src/HOL/Tools/inductive_package.ML
author wenzelm
Sat May 17 13:54:30 2008 +0200 (2008-05-17)
changeset 26928 ca87aff1ad2d
parent 26736 e6091328718f
child 26988 742e26213212
permissions -rw-r--r--
structure Display: less pervasive operations;
     1 (*  Title:      HOL/Tools/inductive_package.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
     5 
     6 (Co)Inductive Definition module for HOL.
     7 
     8 Features:
     9   * least or greatest fixedpoints
    10   * mutually recursive definitions
    11   * definitions involving arbitrary monotone operators
    12   * automatically proves introduction and elimination rules
    13 
    14   Introduction rules have the form
    15   [| M Pj ti, ..., Q x, ... |] ==> Pk t
    16   where M is some monotone operator (usually the identity)
    17   Q x is any side condition on the free variables
    18   ti, t are any terms
    19   Pj, Pk are two of the predicates being defined in mutual recursion
    20 *)
    21 
    22 signature BASIC_INDUCTIVE_PACKAGE =
    23 sig
    24   type inductive_result
    25   val morph_result: morphism -> inductive_result -> inductive_result
    26   type inductive_info
    27   val the_inductive: Proof.context -> string -> inductive_info
    28   val print_inductives: Proof.context -> unit
    29   val mono_add: attribute
    30   val mono_del: attribute
    31   val get_monos: Proof.context -> thm list
    32   val mk_cases: Proof.context -> term -> thm
    33   val inductive_forall_name: string
    34   val inductive_forall_def: thm
    35   val rulify: thm -> thm
    36   val inductive_cases: ((bstring * Attrib.src list) * string list) list ->
    37     Proof.context -> thm list list * local_theory
    38   val inductive_cases_i: ((bstring * Attrib.src list) * term list) list ->
    39     Proof.context -> thm list list * local_theory
    40   type inductive_flags
    41   val add_inductive_i:
    42     inductive_flags -> ((string * typ) * mixfix) list ->
    43     (string * typ) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
    44       local_theory -> inductive_result * local_theory
    45   val add_inductive: bool -> bool -> (string * string option * mixfix) list ->
    46     (string * string option * mixfix) list ->
    47     ((bstring * Attrib.src list) * string) list -> (Facts.ref * Attrib.src list) list ->
    48     local_theory -> inductive_result * local_theory
    49   val add_inductive_global: string -> inductive_flags ->
    50     ((string * typ) * mixfix) list -> (string * typ) list ->
    51     ((bstring * Attrib.src list) * term) list -> thm list -> theory -> inductive_result * theory
    52   val arities_of: thm -> (string * int) list
    53   val params_of: thm -> term list
    54   val partition_rules: thm -> thm list -> (string * thm list) list
    55   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
    56   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    57   val infer_intro_vars: thm -> int -> thm list -> term list list
    58   val setup: theory -> theory
    59 end;
    60 
    61 signature INDUCTIVE_PACKAGE =
    62 sig
    63   include BASIC_INDUCTIVE_PACKAGE
    64   type add_ind_def
    65   val declare_rules: string -> bstring -> bool -> bool -> string list ->
    66     thm list -> bstring list -> Attrib.src list list -> (thm * string list) list ->
    67     thm -> local_theory -> thm list * thm list * thm * local_theory
    68   val add_ind_def: add_ind_def
    69   val gen_add_inductive_i: add_ind_def ->
    70     inductive_flags -> ((string * typ) * mixfix) list ->
    71     (string * typ) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
    72       local_theory -> inductive_result * local_theory
    73   val gen_add_inductive: add_ind_def ->
    74     bool -> bool -> (string * string option * mixfix) list ->
    75     (string * string option * mixfix) list ->
    76     ((bstring * Attrib.src list) * string) list -> (Facts.ref * Attrib.src list) list ->
    77     local_theory -> inductive_result * local_theory
    78   val gen_ind_decl: add_ind_def ->
    79     bool -> OuterParse.token list ->
    80     (Toplevel.transition -> Toplevel.transition) * OuterParse.token list
    81 end;
    82 
    83 structure InductivePackage: INDUCTIVE_PACKAGE =
    84 struct
    85 
    86 
    87 (** theory context references **)
    88 
    89 val inductive_forall_name = "HOL.induct_forall";
    90 val inductive_forall_def = thm "induct_forall_def";
    91 val inductive_conj_name = "HOL.induct_conj";
    92 val inductive_conj_def = thm "induct_conj_def";
    93 val inductive_conj = thms "induct_conj";
    94 val inductive_atomize = thms "induct_atomize";
    95 val inductive_rulify = thms "induct_rulify";
    96 val inductive_rulify_fallback = thms "induct_rulify_fallback";
    97 
    98 val notTrueE = TrueI RSN (2, notE);
    99 val notFalseI = Seq.hd (atac 1 notI);
