src/HOL/Tools/inductive.ML
author wenzelm
Wed Mar 27 14:19:18 2013 +0100 (2013-03-27)
changeset 51551 88d1d19fb74f
parent 50771 2852f997bfb5
child 51580 64ef8260dc60
permissions -rw-r--r--
tuned signature and module arrangement;
     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: Proof.context -> term -> thm
    34   val inductive_forall_def: thm
    35   val rulify: thm -> thm
    36   val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
    37     thm list list * local_theory
    38   val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
    39     thm list list * local_theory
    40   type inductive_flags =
    41     {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
    42       no_elim: bool, no_ind: bool, skip_mono: bool}
    43   val add_inductive_i:
    44     inductive_flags -> ((binding * typ) * mixfix) list ->
    45     (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
    46     inductive_result * local_theory
    47   val add_inductive: bool -> bool ->
    48     (binding * string option * mixfix) list ->
    49     (binding * string option * mixfix) list ->
    50     (Attrib.binding * string) list ->
    51     (Facts.ref * Attrib.src list) list ->
    52     local_theory -> inductive_result * local_theory
    53   val add_inductive_global: inductive_flags ->
    54     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    55     thm list -> theory -> inductive_result * theory
    56   val arities_of: thm -> (string * int) list
    57   val params_of: thm -> term list
    58   val partition_rules: thm -> thm list -> (string * thm list) list
    59   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
    60   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    61   val infer_intro_vars: thm -> int -> thm list -> term list list
    62   val setup: theory -> theory
    63 end;
    64 
    65 signature INDUCTIVE =
    66 sig
    67   include BASIC_INDUCTIVE
    68   type add_ind_def =
    69     inductive_flags ->
    70     term list -> (Attrib.binding * term) list -> thm list ->
    71     term list -> (binding * mixfix) list ->
    72     local_theory -> inductive_result * local_theory
    73   val declare_rules: binding -> bool -> bool -> string list -> term list ->
    74     thm list -> binding list -> Attrib.src list list -> (thm * string list * int) list ->
    75     thm list -> thm -> local_theory -> thm list * thm list * thm list * thm * thm list * local_theory
    76   val add_ind_def: add_ind_def
    77   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
    78     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    79     thm list -> local_theory -> inductive_result * local_theory
    80   val gen_add_inductive: add_ind_def -> bool -> bool ->
    81     (binding * string option * mixfix) list ->
    82     (binding * string option * mixfix) list ->
    83     (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
    84     local_theory -> inductive_result * local_theory
    85   val gen_ind_decl: add_ind_def -> bool -> (local_theory -> local_theory) parser
    86 end;
    87 
    88 structure Inductive: INDUCTIVE =
    89 struct
    90 
    91 (** theory context references **)
    92 
    93 val inductive_forall_def = @{thm induct_forall_def};
    94 val inductive_conj_name = "HOL.induct_conj";
    95 val inductive_conj_def = @{thm induct_conj_def};
    96 val inductive_conj = @{thms induct_conj};
    97 val inductive_atomize = @{thms induct_atomize};
    98 val inductive_rulify = @{thms induct_rulify};
    99 val inductive_rulify_fallback = @{thms induct_rulify_fallback};
   100 
   101 val simp_thms1 =
   102   map mk_meta_eq
   103     @{lemma "(~ True) = False" "(~ False) = True"
   104         "(True --> P) = P" "(False --> P) = True"
   105         "(P & True) = P" "(True & P) = P"
   106       by (fact simp_thms)+};
   107 
   108 val simp_thms2 =
   109   map mk_meta_eq [@{thm inf_fun_def}, @{thm inf_bool_def}] @ simp_thms1;
   110 
   111 val simp_thms3 =
   112   map mk_meta_eq [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_def}, @{thm sup_bool_def}];
   113 
   114 
   115 
   116 (** misc utilities **)
   117 
   118 fun message quiet_mode s = if quiet_mode then () else writeln s;
   119 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
   120 
   121 fun coind_prefix true = "co"
   122   | coind_prefix false = "";
   123 
   124 fun log (b: int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   125 
   126 fun make_bool_args f g [] i = []
   127   | make_bool_args f g (x :: xs) i =
   128       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   129 
   130 fun make_bool_args' xs =
   131   make_bool_args (K @{term False}) (K @{term True}) xs;
   132 
   133 fun arg_types_of k c = drop k (binder_types (fastype_of c));
   134 
   135 fun find_arg T x [] = raise Fail "find_arg"
   136   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   137       apsnd (cons p) (find_arg T x ps)
   138   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   139       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   140       else apsnd (cons p) (find_arg T x ps);
   141 
   142 fun make_args Ts xs =
   143   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
   144     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   145 
   146 fun make_args' Ts xs Us =
   147   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   148 
   149 fun dest_predicate cs params t =
   150   let
   151     val k = length params;
   152     val (c, ts) = strip_comb t;
   153     val (xs, ys) = chop k ts;
   154     val i = find_index (fn c' => c' = c) cs;
   155   in
   156     if xs = params andalso i >= 0 then
   157       SOME (c, i, ys, chop (length ys) (arg_types_of k c))
   158     else NONE
   159   end;
   160 
   161 fun mk_names a 0 = []
   162   | mk_names a 1 = [a]
   163   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   164 
   165 fun select_disj 1 1 = []
   166   | select_disj _ 1 = [rtac disjI1]
   167   | select_disj n i = rtac disjI2 :: select_disj (n - 1) (i - 1);
   168 
   169 
   170 
   171 (** context data **)
   172 
   173 type inductive_result =
