src/HOL/Tools/inductive.ML
author wenzelm
Sat Dec 14 17:28:05 2013 +0100 (2013-12-14)
changeset 54742 7a86358a3c0b
parent 53995 1d457fc83f5c
child 54883 dd04a8b654fc
permissions -rw-r--r--
proper context for basic Simplifier operations: rewrite_rule, rewrite_goals_rule, rewrite_goals_tac etc.;
clarified tool context in some boundary cases;
     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     (string * thm list) list * local_theory
    39   val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
    40     (string * thm list) list * local_theory
    41   val inductive_simps: (Attrib.binding * string list) list -> local_theory ->
    42     (string * thm list) list * local_theory
    43   val inductive_simps_i: (Attrib.binding * term list) list -> local_theory ->
    44     (string * thm list) list * local_theory
    45   type inductive_flags =
    46     {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
    47       no_elim: bool, no_ind: bool, skip_mono: bool}
    48   val add_inductive_i:
    49     inductive_flags -> ((binding * typ) * mixfix) list ->
    50     (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
    51     inductive_result * local_theory
    52   val add_inductive: bool -> bool ->
    53     (binding * string option * mixfix) list ->
    54     (binding * string option * mixfix) list ->
    55     (Attrib.binding * string) list ->
    56     (Facts.ref * Attrib.src list) list ->
    57     local_theory -> inductive_result * local_theory
    58   val add_inductive_global: inductive_flags ->
    59     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    60     thm list -> theory -> inductive_result * theory
    61   val arities_of: thm -> (string * int) list
    62   val params_of: thm -> term list
    63   val partition_rules: thm -> thm list -> (string * thm list) list
    64   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
    65   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    66   val infer_intro_vars: thm -> int -> thm list -> term list list
    67   val setup: theory -> theory
    68 end;
    69 
    70 signature INDUCTIVE =
    71 sig
    72   include BASIC_INDUCTIVE
    73   type add_ind_def =
    74     inductive_flags ->
    75     term list -> (Attrib.binding * term) list -> thm list ->
    76     term list -> (binding * mixfix) list ->
    77     local_theory -> inductive_result * local_theory
    78   val declare_rules: binding -> bool -> bool -> string list -> term list ->
    79     thm list -> binding list -> Attrib.src list list -> (thm * string list * int) list ->
    80     thm list -> thm -> local_theory -> thm list * thm list * thm list * thm * thm list * local_theory
    81   val add_ind_def: add_ind_def
    82   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
    83     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    84     thm list -> local_theory -> inductive_result * local_theory
    85   val gen_add_inductive: add_ind_def -> bool -> bool ->
    86     (binding * string option * mixfix) list ->
    87     (binding * string option * mixfix) list ->
    88     (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
    89     local_theory -> inductive_result * local_theory
    90   val gen_ind_decl: add_ind_def -> bool -> (local_theory -> local_theory) parser
    91 end;
    92 
    93 structure Inductive: INDUCTIVE =
    94 struct
    95 
    96 (** theory context references **)
    97 
    98 val inductive_forall_def = @{thm induct_forall_def};
    99 val inductive_conj_name = "HOL.induct_conj";
   100 val inductive_conj_def = @{thm induct_conj_def};
   101 val inductive_conj = @{thms induct_conj};
   102 val inductive_atomize = @{thms induct_atomize};
   103 val inductive_rulify = @{thms induct_rulify};
   104 val inductive_rulify_fallback = @{thms induct_rulify_fallback};
   105 
   106 val simp_thms1 =
   107   map mk_meta_eq
   108     @{lemma "(~ True) = False" "(~ False) = True"
   109         "(True --> P) = P" "(False --> P) = True"
   110         "(P & True) = P" "(True & P) = P"
   111       by (fact simp_thms)+};
   112 
   113 val simp_thms2 =
   114   map mk_meta_eq [@{thm inf_fun_def}, @{thm inf_bool_def}] @ simp_thms1;
   115 
   116 val simp_thms3 =
   117   map mk_meta_eq [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_def}, @{thm sup_bool_def}];
   118 
   119 
   120 
   121 (** misc utilities **)
   122 
   123 fun message quiet_mode s = if quiet_mode then () else writeln s;
   124 
   125 fun clean_message ctxt quiet_mode s =
   126   if Config.get ctxt quick_and_dirty then () else message quiet_mode s;
   127 
   128 fun coind_prefix true = "co"
   129   | coind_prefix false = "";
   130 
   131 fun log (b: int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   132 
   133 fun make_bool_args f g [] i = []
   134   | make_bool_args f g (x :: xs) i =
   135       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   136 
   137 fun make_bool_args' xs =
   138   make_bool_args (K @{term False}) (K @{term True}) xs;
   139 
   140 fun arg_types_of k c = drop k (binder_types (fastype_of c));
   141 
   142 fun find_arg T x [] = raise Fail "find_arg"
   143   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   144       apsnd (cons p) (find_arg T x ps)
   145   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   146       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   147       else apsnd (cons p) (find_arg T x ps);
   148 
   149 fun make_args Ts xs =
   150   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
   151     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   152 
   153 fun make_args' Ts xs Us =
   154   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   155 
   156 fun dest_predicate cs params t =
   157   let
   158     val k = length params;
   159     val (c, ts) = strip_comb t;
   160     val (xs, ys) = chop k ts;
   161     val i = find_index (fn c' => c' = c) cs;
   162   in
   163     if xs = params andalso i >= 0 then
   164       SOME (c, i, ys, chop (length ys) (arg_types_of k c))
   165     else NONE
   166   end;
   167 
   168 fun mk_names a 0 = []
   169   | mk_names a 1 = [a]
   170   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   171 
   172 fun select_disj 1 1 = []
