src/HOL/Tools/inductive.ML
author wenzelm
Mon Sep 30 13:45:17 2013 +0200 (2013-09-30)
changeset 53994 4237859c186d
parent 52732 b4da1f2ec73f
child 53995 1d457fc83f5c
permissions -rw-r--r--
eliminated clone of Inductive.mk_cases_tac;
     1 (*  Title:      HOL/Tools/inductive.ML
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
     4 
     5 (Co)Inductive Definition module for HOL.
     6 
     7 Features:
     8   * least or greatest fixedpoints
     9   * mutually recursive definitions
    10   * definitions involving arbitrary monotone operators
    11   * automatically proves introduction and elimination rules
    12 
    13   Introduction rules have the form
    14   [| M Pj ti, ..., Q x, ... |] ==> Pk t
    15   where M is some monotone operator (usually the identity)
    16   Q x is any side condition on the free variables
    17   ti, t are any terms
    18   Pj, Pk are two of the predicates being defined in mutual recursion
    19 *)
    20 
    21 signature BASIC_INDUCTIVE =
    22 sig
    23   type inductive_result =
    24     {preds: term list, elims: thm list, raw_induct: thm,
    25      induct: thm, inducts: thm list, intrs: thm list, eqs: thm list}
    26   val transform_result: morphism -> inductive_result -> inductive_result
    27   type inductive_info = {names: string list, coind: bool} * inductive_result
    28   val the_inductive: Proof.context -> string -> inductive_info
    29   val print_inductives: Proof.context -> unit
    30   val get_monos: Proof.context -> thm list
    31   val mono_add: attribute
    32   val mono_del: attribute
    33   val mk_cases_tac: Proof.context -> tactic
    34   val mk_cases: Proof.context -> term -> thm
    35   val inductive_forall_def: thm
    36   val rulify: Proof.context -> thm -> thm
    37   val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
    38     thm list list * local_theory
    39   val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
    40     thm list list * local_theory
    41   type inductive_flags =
    42     {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
    43       no_elim: bool, no_ind: bool, skip_mono: bool}
    44   val add_inductive_i:
    45     inductive_flags -> ((binding * typ) * mixfix) list ->
    46     (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
    47     inductive_result * local_theory
    48   val add_inductive: bool -> bool ->
    49     (binding * string option * mixfix) list ->
    50     (binding * string option * mixfix) list ->
    51     (Attrib.binding * string) list ->
    52     (Facts.ref * Attrib.src list) list ->
    53     local_theory -> inductive_result * local_theory
    54   val add_inductive_global: inductive_flags ->
    55     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    56     thm list -> theory -> inductive_result * theory
    57   val arities_of: thm -> (string * int) list
    58   val params_of: thm -> term list
    59   val partition_rules: thm -> thm list -> (string * thm list) list
    60   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
    61   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    62   val infer_intro_vars: thm -> int -> thm list -> term list list
    63   val setup: theory -> theory
    64 end;
    65 
    66 signature INDUCTIVE =
    67 sig
    68   include BASIC_INDUCTIVE
    69   type add_ind_def =
    70     inductive_flags ->
    71     term list -> (Attrib.binding * term) list -> thm list ->
    72     term list -> (binding * mixfix) list ->
    73     local_theory -> inductive_result * local_theory
    74   val declare_rules: binding -> bool -> bool -> string list -> term list ->
    75     thm list -> binding list -> Attrib.src list list -> (thm * string list * int) list ->
    76     thm list -> thm -> local_theory -> thm list * thm list * thm list * thm * thm list * local_theory
    77   val add_ind_def: add_ind_def
    78   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
    79     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    80     thm list -> local_theory -> inductive_result * local_theory
    81   val gen_add_inductive: add_ind_def -> bool -> bool ->
    82     (binding * string option * mixfix) list ->
    83     (binding * string option * mixfix) list ->
    84     (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
    85     local_theory -> inductive_result * local_theory
    86   val gen_ind_decl: add_ind_def -> bool -> (local_theory -> local_theory) parser
    87 end;
    88 
    89 structure Inductive: INDUCTIVE =
    90 struct
    91 
    92 (** theory context references **)
    93 
    94 val inductive_forall_def = @{thm induct_forall_def};
    95 val inductive_conj_name = "HOL.induct_conj";
    96 val inductive_conj_def = @{thm induct_conj_def};
    97 val inductive_conj = @{thms induct_conj};
    98 val inductive_atomize = @{thms induct_atomize};
    99 val inductive_rulify = @{thms induct_rulify};
   100 val inductive_rulify_fallback = @{thms induct_rulify_fallback};
   101 
   102 val simp_thms1 =
   103   map mk_meta_eq
   104     @{lemma "(~ True) = False" "(~ False) = True"
   105         "(True --> P) = P" "(False --> P) = True"
   106         "(P & True) = P" "(True & P) = P"
   107       by (fact simp_thms)+};
   108 
   109 val simp_thms2 =
   110   map mk_meta_eq [@{thm inf_fun_def}, @{thm inf_bool_def}] @ simp_thms1;
   111 
   112 val simp_thms3 =
   113   map mk_meta_eq [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_def}, @{thm sup_bool_def}];
   114 
   115 
   116 
   117 (** misc utilities **)
   118 
   119 fun message quiet_mode s = if quiet_mode then () else writeln s;
   120 
   121 fun clean_message ctxt quiet_mode s =
   122   if Config.get ctxt quick_and_dirty then () else message quiet_mode s;
   123 
   124 fun coind_prefix true = "co"
   125   | coind_prefix false = "";
   126 
   127 fun log (b: int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   128 
   129 fun make_bool_args f g [] i = []
   130   | make_bool_args f g (x :: xs) i =
   131       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   132 
   133 fun make_bool_args' xs =
   134   make_bool_args (K @{term False}) (K @{term True}) xs;
   135 
   136 fun arg_types_of k c = drop k (binder_types (fastype_of c));
   137 
   138 fun find_arg T x [] = raise Fail "find_arg"
   139   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   140       apsnd (cons p) (find_arg T x ps)
