src/HOL/Tools/inductive.ML
author wenzelm
Sun Apr 17 21:42:47 2011 +0200 (2011-04-17)
changeset 42381 309ec68442c6
parent 42364 8c674b3b8e44
child 42439 9efdd0af15ac
permissions -rw-r--r--
added Binding.print convenience, which includes quote already;
     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 morph_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 mono_add: attribute
    31   val mono_del: attribute
    32   val get_monos: Proof.context -> thm list
    33   val mk_cases: Proof.context -> term -> thm
    34   val inductive_forall_name: string
    35   val inductive_forall_def: thm
    36   val rulify: thm -> thm
    37   val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
    38     thm list list * local_theory
    39   val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
    40     thm list list * local_theory
    41   type inductive_flags =
    42     {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
    43       no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool}
    44   val add_inductive_i:
    45     inductive_flags -> ((binding * typ) * mixfix) list ->
    46     (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
    47     inductive_result * local_theory
    48   val add_inductive: bool -> bool ->
    49     (binding * string option * mixfix) list ->
    50     (binding * string option * mixfix) list ->
    51     (Attrib.binding * string) list ->
    52     (Facts.ref * Attrib.src list) list ->
    53     bool -> local_theory -> inductive_result * local_theory
    54   val add_inductive_global: inductive_flags ->
    55     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    56     thm list -> theory -> inductive_result * theory
    57   val arities_of: thm -> (string * int) list
    58   val params_of: thm -> term list
    59   val partition_rules: thm -> thm list -> (string * thm list) list
    60   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
    61   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    62   val infer_intro_vars: thm -> int -> thm list -> term list list
    63   val setup: theory -> theory
    64 end;
    65 
    66 signature INDUCTIVE =
    67 sig
    68   include BASIC_INDUCTIVE
    69   type add_ind_def =
    70     inductive_flags ->
    71     term list -> (Attrib.binding * term) list -> thm list ->
    72     term list -> (binding * mixfix) list ->
    73     local_theory -> inductive_result * local_theory
    74   val declare_rules: binding -> bool -> bool -> string list -> term list ->
    75     thm list -> binding list -> Attrib.src list list -> (thm * string list * int) list ->
    76     thm list -> thm -> local_theory -> thm list * thm list * thm list * thm * thm list * local_theory
    77   val add_ind_def: add_ind_def
    78   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
    79     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    80     thm list -> local_theory -> inductive_result * local_theory
    81   val gen_add_inductive: add_ind_def -> bool -> bool ->
    82     (binding * string option * mixfix) list ->
    83     (binding * string option * mixfix) list ->
    84     (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
    85     bool -> local_theory -> inductive_result * local_theory
    86   val gen_ind_decl: add_ind_def -> bool -> (bool -> local_theory -> local_theory) parser
    87 end;
    88 
    89 structure Inductive: INDUCTIVE =
    90 struct
    91 
    92 
    93 (** theory context references **)
    94 
    95 val inductive_forall_name = "HOL.induct_forall";
    96 val inductive_forall_def = @{thm induct_forall_def};
    97 val inductive_conj_name = "HOL.induct_conj";
    98 val inductive_conj_def = @{thm induct_conj_def};
    99 val inductive_conj = @{thms induct_conj};
   100 val inductive_atomize = @{thms induct_atomize};
   101 val inductive_rulify = @{thms induct_rulify};
   102 val inductive_rulify_fallback = @{thms induct_rulify_fallback};
   103 
   104 val notTrueE = TrueI RSN (2, notE);
   105 val notFalseI = Seq.hd (atac 1 notI);
   106 
   107 val simp_thms' = 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_thms'' = map mk_meta_eq [@{thm inf_fun_def}, @{thm inf_bool_def}] @ simp_thms';
   114 
   115 val simp_thms''' = map mk_meta_eq
   116   [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_def}, @{thm sup_bool_def}];
   117 
   118 
   119 (** context data **)
   120 
   121 type inductive_result =
   122   {preds: term list, elims: thm list, raw_induct: thm,
   123    induct: thm, inducts: thm list, intrs: thm list, eqs: thm list};
   124 
   125 fun morph_result phi {preds, elims, raw_induct: thm, induct, inducts, intrs, eqs} =
   126   let
   127     val term = Morphism.term phi;
   128     val thm = Morphism.thm phi;
   129     val fact = Morphism.fact phi;
   130   in
   131    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
   132     induct = thm induct, inducts = fact inducts, intrs = fact intrs, eqs = fact eqs}
   133   end;
   134 
   135 type inductive_info =
   136   {names: string list, coind: bool} * inductive_result;
   137 
   138 structure InductiveData = Generic_Data
   139 (
   140   type T = inductive_info Symtab.table * thm list;
   141   val empty = (Symtab.empty, []);
   142   val extend = I;
   143   fun merge ((tab1, monos1), (tab2, monos2)) : T =
   144     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
   145 );
   146 
   147 val get_inductives = InductiveData.get o Context.Proof;
   148 
   149 fun print_inductives ctxt =
   150   let
   151     val (tab, monos) = get_inductives ctxt;
   152     val space = Consts.space_of (Proof_Context.consts_of ctxt);
   153   in
   154     [Pretty.strs ("(co)inductives:" :: map #1 (Name_Space.extern_table ctxt (space, tab))),
   155      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm ctxt) monos)]
   156     |> Pretty.chunks |> Pretty.writeln
   157   end;
   158 
   159 
