocalTheory.declaration (fn phi =>
```
```   769         let val result' = morph_result phi result;
```
```   770         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
```
```   771   in (result, ctxt3) end;
```
```   772
```
```   773
```
```   774 (* external interfaces *)
```
```   775
```
```   776 fun gen_add_inductive_i mk_def
```
```   777     (flags as {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono})
```
```   778     cnames_syn pnames spec monos lthy =
```
```   779   let
```
```   780     val thy = ProofContext.theory_of lthy;
```
```   781     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
```
```   782
```
```   783
```
```   784     (* abbrevs *)
```
```   785
```
```   786     val (_, ctxt1) = Variable.add_fixes (map (fst o fst) cnames_syn) lthy;
```
```   787
```
```   788     fun get_abbrev ((name, atts), t) =
```
```   789       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
```
```   790         let
```
```   791           val _ = name = "" andalso null atts orelse
```
```   792             error "Abbreviations may not have names or attributes";
```
```   793           val ((x, T), rhs) = LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt1 t));
```
```   794           val mx =
```
```   795             (case find_first (fn ((c, _), _) => c = x) cnames_syn of
```
```   796               NONE => error ("Undeclared head of abbreviation " ^ quote x)
```
```   797             | SOME ((_, T'), mx) =>
```
```   798                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
```
```   799                 else mx);
```
```   800         in SOME ((x, mx), rhs) end
```
```   801       else NONE;
```
```   802
```
```   803     val abbrevs = map_filter get_abbrev spec;
```
```   804     val bs = map (fst o fst) abbrevs;
```
```   805
```
```   806
```
```   807     (* predicates *)
```
```   808
```
```   809     val pre_intros = filter_out (is_some o get_abbrev) spec;
```
```   810     val cnames_syn' = filter_out (member (op =) bs o fst o fst) cnames_syn;
```
```   811     val cs = map (Free o fst) cnames_syn';
```
```   812     val ps = map Free pnames;
```
```   813
```
```   814     val (_, ctxt2) = lthy |> Variable.add_fixes (map (fst o fst) cnames_syn');
```
```   815     val _ = map (fn abbr => LocalDefs.fixed_abbrev abbr ctxt2) abbrevs;
```
```   816     val ctxt3 = ctxt2 |> fold (snd oo LocalDefs.fixed_abbrev) abbrevs;
```
```   817     val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy;
```
```   818
```
```   819     fun close_rule r = list_all_free (rev (fold_aterms
```
```   820       (fn t as Free (v as (s, _)) =>
```
```   821           if Variable.is_fixed ctxt1 s orelse
```
```   822             member (op =) ps t then I else insert (op =) v
```
```   823         | _ => I) r []), r);
```
```   824
```
```   825     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
```
```   826     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
```
```   827   in
```
```   828     lthy
```
```   829     |> mk_def flags cs intros monos ps preds
```
```   830     ||> fold (snd oo LocalTheory.abbrev Syntax.mode_default) abbrevs
```
```   831   end;
```
```   832
```
```   833 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
```
```   834   let
```
```   835     val ((vars, specs), _) = lthy |> ProofContext.set_mode ProofContext.mode_abbrev
```
```   836       |> Specification.read_specification
```
```   837           (cnames_syn @ pnames_syn) (map (fn (a, s) => [(a, [s])]) intro_srcs);
```
```   838     val (cs, ps) = chop (length cnames_syn) vars;
```
```   839     val intrs = map (apsnd the_single) specs;
```
```   840     val monos = Attrib.eval_thms lthy raw_monos;
```
```   841     val flags = {quiet_mode = false, verbose = verbose, kind = Thm.theoremK, alt_name = "",
```
```   842       coind = coind, no_elim = false, no_ind = false, skip_mono = false};
```
```   843   in
```
```   844     lthy
```
```   845     |> LocalTheory.set_group (serial_string ())
```
```   846     |> gen_add_inductive_i mk_def flags cs (map fst ps) intrs monos
```
```   847   end;
```
```   848
```
```   849 val add_inductive_i = gen_add_inductive_i add_ind_def;
```
```   850 val add_inductive = gen_add_inductive add_ind_def;
```
```   851
```
```   852 fun add_inductive_global group flags cnames_syn pnames pre_intros monos thy =
```
```   853   let
```
```   854     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
```
```   855     val ctxt' = thy
```
```   856       |> TheoryTarget.init NONE
```
```   857       |> LocalTheory.set_group group
```
```   858       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
```
```   859       |> LocalTheory.exit;
```
```   860     val info = #2 (the_inductive ctxt' name);
```
```   861   in (info, ProofContext.theory_of ctxt') end;
```
```   862
```
```   863
```
```   864 (* read off arities of inductive predicates from raw induction rule *)
```
```   865 fun arities_of induct =
```
```   866   map (fn (_ \$ t \$ u) =>