   141   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   142       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   143       else apsnd (cons p) (find_arg T x ps);
   144 
   145 fun make_args Ts xs =
   146   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
   147     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   148 
   149 fun make_args' Ts xs Us =
   150   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   151 
   152 fun dest_predicate cs params t =
   153   let
   154     val k = length params;
   155     val (c, ts) = strip_comb t;
   156     val (xs, ys) = chop k ts;
   157     val i = find_index (fn c' => c' = c) cs;
   158   in
   159     if xs = params andalso i >= 0 then
   160       SOME (c, i, ys, chop (length ys) (arg_types_of k c))
   161     else NONE
   162   end;
   163 
   164 fun mk_names a 0 = []
   165   | mk_names a 1 = [a]
   166   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   167 
   168 fun select_disj 1 1 = []
   169   | select_disj _ 1 = [rtac disjI1]
   170   | select_disj n i = rtac disjI2 :: select_disj (n - 1) (i - 1);
   171 
   172 
   173 
   174 (** context data **)
   175 
   176 type inductive_result =
   177   {preds: term list, elims: thm list, raw_induct: thm,
   178    induct: thm, inducts: thm list, intrs: thm list, eqs: thm list};
   179 
   180 fun transform_result phi {preds, elims, raw_induct: thm, induct, inducts, intrs, eqs} =
   181   let
   182     val term = Morphism.term phi;
   183     val thm = Morphism.thm phi;
   184     val fact = Morphism.fact phi;
   185   in
   186    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
   187     induct = thm induct, inducts = fact inducts, intrs = fact intrs, eqs = fact eqs}
   188   end;
   189 
   190 type inductive_info = {names: string list, coind: bool} * inductive_result;
   191 
   192 val empty_equations =
   193   Item_Net.init Thm.eq_thm_prop
   194     (single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
   195 
   196 datatype data = Data of
   197  {infos: inductive_info Symtab.table,
   198   monos: thm list,
   199   equations: thm Item_Net.T};
   200 
   201 fun make_data (infos, monos, equations) =
   202   Data {infos = infos, monos = monos, equations = equations};
   203 
   204 structure Data = Generic_Data
   205 (
   206   type T = data;
   207   val empty = make_data (Symtab.empty, [], empty_equations);
   208   val extend = I;
   209   fun merge (Data {infos = infos1, monos = monos1, equations = equations1},
   210       Data {infos = infos2, monos = monos2, equations = equations2}) =
   211     make_data (Symtab.merge (K true) (infos1, infos2),
   212       Thm.merge_thms (monos1, monos2),
   213       Item_Net.merge (equations1, equations2));
   214 );
   215 
   216 fun map_data f =
   217   Data.map (fn Data {infos, monos, equations} => make_data (f (infos, monos, equations)));
   218 
   219 fun rep_data ctxt = Data.get (Context.Proof ctxt) |> (fn Data rep => rep);
   220 
   221 fun print_inductives ctxt =
   222   let
   223     val {infos, monos, ...} = rep_data ctxt;
   224     val space = Consts.space_of (Proof_Context.consts_of ctxt);
   225   in
   226     [Pretty.block
   227       (Pretty.breaks
   228         (Pretty.str "(co)inductives:" ::
   229           map (Pretty.mark_str o #1) (Name_Space.extern_table ctxt (space, infos)))),
   230      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_item ctxt) monos)]
   231   end |> Pretty.chunks |> Pretty.writeln;
   232 
   233 
   234 (* inductive info *)
   235 
   236 fun the_inductive ctxt name =
   237   (case Symtab.lookup (#infos (rep_data ctxt)) name of
   238     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   239   | SOME info => info);
   240 
   241 fun put_inductives names info =
   242   map_data (fn (infos, monos, equations) =>
   243     (fold (fn name => Symtab.update (name, info)) names infos, monos, equations));
   244 
   245 
   246 (* monotonicity rules *)
   247 
   248 val get_monos = #monos o rep_data;
   249 
   250 fun mk_mono ctxt thm =
   251   let
   252     fun eq_to_mono thm' = thm' RS (thm' RS @{thm eq_to_mono});
   253     fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
   254       handle THM _ => thm RS @{thm le_boolD}
   255   in
   256     (case concl_of thm of
   257       Const ("==", _) $ _ $ _ => eq_to_mono (thm RS meta_eq_to_obj_eq)
   258     | _ $ (Const (@{const_name HOL.eq}, _) $ _ $ _) => eq_to_mono thm
   259     | _ $ (Const (@{const_name Orderings.less_eq}, _) $ _ $ _) =>
   260       dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   261         (resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))
   262     | _ => thm)
   263   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm ctxt thm);
   264 
   265 val mono_add =
   266   Thm.declaration_attribute (fn thm => fn context =>
   267     map_data (fn (infos, monos, equations) =>
   268       (infos, Thm.add_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
   269 
   270 val mono_del =
   271   Thm.declaration_attribute (fn thm => fn context =>
   272     map_data (fn (infos, monos, equations) =>
   273       (infos, Thm.del_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
   274 
   275 
   276 (* equations *)
   277 
   278 val get_equations = #equations o rep_data;
   279 
   280 val equation_add_permissive =
   281   Thm.declaration_attribute (fn thm =>
   282     map_data (fn (infos, monos, equations) =>
   283       (infos, monos, perhaps (try (Item_Net.update thm)) equations)));
   284 
   285 
   286 
   287 (** process rules **)
   288 
   289 local
   290 
   291 fun err_in_rule ctxt name t msg =
   292   error (cat_lines ["Ill-formed introduction rule " ^ Binding.print name,
   293     Syntax.string_of_term ctxt t, msg]);
   294 
   295 fun err_in_prem ctxt name t p msg =
   296   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
   297     "in introduction rule " ^ Binding.print name, Syntax.string_of_term ctxt t, msg]);
   298 
   299 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   300 
   301 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   302 
   303 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   304 
   305 fun atomize_term thy = Raw_Simplifier.rewrite_term thy inductive_atomize [];
   306 
   307 in
   308 