   160 (* get and put data *)
   161 
   162 fun the_inductive ctxt name =
   163   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
   164     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   165   | SOME info => info);
   166 
   167 fun put_inductives names info = InductiveData.map
   168   (apfst (fold (fn name => Symtab.update (name, info)) names));
   169 
   170 
   171 
   172 (** monotonicity rules **)
   173 
   174 val get_monos = #2 o get_inductives;
   175 val map_monos = InductiveData.map o apsnd;
   176 
   177 fun mk_mono thm =
   178   let
   179     fun eq2mono thm' = thm' RS (thm' RS eq_to_mono);
   180     fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
   181       handle THM _ => thm RS @{thm le_boolD}
   182   in
   183     case concl_of thm of
   184       Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq)
   185     | _ $ (Const (@{const_name HOL.eq}, _) $ _ $ _) => eq2mono thm
   186     | _ $ (Const (@{const_name Orderings.less_eq}, _) $ _ $ _) =>
   187       dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   188         (resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))
   189     | _ => thm
   190   end handle THM _ =>
   191     error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm_without_context thm);
   192 
   193 val mono_add = Thm.declaration_attribute (map_monos o Thm.add_thm o mk_mono);
   194 val mono_del = Thm.declaration_attribute (map_monos o Thm.del_thm o mk_mono);
   195 
   196 
   197 
   198 (** equations **)
   199 
   200 structure Equation_Data = Generic_Data
   201 (
   202   type T = thm Item_Net.T;
   203   val empty = Item_Net.init (op aconv o pairself Thm.prop_of)
   204     (single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
   205   val extend = I;
   206   val merge = Item_Net.merge;
   207 );
   208 
   209 val add_equation = Thm.declaration_attribute (Equation_Data.map o Item_Net.update)
   210 
   211 
   212 
   213 (** misc utilities **)
   214 
   215 fun message quiet_mode s = if quiet_mode then () else writeln s;
   216 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
   217 
   218 fun coind_prefix true = "co"
   219   | coind_prefix false = "";
   220 
   221 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   222 
   223 fun make_bool_args f g [] i = []
   224   | make_bool_args f g (x :: xs) i =
   225       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   226 
   227 fun make_bool_args' xs =
   228   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
   229 
   230 fun arg_types_of k c = drop k (binder_types (fastype_of c));
   231 
   232 fun find_arg T x [] = raise Fail "find_arg"
   233   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   234       apsnd (cons p) (find_arg T x ps)
   235   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   236       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   237       else apsnd (cons p) (find_arg T x ps);
   238 
   239 fun make_args Ts xs =
   240   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
   241     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   242 
   243 fun make_args' Ts xs Us =
   244   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   245 
   246 fun dest_predicate cs params t =
   247   let
   248     val k = length params;
   249     val (c, ts) = strip_comb t;
   250     val (xs, ys) = chop k ts;
   251     val i = find_index (fn c' => c' = c) cs;
   252   in
   253     if xs = params andalso i >= 0 then
   254       SOME (c, i, ys, chop (length ys) (arg_types_of k c))
   255     else NONE
   256   end;
   257 
   258 fun mk_names a 0 = []
   259   | mk_names a 1 = [a]
   260   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   261 
   262 fun select_disj 1 1 = []
   263   | select_disj _ 1 = [rtac disjI1]
   264   | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   265 
   266 
   267 (** process rules **)
   268 
   269 local
   270 
   271 fun err_in_rule ctxt name t msg =
   272   error (cat_lines ["Ill-formed introduction rule " ^ Binding.print name,
   273     Syntax.string_of_term ctxt t, msg]);
   274 
   275 fun err_in_prem ctxt name t p msg =
   276   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
   277     "in introduction rule " ^ Binding.print name, Syntax.string_of_term ctxt t, msg]);
   278 
   279 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   280 
   281 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   282 
   283 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   284 
   285 fun atomize_term thy = Raw_Simplifier.rewrite_term thy inductive_atomize [];
   286 
   287 in
   288 
   289 fun check_rule ctxt cs params ((binding, att), rule) =
   290   let
   291     val params' = Term.variant_frees rule (Logic.strip_params rule);
   292     val frees = rev (map Free params');
   293     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   294     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   295     val rule' = Logic.list_implies (prems, concl);
   296     val aprems = map (atomize_term (Proof_Context.theory_of ctxt)) prems;
   297     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
   298 
   299     fun check_ind err t = case dest_predicate cs params t of
   300         NONE => err (bad_app ^
   301           commas (map (Syntax.string_of_term ctxt) params))
   302       | SOME (_, _, ys, _) =>
   303           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   304           then err bad_ind_occ else ();
   305 
   306     fun check_prem' prem t =
   307       if member (op =) cs (head_of t) then
   308         check_ind (err_in_prem ctxt binding rule prem) t
   309       else (case t of
   310           Abs (_, _, t) => check_prem' prem t
   311         | t $ u => (check_prem' prem t; check_prem' prem u)
   312         | _ => ());
   313 
   314     fun check_prem (prem, aprem) =