   174   {preds: term list, elims: thm list, raw_induct: thm,
   175    induct: thm, inducts: thm list, intrs: thm list, eqs: thm list};
   176 
   177 fun transform_result phi {preds, elims, raw_induct: thm, induct, inducts, intrs, eqs} =
   178   let
   179     val term = Morphism.term phi;
   180     val thm = Morphism.thm phi;
   181     val fact = Morphism.fact phi;
   182   in
   183    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
   184     induct = thm induct, inducts = fact inducts, intrs = fact intrs, eqs = fact eqs}
   185   end;
   186 
   187 type inductive_info = {names: string list, coind: bool} * inductive_result;
   188 
   189 val empty_equations =
   190   Item_Net.init Thm.eq_thm_prop
   191     (single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
   192 
   193 datatype data = Data of
   194  {infos: inductive_info Symtab.table,
   195   monos: thm list,
   196   equations: thm Item_Net.T};
   197 
   198 fun make_data (infos, monos, equations) =
   199   Data {infos = infos, monos = monos, equations = equations};
   200 
   201 structure Data = Generic_Data
   202 (
   203   type T = data;
   204   val empty = make_data (Symtab.empty, [], empty_equations);
   205   val extend = I;
   206   fun merge (Data {infos = infos1, monos = monos1, equations = equations1},
   207       Data {infos = infos2, monos = monos2, equations = equations2}) =
   208     make_data (Symtab.merge (K true) (infos1, infos2),
   209       Thm.merge_thms (monos1, monos2),
   210       Item_Net.merge (equations1, equations2));
   211 );
   212 
   213 fun map_data f =
   214   Data.map (fn Data {infos, monos, equations} => make_data (f (infos, monos, equations)));
   215 
   216 fun rep_data ctxt = Data.get (Context.Proof ctxt) |> (fn Data rep => rep);
   217 
   218 fun print_inductives ctxt =
   219   let
   220     val {infos, monos, ...} = rep_data ctxt;
   221     val space = Consts.space_of (Proof_Context.consts_of ctxt);
   222   in
   223     [Pretty.block
   224       (Pretty.breaks
   225         (Pretty.str "(co)inductives:" ::
   226           map (Pretty.mark_str o #1) (Name_Space.extern_table ctxt (space, infos)))),
   227      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm ctxt) monos)]
   228   end |> Pretty.chunks |> Pretty.writeln;
   229 
   230 
   231 (* inductive info *)
   232 
   233 fun the_inductive ctxt name =
   234   (case Symtab.lookup (#infos (rep_data ctxt)) name of
   235     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   236   | SOME info => info);
   237 
   238 fun put_inductives names info =
   239   map_data (fn (infos, monos, equations) =>
   240     (fold (fn name => Symtab.update (name, info)) names infos, monos, equations));
   241 
   242 
   243 (* monotonicity rules *)
   244 
   245 val get_monos = #monos o rep_data;
   246 
   247 fun mk_mono ctxt thm =
   248   let
   249     fun eq_to_mono thm' = thm' RS (thm' RS @{thm eq_to_mono});
   250     fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
   251       handle THM _ => thm RS @{thm le_boolD}
   252   in
   253     (case concl_of thm of
   254       Const ("==", _) $ _ $ _ => eq_to_mono (thm RS meta_eq_to_obj_eq)
   255     | _ $ (Const (@{const_name HOL.eq}, _) $ _ $ _) => eq_to_mono thm
   256     | _ $ (Const (@{const_name Orderings.less_eq}, _) $ _ $ _) =>
   257       dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   258         (resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))
   259     | _ => thm)
   260   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm ctxt thm);
   261 
   262 val mono_add =
   263   Thm.declaration_attribute (fn thm => fn context =>
   264     map_data (fn (infos, monos, equations) =>
   265       (infos, Thm.add_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
   266 
   267 val mono_del =
   268   Thm.declaration_attribute (fn thm => fn context =>
   269     map_data (fn (infos, monos, equations) =>
   270       (infos, Thm.del_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
   271 
   272 
   273 (* equations *)
   274 
   275 val get_equations = #equations o rep_data;
   276 
   277 val equation_add_permissive =
   278   Thm.declaration_attribute (fn thm =>
   279     map_data (fn (infos, monos, equations) =>
   280       (infos, monos, perhaps (try (Item_Net.update thm)) equations)));
   281 
   282 
   283 
   284 (** process rules **)
   285 
   286 local
   287 
   288 fun err_in_rule ctxt name t msg =
   289   error (cat_lines ["Ill-formed introduction rule " ^ Binding.print name,
   290     Syntax.string_of_term ctxt t, msg]);
   291 
   292 fun err_in_prem ctxt name t p msg =
   293   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
   294     "in introduction rule " ^ Binding.print name, Syntax.string_of_term ctxt t, msg]);
   295 
   296 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   297 
   298 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   299 
   300 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   301 
   302 fun atomize_term thy = Raw_Simplifier.rewrite_term thy inductive_atomize [];
   303 
   304 in
   305 
   306 fun check_rule ctxt cs params ((binding, att), rule) =
   307   let
   308     val params' = Term.variant_frees rule (Logic.strip_params rule);
   309     val frees = rev (map Free params');
   310     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   311     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   312     val rule' = Logic.list_implies (prems, concl);
   313     val aprems = map (atomize_term (Proof_Context.theory_of ctxt)) prems;
   314     val arule = fold_rev (Logic.all o Free) params' (Logic.list_implies (aprems, concl));
   315 
   316     fun check_ind err t =
   317       (case dest_predicate cs params t of
   318         NONE => err (bad_app ^
   319           commas (map (Syntax.string_of_term ctxt) params))
   320       | SOME (_, _, ys, _) =>
   321           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   322           then err bad_ind_occ else ());
   323 
   324     fun check_prem' prem t =
   325       if member (op =) cs (head_of t) then
   326         check_ind (err_in_prem ctxt binding rule prem) t