   309 fun check_rule ctxt cs params ((binding, att), rule) =
   310   let
   311     val params' = Term.variant_frees rule (Logic.strip_params rule);
   312     val frees = rev (map Free params');
   313     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   314     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   315     val rule' = Logic.list_implies (prems, concl);
   316     val aprems = map (atomize_term (Proof_Context.theory_of ctxt)) prems;
   317     val arule = fold_rev (Logic.all o Free) params' (Logic.list_implies (aprems, concl));
   318 
   319     fun check_ind err t =
   320       (case dest_predicate cs params t of
   321         NONE => err (bad_app ^
   322           commas (map (Syntax.string_of_term ctxt) params))
   323       | SOME (_, _, ys, _) =>
   324           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   325           then err bad_ind_occ else ());
   326 
   327     fun check_prem' prem t =
   328       if member (op =) cs (head_of t) then
   329         check_ind (err_in_prem ctxt binding rule prem) t
   330       else
   331         (case t of
   332           Abs (_, _, t) => check_prem' prem t
   333         | t $ u => (check_prem' prem t; check_prem' prem u)
   334         | _ => ());
   335 
   336     fun check_prem (prem, aprem) =
   337       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   338       else err_in_prem ctxt binding rule prem "Non-atomic premise";
   339 
   340     val _ =
   341       (case concl of
   342         Const (@{const_name Trueprop}, _) $ t =>
   343           if member (op =) cs (head_of t) then
   344            (check_ind (err_in_rule ctxt binding rule') t;
   345             List.app check_prem (prems ~~ aprems))
   346           else err_in_rule ctxt binding rule' bad_concl
   347        | _ => err_in_rule ctxt binding rule' bad_concl);
   348   in
   349     ((binding, att), arule)
   350   end;
   351 
   352 fun rulify ctxt =
   353   hol_simplify ctxt inductive_conj
   354   #> hol_simplify ctxt inductive_rulify
   355   #> hol_simplify ctxt inductive_rulify_fallback
   356   #> Simplifier.norm_hhf;
   357 
   358 end;
   359 
   360 
   361 
   362 (** proofs for (co)inductive predicates **)
   363 
   364 (* prove monotonicity *)
   365 
   366 fun prove_mono quiet_mode skip_mono predT fp_fun monos ctxt =
   367  (message (quiet_mode orelse skip_mono andalso Config.get ctxt quick_and_dirty)
   368     "  Proving monotonicity ...";
   369   (if skip_mono then Goal.prove_sorry else Goal.prove_future) ctxt
   370     [] []
   371     (HOLogic.mk_Trueprop
   372       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   373     (fn _ => EVERY [rtac @{thm monoI} 1,
   374       REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
   375       REPEAT (FIRST
   376         [atac 1,
   377          resolve_tac (map (mk_mono ctxt) monos @ get_monos ctxt) 1,
   378          etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
   379 
   380 
   381 (* prove introduction rules *)
   382 
   383 fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
   384   let
   385     val _ = clean_message ctxt quiet_mode "  Proving the introduction rules ...";
   386 
   387     val unfold = funpow k (fn th => th RS fun_cong)
   388       (mono RS (fp_def RS
   389         (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
   390 
   391     val rules = [refl, TrueI, @{lemma "~ False" by (rule notI)}, exI, conjI];
   392 
   393     val intrs = map_index (fn (i, intr) =>
   394       Goal.prove_sorry ctxt [] [] intr (fn _ => EVERY
   395        [rewrite_goals_tac rec_preds_defs,
   396         rtac (unfold RS iffD2) 1,
   397         EVERY1 (select_disj (length intr_ts) (i + 1)),
   398         (*Not ares_tac, since refl must be tried before any equality assumptions;
   399           backtracking may occur if the premises have extra variables!*)
   400         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)])
   401        |> singleton (Proof_Context.export ctxt ctxt')) intr_ts
   402 
   403   in (intrs, unfold) end;
   404 
   405 
   406 (* prove elimination rules *)
   407 
   408 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
   409   let
   410     val _ = clean_message ctxt quiet_mode "  Proving the elimination rules ...";
   411 
   412     val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
   413     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   414 
   415     fun dest_intr r =
   416       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   417        Logic.strip_assums_hyp r, Logic.strip_params r);
   418 
   419     val intrs = map dest_intr intr_ts ~~ intr_names;
   420 
   421     val rules1 = [disjE, exE, FalseE];
   422     val rules2 = [conjE, FalseE, @{lemma "~ True ==> R" by (rule notE [OF _ TrueI])}];
   423 
   424     fun prove_elim c =
   425       let
   426         val Ts = arg_types_of (length params) c;
   427         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   428         val frees = map Free (anames ~~ Ts);
   429 
   430         fun mk_elim_prem ((_, _, us, _), ts, params') =
   431           Logic.list_all (params',
   432             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   433               (frees ~~ us) @ ts, P));
   434         val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
   435         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   436            map mk_elim_prem (map #1 c_intrs)
   437       in
   438         (Goal.prove_sorry ctxt'' [] prems P
   439           (fn {prems, ...} => EVERY
   440             [cut_tac (hd prems) 1,
   441              rewrite_goals_tac rec_preds_defs,
   442              dtac (unfold RS iffD1) 1,
   443              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   444              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   445              EVERY (map (fn prem =>
   446                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
   447           |> singleton (Proof_Context.export ctxt'' ctxt'''),
   448          map #2 c_intrs, length Ts)
   449       end
   450 
   451    in map prove_elim cs end;
   452 
   453 
   454 (* prove simplification equations *)
   455 
   456 fun prove_eqs quiet_mode cs params intr_ts intrs
   457     (elims: (thm * bstring list * int) list) ctxt ctxt'' =  (* FIXME ctxt'' ?? *)