   173   | select_disj _ 1 = [rtac disjI1]
   174   | select_disj n i = rtac disjI2 :: select_disj (n - 1) (i - 1);
   175 
   176 
   177 
   178 (** context data **)
   179 
   180 type inductive_result =
   181   {preds: term list, elims: thm list, raw_induct: thm,
   182    induct: thm, inducts: thm list, intrs: thm list, eqs: thm list};
   183 
   184 fun transform_result phi {preds, elims, raw_induct: thm, induct, inducts, intrs, eqs} =
   185   let
   186     val term = Morphism.term phi;
   187     val thm = Morphism.thm phi;
   188     val fact = Morphism.fact phi;
   189   in
   190    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
   191     induct = thm induct, inducts = fact inducts, intrs = fact intrs, eqs = fact eqs}
   192   end;
   193 
   194 type inductive_info = {names: string list, coind: bool} * inductive_result;
   195 
   196 val empty_equations =
   197   Item_Net.init Thm.eq_thm_prop
   198     (single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
   199 
   200 datatype data = Data of
   201  {infos: inductive_info Symtab.table,
   202   monos: thm list,
   203   equations: thm Item_Net.T};
   204 
   205 fun make_data (infos, monos, equations) =
   206   Data {infos = infos, monos = monos, equations = equations};
   207 
   208 structure Data = Generic_Data
   209 (
   210   type T = data;
   211   val empty = make_data (Symtab.empty, [], empty_equations);
   212   val extend = I;
   213   fun merge (Data {infos = infos1, monos = monos1, equations = equations1},
   214       Data {infos = infos2, monos = monos2, equations = equations2}) =
   215     make_data (Symtab.merge (K true) (infos1, infos2),
   216       Thm.merge_thms (monos1, monos2),
   217       Item_Net.merge (equations1, equations2));
   218 );
   219 
   220 fun map_data f =
   221   Data.map (fn Data {infos, monos, equations} => make_data (f (infos, monos, equations)));
   222 
   223 fun rep_data ctxt = Data.get (Context.Proof ctxt) |> (fn Data rep => rep);
   224 
   225 fun print_inductives ctxt =
   226   let
   227     val {infos, monos, ...} = rep_data ctxt;
   228     val space = Consts.space_of (Proof_Context.consts_of ctxt);
   229   in
   230     [Pretty.block
   231       (Pretty.breaks
   232         (Pretty.str "(co)inductives:" ::
   233           map (Pretty.mark_str o #1) (Name_Space.extern_table ctxt (space, infos)))),
   234      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_item ctxt) monos)]
   235   end |> Pretty.chunks |> Pretty.writeln;
   236 
   237 
   238 (* inductive info *)
   239 
   240 fun the_inductive ctxt name =
   241   (case Symtab.lookup (#infos (rep_data ctxt)) name of
   242     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   243   | SOME info => info);
   244 
   245 fun put_inductives names info =
   246   map_data (fn (infos, monos, equations) =>
   247     (fold (fn name => Symtab.update (name, info)) names infos, monos, equations));
   248 
   249 
   250 (* monotonicity rules *)
   251 
   252 val get_monos = #monos o rep_data;
   253 
   254 fun mk_mono ctxt thm =
   255   let
   256     fun eq_to_mono thm' = thm' RS (thm' RS @{thm eq_to_mono});
   257     fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
   258       handle THM _ => thm RS @{thm le_boolD}
   259   in
   260     (case concl_of thm of
   261       Const ("==", _) $ _ $ _ => eq_to_mono (thm RS meta_eq_to_obj_eq)
   262     | _ $ (Const (@{const_name HOL.eq}, _) $ _ $ _) => eq_to_mono thm
   263     | _ $ (Const (@{const_name Orderings.less_eq}, _) $ _ $ _) =>
   264       dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   265         (resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))
   266     | _ => thm)
   267   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm ctxt thm);
   268 
   269 val mono_add =
   270   Thm.declaration_attribute (fn thm => fn context =>
   271     map_data (fn (infos, monos, equations) =>
   272       (infos, Thm.add_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
   273 
   274 val mono_del =
   275   Thm.declaration_attribute (fn thm => fn context =>
   276     map_data (fn (infos, monos, equations) =>
   277       (infos, Thm.del_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
   278 
   279 
   280 (* equations *)
   281 
   282 val get_equations = #equations o rep_data;
   283 
   284 val equation_add_permissive =
   285   Thm.declaration_attribute (fn thm =>
   286     map_data (fn (infos, monos, equations) =>
   287       (infos, monos, perhaps (try (Item_Net.update thm)) equations)));
   288 
   289 
   290 
   291 (** process rules **)
   292 
   293 local
   294 
   295 fun err_in_rule ctxt name t msg =
   296   error (cat_lines ["Ill-formed introduction rule " ^ Binding.print name,
   297     Syntax.string_of_term ctxt t, msg]);
   298 
   299 fun err_in_prem ctxt name t p msg =
   300   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
   301     "in introduction rule " ^ Binding.print name, Syntax.string_of_term ctxt t, msg]);
   302 
   303 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   304 
   305 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   306 
   307 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   308 
   309 fun atomize_term thy = Raw_Simplifier.rewrite_term thy inductive_atomize [];
   310 
   311 in
   312 
   313 fun check_rule ctxt cs params ((binding, att), rule) =
   314   let
   315     val params' = Term.variant_frees rule (Logic.strip_params rule);
   316     val frees = rev (map Free params');
   317     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   318     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   319     val rule' = Logic.list_implies (prems, concl);
   320     val aprems = map (atomize_term (Proof_Context.theory_of ctxt)) prems;
   321     val arule = fold_rev (Logic.all o Free) params' (Logic.list_implies (aprems, concl));
   322 
   323     fun check_ind err t =
   324       (case dest_predicate cs params t of
   325         NONE => err (bad_app ^
   326           commas (map (Syntax.string_of_term ctxt) params))
   327       | SOME (_, _, ys, _) =>
   328           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   329           then err bad_ind_occ else ());