   100 val simp_thms' = map (fn s => mk_meta_eq (the (find_first
   101   (equal (Sign.read_prop HOL.thy s) o prop_of) simp_thms)))
   102   ["(~True) = False", "(~False) = True",
   103    "(True --> ?P) = ?P", "(False --> ?P) = True",
   104    "(?P & True) = ?P", "(True & ?P) = ?P"];
   105 
   106 
   107 
   108 (** context data **)
   109 
   110 type inductive_result =
   111   {preds: term list, elims: thm list, raw_induct: thm,
   112    induct: thm, intrs: thm list};
   113 
   114 fun morph_result phi {preds, elims, raw_induct: thm, induct, intrs} =
   115   let
   116     val term = Morphism.term phi;
   117     val thm = Morphism.thm phi;
   118     val fact = Morphism.fact phi;
   119   in
   120    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
   121     induct = thm induct, intrs = fact intrs}
   122   end;
   123 
   124 type inductive_info =
   125   {names: string list, coind: bool} * inductive_result;
   126 
   127 structure InductiveData = GenericDataFun
   128 (
   129   type T = inductive_info Symtab.table * thm list;
   130   val empty = (Symtab.empty, []);
   131   val extend = I;
   132   fun merge _ ((tab1, monos1), (tab2, monos2)) =
   133     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
   134 );
   135 
   136 val get_inductives = InductiveData.get o Context.Proof;
   137 
   138 fun print_inductives ctxt =
   139   let
   140     val (tab, monos) = get_inductives ctxt;
   141     val space = Consts.space_of (ProofContext.consts_of ctxt);
   142   in
   143     [Pretty.strs ("(co)inductives:" :: map #1 (NameSpace.extern_table (space, tab))),
   144      Pretty.big_list "monotonicity rules:" (map (ProofContext.pretty_thm ctxt) monos)]
   145     |> Pretty.chunks |> Pretty.writeln
   146   end;
   147 
   148 
   149 (* get and put data *)
   150 
   151 fun the_inductive ctxt name =
   152   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
   153     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   154   | SOME info => info);
   155 
   156 fun put_inductives names info = InductiveData.map
   157   (apfst (fold (fn name => Symtab.update (name, info)) names));
   158 
   159 
   160 
   161 (** monotonicity rules **)
   162 
   163 val get_monos = #2 o get_inductives;
   164 val map_monos = InductiveData.map o apsnd;
   165 
   166 fun mk_mono thm =
   167   let
   168     val concl = concl_of thm;
   169     fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @
   170       (case concl of
   171           (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
   172         | _ => [thm' RS (thm' RS eq_to_mono2)]);
   173     fun dest_less_concl thm = dest_less_concl (thm RS le_funD)
   174       handle THM _ => thm RS le_boolD
   175   in
   176     case concl of
   177       Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq)
   178     | _ $ (Const ("op =", _) $ _ $ _) => eq2mono thm
   179     | _ $ (Const ("HOL.ord_class.less_eq", _) $ _ $ _) =>
   180       [dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   181          (resolve_tac [le_funI, le_boolI'])) thm))]
   182     | _ => [thm]
   183   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm thm);
   184 
   185 val mono_add = Thm.declaration_attribute (map_monos o fold Thm.add_thm o mk_mono);
   186 val mono_del = Thm.declaration_attribute (map_monos o fold Thm.del_thm o mk_mono);
   187 
   188 
   189 
   190 (** misc utilities **)
   191 
   192 fun message quiet_mode s = if quiet_mode then () else writeln s;
   193 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
   194 
   195 fun coind_prefix true = "co"
   196   | coind_prefix false = "";
   197 
   198 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   199 
   200 fun make_bool_args f g [] i = []
   201   | make_bool_args f g (x :: xs) i =
   202       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   203 
   204 fun make_bool_args' xs =
   205   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
   206 
   207 fun find_arg T x [] = sys_error "find_arg"
   208   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   209       apsnd (cons p) (find_arg T x ps)
   210   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   211       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   212       else apsnd (cons p) (find_arg T x ps);
   213 
   214 fun make_args Ts xs =
   215   map (fn (T, (NONE, ())) => Const ("arbitrary", T) | (_, (SOME t, ())) => t)
   216     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   217 
   218 fun make_args' Ts xs Us =
   219   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   220 
   221 fun dest_predicate cs params t =
   222   let
   223     val k = length params;
   224     val (c, ts) = strip_comb t;
   225     val (xs, ys) = chop k ts;
   226     val i = find_index_eq c cs;
   227   in
   228     if xs = params andalso i >= 0 then
   229       SOME (c, i, ys, chop (length ys)