   458   let
   459     val _ = clean_message ctxt quiet_mode "  Proving the simplification rules ...";
   460 
   461     fun dest_intr r =
   462       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   463        Logic.strip_assums_hyp r, Logic.strip_params r);
   464     val intr_ts' = map dest_intr intr_ts;
   465 
   466     fun prove_eq c (elim: thm * 'a * 'b) =
   467       let
   468         val Ts = arg_types_of (length params) c;
   469         val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
   470         val frees = map Free (anames ~~ Ts);
   471         val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
   472         fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
   473           let
   474             fun list_ex ([], t) = t
   475               | list_ex ((a, T) :: vars, t) =
   476                   HOLogic.exists_const T $ Abs (a, T, list_ex (vars, t));
   477             val conjs = map2 (curry HOLogic.mk_eq) frees us @ map HOLogic.dest_Trueprop ts;
   478           in
   479             list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
   480           end;
   481         val lhs = list_comb (c, params @ frees);
   482         val rhs =
   483           if null c_intrs then @{term False}
   484           else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs);
   485         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
   486         fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
   487             EVERY1 (select_disj (length c_intrs) (i + 1)) THEN
   488             EVERY (replicate (length params) (rtac @{thm exI} 1)) THEN
   489             (if null prems then rtac @{thm TrueI} 1
   490              else
   491               let
   492                 val (prems', last_prem) = split_last prems;
   493               in
   494                 EVERY (map (fn prem => (rtac @{thm conjI} 1 THEN rtac prem 1)) prems') THEN
   495                 rtac last_prem 1
   496               end)) ctxt' 1;
   497         fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
   498           EVERY (replicate (length params') (etac @{thm exE} 1)) THEN
   499           (if null ts andalso null us then rtac intr 1
   500            else
   501             EVERY (replicate (length ts + length us - 1) (etac @{thm conjE} 1)) THEN
   502             Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
   503               let
   504                 val (eqs, prems') = chop (length us) prems;
   505                 val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs;
   506               in
   507                 rewrite_goal_tac rew_thms 1 THEN
   508                 rtac intr 1 THEN
   509                 EVERY (map (fn p => rtac p 1) prems')
   510               end) ctxt' 1);
   511       in
   512         Goal.prove_sorry ctxt' [] [] eq (fn _ =>
   513           rtac @{thm iffI} 1 THEN etac (#1 elim) 1 THEN
   514           EVERY (map_index prove_intr1 c_intrs) THEN
   515           (if null c_intrs then etac @{thm FalseE} 1
   516            else
   517             let val (c_intrs', last_c_intr) = split_last c_intrs in
   518               EVERY (map (fn ci => etac @{thm disjE} 1 THEN prove_intr2 ci) c_intrs') THEN
   519               prove_intr2 last_c_intr
   520             end))
   521         |> rulify ctxt'
   522         |> singleton (Proof_Context.export ctxt' ctxt'')
   523       end;
   524   in
   525     map2 prove_eq cs elims
   526   end;
   527 
   528 
   529 (* derivation of simplified elimination rules *)
   530 
   531 local
   532 
   533 (*delete needless equality assumptions*)
   534 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
   535   (fn _ => assume_tac 1);
   536 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   537 val elim_tac = REPEAT o eresolve_tac elim_rls;
   538 
   539 fun simp_case_tac ctxt i =
   540   EVERY' [elim_tac, asm_full_simp_tac ctxt, elim_tac, REPEAT o bound_hyp_subst_tac ctxt] i;
   541 
   542 in
   543 
   544 fun mk_cases_tac ctxt = ALLGOALS (simp_case_tac ctxt) THEN prune_params_tac;
   545 
   546 fun mk_cases ctxt prop =
   547   let
   548     val thy = Proof_Context.theory_of ctxt;
   549 
   550     fun err msg =
   551       error (Pretty.string_of (Pretty.block
   552         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
   553 
   554     val elims = Induct.find_casesP ctxt prop;
   555 
   556     val cprop = Thm.cterm_of thy prop;
   557     fun mk_elim rl =
   558       Thm.implies_intr cprop
   559         (Tactic.rule_by_tactic ctxt (mk_cases_tac ctxt) (Thm.assume cprop RS rl))
   560       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   561   in
   562     (case get_first (try mk_elim) elims of
   563       SOME r => r
   564     | NONE => err "Proposition not an inductive predicate:")
   565   end;
   566 
   567 end;
   568 
   569 
   570 (* inductive_cases *)
   571 
   572 fun gen_inductive_cases prep_att prep_prop args lthy =
   573   let
   574     val thy = Proof_Context.theory_of lthy;
   575     val thmss =
   576       map snd args
   577       |> burrow (grouped 10 Par_List.map (mk_cases lthy o prep_prop lthy));
   578     val facts =
   579       map2 (fn ((a, atts), _) => fn thms => ((a, map (prep_att thy) atts), [(thms, [])]))
   580         args thmss;
   581   in lthy |> Local_Theory.notes facts |>> map snd end;
   582 
   583 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
   584 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   585 
   586 
   587 val ind_cases_setup =
   588   Method.setup @{binding ind_cases}
   589     (Scan.lift (Scan.repeat1 Args.name_source --
   590       Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.binding) []) >>
   591       (fn (raw_props, fixes) => fn ctxt =>
   592         let
   593           val (_, ctxt') = Variable.add_fixes_binding fixes ctxt;
   594           val props = Syntax.read_props ctxt' raw_props;
   595           val ctxt'' = fold Variable.declare_term props ctxt';
   596           val rules = Proof_Context.export ctxt'' ctxt (map (mk_cases ctxt'') props)
   597         in Method.erule 0 rules end))
   598     "dynamic case analysis on predicates";
   599 
   600 
   601 (* derivation of simplified equation *)
   602 
   603 fun mk_simp_eq ctxt prop =
   604   let
   605     val thy = Proof_Context.theory_of ctxt;