   327       else
   328         (case t of
   329           Abs (_, _, t) => check_prem' prem t
   330         | t $ u => (check_prem' prem t; check_prem' prem u)
   331         | _ => ());
   332 
   333     fun check_prem (prem, aprem) =
   334       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   335       else err_in_prem ctxt binding rule prem "Non-atomic premise";
   336 
   337     val _ =
   338       (case concl of
   339         Const (@{const_name Trueprop}, _) $ t =>
   340           if member (op =) cs (head_of t) then
   341            (check_ind (err_in_rule ctxt binding rule') t;
   342             List.app check_prem (prems ~~ aprems))
   343           else err_in_rule ctxt binding rule' bad_concl
   344        | _ => err_in_rule ctxt binding rule' bad_concl);
   345   in
   346     ((binding, att), arule)
   347   end;
   348 
   349 val rulify =
   350   hol_simplify inductive_conj
   351   #> hol_simplify inductive_rulify
   352   #> hol_simplify inductive_rulify_fallback
   353   #> Simplifier.norm_hhf;
   354 
   355 end;
   356 
   357 
   358 
   359 (** proofs for (co)inductive predicates **)
   360 
   361 (* prove monotonicity *)
   362 
   363 fun prove_mono quiet_mode skip_mono predT fp_fun monos ctxt =
   364  (message (quiet_mode orelse skip_mono andalso ! quick_and_dirty)
   365     "  Proving monotonicity ...";
   366   (if skip_mono then Goal.prove_sorry else Goal.prove_future) ctxt
   367     [] []
   368     (HOLogic.mk_Trueprop
   369       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   370     (fn _ => EVERY [rtac @{thm monoI} 1,
   371       REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
   372       REPEAT (FIRST
   373         [atac 1,
   374          resolve_tac (map (mk_mono ctxt) monos @ get_monos ctxt) 1,
   375          etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
   376 
   377 
   378 (* prove introduction rules *)
   379 
   380 fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
   381   let
   382     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
   383 
   384     val unfold = funpow k (fn th => th RS fun_cong)
   385       (mono RS (fp_def RS
   386         (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
   387 
   388     val rules = [refl, TrueI, @{lemma "~ False" by (rule notI)}, exI, conjI];
   389 
   390     val intrs = map_index (fn (i, intr) =>
   391       Goal.prove_sorry ctxt [] [] intr (fn _ => EVERY
   392        [rewrite_goals_tac rec_preds_defs,
   393         rtac (unfold RS iffD2) 1,
   394         EVERY1 (select_disj (length intr_ts) (i + 1)),
   395         (*Not ares_tac, since refl must be tried before any equality assumptions;
   396           backtracking may occur if the premises have extra variables!*)
   397         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)])
   398        |> singleton (Proof_Context.export ctxt ctxt')) intr_ts
   399 
   400   in (intrs, unfold) end;
   401 
   402 
   403 (* prove elimination rules *)
   404 
   405 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
   406   let
   407     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
   408 
   409     val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
   410     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   411 
   412     fun dest_intr r =
   413       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   414        Logic.strip_assums_hyp r, Logic.strip_params r);
   415 
   416     val intrs = map dest_intr intr_ts ~~ intr_names;
   417 
   418     val rules1 = [disjE, exE, FalseE];
   419     val rules2 = [conjE, FalseE, @{lemma "~ True ==> R" by (rule notE [OF _ TrueI])}];
   420 
   421     fun prove_elim c =
   422       let
   423         val Ts = arg_types_of (length params) c;
   424         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   425         val frees = map Free (anames ~~ Ts);
   426 
   427         fun mk_elim_prem ((_, _, us, _), ts, params') =
   428           Logic.list_all (params',
   429             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   430               (frees ~~ us) @ ts, P));
   431         val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
   432         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   433            map mk_elim_prem (map #1 c_intrs)
   434       in
   435         (Goal.prove_sorry ctxt'' [] prems P
   436           (fn {prems, ...} => EVERY
   437             [cut_tac (hd prems) 1,
   438              rewrite_goals_tac rec_preds_defs,
   439              dtac (unfold RS iffD1) 1,
   440              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   441              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   442              EVERY (map (fn prem =>
   443                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
   444           |> singleton (Proof_Context.export ctxt'' ctxt'''),
   445          map #2 c_intrs, length Ts)
   446       end
   447 
   448    in map prove_elim cs end;
   449 
   450 
   451 (* prove simplification equations *)
   452 
   453 fun prove_eqs quiet_mode cs params intr_ts intrs
   454     (elims: (thm * bstring list * int) list) ctxt ctxt'' =  (* FIXME ctxt'' ?? *)
   455   let
   456     val _ = clean_message quiet_mode "  Proving the simplification rules ...";
   457 
   458     fun dest_intr r =
   459       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   460        Logic.strip_assums_hyp r, Logic.strip_params r);
   461     val intr_ts' = map dest_intr intr_ts;
   462 
   463     fun prove_eq c (elim: thm * 'a * 'b) =
   464       let
   465         val Ts = arg_types_of (length params) c;
   466         val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
   467         val frees = map Free (anames ~~ Ts);
   468         val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
   469         fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
   470           let
   471             fun list_ex ([], t) = t
   472               | list_ex ((a, T) :: vars, t) =
   473                   HOLogic.exists_const T $ Abs (a, T, list_ex (vars, t));