   230         (List.drop (binder_types (fastype_of c), k)))
   231     else NONE
   232   end;
   233 
   234 fun mk_names a 0 = []
   235   | mk_names a 1 = [a]
   236   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   237 
   238 
   239 
   240 (** process rules **)
   241 
   242 local
   243 
   244 fun err_in_rule ctxt name t msg =
   245   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
   246     Syntax.string_of_term ctxt t, msg]);
   247 
   248 fun err_in_prem ctxt name t p msg =
   249   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
   250     "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
   251 
   252 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   253 
   254 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   255 
   256 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   257 
   258 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
   259 
   260 in
   261 
   262 fun check_rule ctxt cs params ((name, att), rule) =
   263   let
   264     val params' = Term.variant_frees rule (Logic.strip_params rule);
   265     val frees = rev (map Free params');
   266     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   267     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   268     val rule' = Logic.list_implies (prems, concl);
   269     val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
   270     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
   271 
   272     fun check_ind err t = case dest_predicate cs params t of
   273         NONE => err (bad_app ^
   274           commas (map (Syntax.string_of_term ctxt) params))
   275       | SOME (_, _, ys, _) =>
   276           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   277           then err bad_ind_occ else ();
   278 
   279     fun check_prem' prem t =
   280       if head_of t mem cs then
   281         check_ind (err_in_prem ctxt name rule prem) t
   282       else (case t of
   283           Abs (_, _, t) => check_prem' prem t
   284         | t $ u => (check_prem' prem t; check_prem' prem u)
   285         | _ => ());
   286 
   287     fun check_prem (prem, aprem) =
   288       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   289       else err_in_prem ctxt name rule prem "Non-atomic premise";
   290   in
   291     (case concl of
   292        Const ("Trueprop", _) $ t =>
   293          if head_of t mem cs then
   294            (check_ind (err_in_rule ctxt name rule') t;
   295             List.app check_prem (prems ~~ aprems))
   296          else err_in_rule ctxt name rule' bad_concl
   297      | _ => err_in_rule ctxt name rule' bad_concl);
   298     ((name, att), arule)
   299   end;
   300 
   301 val rulify =
   302   hol_simplify inductive_conj
   303   #> hol_simplify inductive_rulify
   304   #> hol_simplify inductive_rulify_fallback
   305   #> MetaSimplifier.norm_hhf;
   306 
   307 end;
   308 
   309 
   310 
   311 (** proofs for (co)inductive predicates **)
   312 
   313 (* prove monotonicity *)
   314 
   315 fun prove_mono quiet_mode skip_mono predT fp_fun monos ctxt =
   316  (message (quiet_mode orelse skip_mono andalso !quick_and_dirty)
   317     "  Proving monotonicity ...";
   318   (if skip_mono then SkipProof.prove else Goal.prove) ctxt [] []
   319     (HOLogic.mk_Trueprop
   320       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   321     (fn _ => EVERY [rtac @{thm monoI} 1,
   322       REPEAT (resolve_tac [le_funI, le_boolI'] 1),
   323       REPEAT (FIRST
   324         [atac 1,
   325          resolve_tac (List.concat (map mk_mono monos) @ get_monos ctxt) 1,
   326          etac le_funE 1, dtac le_boolD 1])]));
   327 
   328 
   329 (* prove introduction rules *)
   330 
   331 fun prove_intrs quiet_mode coind mono fp_def k params intr_ts rec_preds_defs ctxt =
   332   let
   333     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
   334 
   335     val unfold = funpow k (fn th => th RS fun_cong)
   336       (mono RS (fp_def RS
   337         (if coind then def_gfp_unfold else def_lfp_unfold)));
   338 
   339     fun select_disj 1 1 = []
   340       | select_disj _ 1 = [rtac disjI1]
   341       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   342 
   343     val rules = [refl, TrueI, notFalseI, exI, conjI];
   344 
   345     val intrs = map_index (fn (i, intr) => rulify
   346       (SkipProof.prove ctxt (map (fst o dest_Free) params) [] intr (fn _ => EVERY
   347        [rewrite_goals_tac rec_preds_defs,
   348         rtac (unfold RS iffD2) 1,
   349         EVERY1 (select_disj (length intr_ts) (i + 1)),
   350         (*Not ares_tac, since refl must be tried before any equality assumptions;
   351           backtracking may occur if the premises have extra variables!*)
   352         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts
   353 
   354   in (intrs, unfold) end;
   355 
   356 