   606     val ctxt' = Variable.auto_fixes prop ctxt;
   607     val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of;
   608     val substs =
   609       Item_Net.retrieve (get_equations ctxt) (HOLogic.dest_Trueprop prop)
   610       |> map_filter
   611         (fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
   612             (Vartab.empty, Vartab.empty), eq)
   613           handle Pattern.MATCH => NONE);
   614     val (subst, eq) =
   615       (case substs of
   616         [s] => s
   617       | _ => error
   618         ("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique"));
   619     val inst =
   620       map (fn v => (cterm_of thy (Var v), cterm_of thy (Envir.subst_term subst (Var v))))
   621         (Term.add_vars (lhs_of eq) []);
   622   in
   623     Drule.cterm_instantiate inst eq
   624     |> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv (Simplifier.full_rewrite ctxt)))
   625     |> singleton (Variable.export ctxt' ctxt)
   626   end
   627 
   628 
   629 (* inductive simps *)
   630 
   631 fun gen_inductive_simps prep_att prep_prop args lthy =
   632   let
   633     val thy = Proof_Context.theory_of lthy;
   634     val facts = args |> map (fn ((a, atts), props) =>
   635       ((a, map (prep_att thy) atts),
   636         map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
   637   in lthy |> Local_Theory.notes facts |>> map snd end;
   638 
   639 val inductive_simps = gen_inductive_simps Attrib.intern_src Syntax.read_prop;
   640 val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
   641 
   642 
   643 (* prove induction rule *)
   644 
   645 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
   646     fp_def rec_preds_defs ctxt ctxt''' =  (* FIXME ctxt''' ?? *)
   647   let
   648     val _ = clean_message ctxt quiet_mode "  Proving the induction rule ...";
   649 
   650     (* predicates for induction rule *)
   651 
   652     val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
   653     val preds =
   654       map2 (curry Free) pnames
   655         (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
   656 
   657     (* transform an introduction rule into a premise for induction rule *)
   658 
   659     fun mk_ind_prem r =
   660       let
   661         fun subst s =
   662           (case dest_predicate cs params s of
   663             SOME (_, i, ys, (_, Ts)) =>
   664               let
   665                 val k = length Ts;
   666                 val bs = map Bound (k - 1 downto 0);
   667                 val P = list_comb (nth preds i, map (incr_boundvars k) ys @ bs);
   668                 val Q =
   669                   fold_rev Term.abs (mk_names "x" k ~~ Ts)
   670                     (HOLogic.mk_binop inductive_conj_name
   671                       (list_comb (incr_boundvars k s, bs), P));
   672               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   673           | NONE =>
   674               (case s of
   675                 t $ u => (fst (subst t) $ fst (subst u), NONE)
   676               | Abs (a, T, t) => (Abs (a, T, fst (subst t)), NONE)
   677               | _ => (s, NONE)));
   678 
   679         fun mk_prem s prems =
   680           (case subst s of
   681             (_, SOME (t, u)) => t :: u :: prems
   682           | (t, _) => t :: prems);
   683 
   684         val SOME (_, i, ys, _) =
   685           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   686       in
   687         fold_rev (Logic.all o Free) (Logic.strip_params r)
   688           (Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
   689             (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
   690               HOLogic.mk_Trueprop (list_comb (nth preds i, ys))))
   691       end;
   692 
   693     val ind_prems = map mk_ind_prem intr_ts;
   694 
   695 
   696     (* make conclusions for induction rules *)
   697 
   698     val Tss = map (binder_types o fastype_of) preds;
   699     val (xnames, ctxt'') = Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   700     val mutual_ind_concl =
   701       HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   702         (map (fn (((xnames, Ts), c), P) =>
   703           let val frees = map Free (xnames ~~ Ts)
   704           in HOLogic.mk_imp (list_comb (c, params @ frees), list_comb (P, frees)) end)
   705         (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   706 
   707 
   708     (* make predicate for instantiation of abstract induction rule *)
   709 
   710     val ind_pred =
   711       fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   712         (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
   713            (make_bool_args HOLogic.mk_not I bs i)
   714            (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
   715 
   716     val ind_concl =
   717       HOLogic.mk_Trueprop
   718         (HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
   719 
   720     val raw_fp_induct = mono RS (fp_def RS @{thm def_lfp_induct});
   721 
   722     val induct = Goal.prove_sorry ctxt'' [] ind_prems ind_concl
   723       (fn {prems, ...} => EVERY
   724         [rewrite_goals_tac [inductive_conj_def],
   725          DETERM (rtac raw_fp_induct 1),
   726          REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
   727          rewrite_goals_tac simp_thms2,
   728          (*This disjE separates out the introduction rules*)
   729          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
   730          (*Now break down the individual cases.  No disjE here in case
   731            some premise involves disjunction.*)
   732          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac ctxt'')),
   733          REPEAT (FIRSTGOAL
   734            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
   735          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
   736              (inductive_conj_def :: rec_preds_defs @ simp_thms2) prem,
   737            conjI, refl] 1)) prems)]);
   738 
   739     val lemma = Goal.prove_sorry ctxt'' [] []
   740       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
   741         [rewrite_goals_tac rec_preds_defs,
   742          REPEAT (EVERY
   743            [REPEAT (resolve_tac [conjI, impI] 1),
   744             REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
   745             atac 1,