   330 
   331     fun check_prem' prem t =
   332       if member (op =) cs (head_of t) then
   333         check_ind (err_in_prem ctxt binding rule prem) t
   334       else
   335         (case t of
   336           Abs (_, _, t) => check_prem' prem t
   337         | t $ u => (check_prem' prem t; check_prem' prem u)
   338         | _ => ());
   339 
   340     fun check_prem (prem, aprem) =
   341       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   342       else err_in_prem ctxt binding rule prem "Non-atomic premise";
   343 
   344     val _ =
   345       (case concl of
   346         Const (@{const_name Trueprop}, _) $ t =>
   347           if member (op =) cs (head_of t) then
   348            (check_ind (err_in_rule ctxt binding rule') t;
   349             List.app check_prem (prems ~~ aprems))
   350           else err_in_rule ctxt binding rule' bad_concl
   351        | _ => err_in_rule ctxt binding rule' bad_concl);
   352   in
   353     ((binding, att), arule)
   354   end;
   355 
   356 fun rulify ctxt =
   357   hol_simplify ctxt inductive_conj
   358   #> hol_simplify ctxt inductive_rulify
   359   #> hol_simplify ctxt inductive_rulify_fallback
   360   #> Simplifier.norm_hhf;
   361 
   362 end;
   363 
   364 
   365 
   366 (** proofs for (co)inductive predicates **)
   367 
   368 (* prove monotonicity *)
   369 
   370 fun prove_mono quiet_mode skip_mono predT fp_fun monos ctxt =
   371  (message (quiet_mode orelse skip_mono andalso Config.get ctxt quick_and_dirty)
   372     "  Proving monotonicity ...";
   373   (if skip_mono then Goal.prove_sorry else Goal.prove_future) ctxt
   374     [] []
   375     (HOLogic.mk_Trueprop
   376       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   377     (fn _ => EVERY [rtac @{thm monoI} 1,
   378       REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
   379       REPEAT (FIRST
   380         [atac 1,
   381          resolve_tac (map (mk_mono ctxt) monos @ get_monos ctxt) 1,
   382          etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
   383 
   384 
   385 (* prove introduction rules *)
   386 
   387 fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
   388   let
   389     val _ = clean_message ctxt quiet_mode "  Proving the introduction rules ...";
   390 
   391     val unfold = funpow k (fn th => th RS fun_cong)
   392       (mono RS (fp_def RS
   393         (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
   394 
   395     val rules = [refl, TrueI, @{lemma "~ False" by (rule notI)}, exI, conjI];
   396 
   397     val intrs = map_index (fn (i, intr) =>
   398       Goal.prove_sorry ctxt [] [] intr (fn _ => EVERY
   399        [rewrite_goals_tac ctxt rec_preds_defs,
   400         rtac (unfold RS iffD2) 1,
   401         EVERY1 (select_disj (length intr_ts) (i + 1)),
   402         (*Not ares_tac, since refl must be tried before any equality assumptions;
   403           backtracking may occur if the premises have extra variables!*)
   404         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)])
   405        |> singleton (Proof_Context.export ctxt ctxt')) intr_ts
   406 
   407   in (intrs, unfold) end;
   408 
   409 
   410 (* prove elimination rules *)
   411 
   412 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
   413   let
   414     val _ = clean_message ctxt quiet_mode "  Proving the elimination rules ...";
   415 
   416     val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
   417     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   418 
   419     fun dest_intr r =
   420       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   421        Logic.strip_assums_hyp r, Logic.strip_params r);
   422 
   423     val intrs = map dest_intr intr_ts ~~ intr_names;
   424 
   425     val rules1 = [disjE, exE, FalseE];
   426     val rules2 = [conjE, FalseE, @{lemma "~ True ==> R" by (rule notE [OF _ TrueI])}];
   427 
   428     fun prove_elim c =
   429       let
   430         val Ts = arg_types_of (length params) c;
   431         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   432         val frees = map Free (anames ~~ Ts);
   433 
   434         fun mk_elim_prem ((_, _, us, _), ts, params') =
   435           Logic.list_all (params',
   436             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   437               (frees ~~ us) @ ts, P));
   438         val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
   439         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   440            map mk_elim_prem (map #1 c_intrs)
   441       in
   442         (Goal.prove_sorry ctxt'' [] prems P
   443           (fn {context = ctxt4, prems} => EVERY
   444             [cut_tac (hd prems) 1,
   445              rewrite_goals_tac ctxt4 rec_preds_defs,
   446              dtac (unfold RS iffD1) 1,
   447              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   448              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   449              EVERY (map (fn prem =>
   450                DEPTH_SOLVE_1 (ares_tac [rewrite_rule ctxt4 rec_preds_defs prem, conjI] 1))
   451                 (tl prems))])
   452           |> singleton (Proof_Context.export ctxt'' ctxt'''),
   453          map #2 c_intrs, length Ts)
   454       end
   455 
   456    in map prove_elim cs end;
   457 
   458 
   459 (* prove simplification equations *)
   460 
   461 fun prove_eqs quiet_mode cs params intr_ts intrs
   462     (elims: (thm * bstring list * int) list) ctxt ctxt'' =  (* FIXME ctxt'' ?? *)
   463   let
   464     val _ = clean_message ctxt quiet_mode "  Proving the simplification rules ...";
   465 
   466     fun dest_intr r =
   467       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   468        Logic.strip_assums_hyp r, Logic.strip_params r);
   469     val intr_ts' = map dest_intr intr_ts;
   470 
   471     fun prove_eq c (elim: thm * 'a * 'b) =
   472       let
   473         val Ts = arg_types_of (length params) c;
   474         val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
   475         val frees = map Free (anames ~~ Ts);
   476         val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
   477         fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