   357 (* prove elimination rules *)
   358 
   359 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt =
   360   let
   361     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
   362 
   363     val ([pname], ctxt') = ctxt |>
   364       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   365       Variable.variant_fixes ["P"];
   366     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   367 
   368     fun dest_intr r =
   369       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   370        Logic.strip_assums_hyp r, Logic.strip_params r);
   371 
   372     val intrs = map dest_intr intr_ts ~~ intr_names;
   373 
   374     val rules1 = [disjE, exE, FalseE];
   375     val rules2 = [conjE, FalseE, notTrueE];
   376 
   377     fun prove_elim c =
   378       let
   379         val Ts = List.drop (binder_types (fastype_of c), length params);
   380         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   381         val frees = map Free (anames ~~ Ts);
   382 
   383         fun mk_elim_prem ((_, _, us, _), ts, params') =
   384           list_all (params',
   385             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   386               (frees ~~ us) @ ts, P));
   387         val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs);
   388         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   389            map mk_elim_prem (map #1 c_intrs)
   390       in
   391         (SkipProof.prove ctxt'' [] prems P
   392           (fn {prems, ...} => EVERY
   393             [cut_facts_tac [hd prems] 1,
   394              rewrite_goals_tac rec_preds_defs,
   395              dtac (unfold RS iffD1) 1,
   396              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   397              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   398              EVERY (map (fn prem =>
   399                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
   400           |> rulify
   401           |> singleton (ProofContext.export ctxt'' ctxt),
   402          map #2 c_intrs)
   403       end
   404 
   405    in map prove_elim cs end;
   406 
   407 
   408 (* derivation of simplified elimination rules *)
   409 
   410 local
   411 
   412 (*delete needless equality assumptions*)
   413 val refl_thin = Goal.prove_global HOL.thy [] [] @{prop "!!P. a = a ==> P ==> P"}
   414   (fn _ => assume_tac 1);
   415 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   416 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   417 
   418 fun simp_case_tac ss i =
   419   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
   420 
   421 in
   422 
   423 fun mk_cases ctxt prop =
   424   let
   425     val thy = ProofContext.theory_of ctxt;
   426     val ss = Simplifier.local_simpset_of ctxt;
   427 
   428     fun err msg =
   429       error (Pretty.string_of (Pretty.block
   430         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
   431 
   432     val elims = Induct.find_casesP ctxt prop;
   433 
   434     val cprop = Thm.cterm_of thy prop;
   435     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
   436     fun mk_elim rl =
   437       Thm.implies_intr cprop (Tactic.rule_by_tactic tac (Thm.assume cprop RS rl))
   438       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   439   in
   440     (case get_first (try mk_elim) elims of
   441       SOME r => r
   442     | NONE => err "Proposition not an inductive predicate:")
   443   end;
   444 
   445 end;
   446 
   447 
   448 (* inductive_cases *)
   449 
   450 fun gen_inductive_cases prep_att prep_prop args lthy =
   451   let
   452     val thy = ProofContext.theory_of lthy;
   453     val facts = args |> map (fn ((a, atts), props) =>
   454       ((a, map (prep_att thy) atts),
   455         map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
   456   in lthy |> LocalTheory.notes Thm.theoremK facts |>> map snd end;
   457 
   458 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
   459 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   460 
   461 
   462 fun ind_cases src = Method.syntax (Scan.lift (Scan.repeat1 Args.name --
   463     Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.name) [])) src
   464   #> (fn ((raw_props, fixes), ctxt) =>
   465     let
   466       val (_, ctxt') = Variable.add_fixes fixes ctxt;
   467       val props = Syntax.read_props ctxt' raw_props;
   468       val ctxt'' = fold Variable.declare_term props ctxt';
   469       val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props)
   470     in Method.erule 0 rules end);
   471 
   472 
   473 
   474 (* prove induction rule *)
   475 
   476 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
   477     fp_def rec_preds_defs ctxt =
   478   let
   479     val _ = clean_message quiet_mode "  Proving the induction rule ...";
   480     val thy = ProofContext.theory_of ctxt;