   746             rewrite_goals_tac simp_thms1,
   747             atac 1])]);
   748 
   749   in singleton (Proof_Context.export ctxt'' ctxt''') (induct RS lemma) end;
   750 
   751 
   752 
   753 (** specification of (co)inductive predicates **)
   754 
   755 fun mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts monos params cnames_syn lthy =
   756   let
   757     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
   758 
   759     val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
   760     val k = log 2 1 (length cs);
   761     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   762     val p :: xs =
   763       map Free (Variable.variant_frees lthy intr_ts
   764         (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   765     val bs =
   766       map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
   767         (map (rpair HOLogic.boolT) (mk_names "b" k)));
   768 
   769     fun subst t =
   770       (case dest_predicate cs params t of
   771         SOME (_, i, ts, (Ts, Us)) =>
   772           let
   773             val l = length Us;
   774             val zs = map Bound (l - 1 downto 0);
   775           in
   776             fold_rev (Term.abs o pair "z") Us
   777               (list_comb (p,
   778                 make_bool_args' bs i @ make_args argTs
   779                   ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   780           end
   781       | NONE =>
   782           (case t of
   783             t1 $ t2 => subst t1 $ subst t2
   784           | Abs (x, T, u) => Abs (x, T, subst u)
   785           | _ => t));
   786 
   787     (* transform an introduction rule into a conjunction  *)
   788     (*   [| p_i t; ... |] ==> p_j u                       *)
   789     (* is transformed into                                *)
   790     (*   b_j & x_j = u & p b_j t & ...                    *)
   791 
   792     fun transform_rule r =
   793       let
   794         val SOME (_, i, ts, (Ts, _)) =
   795           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   796         val ps =
   797           make_bool_args HOLogic.mk_not I bs i @
   798           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   799           map (subst o HOLogic.dest_Trueprop) (Logic.strip_assums_hyp r);
   800       in
   801         fold_rev (fn (x, T) => fn P => HOLogic.exists_const T $ Abs (x, T, P))
   802           (Logic.strip_params r)
   803           (if null ps then @{term True} else foldr1 HOLogic.mk_conj ps)
   804       end;
   805 
   806     (* make a disjunction of all introduction rules *)
   807 
   808     val fp_fun =
   809       fold_rev lambda (p :: bs @ xs)
   810         (if null intr_ts then @{term False}
   811          else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   812 
   813     (* add definiton of recursive predicates to theory *)
   814 
   815     val rec_name =
   816       if Binding.is_empty alt_name then
   817         Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
   818       else alt_name;
   819 
   820     val ((rec_const, (_, fp_def)), lthy') = lthy
   821       |> Local_Theory.conceal
   822       |> Local_Theory.define
   823         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   824          ((Thm.def_binding rec_name, @{attributes [nitpick_unfold]}),
   825            fold_rev lambda params
   826              (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)))
   827       ||> Local_Theory.restore_naming lthy;
   828     val fp_def' =
   829       Simplifier.rewrite (put_simpset HOL_basic_ss lthy' addsimps [fp_def])
   830         (cterm_of (Proof_Context.theory_of lthy') (list_comb (rec_const, params)));
   831     val specs =
   832       if length cs < 2 then []
   833       else
   834         map_index (fn (i, (name_mx, c)) =>
   835           let
   836             val Ts = arg_types_of (length params) c;
   837             val xs =
   838               map Free (Variable.variant_frees lthy intr_ts (mk_names "x" (length Ts) ~~ Ts));
   839           in
   840             (name_mx, (apfst Binding.conceal Attrib.empty_binding, fold_rev lambda (params @ xs)
   841               (list_comb (rec_const, params @ make_bool_args' bs i @
   842                 make_args argTs (xs ~~ Ts)))))
   843           end) (cnames_syn ~~ cs);
   844     val (consts_defs, lthy'') = lthy'
   845       |> fold_map Local_Theory.define specs;
   846     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   847 
   848     val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
   849     val mono = prove_mono quiet_mode skip_mono predT fp_fun monos lthy''';
   850     val (_, lthy'''') =
   851       Local_Theory.note (apfst Binding.conceal Attrib.empty_binding,
   852         Proof_Context.export lthy''' lthy'' [mono]) lthy'';
   853 
   854   in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
   855     list_comb (rec_const, params), preds, argTs, bs, xs)
   856   end;
   857 
   858 fun declare_rules rec_binding coind no_ind cnames
   859     preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
   860   let
   861     val rec_name = Binding.name_of rec_binding;
   862     fun rec_qualified qualified = Binding.qualify qualified rec_name;
   863     val intr_names = map Binding.name_of intr_bindings;
   864     val ind_case_names = Rule_Cases.case_names intr_names;
   865     val induct =
   866       if coind then
   867         (raw_induct,
   868          [Rule_Cases.case_names [rec_name],
   869           Rule_Cases.case_conclusion (rec_name, intr_names),
   870           Rule_Cases.consumes (1 - Thm.nprems_of raw_induct),
   871           Induct.coinduct_pred (hd cnames)])
   872       else if no_ind orelse length cnames > 1 then
   873         (raw_induct,
   874           [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of raw_induct))])
   875       else
   876         (raw_induct RSN (2, rev_mp),
   877           [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of raw_induct))]);
   878 
   879     val (intrs', lthy1) =
   880       lthy |>
   881       Spec_Rules.add
   882         (if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
   883       Local_Theory.notes
   884         (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
   885           map (fn th => [([th],
   886            [Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