   315       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   316       else err_in_prem ctxt binding rule prem "Non-atomic premise";
   317   in
   318     (case concl of
   319        Const (@{const_name Trueprop}, _) $ t =>
   320          if member (op =) cs (head_of t) then
   321            (check_ind (err_in_rule ctxt binding rule') t;
   322             List.app check_prem (prems ~~ aprems))
   323          else err_in_rule ctxt binding rule' bad_concl
   324      | _ => err_in_rule ctxt binding rule' bad_concl);
   325     ((binding, att), arule)
   326   end;
   327 
   328 val rulify =
   329   hol_simplify inductive_conj
   330   #> hol_simplify inductive_rulify
   331   #> hol_simplify inductive_rulify_fallback
   332   #> Simplifier.norm_hhf;
   333 
   334 end;
   335 
   336 
   337 
   338 (** proofs for (co)inductive predicates **)
   339 
   340 (* prove monotonicity *)
   341 
   342 fun prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt =
   343  (message (quiet_mode orelse skip_mono andalso !quick_and_dirty orelse fork_mono)
   344     "  Proving monotonicity ...";
   345   (if skip_mono then Skip_Proof.prove else if fork_mono then Goal.prove_future else Goal.prove) ctxt
   346     [] []
   347     (HOLogic.mk_Trueprop
   348       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   349     (fn _ => EVERY [rtac @{thm monoI} 1,
   350       REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
   351       REPEAT (FIRST
   352         [atac 1,
   353          resolve_tac (map mk_mono monos @ get_monos ctxt) 1,
   354          etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
   355 
   356 
   357 (* prove introduction rules *)
   358 
   359 fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
   360   let
   361     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
   362 
   363     val unfold = funpow k (fn th => th RS fun_cong)
   364       (mono RS (fp_def RS
   365         (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
   366 
   367     val rules = [refl, TrueI, notFalseI, exI, conjI];
   368 
   369     val intrs = map_index (fn (i, intr) =>
   370       Skip_Proof.prove ctxt [] [] intr (fn _ => EVERY
   371        [rewrite_goals_tac rec_preds_defs,
   372         rtac (unfold RS iffD2) 1,
   373         EVERY1 (select_disj (length intr_ts) (i + 1)),
   374         (*Not ares_tac, since refl must be tried before any equality assumptions;
   375           backtracking may occur if the premises have extra variables!*)
   376         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)])
   377        |> singleton (Proof_Context.export ctxt ctxt')) intr_ts
   378 
   379   in (intrs, unfold) end;
   380 
   381 
   382 (* prove elimination rules *)
   383 
   384 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
   385   let
   386     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
   387 
   388     val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
   389     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   390 
   391     fun dest_intr r =
   392       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   393        Logic.strip_assums_hyp r, Logic.strip_params r);
   394 
   395     val intrs = map dest_intr intr_ts ~~ intr_names;
   396 
   397     val rules1 = [disjE, exE, FalseE];
   398     val rules2 = [conjE, FalseE, notTrueE];
   399 
   400     fun prove_elim c =
   401       let
   402         val Ts = arg_types_of (length params) c;
   403         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   404         val frees = map Free (anames ~~ Ts);
   405 
   406         fun mk_elim_prem ((_, _, us, _), ts, params') =
   407           list_all (params',
   408             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   409               (frees ~~ us) @ ts, P));
   410         val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
   411         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   412            map mk_elim_prem (map #1 c_intrs)
   413       in
   414         (Skip_Proof.prove ctxt'' [] prems P
   415           (fn {prems, ...} => EVERY
   416             [cut_facts_tac [hd prems] 1,
   417              rewrite_goals_tac rec_preds_defs,
   418              dtac (unfold RS iffD1) 1,
   419              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   420              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   421              EVERY (map (fn prem =>
   422                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
   423           |> singleton (Proof_Context.export ctxt'' ctxt'''),
   424          map #2 c_intrs, length Ts)
   425       end
   426 
   427    in map prove_elim cs end;
   428 
   429 (* prove simplification equations *)
   430 
   431 fun prove_eqs quiet_mode cs params intr_ts intrs (elims: (thm * bstring list * int) list) ctxt ctxt'' =
   432   let
   433     val _ = clean_message quiet_mode "  Proving the simplification rules ...";
   434     
   435     fun dest_intr r =
   436       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   437        Logic.strip_assums_hyp r, Logic.strip_params r);
   438     val intr_ts' = map dest_intr intr_ts;
   439     fun prove_eq c (elim: thm * 'a * 'b) =
   440       let
   441         val Ts = arg_types_of (length params) c;
   442         val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
   443         val frees = map Free (anames ~~ Ts);
   444         val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
   445         fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
   446           let
   447             fun list_ex ([], t) = t
   448               | list_ex ((a,T)::vars, t) =
   449                  (HOLogic.exists_const T) $ (Abs(a, T, list_ex(vars,t)));
   450             val conjs = map2 (curry HOLogic.mk_eq) frees us @ (map HOLogic.dest_Trueprop ts)
   451           in
   452             list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