   474             val conjs = map2 (curry HOLogic.mk_eq) frees us @ map HOLogic.dest_Trueprop ts;
   475           in
   476             list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
   477           end;
   478         val lhs = list_comb (c, params @ frees);
   479         val rhs =
   480           if null c_intrs then @{term False}
   481           else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs);
   482         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
   483         fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
   484             EVERY1 (select_disj (length c_intrs) (i + 1)) THEN
   485             EVERY (replicate (length params) (rtac @{thm exI} 1)) THEN
   486             (if null prems then rtac @{thm TrueI} 1
   487              else
   488               let
   489                 val (prems', last_prem) = split_last prems;
   490               in
   491                 EVERY (map (fn prem => (rtac @{thm conjI} 1 THEN rtac prem 1)) prems') THEN
   492                 rtac last_prem 1
   493               end)) ctxt' 1;
   494         fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
   495           EVERY (replicate (length params') (etac @{thm exE} 1)) THEN
   496           (if null ts andalso null us then rtac intr 1
   497            else
   498             EVERY (replicate (length ts + length us - 1) (etac @{thm conjE} 1)) THEN
   499             Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
   500               let
   501                 val (eqs, prems') = chop (length us) prems;
   502                 val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs;
   503               in
   504                 rewrite_goal_tac rew_thms 1 THEN
   505                 rtac intr 1 THEN
   506                 EVERY (map (fn p => rtac p 1) prems')
   507               end) ctxt' 1);
   508       in
   509         Goal.prove_sorry ctxt' [] [] eq (fn _ =>
   510           rtac @{thm iffI} 1 THEN etac (#1 elim) 1 THEN
   511           EVERY (map_index prove_intr1 c_intrs) THEN
   512           (if null c_intrs then etac @{thm FalseE} 1
   513            else
   514             let val (c_intrs', last_c_intr) = split_last c_intrs in
   515               EVERY (map (fn ci => etac @{thm disjE} 1 THEN prove_intr2 ci) c_intrs') THEN
   516               prove_intr2 last_c_intr
   517             end))
   518         |> rulify
   519         |> singleton (Proof_Context.export ctxt' ctxt'')
   520       end;
   521   in
   522     map2 prove_eq cs elims
   523   end;
   524 
   525 
   526 (* derivation of simplified elimination rules *)
   527 
   528 local
   529 
   530 (*delete needless equality assumptions*)
   531 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
   532   (fn _ => assume_tac 1);
   533 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   534 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   535 
   536 fun simp_case_tac ss i =
   537   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
   538 
   539 in
   540 
   541 fun mk_cases ctxt prop =
   542   let
   543     val thy = Proof_Context.theory_of ctxt;
   544     val ss = simpset_of ctxt;
   545 
   546     fun err msg =
   547       error (Pretty.string_of (Pretty.block
   548         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
   549 
   550     val elims = Induct.find_casesP ctxt prop;
   551 
   552     val cprop = Thm.cterm_of thy prop;
   553     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
   554     fun mk_elim rl =
   555       Thm.implies_intr cprop (Tactic.rule_by_tactic ctxt tac (Thm.assume cprop RS rl))
   556       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   557   in
   558     (case get_first (try mk_elim) elims of
   559       SOME r => r
   560     | NONE => err "Proposition not an inductive predicate:")
   561   end;
   562 
   563 end;
   564 
   565 
   566 (* inductive_cases *)
   567 
   568 fun gen_inductive_cases prep_att prep_prop args lthy =
   569   let
   570     val thy = Proof_Context.theory_of lthy;
   571     val thmss =
   572       map snd args
   573       |> burrow (grouped 10 Par_List.map (mk_cases lthy o prep_prop lthy));
   574     val facts =
   575       map2 (fn ((a, atts), _) => fn thms => ((a, map (prep_att thy) atts), [(thms, [])]))
   576         args thmss;
   577   in lthy |> Local_Theory.notes facts |>> map snd end;
   578 
   579 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
   580 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   581 
   582 
   583 val ind_cases_setup =
   584   Method.setup @{binding ind_cases}
   585     (Scan.lift (Scan.repeat1 Args.name_source --
   586       Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.binding) []) >>
   587       (fn (raw_props, fixes) => fn ctxt =>
   588         let
   589           val (_, ctxt') = Variable.add_fixes_binding fixes ctxt;
   590           val props = Syntax.read_props ctxt' raw_props;
   591           val ctxt'' = fold Variable.declare_term props ctxt';
   592           val rules = Proof_Context.export ctxt'' ctxt (map (mk_cases ctxt'') props)
   593         in Method.erule 0 rules end))
   594     "dynamic case analysis on predicates";
   595 
   596 
   597 (* derivation of simplified equation *)
   598 
   599 fun mk_simp_eq ctxt prop =
   600   let
   601     val thy = Proof_Context.theory_of ctxt;
   602     val ctxt' = Variable.auto_fixes prop ctxt;
   603     val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of;
   604     val substs =
   605       Item_Net.retrieve (get_equations ctxt) (HOLogic.dest_Trueprop prop)
   606       |> map_filter
   607         (fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
   608             (Vartab.empty, Vartab.empty), eq)
   609           handle Pattern.MATCH => NONE);
   610     val (subst, eq) =
   611       (case substs of
   612         [s] => s
   613       | _ => error
   614         ("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique"));
   615     val inst =
   616       map (fn v => (cterm_of thy (Var v), cterm_of thy (Envir.subst_term subst (Var v))))
   617         (Term.add_vars (lhs_of eq) []);
   618   in