   887       map (hd o snd);
   888     val (((_, elims'), (_, [induct'])), lthy2) =
   889       lthy1 |>
   890       Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
   891       fold_map (fn (name, (elim, cases, k)) =>
   892         Local_Theory.note
   893           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
   894             [Attrib.internal (K (Rule_Cases.case_names cases)),
   895              Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of elim))),
   896              Attrib.internal (K (Rule_Cases.constraints k)),
   897              Attrib.internal (K (Induct.cases_pred name)),
   898              Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
   899         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   900       Local_Theory.note
   901         ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
   902           map (Attrib.internal o K) (#2 induct)), [rulify lthy1 (#1 induct)]);
   903 
   904     val (eqs', lthy3) = lthy2 |>
   905       fold_map (fn (name, eq) => Local_Theory.note
   906           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
   907             [Attrib.internal (K equation_add_permissive)]), [eq])
   908           #> apfst (hd o snd))
   909         (if null eqs then [] else (cnames ~~ eqs))
   910     val (inducts, lthy4) =
   911       if no_ind orelse coind then ([], lthy3)
   912       else
   913         let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
   914           lthy3 |>
   915           Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
   916             inducts |> map (fn (name, th) => ([th],
   917               [Attrib.internal (K ind_case_names),
   918                Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of th))),
   919                Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
   920         end;
   921   in (intrs', elims', eqs', induct', inducts, lthy4) end;
   922 
   923 type inductive_flags =
   924   {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
   925     no_elim: bool, no_ind: bool, skip_mono: bool};
   926 
   927 type add_ind_def =
   928   inductive_flags ->
   929   term list -> (Attrib.binding * term) list -> thm list ->
   930   term list -> (binding * mixfix) list ->
   931   local_theory -> inductive_result * local_theory;
   932 
   933 fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono}
   934     cs intros monos params cnames_syn lthy =
   935   let
   936     val _ = null cnames_syn andalso error "No inductive predicates given";
   937     val names = map (Binding.name_of o fst) cnames_syn;
   938     val _ = message (quiet_mode andalso not verbose)
   939       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
   940 
   941     val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
   942     val ((intr_names, intr_atts), intr_ts) =
   943       apfst split_list (split_list (map (check_rule lthy cs params) intros));
   944 
   945     val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   946       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts
   947         monos params cnames_syn lthy;
   948 
   949     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
   950       intr_ts rec_preds_defs lthy2 lthy1;
   951     val elims =
   952       if no_elim then []
   953       else
   954         prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
   955           unfold rec_preds_defs lthy2 lthy1;
   956     val raw_induct = zero_var_indexes
   957       (if no_ind then Drule.asm_rl
   958        else if coind then
   959          singleton (Proof_Context.export lthy2 lthy1)
   960            (rotate_prems ~1 (Object_Logic.rulify
   961              (fold_rule rec_preds_defs
   962                (rewrite_rule simp_thms3
   963                 (mono RS (fp_def RS @{thm def_coinduct}))))))
   964        else
   965          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
   966            rec_preds_defs lthy2 lthy1);
   967     val eqs =
   968       if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1;
   969 
   970     val elims' = map (fn (th, ns, i) => (rulify lthy1 th, ns, i)) elims;
   971     val intrs' = map (rulify lthy1) intrs;
   972 
   973     val (intrs'', elims'', eqs', induct, inducts, lthy3) =
   974       declare_rules rec_name coind no_ind
   975         cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
   976 
   977     val result =
   978       {preds = preds,
   979        intrs = intrs'',
   980        elims = elims'',
   981        raw_induct = rulify lthy3 raw_induct,
   982        induct = induct,
   983        inducts = inducts,
   984        eqs = eqs'};
   985 
   986     val lthy4 = lthy3
   987       |> Local_Theory.declaration {syntax = false, pervasive = false} (fn phi =>
   988         let val result' = transform_result phi result;
   989         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
   990   in (result, lthy4) end;
   991 
   992 
   993 (* external interfaces *)
   994 
   995 fun gen_add_inductive_i mk_def
   996     (flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono})
   997     cnames_syn pnames spec monos lthy =
   998   let
   999     val thy = Proof_Context.theory_of lthy;
  1000     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
  1001 
  1002 
  1003     (* abbrevs *)
  1004 
  1005     val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
  1006 
  1007     fun get_abbrev ((name, atts), t) =
  1008       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
  1009         let
  1010           val _ = Binding.is_empty name andalso null atts orelse
  1011             error "Abbreviations may not have names or attributes";
  1012           val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
  1013           val var =
  1014             (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
  1015               NONE => error ("Undeclared head of abbreviation " ^ quote x)
  1016             | SOME ((b, T'), mx) =>
  1017                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
  1018                 else (b, mx));
  1019         in SOME (var, rhs) end
  1020       else NONE;
  1021 
  1022     val abbrevs = map_filter get_abbrev spec;
  1023     val bs = map (Binding.name_of o fst o fst) abbrevs;
  1024 
  1025 
  1026     (* predicates *)
  1027 