   453           end;
   454         val lhs = list_comb (c, params @ frees)
   455         val rhs =
   456           if null c_intrs then @{term False} else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs)
   457         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
   458         fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
   459             let
   460               val (prems', last_prem) = split_last prems
   461             in
   462               EVERY1 (select_disj (length c_intrs) (i + 1))
   463               THEN EVERY (replicate (length params) (rtac @{thm exI} 1))
   464               THEN EVERY (map (fn prem => (rtac @{thm conjI} 1 THEN rtac prem 1)) prems')
   465               THEN rtac last_prem 1
   466             end) ctxt' 1
   467         fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
   468           EVERY (replicate (length params') (etac @{thm exE} 1))
   469           THEN EVERY (replicate (length ts + length us - 1) (etac @{thm conjE} 1))
   470           THEN Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
   471             let
   472               val (eqs, prems') = chop (length us) prems
   473               val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs
   474             in
   475               rewrite_goal_tac rew_thms 1
   476               THEN rtac intr 1
   477               THEN (EVERY (map (fn p => rtac p 1) prems'))              
   478             end) ctxt' 1 
   479       in
   480         Skip_Proof.prove ctxt' [] [] eq (fn {...} =>
   481           rtac @{thm iffI} 1 THEN etac (#1 elim) 1
   482           THEN EVERY (map_index prove_intr1 c_intrs)
   483           THEN (if null c_intrs then etac @{thm FalseE} 1 else
   484             let val (c_intrs', last_c_intr) = split_last c_intrs in
   485               EVERY (map (fn ci => etac @{thm disjE} 1 THEN prove_intr2 ci) c_intrs')
   486               THEN prove_intr2 last_c_intr
   487             end))
   488         |> rulify
   489         |> singleton (Proof_Context.export ctxt' ctxt'')
   490       end;  
   491   in
   492     map2 prove_eq cs elims
   493   end;
   494   
   495 (* derivation of simplified elimination rules *)
   496 
   497 local
   498 
   499 (*delete needless equality assumptions*)
   500 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
   501   (fn _ => assume_tac 1);
   502 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   503 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   504 
   505 fun simp_case_tac ss i =
   506   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
   507 
   508 in
   509 
   510 fun mk_cases ctxt prop =
   511   let
   512     val thy = Proof_Context.theory_of ctxt;
   513     val ss = simpset_of ctxt;
   514 
   515     fun err msg =
   516       error (Pretty.string_of (Pretty.block
   517         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
   518 
   519     val elims = Induct.find_casesP ctxt prop;
   520 
   521     val cprop = Thm.cterm_of thy prop;
   522     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
   523     fun mk_elim rl =
   524       Thm.implies_intr cprop (Tactic.rule_by_tactic ctxt tac (Thm.assume cprop RS rl))
   525       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   526   in
   527     (case get_first (try mk_elim) elims of
   528       SOME r => r
   529     | NONE => err "Proposition not an inductive predicate:")
   530   end;
   531 
   532 end;
   533 
   534 (* inductive_cases *)
   535 
   536 fun gen_inductive_cases prep_att prep_prop args lthy =
   537   let
   538     val thy = Proof_Context.theory_of lthy;
   539     val facts = args |> Par_List.map (fn ((a, atts), props) =>
   540       ((a, map (prep_att thy) atts),
   541         Par_List.map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
   542   in lthy |> Local_Theory.notes facts |>> map snd end;
   543 
   544 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
   545 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   546 
   547 
   548 val ind_cases_setup =
   549   Method.setup @{binding ind_cases}
   550     (Scan.lift (Scan.repeat1 Args.name_source --
   551       Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.name) []) >>
   552       (fn (raw_props, fixes) => fn ctxt =>
   553         let
   554           val (_, ctxt') = Variable.add_fixes fixes ctxt;
   555           val props = Syntax.read_props ctxt' raw_props;
   556           val ctxt'' = fold Variable.declare_term props ctxt';
   557           val rules = Proof_Context.export ctxt'' ctxt (map (mk_cases ctxt'') props)
   558         in Method.erule 0 rules end))
   559     "dynamic case analysis on predicates";
   560 
   561 (* derivation of simplified equation *)
   562 
   563 fun mk_simp_eq ctxt prop =
   564   let
   565     val thy = Proof_Context.theory_of ctxt
   566     val ctxt' = Variable.auto_fixes prop ctxt
   567     val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of
   568     val substs = Item_Net.retrieve (Equation_Data.get (Context.Proof ctxt)) (HOLogic.dest_Trueprop prop) 
   569       |> map_filter
   570         (fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
   571             (Vartab.empty, Vartab.empty), eq)
   572           handle Pattern.MATCH => NONE)
   573     val (subst, eq) = case substs of
   574         [s] => s
   575       | _ => error
   576         ("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique")
   577     val inst = map (fn v => (cterm_of thy (Var v), cterm_of thy (Envir.subst_term subst (Var v))))
   578       (Term.add_vars (lhs_of eq) [])
   579    in
   580     cterm_instantiate inst eq
   581     |> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv
   582       (Simplifier.full_rewrite (simpset_of ctxt))))
   583     |> singleton (Variable.export ctxt' ctxt)
   584   end
   585 
   586 (* inductive simps *)
   587 
   588 fun gen_inductive_simps prep_att prep_prop args lthy =