   478           let
   479             fun list_ex ([], t) = t
   480               | list_ex ((a, T) :: vars, t) =
   481                   HOLogic.exists_const T $ Abs (a, T, list_ex (vars, t));
   482             val conjs = map2 (curry HOLogic.mk_eq) frees us @ map HOLogic.dest_Trueprop ts;
   483           in
   484             list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
   485           end;
   486         val lhs = list_comb (c, params @ frees);
   487         val rhs =
   488           if null c_intrs then @{term False}
   489           else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs);
   490         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
   491         fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
   492             EVERY1 (select_disj (length c_intrs) (i + 1)) THEN
   493             EVERY (replicate (length params) (rtac @{thm exI} 1)) THEN
   494             (if null prems then rtac @{thm TrueI} 1
   495              else
   496               let
   497                 val (prems', last_prem) = split_last prems;
   498               in
   499                 EVERY (map (fn prem => (rtac @{thm conjI} 1 THEN rtac prem 1)) prems') THEN
   500                 rtac last_prem 1
   501               end)) ctxt' 1;
   502         fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
   503           EVERY (replicate (length params') (etac @{thm exE} 1)) THEN
   504           (if null ts andalso null us then rtac intr 1
   505            else
   506             EVERY (replicate (length ts + length us - 1) (etac @{thm conjE} 1)) THEN
   507             Subgoal.FOCUS_PREMS (fn {context = ctxt'', params, prems, ...} =>
   508               let
   509                 val (eqs, prems') = chop (length us) prems;
   510                 val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs;
   511               in
   512                 rewrite_goal_tac ctxt'' rew_thms 1 THEN
   513                 rtac intr 1 THEN
   514                 EVERY (map (fn p => rtac p 1) prems')
   515               end) ctxt' 1);
   516       in
   517         Goal.prove_sorry ctxt' [] [] eq (fn _ =>
   518           rtac @{thm iffI} 1 THEN etac (#1 elim) 1 THEN
   519           EVERY (map_index prove_intr1 c_intrs) THEN
   520           (if null c_intrs then etac @{thm FalseE} 1
   521            else
   522             let val (c_intrs', last_c_intr) = split_last c_intrs in
   523               EVERY (map (fn ci => etac @{thm disjE} 1 THEN prove_intr2 ci) c_intrs') THEN
   524               prove_intr2 last_c_intr
   525             end))
   526         |> rulify ctxt'
   527         |> singleton (Proof_Context.export ctxt' ctxt'')
   528       end;
   529   in
   530     map2 prove_eq cs elims
   531   end;
   532 
   533 
   534 (* derivation of simplified elimination rules *)
   535 
   536 local
   537 
   538 (*delete needless equality assumptions*)
   539 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
   540   (fn _ => assume_tac 1);
   541 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   542 val elim_tac = REPEAT o eresolve_tac elim_rls;
   543 
   544 fun simp_case_tac ctxt i =
   545   EVERY' [elim_tac, asm_full_simp_tac ctxt, elim_tac, REPEAT o bound_hyp_subst_tac ctxt] i;
   546 
   547 in
   548 
   549 fun mk_cases_tac ctxt = ALLGOALS (simp_case_tac ctxt) THEN prune_params_tac ctxt;
   550 
   551 fun mk_cases ctxt prop =
   552   let
   553     val thy = Proof_Context.theory_of ctxt;
   554 
   555     fun err msg =
   556       error (Pretty.string_of (Pretty.block
   557         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
   558 
   559     val elims = Induct.find_casesP ctxt prop;
   560 
   561     val cprop = Thm.cterm_of thy prop;
   562     fun mk_elim rl =
   563       Thm.implies_intr cprop
   564         (Tactic.rule_by_tactic ctxt (mk_cases_tac ctxt) (Thm.assume cprop RS rl))
   565       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   566   in
   567     (case get_first (try mk_elim) elims of
   568       SOME r => r
   569     | NONE => err "Proposition not an inductive predicate:")
   570   end;
   571 
   572 end;
   573 
   574 
   575 (* inductive_cases *)
   576 
   577 fun gen_inductive_cases prep_att prep_prop args lthy =
   578   let
   579     val thy = Proof_Context.theory_of lthy;
   580     val thmss =
   581       map snd args
   582       |> burrow (grouped 10 Par_List.map (mk_cases lthy o prep_prop lthy));
   583     val facts =
   584       map2 (fn ((a, atts), _) => fn thms => ((a, map (prep_att thy) atts), [(thms, [])]))
   585         args thmss;
   586   in lthy |> Local_Theory.notes facts end;
   587 
   588 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
   589 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   590 
   591 
   592 val ind_cases_setup =
   593   Method.setup @{binding ind_cases}
   594     (Scan.lift (Scan.repeat1 Args.name_source --
   595       Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.binding) []) >>
   596       (fn (raw_props, fixes) => fn ctxt =>
   597         let
   598           val (_, ctxt') = Variable.add_fixes_binding fixes ctxt;
   599           val props = Syntax.read_props ctxt' raw_props;
   600           val ctxt'' = fold Variable.declare_term props ctxt';
   601           val rules = Proof_Context.export ctxt'' ctxt (map (mk_cases ctxt'') props)
   602         in Method.erule ctxt 0 rules end))
   603     "dynamic case analysis on predicates";
   604 
   605 
   606 (* derivation of simplified equation *)
   607 
   608 fun mk_simp_eq ctxt prop =
   609   let
   610     val thy = Proof_Context.theory_of ctxt;
   611     val ctxt' = Variable.auto_fixes prop ctxt;
   612     val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of;
   613     val substs =
   614       Item_Net.retrieve (get_equations ctxt) (HOLogic.dest_Trueprop prop)
   615       |> map_filter
   616         (fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
   617             (Vartab.empty, Vartab.empty), eq)
   618           handle Pattern.MATCH => NONE);
   619     val (subst, eq) =
   620       (case substs of
   621         [s] => s
   622       | _ => error
   623         ("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique"));