   481 
   482     (* predicates for induction rule *)
   483 
   484     val (pnames, ctxt') = ctxt |>
   485       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   486       Variable.variant_fixes (mk_names "P" (length cs));
   487     val preds = map Free (pnames ~~
   488       map (fn c => List.drop (binder_types (fastype_of c), length params) --->
   489         HOLogic.boolT) cs);
   490 
   491     (* transform an introduction rule into a premise for induction rule *)
   492 
   493     fun mk_ind_prem r =
   494       let
   495         fun subst s = (case dest_predicate cs params s of
   496             SOME (_, i, ys, (_, Ts)) =>
   497               let
   498                 val k = length Ts;
   499                 val bs = map Bound (k - 1 downto 0);
   500                 val P = list_comb (List.nth (preds, i),
   501                   map (incr_boundvars k) ys @ bs);
   502                 val Q = list_abs (mk_names "x" k ~~ Ts,
   503                   HOLogic.mk_binop inductive_conj_name
   504                     (list_comb (incr_boundvars k s, bs), P))
   505               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   506           | NONE => (case s of
   507               (t $ u) => (fst (subst t) $ fst (subst u), NONE)
   508             | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
   509             | _ => (s, NONE)));
   510 
   511         fun mk_prem (s, prems) = (case subst s of
   512               (_, SOME (t, u)) => t :: u :: prems
   513             | (t, _) => t :: prems);
   514 
   515         val SOME (_, i, ys, _) = dest_predicate cs params
   516           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
   517 
   518       in list_all_free (Logic.strip_params r,
   519         Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
   520           [] (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r))),
   521             HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
   522       end;
   523 
   524     val ind_prems = map mk_ind_prem intr_ts;
   525 
   526 
   527     (* make conclusions for induction rules *)
   528 
   529     val Tss = map (binder_types o fastype_of) preds;
   530     val (xnames, ctxt'') =
   531       Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   532     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   533         (map (fn (((xnames, Ts), c), P) =>
   534            let val frees = map Free (xnames ~~ Ts)
   535            in HOLogic.mk_imp
   536              (list_comb (c, params @ frees), list_comb (P, frees))
   537            end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   538 
   539 
   540     (* make predicate for instantiation of abstract induction rule *)
   541 
   542     val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   543       (map_index (fn (i, P) => foldr HOLogic.mk_imp
   544          (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))
   545          (make_bool_args HOLogic.mk_not I bs i)) preds));
   546 
   547     val ind_concl = HOLogic.mk_Trueprop
   548       (HOLogic.mk_binrel "HOL.ord_class.less_eq" (rec_const, ind_pred));
   549 
   550     val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));
   551 
   552     val induct = SkipProof.prove ctxt'' [] ind_prems ind_concl
   553       (fn {prems, ...} => EVERY
   554         [rewrite_goals_tac [inductive_conj_def],
   555          DETERM (rtac raw_fp_induct 1),
   556          REPEAT (resolve_tac [le_funI, le_boolI] 1),
   557          rewrite_goals_tac (inf_fun_eq :: inf_bool_eq :: simp_thms'),
   558          (*This disjE separates out the introduction rules*)
   559          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
   560          (*Now break down the individual cases.  No disjE here in case
   561            some premise involves disjunction.*)
   562          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
   563          REPEAT (FIRSTGOAL
   564            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
   565          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
   566              (inductive_conj_def :: rec_preds_defs @ simp_thms') prem,
   567            conjI, refl] 1)) prems)]);
   568 
   569     val lemma = SkipProof.prove ctxt'' [] []
   570       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
   571         [rewrite_goals_tac rec_preds_defs,
   572          REPEAT (EVERY
   573            [REPEAT (resolve_tac [conjI, impI] 1),
   574             REPEAT (eresolve_tac [le_funE, le_boolE] 1),
   575             atac 1,
   576             rewrite_goals_tac simp_thms',
   577             atac 1])])
   578 
   579   in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end;
   580 
   581 
   582 
   583 (** specification of (co)inductive predicates **)
   584 
   585 fun mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts monos params cnames_syn ctxt =