   589   let
   590     val thy = Proof_Context.theory_of lthy;
   591     val facts = args |> map (fn ((a, atts), props) =>
   592       ((a, map (prep_att thy) atts),
   593         map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
   594   in lthy |> Local_Theory.notes facts |>> map snd end;
   595 
   596 val inductive_simps = gen_inductive_simps Attrib.intern_src Syntax.read_prop;
   597 val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
   598 
   599 (* prove induction rule *)
   600 
   601 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
   602     fp_def rec_preds_defs ctxt ctxt''' =
   603   let
   604     val _ = clean_message quiet_mode "  Proving the induction rule ...";
   605     val thy = Proof_Context.theory_of ctxt;
   606 
   607     (* predicates for induction rule *)
   608 
   609     val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
   610     val preds = map2 (curry Free) pnames
   611       (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
   612 
   613     (* transform an introduction rule into a premise for induction rule *)
   614 
   615     fun mk_ind_prem r =
   616       let
   617         fun subst s =
   618           (case dest_predicate cs params s of
   619             SOME (_, i, ys, (_, Ts)) =>
   620               let
   621                 val k = length Ts;
   622                 val bs = map Bound (k - 1 downto 0);
   623                 val P = list_comb (nth preds i, map (incr_boundvars k) ys @ bs);
   624                 val Q = list_abs (mk_names "x" k ~~ Ts,
   625                   HOLogic.mk_binop inductive_conj_name
   626                     (list_comb (incr_boundvars k s, bs), P))
   627               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   628           | NONE =>
   629               (case s of
   630                 (t $ u) => (fst (subst t) $ fst (subst u), NONE)
   631               | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
   632               | _ => (s, NONE)));
   633 
   634         fun mk_prem s prems =
   635           (case subst s of
   636             (_, SOME (t, u)) => t :: u :: prems
   637           | (t, _) => t :: prems);
   638 
   639         val SOME (_, i, ys, _) = dest_predicate cs params
   640           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
   641 
   642       in
   643         list_all_free (Logic.strip_params r,
   644           Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
   645             (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
   646               HOLogic.mk_Trueprop (list_comb (nth preds i, ys))))
   647       end;
   648 
   649     val ind_prems = map mk_ind_prem intr_ts;
   650 
   651 
   652     (* make conclusions for induction rules *)
   653 
   654     val Tss = map (binder_types o fastype_of) preds;
   655     val (xnames, ctxt'') =
   656       Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   657     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   658         (map (fn (((xnames, Ts), c), P) =>
   659            let val frees = map Free (xnames ~~ Ts)
   660            in HOLogic.mk_imp
   661              (list_comb (c, params @ frees), list_comb (P, frees))
   662            end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   663 
   664 
   665     (* make predicate for instantiation of abstract induction rule *)
   666 
   667     val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   668       (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
   669          (make_bool_args HOLogic.mk_not I bs i)
   670          (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
   671 
   672     val ind_concl = HOLogic.mk_Trueprop
   673       (HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
   674 
   675     val raw_fp_induct = (mono RS (fp_def RS @{thm def_lfp_induct}));
   676 
   677     val induct = Skip_Proof.prove ctxt'' [] ind_prems ind_concl
   678       (fn {prems, ...} => EVERY
   679         [rewrite_goals_tac [inductive_conj_def],
   680          DETERM (rtac raw_fp_induct 1),
   681          REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
   682          rewrite_goals_tac simp_thms'',
   683          (*This disjE separates out the introduction rules*)
   684          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
   685          (*Now break down the individual cases.  No disjE here in case
   686            some premise involves disjunction.*)
   687          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
   688          REPEAT (FIRSTGOAL
   689            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
   690          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
   691              (inductive_conj_def :: rec_preds_defs @ simp_thms'') prem,
   692            conjI, refl] 1)) prems)]);
   693 
   694     val lemma = Skip_Proof.prove ctxt'' [] []
   695       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
   696         [rewrite_goals_tac rec_preds_defs,
   697          REPEAT (EVERY
   698            [REPEAT (resolve_tac [conjI, impI] 1),
   699             REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
   700             atac 1,
   701             rewrite_goals_tac simp_thms',
   702             atac 1])])
   703 
   704   in singleton (Proof_Context.export ctxt'' ctxt''') (induct RS lemma) end;
   705 
   706 
   707 
   708 (** specification of (co)inductive predicates **)
   709 
   710 fun mk_ind_def quiet_mode skip_mono fork_mono alt_name coind
   711     cs intr_ts monos params cnames_syn lthy =
   712   let
   713     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
   714 
   715     val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
   716     val k = log 2 1 (length cs);
   717     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   718     val p :: xs = map Free (Variable.variant_frees lthy intr_ts