   619     Drule.cterm_instantiate inst eq
   620     |> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv (Simplifier.full_rewrite (simpset_of ctxt))))
   621     |> singleton (Variable.export ctxt' ctxt)
   622   end
   623 
   624 
   625 (* inductive simps *)
   626 
   627 fun gen_inductive_simps prep_att prep_prop args lthy =
   628   let
   629     val thy = Proof_Context.theory_of lthy;
   630     val facts = args |> map (fn ((a, atts), props) =>
   631       ((a, map (prep_att thy) atts),
   632         map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
   633   in lthy |> Local_Theory.notes facts |>> map snd end;
   634 
   635 val inductive_simps = gen_inductive_simps Attrib.intern_src Syntax.read_prop;
   636 val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
   637 
   638 
   639 (* prove induction rule *)
   640 
   641 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
   642     fp_def rec_preds_defs ctxt ctxt''' =  (* FIXME ctxt''' ?? *)
   643   let
   644     val _ = clean_message quiet_mode "  Proving the induction rule ...";
   645 
   646     (* predicates for induction rule *)
   647 
   648     val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
   649     val preds =
   650       map2 (curry Free) pnames
   651         (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
   652 
   653     (* transform an introduction rule into a premise for induction rule *)
   654 
   655     fun mk_ind_prem r =
   656       let
   657         fun subst s =
   658           (case dest_predicate cs params s of
   659             SOME (_, i, ys, (_, Ts)) =>
   660               let
   661                 val k = length Ts;
   662                 val bs = map Bound (k - 1 downto 0);
   663                 val P = list_comb (nth preds i, map (incr_boundvars k) ys @ bs);
   664                 val Q =
   665                   fold_rev Term.abs (mk_names "x" k ~~ Ts)
   666                     (HOLogic.mk_binop inductive_conj_name
   667                       (list_comb (incr_boundvars k s, bs), P));
   668               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   669           | NONE =>
   670               (case s of
   671                 t $ u => (fst (subst t) $ fst (subst u), NONE)
   672               | Abs (a, T, t) => (Abs (a, T, fst (subst t)), NONE)
   673               | _ => (s, NONE)));
   674 
   675         fun mk_prem s prems =
   676           (case subst s of
   677             (_, SOME (t, u)) => t :: u :: prems
   678           | (t, _) => t :: prems);
   679 
   680         val SOME (_, i, ys, _) =
   681           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   682       in
   683         fold_rev (Logic.all o Free) (Logic.strip_params r)
   684           (Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
   685             (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
   686               HOLogic.mk_Trueprop (list_comb (nth preds i, ys))))
   687       end;
   688 
   689     val ind_prems = map mk_ind_prem intr_ts;
   690 
   691 
   692     (* make conclusions for induction rules *)
   693 
   694     val Tss = map (binder_types o fastype_of) preds;
   695     val (xnames, ctxt'') = Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   696     val mutual_ind_concl =
   697       HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   698         (map (fn (((xnames, Ts), c), P) =>
   699           let val frees = map Free (xnames ~~ Ts)
   700           in HOLogic.mk_imp (list_comb (c, params @ frees), list_comb (P, frees)) end)
   701         (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   702 
   703 
   704     (* make predicate for instantiation of abstract induction rule *)
   705 
   706     val ind_pred =
   707       fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   708         (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
   709            (make_bool_args HOLogic.mk_not I bs i)
   710            (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
   711 
   712     val ind_concl =
   713       HOLogic.mk_Trueprop
   714         (HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
   715 
   716     val raw_fp_induct = mono RS (fp_def RS @{thm def_lfp_induct});
   717 
   718     val induct = Goal.prove_sorry ctxt'' [] ind_prems ind_concl
   719       (fn {prems, ...} => EVERY
   720         [rewrite_goals_tac [inductive_conj_def],
   721          DETERM (rtac raw_fp_induct 1),
   722          REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
   723          rewrite_goals_tac simp_thms2,
   724          (*This disjE separates out the introduction rules*)
   725          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
   726          (*Now break down the individual cases.  No disjE here in case
   727            some premise involves disjunction.*)
   728          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
   729          REPEAT (FIRSTGOAL
   730            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
   731          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
   732              (inductive_conj_def :: rec_preds_defs @ simp_thms2) prem,
   733            conjI, refl] 1)) prems)]);
   734 
   735     val lemma = Goal.prove_sorry ctxt'' [] []
   736       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
   737         [rewrite_goals_tac rec_preds_defs,
   738          REPEAT (EVERY
   739            [REPEAT (resolve_tac [conjI, impI] 1),
   740             REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
   741             atac 1,
   742             rewrite_goals_tac simp_thms1,
   743             atac 1])]);
   744 
   745   in singleton (Proof_Context.export ctxt'' ctxt''') (induct RS lemma) end;
   746 
   747 
   748 
   749 (** specification of (co)inductive predicates **)
   750 
   751 fun mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts monos params cnames_syn lthy =
   752   let
   753     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
   754 
   755     val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
   756     val k = log 2 1 (length cs);
   757     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   758     val p :: xs =