   624     val inst =
   625       map (fn v => (cterm_of thy (Var v), cterm_of thy (Envir.subst_term subst (Var v))))
   626         (Term.add_vars (lhs_of eq) []);
   627   in
   628     Drule.cterm_instantiate inst eq
   629     |> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv (Simplifier.full_rewrite ctxt)))
   630     |> singleton (Variable.export ctxt' ctxt)
   631   end
   632 
   633 
   634 (* inductive simps *)
   635 
   636 fun gen_inductive_simps prep_att prep_prop args lthy =
   637   let
   638     val thy = Proof_Context.theory_of lthy;
   639     val facts = args |> map (fn ((a, atts), props) =>
   640       ((a, map (prep_att thy) atts),
   641         map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
   642   in lthy |> Local_Theory.notes facts end;
   643 
   644 val inductive_simps = gen_inductive_simps Attrib.intern_src Syntax.read_prop;
   645 val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
   646 
   647 
   648 (* prove induction rule *)
   649 
   650 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
   651     fp_def rec_preds_defs ctxt ctxt''' =  (* FIXME ctxt''' ?? *)
   652   let
   653     val _ = clean_message ctxt quiet_mode "  Proving the induction rule ...";
   654 
   655     (* predicates for induction rule *)
   656 
   657     val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
   658     val preds =
   659       map2 (curry Free) pnames
   660         (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
   661 
   662     (* transform an introduction rule into a premise for induction rule *)
   663 
   664     fun mk_ind_prem r =
   665       let
   666         fun subst s =
   667           (case dest_predicate cs params s of
   668             SOME (_, i, ys, (_, Ts)) =>
   669               let
   670                 val k = length Ts;
   671                 val bs = map Bound (k - 1 downto 0);
   672                 val P = list_comb (nth preds i, map (incr_boundvars k) ys @ bs);
   673                 val Q =
   674                   fold_rev Term.abs (mk_names "x" k ~~ Ts)
   675                     (HOLogic.mk_binop inductive_conj_name
   676                       (list_comb (incr_boundvars k s, bs), P));
   677               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   678           | NONE =>
   679               (case s of
   680                 t $ u => (fst (subst t) $ fst (subst u), NONE)
   681               | Abs (a, T, t) => (Abs (a, T, fst (subst t)), NONE)
   682               | _ => (s, NONE)));
   683 
   684         fun mk_prem s prems =
   685           (case subst s of
   686             (_, SOME (t, u)) => t :: u :: prems
   687           | (t, _) => t :: prems);
   688 
   689         val SOME (_, i, ys, _) =
   690           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   691       in
   692         fold_rev (Logic.all o Free) (Logic.strip_params r)
   693           (Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
   694             (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
   695               HOLogic.mk_Trueprop (list_comb (nth preds i, ys))))
   696       end;
   697 
   698     val ind_prems = map mk_ind_prem intr_ts;
   699 
   700 
   701     (* make conclusions for induction rules *)
   702 
   703     val Tss = map (binder_types o fastype_of) preds;
   704     val (xnames, ctxt'') = Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   705     val mutual_ind_concl =
   706       HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   707         (map (fn (((xnames, Ts), c), P) =>
   708           let val frees = map Free (xnames ~~ Ts)
   709           in HOLogic.mk_imp (list_comb (c, params @ frees), list_comb (P, frees)) end)
   710         (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   711 
   712 
   713     (* make predicate for instantiation of abstract induction rule *)
   714 
   715     val ind_pred =
   716       fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   717         (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
   718            (make_bool_args HOLogic.mk_not I bs i)
   719            (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
   720 
   721     val ind_concl =
   722       HOLogic.mk_Trueprop
   723         (HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
   724 
   725     val raw_fp_induct = mono RS (fp_def RS @{thm def_lfp_induct});
   726 
   727     val induct = Goal.prove_sorry ctxt'' [] ind_prems ind_concl
   728       (fn {context = ctxt3, prems} => EVERY
   729         [rewrite_goals_tac ctxt3 [inductive_conj_def],
   730          DETERM (rtac raw_fp_induct 1),
   731          REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
   732          rewrite_goals_tac ctxt3 simp_thms2,
   733          (*This disjE separates out the introduction rules*)
   734          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
   735          (*Now break down the individual cases.  No disjE here in case
   736            some premise involves disjunction.*)
   737          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac ctxt3)),
   738          REPEAT (FIRSTGOAL
   739            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
   740          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule ctxt3
   741              (inductive_conj_def :: rec_preds_defs @ simp_thms2) prem,
   742            conjI, refl] 1)) prems)]);
   743 
   744     val lemma = Goal.prove_sorry ctxt'' [] []
   745       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn {context = ctxt3, ...} => EVERY
   746         [rewrite_goals_tac ctxt3 rec_preds_defs,
   747          REPEAT (EVERY
   748            [REPEAT (resolve_tac [conjI, impI] 1),
   749             REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
   750             atac 1,
   751             rewrite_goals_tac ctxt3 simp_thms1,
   752             atac 1])]);
   753 
   754   in singleton (Proof_Context.export ctxt'' ctxt''') (induct RS lemma) end;
   755 
   756 
   757 
   758 (** specification of (co)inductive predicates **)
   759 
   760 fun mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts monos params cnames_syn lthy =
   761   let