   719       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   720     val bs = map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
   721       (map (rpair HOLogic.boolT) (mk_names "b" k)));
   722 
   723     fun subst t =
   724       (case dest_predicate cs params t of
   725         SOME (_, i, ts, (Ts, Us)) =>
   726           let
   727             val l = length Us;
   728             val zs = map Bound (l - 1 downto 0);
   729           in
   730             list_abs (map (pair "z") Us, list_comb (p,
   731               make_bool_args' bs i @ make_args argTs
   732                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   733           end
   734       | NONE =>
   735           (case t of
   736             t1 $ t2 => subst t1 $ subst t2
   737           | Abs (x, T, u) => Abs (x, T, subst u)
   738           | _ => t));
   739 
   740     (* transform an introduction rule into a conjunction  *)
   741     (*   [| p_i t; ... |] ==> p_j u                       *)
   742     (* is transformed into                                *)
   743     (*   b_j & x_j = u & p b_j t & ...                    *)
   744 
   745     fun transform_rule r =
   746       let
   747         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
   748           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   749         val ps = make_bool_args HOLogic.mk_not I bs i @
   750           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   751           map (subst o HOLogic.dest_Trueprop)
   752             (Logic.strip_assums_hyp r)
   753       in
   754         fold_rev (fn (x, T) => fn P => HOLogic.exists_const T $ Abs (x, T, P))
   755           (Logic.strip_params r)
   756           (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
   757       end
   758 
   759     (* make a disjunction of all introduction rules *)
   760 
   761     val fp_fun = fold_rev lambda (p :: bs @ xs)
   762       (if null intr_ts then HOLogic.false_const
   763        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   764 
   765     (* add definiton of recursive predicates to theory *)
   766 
   767     val rec_name =
   768       if Binding.is_empty alt_name then
   769         Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
   770       else alt_name;
   771 
   772     val ((rec_const, (_, fp_def)), lthy') = lthy
   773       |> Local_Theory.conceal
   774       |> Local_Theory.define
   775         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   776          ((Binding.empty, [Attrib.internal (K Nitpick_Unfolds.add)]),
   777          fold_rev lambda params
   778            (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)))
   779       ||> Local_Theory.restore_naming lthy;
   780     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
   781       (cterm_of (Proof_Context.theory_of lthy') (list_comb (rec_const, params)));
   782     val specs =
   783       if length cs < 2 then []
   784       else
   785         map_index (fn (i, (name_mx, c)) =>
   786           let
   787             val Ts = arg_types_of (length params) c;
   788             val xs = map Free (Variable.variant_frees lthy intr_ts
   789               (mk_names "x" (length Ts) ~~ Ts))
   790           in
   791             (name_mx, (apfst Binding.conceal Attrib.empty_binding, fold_rev lambda (params @ xs)
   792               (list_comb (rec_const, params @ make_bool_args' bs i @
   793                 make_args argTs (xs ~~ Ts)))))
   794           end) (cnames_syn ~~ cs);
   795     val (consts_defs, lthy'') = lthy'
   796       |> fold_map Local_Theory.define specs;
   797     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   798 
   799     val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
   800     val mono = prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos lthy''';
   801     val (_, lthy'''') =
   802       Local_Theory.note (apfst Binding.conceal Attrib.empty_binding,
   803         Proof_Context.export lthy''' lthy'' [mono]) lthy'';
   804 
   805   in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
   806     list_comb (rec_const, params), preds, argTs, bs, xs)
   807   end;
   808 
   809 fun declare_rules rec_binding coind no_ind cnames
   810     preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
   811   let
   812     val rec_name = Binding.name_of rec_binding;
   813     fun rec_qualified qualified = Binding.qualify qualified rec_name;
   814     val intr_names = map Binding.name_of intr_bindings;
   815     val ind_case_names = Rule_Cases.case_names intr_names;
   816     val induct =
   817       if coind then
   818         (raw_induct, [Rule_Cases.case_names [rec_name],
   819           Rule_Cases.case_conclusion (rec_name, intr_names),
   820           Rule_Cases.consumes 1, Induct.coinduct_pred (hd cnames)])
   821       else if no_ind orelse length cnames > 1 then
   822         (raw_induct, [ind_case_names, Rule_Cases.consumes 0])
   823       else (raw_induct RSN (2, rev_mp), [ind_case_names, Rule_Cases.consumes 1]);
   824 
   825     val (intrs', lthy1) =
   826       lthy |>
   827       Spec_Rules.add
   828         (if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
   829       Local_Theory.notes
   830         (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
   831           map (fn th => [([th],
   832            [Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
   833       map (hd o snd);
   834     val (((_, elims'), (_, [induct'])), lthy2) =
   835       lthy1 |>
   836       Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
   837       fold_map (fn (name, (elim, cases, k)) =>
   838         Local_Theory.note
   839           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
   840             [Attrib.internal (K (Rule_Cases.case_names cases)),
   841              Attrib.internal (K (Rule_Cases.consumes 1)),