   759       map Free (Variable.variant_frees lthy intr_ts
   760         (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   761     val bs =
   762       map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
   763         (map (rpair HOLogic.boolT) (mk_names "b" k)));
   764 
   765     fun subst t =
   766       (case dest_predicate cs params t of
   767         SOME (_, i, ts, (Ts, Us)) =>
   768           let
   769             val l = length Us;
   770             val zs = map Bound (l - 1 downto 0);
   771           in
   772             fold_rev (Term.abs o pair "z") Us
   773               (list_comb (p,
   774                 make_bool_args' bs i @ make_args argTs
   775                   ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   776           end
   777       | NONE =>
   778           (case t of
   779             t1 $ t2 => subst t1 $ subst t2
   780           | Abs (x, T, u) => Abs (x, T, subst u)
   781           | _ => t));
   782 
   783     (* transform an introduction rule into a conjunction  *)
   784     (*   [| p_i t; ... |] ==> p_j u                       *)
   785     (* is transformed into                                *)
   786     (*   b_j & x_j = u & p b_j t & ...                    *)
   787 
   788     fun transform_rule r =
   789       let
   790         val SOME (_, i, ts, (Ts, _)) =
   791           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   792         val ps =
   793           make_bool_args HOLogic.mk_not I bs i @
   794           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   795           map (subst o HOLogic.dest_Trueprop) (Logic.strip_assums_hyp r);
   796       in
   797         fold_rev (fn (x, T) => fn P => HOLogic.exists_const T $ Abs (x, T, P))
   798           (Logic.strip_params r)
   799           (if null ps then @{term True} else foldr1 HOLogic.mk_conj ps)
   800       end;
   801 
   802     (* make a disjunction of all introduction rules *)
   803 
   804     val fp_fun =
   805       fold_rev lambda (p :: bs @ xs)
   806         (if null intr_ts then @{term False}
   807          else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   808 
   809     (* add definiton of recursive predicates to theory *)
   810 
   811     val rec_name =
   812       if Binding.is_empty alt_name then
   813         Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
   814       else alt_name;
   815 
   816     val ((rec_const, (_, fp_def)), lthy') = lthy
   817       |> Local_Theory.conceal
   818       |> Local_Theory.define
   819         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   820          ((Thm.def_binding rec_name, @{attributes [nitpick_unfold]}),
   821            fold_rev lambda params
   822              (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)))
   823       ||> Local_Theory.restore_naming lthy;
   824     val fp_def' =
   825       Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
   826         (cterm_of (Proof_Context.theory_of lthy') (list_comb (rec_const, params)));
   827     val specs =
   828       if length cs < 2 then []
   829       else
   830         map_index (fn (i, (name_mx, c)) =>
   831           let
   832             val Ts = arg_types_of (length params) c;
   833             val xs =
   834               map Free (Variable.variant_frees lthy intr_ts (mk_names "x" (length Ts) ~~ Ts));
   835           in
   836             (name_mx, (apfst Binding.conceal Attrib.empty_binding, fold_rev lambda (params @ xs)
   837               (list_comb (rec_const, params @ make_bool_args' bs i @
   838                 make_args argTs (xs ~~ Ts)))))
   839           end) (cnames_syn ~~ cs);
   840     val (consts_defs, lthy'') = lthy'
   841       |> fold_map Local_Theory.define specs;
   842     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   843 
   844     val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
   845     val mono = prove_mono quiet_mode skip_mono predT fp_fun monos lthy''';
   846     val (_, lthy'''') =
   847       Local_Theory.note (apfst Binding.conceal Attrib.empty_binding,
   848         Proof_Context.export lthy''' lthy'' [mono]) lthy'';
   849 
   850   in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
   851     list_comb (rec_const, params), preds, argTs, bs, xs)
   852   end;
   853 
   854 fun declare_rules rec_binding coind no_ind cnames
   855     preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
   856   let
   857     val rec_name = Binding.name_of rec_binding;
   858     fun rec_qualified qualified = Binding.qualify qualified rec_name;
   859     val intr_names = map Binding.name_of intr_bindings;
   860     val ind_case_names = Rule_Cases.case_names intr_names;
   861     val induct =
   862       if coind then
   863         (raw_induct,
   864          [Rule_Cases.case_names [rec_name],
   865           Rule_Cases.case_conclusion (rec_name, intr_names),
   866           Rule_Cases.consumes (1 - Thm.nprems_of raw_induct),
   867           Induct.coinduct_pred (hd cnames)])
   868       else if no_ind orelse length cnames > 1 then
   869         (raw_induct,
   870           [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of raw_induct))])
   871       else
   872         (raw_induct RSN (2, rev_mp),
   873           [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of raw_induct))]);
   874 
   875     val (intrs', lthy1) =
   876       lthy |>
   877       Spec_Rules.add
   878         (if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
   879       Local_Theory.notes
   880         (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
   881           map (fn th => [([th],
   882            [Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
   883       map (hd o snd);
   884     val (((_, elims'), (_, [induct'])), lthy2) =
   885       lthy1 |>
   886       Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
   887       fold_map (fn (name, (elim, cases, k)) =>
   888         Local_Theory.note
   889           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
   890             [Attrib.internal (K (Rule_Cases.case_names cases)),