   762     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
   763 
   764     val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
   765     val k = log 2 1 (length cs);
   766     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   767     val p :: xs =
   768       map Free (Variable.variant_frees lthy intr_ts
   769         (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   770     val bs =
   771       map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
   772         (map (rpair HOLogic.boolT) (mk_names "b" k)));
   773 
   774     fun subst t =
   775       (case dest_predicate cs params t of
   776         SOME (_, i, ts, (Ts, Us)) =>
   777           let
   778             val l = length Us;
   779             val zs = map Bound (l - 1 downto 0);
   780           in
   781             fold_rev (Term.abs o pair "z") Us
   782               (list_comb (p,
   783                 make_bool_args' bs i @ make_args argTs
   784                   ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   785           end
   786       | NONE =>
   787           (case t of
   788             t1 $ t2 => subst t1 $ subst t2
   789           | Abs (x, T, u) => Abs (x, T, subst u)
   790           | _ => t));
   791 
   792     (* transform an introduction rule into a conjunction  *)
   793     (*   [| p_i t; ... |] ==> p_j u                       *)
   794     (* is transformed into                                *)
   795     (*   b_j & x_j = u & p b_j t & ...                    *)
   796 
   797     fun transform_rule r =
   798       let
   799         val SOME (_, i, ts, (Ts, _)) =
   800           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   801         val ps =
   802           make_bool_args HOLogic.mk_not I bs i @
   803           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   804           map (subst o HOLogic.dest_Trueprop) (Logic.strip_assums_hyp r);
   805       in
   806         fold_rev (fn (x, T) => fn P => HOLogic.exists_const T $ Abs (x, T, P))
   807           (Logic.strip_params r)
   808           (if null ps then @{term True} else foldr1 HOLogic.mk_conj ps)
   809       end;
   810 
   811     (* make a disjunction of all introduction rules *)
   812 
   813     val fp_fun =
   814       fold_rev lambda (p :: bs @ xs)
   815         (if null intr_ts then @{term False}
   816          else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   817 
   818     (* add definiton of recursive predicates to theory *)
   819 
   820     val rec_name =
   821       if Binding.is_empty alt_name then
   822         Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
   823       else alt_name;
   824 
   825     val ((rec_const, (_, fp_def)), lthy') = lthy
   826       |> Local_Theory.conceal
   827       |> Local_Theory.define
   828         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   829          ((Thm.def_binding rec_name, @{attributes [nitpick_unfold]}),
   830            fold_rev lambda params
   831              (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)))
   832       ||> Local_Theory.restore_naming lthy;
   833     val fp_def' =
   834       Simplifier.rewrite (put_simpset HOL_basic_ss lthy' addsimps [fp_def])
   835         (cterm_of (Proof_Context.theory_of lthy') (list_comb (rec_const, params)));
   836     val specs =
   837       if length cs < 2 then []
   838       else
   839         map_index (fn (i, (name_mx, c)) =>
   840           let
   841             val Ts = arg_types_of (length params) c;
   842             val xs =
   843               map Free (Variable.variant_frees lthy intr_ts (mk_names "x" (length Ts) ~~ Ts));
   844           in
   845             (name_mx, (apfst Binding.conceal Attrib.empty_binding, fold_rev lambda (params @ xs)
   846               (list_comb (rec_const, params @ make_bool_args' bs i @
   847                 make_args argTs (xs ~~ Ts)))))
   848           end) (cnames_syn ~~ cs);
   849     val (consts_defs, lthy'') = lthy'
   850       |> fold_map Local_Theory.define specs;
   851     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   852 
   853     val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
   854     val mono = prove_mono quiet_mode skip_mono predT fp_fun monos lthy''';
   855     val (_, lthy'''') =
   856       Local_Theory.note (apfst Binding.conceal Attrib.empty_binding,
   857         Proof_Context.export lthy''' lthy'' [mono]) lthy'';
   858 
   859   in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
   860     list_comb (rec_const, params), preds, argTs, bs, xs)
   861   end;
   862 
   863 fun declare_rules rec_binding coind no_ind cnames
   864     preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
   865   let
   866     val rec_name = Binding.name_of rec_binding;
   867     fun rec_qualified qualified = Binding.qualify qualified rec_name;
   868     val intr_names = map Binding.name_of intr_bindings;
   869     val ind_case_names = Rule_Cases.case_names intr_names;
   870     val induct =
   871       if coind then
   872         (raw_induct,
   873          [Rule_Cases.case_names [rec_name],
   874           Rule_Cases.case_conclusion (rec_name, intr_names),
   875           Rule_Cases.consumes (1 - Thm.nprems_of raw_induct),
   876           Induct.coinduct_pred (hd cnames)])
   877       else if no_ind orelse length cnames > 1 then
   878         (raw_induct,
   879           [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of raw_induct))])
   880       else
   881         (raw_induct RSN (2, rev_mp),
   882           [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of raw_induct))]);
   883 
   884     val (intrs', lthy1) =
   885       lthy |>
   886       Spec_Rules.add
   887         (if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
   888       Local_Theory.notes
   889         (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
   890           map (fn th => [([th],
   891            [Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
   892       map (hd o snd);
   893     val (((_, elims'), (_, [induct'])), lthy2) =
   894       lthy1 |>
   895       Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
   896       fold_map (fn (name, (elim, cases, k)) =>