   842              Attrib.internal (K (Rule_Cases.constraints k)),
   843              Attrib.internal (K (Induct.cases_pred name)),
   844              Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
   845         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   846       Local_Theory.note
   847         ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
   848           map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
   849 
   850     val (eqs', lthy3) = lthy2 |> 
   851       fold_map (fn (name, eq) => Local_Theory.note
   852           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
   853             [Attrib.internal (K add_equation)]), [eq])
   854           #> apfst (hd o snd))
   855         (if null eqs then [] else (cnames ~~ eqs))
   856     val (inducts, lthy4) =
   857       if no_ind orelse coind then ([], lthy3)
   858       else
   859         let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
   860           lthy3 |>
   861           Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
   862             inducts |> map (fn (name, th) => ([th],
   863               [Attrib.internal (K ind_case_names),
   864                Attrib.internal (K (Rule_Cases.consumes 1)),
   865                Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
   866         end;
   867   in (intrs', elims', eqs', induct', inducts, lthy4) end;
   868 
   869 type inductive_flags =
   870   {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
   871     no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool};
   872 
   873 type add_ind_def =
   874   inductive_flags ->
   875   term list -> (Attrib.binding * term) list -> thm list ->
   876   term list -> (binding * mixfix) list ->
   877   local_theory -> inductive_result * local_theory;
   878 
   879 fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}
   880     cs intros monos params cnames_syn lthy =
   881   let
   882     val _ = null cnames_syn andalso error "No inductive predicates given";
   883     val names = map (Binding.name_of o fst) cnames_syn;
   884     val _ = message (quiet_mode andalso not verbose)
   885       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
   886 
   887     val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
   888     val ((intr_names, intr_atts), intr_ts) =
   889       apfst split_list (split_list (map (check_rule lthy cs params) intros));
   890 
   891     val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   892       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts
   893         monos params cnames_syn lthy;
   894 
   895     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
   896       intr_ts rec_preds_defs lthy2 lthy1;
   897     val elims =
   898       if no_elim then []
   899       else
   900         prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
   901           unfold rec_preds_defs lthy2 lthy1;
   902     val raw_induct = zero_var_indexes
   903       (if no_ind then Drule.asm_rl
   904        else if coind then
   905          singleton (Proof_Context.export lthy2 lthy1)
   906            (rotate_prems ~1 (Object_Logic.rulify
   907              (fold_rule rec_preds_defs
   908                (rewrite_rule simp_thms'''
   909                 (mono RS (fp_def RS @{thm def_coinduct}))))))
   910        else
   911          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
   912            rec_preds_defs lthy2 lthy1);
   913     val eqs =
   914       if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1
   915 
   916     val elims' = map (fn (th, ns, i) => (rulify th, ns, i)) elims
   917     val intrs' = map rulify intrs
   918 
   919     val (intrs'', elims'', eqs', induct, inducts, lthy3) = declare_rules rec_name coind no_ind
   920       cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
   921 
   922     val result =
   923       {preds = preds,
   924        intrs = intrs'',
   925        elims = elims'',
   926        raw_induct = rulify raw_induct,
   927        induct = induct,
   928        inducts = inducts,
   929        eqs = eqs'};
   930 
   931     val lthy4 = lthy3
   932       |> Local_Theory.declaration false (fn phi =>
   933         let val result' = morph_result phi result;
   934         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
   935   in (result, lthy4) end;
   936 
   937 
   938 (* external interfaces *)
   939 
   940 fun gen_add_inductive_i mk_def
   941     (flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono})
   942     cnames_syn pnames spec monos lthy =
   943   let
   944     val thy = Proof_Context.theory_of lthy;
   945     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   946 
   947 
   948     (* abbrevs *)
   949 
   950     val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
   951 
   952     fun get_abbrev ((name, atts), t) =
   953       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
   954         let
   955           val _ = Binding.is_empty name andalso null atts orelse
   956             error "Abbreviations may not have names or attributes";
   957           val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
   958           val var =
   959             (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
   960               NONE => error ("Undeclared head of abbreviation " ^ quote x)
   961             | SOME ((b, T'), mx) =>
   962                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
   963                 else (b, mx));
   964         in SOME (var, rhs) end
   965       else NONE;
   966 
   967     val abbrevs = map_filter get_abbrev spec;
   968     val bs = map (Binding.name_of o fst o fst) abbrevs;
   969 
   970 
   971     (* predicates *)
   972 
   973     val pre_intros = filter_out (is_some o get_abbrev) spec;
   974     val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
   975     val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