```
```   867       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
```
```   868     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
```
```   869
```
```   870 (* read off parameters of inductive predicate from raw induction rule *)
```
```   871 fun params_of induct =
```
```   872   let
```
```   873     val (_ \$ t \$ u :: _) =
```
```   874       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
```
```   875     val (_, ts) = strip_comb t;
```
```   876     val (_, us) = strip_comb u
```
```   877   in
```
```   878     List.take (ts, length ts - length us)
```
```   879   end;
```
```   880
```
```   881 val pname_of_intr =
```
```   882   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
```
```   883
```
```   884 (* partition introduction rules according to predicate name *)
```
```   885 fun gen_partition_rules f induct intros =
```
```   886   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
```
```   887     (map (rpair [] o fst) (arities_of induct));
```
```   888
```
```   889 val partition_rules = gen_partition_rules I;
```
```   890 fun partition_rules' induct = gen_partition_rules fst induct;
```
```   891
```
```   892 fun unpartition_rules intros xs =
```
```   893   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
```
```   894     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
```
```   895
```
```   896 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
```
```   897 fun infer_intro_vars elim arity intros =
```
```   898   let
```
```   899     val thy = theory_of_thm elim;
```
```   900     val _ :: cases = prems_of elim;
```
```   901     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
```
```   902     fun mtch (t, u) =
```
```   903       let
```
```   904         val params = Logic.strip_params t;
```
```   905         val vars = map (Var o apfst (rpair 0))
```
```   906           (Name.variant_list used (map fst params) ~~ map snd params);
```
```   907         val ts = map (curry subst_bounds (rev vars))
```
```   908           (List.drop (Logic.strip_assums_hyp t, arity));
```
```   909         val us = Logic.strip_imp_prems u;
```
```   910         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
```
```   911           (Vartab.empty, Vartab.empty);
```
```   912       in
```
```   913         map (Envir.subst_vars tab) vars
```
```   914       end
```
```   915   in
```
```   916     map (mtch o apsnd prop_of) (cases ~~ intros)
```
```   917   end;
```
```   918
```
```   919
```
```   920
```
```   921 (** package setup **)
```
```   922
```
```   923 (* setup theory *)
```
```   924
```
```   925 val setup =
```
```   926   Method.add_methods [("ind_cases", ind_cases,
```
```   927     "dynamic case analysis on predicates")] #>
```
```   928   Attrib.add_attributes [("mono", Attrib.add_del_args mono_add mono_del,
```
```   929     "declaration of monotonicity rule")];
```
```   930
```
```   931
```
```   932 (* outer syntax *)
```
```   933
```
```   934 local structure P = OuterParse and K = OuterKeyword in
```
```   935
```
```   936 val _ = OuterSyntax.keywords ["monos"];
```
```   937
```
```   938 fun flatten_specification specs = specs |> maps
```
```   939   (fn (a, (concl, [])) => concl |> map
```
```   940         (fn ((b, atts), [B]) =>
```
```   941               if a = "" then ((b, atts), B)
```
```   942               else if b = "" then ((a, atts), B)
```
```   943               else error ("Illegal nested case names " ^ quote (NameSpace.append a b))
```
```   944           | ((b, _), _) => error ("Illegal simultaneous specification " ^ quote b))
```
```   945     | (a, _) => error ("Illegal local specification parameters for " ^ quote a));
```
```   946
```
```   947 fun gen_ind_decl mk_def coind =
```
```   948   P.opt_target --
```
```   949   P.fixes -- P.for_fixes --
```
```   950   Scan.optional (P.\$\$\$ "where" |-- P.!!! SpecParse.specification) [] --
```
```   951   Scan.optional (P.\$\$\$ "monos" |-- P.!!! SpecParse.xthms1) []
```
```   952   >> (fn ((((loc, preds), params), specs), monos) =>
```
```   953     Toplevel.local_theory loc
```
```   954       (fn lthy => lthy |> gen_add_inductive mk_def true coind preds params
```
```   955          (flatten_specification specs) monos |> snd));
```
```   956
```
```   957 val ind_decl = gen_ind_decl add_ind_def;
```
```   958
```
```   959 val _ = OuterSyntax.command "inductive" "define inductive predicates" K.thy_decl (ind_decl false);
```
```   960 val _ = OuterSyntax.command "coinductive" "define coinductive predicates" K.thy_decl (ind_decl true);
```
```   961
```
```   962 val _ =
```
```   963   OuterSyntax.command "inductive_cases"
```
```   964     "create simplified instances of elimination rules (improper)" K.thy_script
```
```   965     (P.opt_target -- P.and_list1 SpecParse.spec
```
```   966       >> (fn (loc, specs) => Toplevel.local_theory loc (snd o inductive_cases specs)));
```
```   967
```
```   968 end;
```
```   969
```
```   970 end;
```