   897         Local_Theory.note
   898           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
   899             [Attrib.internal (K (Rule_Cases.case_names cases)),
   900              Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of elim))),
   901              Attrib.internal (K (Rule_Cases.constraints k)),
   902              Attrib.internal (K (Induct.cases_pred name)),
   903              Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
   904         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   905       Local_Theory.note
   906         ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
   907           map (Attrib.internal o K) (#2 induct)), [rulify lthy1 (#1 induct)]);
   908 
   909     val (eqs', lthy3) = lthy2 |>
   910       fold_map (fn (name, eq) => Local_Theory.note
   911           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
   912             [Attrib.internal (K equation_add_permissive)]), [eq])
   913           #> apfst (hd o snd))
   914         (if null eqs then [] else (cnames ~~ eqs))
   915     val (inducts, lthy4) =
   916       if no_ind orelse coind then ([], lthy3)
   917       else
   918         let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
   919           lthy3 |>
   920           Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
   921             inducts |> map (fn (name, th) => ([th],
   922               [Attrib.internal (K ind_case_names),
   923                Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of th))),
   924                Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
   925         end;
   926   in (intrs', elims', eqs', induct', inducts, lthy4) end;
   927 
   928 type inductive_flags =
   929   {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
   930     no_elim: bool, no_ind: bool, skip_mono: bool};
   931 
   932 type add_ind_def =
   933   inductive_flags ->
   934   term list -> (Attrib.binding * term) list -> thm list ->
   935   term list -> (binding * mixfix) list ->
   936   local_theory -> inductive_result * local_theory;
   937 
   938 fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono}
   939     cs intros monos params cnames_syn lthy =
   940   let
   941     val _ = null cnames_syn andalso error "No inductive predicates given";
   942     val names = map (Binding.name_of o fst) cnames_syn;
   943     val _ = message (quiet_mode andalso not verbose)
   944       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
   945 
   946     val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
   947     val ((intr_names, intr_atts), intr_ts) =
   948       apfst split_list (split_list (map (check_rule lthy cs params) intros));
   949 
   950     val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   951       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts
   952         monos params cnames_syn lthy;
   953 
   954     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
   955       intr_ts rec_preds_defs lthy2 lthy1;
   956     val elims =
   957       if no_elim then []
   958       else
   959         prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
   960           unfold rec_preds_defs lthy2 lthy1;
   961     val raw_induct = zero_var_indexes
   962       (if no_ind then Drule.asm_rl
   963        else if coind then
   964          singleton (Proof_Context.export lthy2 lthy1)
   965            (rotate_prems ~1 (Object_Logic.rulify lthy2
   966              (fold_rule lthy2 rec_preds_defs
   967                (rewrite_rule lthy2 simp_thms3
   968                 (mono RS (fp_def RS @{thm def_coinduct}))))))
   969        else
   970          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
   971            rec_preds_defs lthy2 lthy1);
   972     val eqs =
   973       if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1;
   974 
   975     val elims' = map (fn (th, ns, i) => (rulify lthy1 th, ns, i)) elims;
   976     val intrs' = map (rulify lthy1) intrs;
   977 
   978     val (intrs'', elims'', eqs', induct, inducts, lthy3) =
   979       declare_rules rec_name coind no_ind
   980         cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
   981 
   982     val result =
   983       {preds = preds,
   984        intrs = intrs'',
   985        elims = elims'',
   986        raw_induct = rulify lthy3 raw_induct,
   987        induct = induct,
   988        inducts = inducts,
   989        eqs = eqs'};
   990 
   991     val lthy4 = lthy3
   992       |> Local_Theory.declaration {syntax = false, pervasive = false} (fn phi =>
   993         let val result' = transform_result phi result;
   994         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
   995   in (result, lthy4) end;
   996 
   997 
   998 (* external interfaces *)
   999 
  1000 fun gen_add_inductive_i mk_def
  1001     (flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono})
  1002     cnames_syn pnames spec monos lthy =
  1003   let
  1004     val thy = Proof_Context.theory_of lthy;
  1005     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
  1006 
  1007 
  1008     (* abbrevs *)
  1009 
  1010     val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
  1011 
  1012     fun get_abbrev ((name, atts), t) =
  1013       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
  1014         let
  1015           val _ = Binding.is_empty name andalso null atts orelse
  1016             error "Abbreviations may not have names or attributes";
  1017           val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
  1018           val var =
  1019             (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
  1020               NONE => error ("Undeclared head of abbreviation " ^ quote x)
  1021             | SOME ((b, T'), mx) =>
  1022                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
  1023                 else (b, mx));
  1024         in SOME (var, rhs) end
  1025       else NONE;
  1026 
  1027     val abbrevs = map_filter get_abbrev spec;
  1028     val bs = map (Binding.name_of o fst o fst) abbrevs;
  1029 
  1030 
  1031     (* predicates *)
  1032 
  1033     val pre_intros = filter_out (is_some o get_abbrev) spec;
  1034     val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