  1028     val pre_intros = filter_out (is_some o get_abbrev) spec;
  1029     val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
  1030     val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
  1031     val ps = map Free pnames;
  1032 
  1033     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
  1034     val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
  1035     val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
  1036     val expand = Assumption.export_term ctxt3 lthy #> Proof_Context.cert_term lthy;
  1037 
  1038     fun close_rule r =
  1039       fold (Logic.all o Free) (fold_aterms
  1040         (fn t as Free (v as (s, _)) =>
  1041             if Variable.is_fixed ctxt1 s orelse
  1042               member (op =) ps t then I else insert (op =) v
  1043           | _ => I) r []) r;
  1044 
  1045     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
  1046     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
  1047   in
  1048     lthy
  1049     |> mk_def flags cs intros monos ps preds
  1050     ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
  1051   end;
  1052 
  1053 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
  1054   let
  1055     val ((vars, intrs), _) = lthy
  1056       |> Proof_Context.set_mode Proof_Context.mode_abbrev
  1057       |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
  1058     val (cs, ps) = chop (length cnames_syn) vars;
  1059     val monos = Attrib.eval_thms lthy raw_monos;
  1060     val flags =
  1061      {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
  1062       coind = coind, no_elim = false, no_ind = false, skip_mono = false};
  1063   in
  1064     lthy
  1065     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
  1066   end;
  1067 
  1068 val add_inductive_i = gen_add_inductive_i add_ind_def;
  1069 val add_inductive = gen_add_inductive add_ind_def;
  1070 
  1071 fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
  1072   let
  1073     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
  1074     val ctxt' = thy
  1075       |> Named_Target.theory_init
  1076       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
  1077       |> Local_Theory.exit;
  1078     val info = #2 (the_inductive ctxt' name);
  1079   in (info, Proof_Context.theory_of ctxt') end;
  1080 
  1081 
  1082 (* read off arities of inductive predicates from raw induction rule *)
  1083 fun arities_of induct =
  1084   map (fn (_ $ t $ u) =>
  1085       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
  1086     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
  1087 
  1088 (* read off parameters of inductive predicate from raw induction rule *)
  1089 fun params_of induct =
  1090   let
  1091     val (_ $ t $ u :: _) = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
  1092     val (_, ts) = strip_comb t;
  1093     val (_, us) = strip_comb u;
  1094   in
  1095     List.take (ts, length ts - length us)
  1096   end;
  1097 
  1098 val pname_of_intr =
  1099   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
  1100 
  1101 (* partition introduction rules according to predicate name *)
  1102 fun gen_partition_rules f induct intros =
  1103   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
  1104     (map (rpair [] o fst) (arities_of induct));
  1105 
  1106 val partition_rules = gen_partition_rules I;
  1107 fun partition_rules' induct = gen_partition_rules fst induct;
  1108 
  1109 fun unpartition_rules intros xs =
  1110   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
  1111     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
  1112 
  1113 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
  1114 fun infer_intro_vars elim arity intros =
  1115   let
  1116     val thy = theory_of_thm elim;
  1117     val _ :: cases = prems_of elim;
  1118     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
  1119     fun mtch (t, u) =
  1120       let
  1121         val params = Logic.strip_params t;
  1122         val vars =
  1123           map (Var o apfst (rpair 0))
  1124             (Name.variant_list used (map fst params) ~~ map snd params);
  1125         val ts =
  1126           map (curry subst_bounds (rev vars))
  1127             (List.drop (Logic.strip_assums_hyp t, arity));
  1128         val us = Logic.strip_imp_prems u;
  1129         val tab =
  1130           fold (Pattern.first_order_match thy) (ts ~~ us) (Vartab.empty, Vartab.empty);
  1131       in
  1132         map (Envir.subst_term tab) vars
  1133       end
  1134   in
  1135     map (mtch o apsnd prop_of) (cases ~~ intros)
  1136   end;
  1137 
  1138 
  1139 
  1140 (** package setup **)
  1141 
  1142 (* setup theory *)
  1143 
  1144 val setup =
  1145   ind_cases_setup #>
  1146   Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
  1147     "declaration of monotonicity rule";
  1148 
  1149 
  1150 (* outer syntax *)
  1151 
  1152 fun gen_ind_decl mk_def coind =
  1153   Parse.fixes -- Parse.for_fixes --
  1154   Scan.optional Parse_Spec.where_alt_specs [] --
  1155   Scan.optional (@{keyword "monos"} |-- Parse.!!! Parse_Spec.xthms1) []
  1156   >> (fn (((preds, params), specs), monos) =>
  1157       (snd o gen_add_inductive mk_def true coind preds params specs monos));
  1158 
  1159 val ind_decl = gen_ind_decl add_ind_def;
  1160 
  1161 val _ =
  1162   Outer_Syntax.local_theory @{command_spec "inductive"} "define inductive predicates"
  1163     (ind_decl false);
  1164 
  1165 val _ =
  1166   Outer_Syntax.local_theory @{command_spec "coinductive"} "define coinductive predicates"
  1167     (ind_decl true);
  1168 
  1169 val _ =
  1170   Outer_Syntax.local_theory @{command_spec "inductive_cases"}
  1171     "create simplified instances of elimination rules"
  1172     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
  1173 
  1174 val _ =
  1175   Outer_Syntax.local_theory @{command_spec "inductive_simps"}
  1176     "create simplification rules for inductive predicates"
  1177     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
  1178 
  1179 val _ =
  1180   Outer_Syntax.improper_command @{command_spec "print_inductives"}
  1181     "print (co)inductive definitions and monotonicity rules"
  1182     (Scan.succeed (Toplevel.unknown_context o
  1183       Toplevel.keep (print_inductives o Toplevel.context_of)));
  1184 
  1185 end;