   891              Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of elim))),
   892              Attrib.internal (K (Rule_Cases.constraints k)),
   893              Attrib.internal (K (Induct.cases_pred name)),
   894              Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
   895         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   896       Local_Theory.note
   897         ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
   898           map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
   899 
   900     val (eqs', lthy3) = lthy2 |>
   901       fold_map (fn (name, eq) => Local_Theory.note
   902           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
   903             [Attrib.internal (K equation_add_permissive)]), [eq])
   904           #> apfst (hd o snd))
   905         (if null eqs then [] else (cnames ~~ eqs))
   906     val (inducts, lthy4) =
   907       if no_ind orelse coind then ([], lthy3)
   908       else
   909         let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
   910           lthy3 |>
   911           Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
   912             inducts |> map (fn (name, th) => ([th],
   913               [Attrib.internal (K ind_case_names),
   914                Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of th))),
   915                Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
   916         end;
   917   in (intrs', elims', eqs', induct', inducts, lthy4) end;
   918 
   919 type inductive_flags =
   920   {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
   921     no_elim: bool, no_ind: bool, skip_mono: bool};
   922 
   923 type add_ind_def =
   924   inductive_flags ->
   925   term list -> (Attrib.binding * term) list -> thm list ->
   926   term list -> (binding * mixfix) list ->
   927   local_theory -> inductive_result * local_theory;
   928 
   929 fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono}
   930     cs intros monos params cnames_syn lthy =
   931   let
   932     val _ = null cnames_syn andalso error "No inductive predicates given";
   933     val names = map (Binding.name_of o fst) cnames_syn;
   934     val _ = message (quiet_mode andalso not verbose)
   935       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
   936 
   937     val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
   938     val ((intr_names, intr_atts), intr_ts) =
   939       apfst split_list (split_list (map (check_rule lthy cs params) intros));
   940 
   941     val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   942       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts
   943         monos params cnames_syn lthy;
   944 
   945     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
   946       intr_ts rec_preds_defs lthy2 lthy1;
   947     val elims =
   948       if no_elim then []
   949       else
   950         prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
   951           unfold rec_preds_defs lthy2 lthy1;
   952     val raw_induct = zero_var_indexes
   953       (if no_ind then Drule.asm_rl
   954        else if coind then
   955          singleton (Proof_Context.export lthy2 lthy1)
   956            (rotate_prems ~1 (Object_Logic.rulify
   957              (fold_rule rec_preds_defs
   958                (rewrite_rule simp_thms3
   959                 (mono RS (fp_def RS @{thm def_coinduct}))))))
   960        else
   961          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
   962            rec_preds_defs lthy2 lthy1);
   963     val eqs =
   964       if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1;
   965 
   966     val elims' = map (fn (th, ns, i) => (rulify th, ns, i)) elims;
   967     val intrs' = map rulify intrs;
   968 
   969     val (intrs'', elims'', eqs', induct, inducts, lthy3) =
   970       declare_rules rec_name coind no_ind
   971         cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
   972 
   973     val result =
   974       {preds = preds,
   975        intrs = intrs'',
   976        elims = elims'',
   977        raw_induct = rulify raw_induct,
   978        induct = induct,
   979        inducts = inducts,
   980        eqs = eqs'};
   981 
   982     val lthy4 = lthy3
   983       |> Local_Theory.declaration {syntax = false, pervasive = false} (fn phi =>
   984         let val result' = transform_result phi result;
   985         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
   986   in (result, lthy4) end;
   987 
   988 
   989 (* external interfaces *)
   990 
   991 fun gen_add_inductive_i mk_def
   992     (flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono})
   993     cnames_syn pnames spec monos lthy =
   994   let
   995     val thy = Proof_Context.theory_of lthy;
   996     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   997 
   998 
   999     (* abbrevs *)
  1000 
  1001     val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
  1002 
  1003     fun get_abbrev ((name, atts), t) =
  1004       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
  1005         let
  1006           val _ = Binding.is_empty name andalso null atts orelse
  1007             error "Abbreviations may not have names or attributes";
  1008           val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
  1009           val var =
  1010             (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
  1011               NONE => error ("Undeclared head of abbreviation " ^ quote x)
  1012             | SOME ((b, T'), mx) =>
  1013                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
  1014                 else (b, mx));
  1015         in SOME (var, rhs) end
  1016       else NONE;
  1017 
  1018     val abbrevs = map_filter get_abbrev spec;
  1019     val bs = map (Binding.name_of o fst o fst) abbrevs;
  1020 
  1021 
  1022     (* predicates *)
  1023 
  1024     val pre_intros = filter_out (is_some o get_abbrev) spec;
  1025     val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
  1026     val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
  1027     val ps = map Free pnames;