  1035     val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
  1036     val ps = map Free pnames;
  1037 
  1038     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
  1039     val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
  1040     val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
  1041     val expand = Assumption.export_term ctxt3 lthy #> Proof_Context.cert_term lthy;
  1042 
  1043     fun close_rule r =
  1044       fold (Logic.all o Free) (fold_aterms
  1045         (fn t as Free (v as (s, _)) =>
  1046             if Variable.is_fixed ctxt1 s orelse
  1047               member (op =) ps t then I else insert (op =) v
  1048           | _ => I) r []) r;
  1049 
  1050     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
  1051     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
  1052   in
  1053     lthy
  1054     |> mk_def flags cs intros monos ps preds
  1055     ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
  1056   end;
  1057 
  1058 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
  1059   let
  1060     val ((vars, intrs), _) = lthy
  1061       |> Proof_Context.set_mode Proof_Context.mode_abbrev
  1062       |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
  1063     val (cs, ps) = chop (length cnames_syn) vars;
  1064     val monos = Attrib.eval_thms lthy raw_monos;
  1065     val flags =
  1066      {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
  1067       coind = coind, no_elim = false, no_ind = false, skip_mono = false};
  1068   in
  1069     lthy
  1070     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
  1071   end;
  1072 
  1073 val add_inductive_i = gen_add_inductive_i add_ind_def;
  1074 val add_inductive = gen_add_inductive add_ind_def;
  1075 
  1076 fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
  1077   let
  1078     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
  1079     val ctxt' = thy
  1080       |> Named_Target.theory_init
  1081       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
  1082       |> Local_Theory.exit;
  1083     val info = #2 (the_inductive ctxt' name);
  1084   in (info, Proof_Context.theory_of ctxt') end;
  1085 
  1086 
  1087 (* read off arities of inductive predicates from raw induction rule *)
  1088 fun arities_of induct =
  1089   map (fn (_ $ t $ u) =>
  1090       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
  1091     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
  1092 
  1093 (* read off parameters of inductive predicate from raw induction rule *)
  1094 fun params_of induct =
  1095   let
  1096     val (_ $ t $ u :: _) = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
  1097     val (_, ts) = strip_comb t;
  1098     val (_, us) = strip_comb u;
  1099   in
  1100     List.take (ts, length ts - length us)
  1101   end;
  1102 
  1103 val pname_of_intr =
  1104   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
  1105 
  1106 (* partition introduction rules according to predicate name *)
  1107 fun gen_partition_rules f induct intros =
  1108   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
  1109     (map (rpair [] o fst) (arities_of induct));
  1110 
  1111 val partition_rules = gen_partition_rules I;
  1112 fun partition_rules' induct = gen_partition_rules fst induct;
  1113 
  1114 fun unpartition_rules intros xs =
  1115   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
  1116     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
  1117 
  1118 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
  1119 fun infer_intro_vars elim arity intros =
  1120   let
  1121     val thy = theory_of_thm elim;
  1122     val _ :: cases = prems_of elim;
  1123     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
  1124     fun mtch (t, u) =
  1125       let
  1126         val params = Logic.strip_params t;
  1127         val vars =
  1128           map (Var o apfst (rpair 0))
  1129             (Name.variant_list used (map fst params) ~~ map snd params);
  1130         val ts =
  1131           map (curry subst_bounds (rev vars))
  1132             (List.drop (Logic.strip_assums_hyp t, arity));
  1133         val us = Logic.strip_imp_prems u;
  1134         val tab =
  1135           fold (Pattern.first_order_match thy) (ts ~~ us) (Vartab.empty, Vartab.empty);
  1136       in
  1137         map (Envir.subst_term tab) vars
  1138       end
  1139   in
  1140     map (mtch o apsnd prop_of) (cases ~~ intros)
  1141   end;
  1142 
  1143 
  1144 
  1145 (** package setup **)
  1146 
  1147 (* setup theory *)
  1148 
  1149 val setup =
  1150   ind_cases_setup #>
  1151   Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
  1152     "declaration of monotonicity rule";
  1153 
  1154 
  1155 (* outer syntax *)
  1156 
  1157 fun gen_ind_decl mk_def coind =
  1158   Parse.fixes -- Parse.for_fixes --
  1159   Scan.optional Parse_Spec.where_alt_specs [] --
  1160   Scan.optional (@{keyword "monos"} |-- Parse.!!! Parse_Spec.xthms1) []
  1161   >> (fn (((preds, params), specs), monos) =>
  1162       (snd o gen_add_inductive mk_def true coind preds params specs monos));
  1163 
  1164 val ind_decl = gen_ind_decl add_ind_def;
  1165 
  1166 val _ =
  1167   Outer_Syntax.local_theory @{command_spec "inductive"} "define inductive predicates"
  1168     (ind_decl false);
  1169 
  1170 val _ =
  1171   Outer_Syntax.local_theory @{command_spec "coinductive"} "define coinductive predicates"
  1172     (ind_decl true);
  1173 
  1174 val _ =
  1175   Outer_Syntax.local_theory @{command_spec "inductive_cases"}
  1176     "create simplified instances of elimination rules"
  1177     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
  1178 
  1179 val _ =
  1180   Outer_Syntax.local_theory @{command_spec "inductive_simps"}
  1181     "create simplification rules for inductive predicates"
  1182     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
  1183 
  1184 val _ =
  1185   Outer_Syntax.improper_command @{command_spec "print_inductives"}
  1186     "print (co)inductive definitions and monotonicity rules"
  1187     (Scan.succeed (Toplevel.unknown_context o
  1188       Toplevel.keep (print_inductives o Toplevel.context_of)));
  1189 
  1190 end;