   976     val ps = map Free pnames;
   977 
   978     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
   979     val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
   980     val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
   981     val expand = Assumption.export_term ctxt3 lthy #> Proof_Context.cert_term lthy;
   982 
   983     fun close_rule r = list_all_free (rev (fold_aterms
   984       (fn t as Free (v as (s, _)) =>
   985           if Variable.is_fixed ctxt1 s orelse
   986             member (op =) ps t then I else insert (op =) v
   987         | _ => I) r []), r);
   988 
   989     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
   990     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
   991   in
   992     lthy
   993     |> mk_def flags cs intros monos ps preds
   994     ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
   995   end;
   996 
   997 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos int lthy =
   998   let
   999     val ((vars, intrs), _) = lthy
  1000       |> Proof_Context.set_mode Proof_Context.mode_abbrev
  1001       |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
  1002     val (cs, ps) = chop (length cnames_syn) vars;
  1003     val monos = Attrib.eval_thms lthy raw_monos;
  1004     val flags = {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
  1005       coind = coind, no_elim = false, no_ind = false, skip_mono = false, fork_mono = not int};
  1006   in
  1007     lthy
  1008     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
  1009   end;
  1010 
  1011 val add_inductive_i = gen_add_inductive_i add_ind_def;
  1012 val add_inductive = gen_add_inductive add_ind_def;
  1013 
  1014 fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
  1015   let
  1016     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
  1017     val ctxt' = thy
  1018       |> Named_Target.theory_init
  1019       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
  1020       |> Local_Theory.exit;
  1021     val info = #2 (the_inductive ctxt' name);
  1022   in (info, Proof_Context.theory_of ctxt') end;
  1023 
  1024 
  1025 (* read off arities of inductive predicates from raw induction rule *)
  1026 fun arities_of induct =
  1027   map (fn (_ $ t $ u) =>
  1028       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
  1029     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
  1030 
  1031 (* read off parameters of inductive predicate from raw induction rule *)
  1032 fun params_of induct =
  1033   let
  1034     val (_ $ t $ u :: _) =
  1035       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
  1036     val (_, ts) = strip_comb t;
  1037     val (_, us) = strip_comb u
  1038   in
  1039     List.take (ts, length ts - length us)
  1040   end;
  1041 
  1042 val pname_of_intr =
  1043   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
  1044 
  1045 (* partition introduction rules according to predicate name *)
  1046 fun gen_partition_rules f induct intros =
  1047   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
  1048     (map (rpair [] o fst) (arities_of induct));
  1049 
  1050 val partition_rules = gen_partition_rules I;
  1051 fun partition_rules' induct = gen_partition_rules fst induct;
  1052 
  1053 fun unpartition_rules intros xs =
  1054   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
  1055     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
  1056 
  1057 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
  1058 fun infer_intro_vars elim arity intros =
  1059   let
  1060     val thy = theory_of_thm elim;
  1061     val _ :: cases = prems_of elim;
  1062     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
  1063     fun mtch (t, u) =
  1064       let
  1065         val params = Logic.strip_params t;
  1066         val vars = map (Var o apfst (rpair 0))
  1067           (Name.variant_list used (map fst params) ~~ map snd params);
  1068         val ts = map (curry subst_bounds (rev vars))
  1069           (List.drop (Logic.strip_assums_hyp t, arity));
  1070         val us = Logic.strip_imp_prems u;
  1071         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
  1072           (Vartab.empty, Vartab.empty);
  1073       in
  1074         map (Envir.subst_term tab) vars
  1075       end
  1076   in
  1077     map (mtch o apsnd prop_of) (cases ~~ intros)
  1078   end;
  1079 
  1080 
  1081 
  1082 (** package setup **)
  1083 
  1084 (* setup theory *)
  1085 
  1086 val setup =
  1087   ind_cases_setup #>
  1088   Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
  1089     "declaration of monotonicity rule";
  1090 
  1091 
  1092 (* outer syntax *)
  1093 
  1094 val _ = Keyword.keyword "monos";
  1095 
  1096 fun gen_ind_decl mk_def coind =
  1097   Parse.fixes -- Parse.for_fixes --
  1098   Scan.optional Parse_Spec.where_alt_specs [] --
  1099   Scan.optional (Parse.$$$ "monos" |-- Parse.!!! Parse_Spec.xthms1) []
  1100   >> (fn (((preds, params), specs), monos) =>
  1101       (snd oo gen_add_inductive mk_def true coind preds params specs monos));
  1102 
  1103 val ind_decl = gen_ind_decl add_ind_def;
  1104 
  1105 val _ =
  1106   Outer_Syntax.local_theory' "inductive" "define inductive predicates" Keyword.thy_decl
  1107     (ind_decl false);
  1108 
  1109 val _ =
  1110   Outer_Syntax.local_theory' "coinductive" "define coinductive predicates" Keyword.thy_decl
  1111     (ind_decl true);
  1112 
  1113 val _ =
  1114   Outer_Syntax.local_theory "inductive_cases"
  1115     "create simplified instances of elimination rules (improper)" Keyword.thy_script
  1116     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
  1117 
  1118 val _ =
  1119   Outer_Syntax.local_theory "inductive_simps"
  1120     "create simplification rules for inductive predicates" Keyword.thy_script
  1121     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
  1122 
  1123 end;