  1028 
  1029     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
  1030     val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
  1031     val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
  1032     val expand = Assumption.export_term ctxt3 lthy #> Proof_Context.cert_term lthy;
  1033 
  1034     fun close_rule r =
  1035       fold (Logic.all o Free) (fold_aterms
  1036         (fn t as Free (v as (s, _)) =>
  1037             if Variable.is_fixed ctxt1 s orelse
  1038               member (op =) ps t then I else insert (op =) v
  1039           | _ => I) r []) r;
  1040 
  1041     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
  1042     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
  1043   in
  1044     lthy
  1045     |> mk_def flags cs intros monos ps preds
  1046     ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
  1047   end;
  1048 
  1049 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
  1050   let
  1051     val ((vars, intrs), _) = lthy
  1052       |> Proof_Context.set_mode Proof_Context.mode_abbrev
  1053       |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
  1054     val (cs, ps) = chop (length cnames_syn) vars;
  1055     val monos = Attrib.eval_thms lthy raw_monos;
  1056     val flags =
  1057      {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
  1058       coind = coind, no_elim = false, no_ind = false, skip_mono = false};
  1059   in
  1060     lthy
  1061     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
  1062   end;
  1063 
  1064 val add_inductive_i = gen_add_inductive_i add_ind_def;
  1065 val add_inductive = gen_add_inductive add_ind_def;
  1066 
  1067 fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
  1068   let
  1069     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
  1070     val ctxt' = thy
  1071       |> Named_Target.theory_init
  1072       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
  1073       |> Local_Theory.exit;
  1074     val info = #2 (the_inductive ctxt' name);
  1075   in (info, Proof_Context.theory_of ctxt') end;
  1076 
  1077 
  1078 (* read off arities of inductive predicates from raw induction rule *)
  1079 fun arities_of induct =
  1080   map (fn (_ $ t $ u) =>
  1081       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
  1082     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
  1083 
  1084 (* read off parameters of inductive predicate from raw induction rule *)
  1085 fun params_of induct =
  1086   let
  1087     val (_ $ t $ u :: _) = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
  1088     val (_, ts) = strip_comb t;
  1089     val (_, us) = strip_comb u;
  1090   in
  1091     List.take (ts, length ts - length us)
  1092   end;
  1093 
  1094 val pname_of_intr =
  1095   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
  1096 
  1097 (* partition introduction rules according to predicate name *)
  1098 fun gen_partition_rules f induct intros =
  1099   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
  1100     (map (rpair [] o fst) (arities_of induct));
  1101 
  1102 val partition_rules = gen_partition_rules I;
  1103 fun partition_rules' induct = gen_partition_rules fst induct;
  1104 
  1105 fun unpartition_rules intros xs =
  1106   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
  1107     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
  1108 
  1109 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
  1110 fun infer_intro_vars elim arity intros =
  1111   let
  1112     val thy = theory_of_thm elim;
  1113     val _ :: cases = prems_of elim;
  1114     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
  1115     fun mtch (t, u) =
  1116       let
  1117         val params = Logic.strip_params t;
  1118         val vars =
  1119           map (Var o apfst (rpair 0))
  1120             (Name.variant_list used (map fst params) ~~ map snd params);
  1121         val ts =
  1122           map (curry subst_bounds (rev vars))
  1123             (List.drop (Logic.strip_assums_hyp t, arity));
  1124         val us = Logic.strip_imp_prems u;
  1125         val tab =
  1126           fold (Pattern.first_order_match thy) (ts ~~ us) (Vartab.empty, Vartab.empty);
  1127       in
  1128         map (Envir.subst_term tab) vars
  1129       end
  1130   in
  1131     map (mtch o apsnd prop_of) (cases ~~ intros)
  1132   end;
  1133 
  1134 
  1135 
  1136 (** package setup **)
  1137 
  1138 (* setup theory *)
  1139 
  1140 val setup =
  1141   ind_cases_setup #>
  1142   Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
  1143     "declaration of monotonicity rule";
  1144 
  1145 
  1146 (* outer syntax *)
  1147 
  1148 fun gen_ind_decl mk_def coind =
  1149   Parse.fixes -- Parse.for_fixes --
  1150   Scan.optional Parse_Spec.where_alt_specs [] --
  1151   Scan.optional (@{keyword "monos"} |-- Parse.!!! Parse_Spec.xthms1) []
  1152   >> (fn (((preds, params), specs), monos) =>
  1153       (snd o gen_add_inductive mk_def true coind preds params specs monos));
  1154 
  1155 val ind_decl = gen_ind_decl add_ind_def;
  1156 
  1157 val _ =
  1158   Outer_Syntax.local_theory @{command_spec "inductive"} "define inductive predicates"
  1159     (ind_decl false);
  1160 
  1161 val _ =
  1162   Outer_Syntax.local_theory @{command_spec "coinductive"} "define coinductive predicates"
  1163     (ind_decl true);
  1164 
  1165 val _ =
  1166   Outer_Syntax.local_theory @{command_spec "inductive_cases"}
  1167     "create simplified instances of elimination rules"
  1168     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
  1169 
  1170 val _ =
  1171   Outer_Syntax.local_theory @{command_spec "inductive_simps"}
  1172     "create simplification rules for inductive predicates"
  1173     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
  1174 
  1175 val _ =
  1176   Outer_Syntax.improper_command @{command_spec "print_inductives"}
  1177     "print (co)inductive definitions and monotonicity rules"
  1178     (Scan.succeed
  1179       (Toplevel.no_timing o Toplevel.unknown_context o
  1180         Toplevel.keep (print_inductives o Toplevel.context_of)));
  1181 
  1182 end;