   586   let
   587     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
   588 
   589     val argTs = fold (fn c => fn Ts => Ts @
   590       (List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs [];
   591     val k = log 2 1 (length cs);
   592     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   593     val p :: xs = map Free (Variable.variant_frees ctxt intr_ts
   594       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   595     val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts)
   596       (map (rpair HOLogic.boolT) (mk_names "b" k)));
   597 
   598     fun subst t = (case dest_predicate cs params t of
   599         SOME (_, i, ts, (Ts, Us)) =>
   600           let
   601             val l = length Us;
   602             val zs = map Bound (l - 1 downto 0)
   603           in
   604             list_abs (map (pair "z") Us, list_comb (p,
   605               make_bool_args' bs i @ make_args argTs
   606                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   607           end
   608       | NONE => (case t of
   609           t1 $ t2 => subst t1 $ subst t2
   610         | Abs (x, T, u) => Abs (x, T, subst u)
   611         | _ => t));
   612 
   613     (* transform an introduction rule into a conjunction  *)
   614     (*   [| p_i t; ... |] ==> p_j u                       *)
   615     (* is transformed into                                *)
   616     (*   b_j & x_j = u & p b_j t & ...                    *)
   617 
   618     fun transform_rule r =
   619       let
   620         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
   621           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   622         val ps = make_bool_args HOLogic.mk_not I bs i @
   623           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   624           map (subst o HOLogic.dest_Trueprop)
   625             (Logic.strip_assums_hyp r)
   626       in foldr (fn ((x, T), P) => HOLogic.exists_const T $ (Abs (x, T, P)))
   627         (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
   628         (Logic.strip_params r)
   629       end
   630 
   631     (* make a disjunction of all introduction rules *)
   632 
   633     val fp_fun = fold_rev lambda (p :: bs @ xs)
   634       (if null intr_ts then HOLogic.false_const
   635        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   636 
   637     (* add definiton of recursive predicates to theory *)
   638 
   639     val rec_name = if alt_name = "" then
   640       space_implode "_" (map fst cnames_syn) else alt_name;
   641 
   642     val ((rec_const, (_, fp_def)), ctxt') = ctxt |>
   643       LocalTheory.define Thm.internalK
   644         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   645          (("", []), fold_rev lambda params
   646            (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)));
   647     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
   648       (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params)));
   649     val specs = if length cs < 2 then [] else
   650       map_index (fn (i, (name_mx, c)) =>
   651         let
   652           val Ts = List.drop (binder_types (fastype_of c), length params);
   653           val xs = map Free (Variable.variant_frees ctxt intr_ts
   654             (mk_names "x" (length Ts) ~~ Ts))
   655         in
   656           (name_mx, (("", []), fold_rev lambda (params @ xs)
   657             (list_comb (rec_const, params @ make_bool_args' bs i @
   658               make_args argTs (xs ~~ Ts)))))
   659         end) (cnames_syn ~~ cs);
   660     val (consts_defs, ctxt'') = fold_map (LocalTheory.define Thm.internalK) specs ctxt';
   661     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   662 
   663     val mono = prove_mono quiet_mode skip_mono predT fp_fun monos ctxt''
   664 
   665   in (ctxt'', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
   666     list_comb (rec_const, params), preds, argTs, bs, xs)
   667   end;
   668 
   669 fun declare_rules kind rec_name coind no_ind cnames intrs intr_names intr_atts
   670       elims raw_induct ctxt =
   671   let
   672     val ind_case_names = RuleCases.case_names intr_names;
   673     val induct =
   674       if coind then
   675         (raw_induct, [RuleCases.case_names [rec_name],
   676           RuleCases.case_conclusion (rec_name, intr_names),
   677           RuleCases.consumes 1, Induct.coinduct_pred (hd cnames)])
   678       else if no_ind orelse length cnames > 1 then
   679         (raw_induct, [ind_case_names, RuleCases.consumes 0])
   680       else (raw_induct RSN (2, rev_mp), [ind_case_names, RuleCases.consumes 1]);
   681 
   682     val (intrs', ctxt1) =
   683       ctxt |>
   684       LocalTheory.notes kind
   685         (map (NameSpace.qualified rec_name) intr_names ~~
   686          intr_atts ~~ map (fn th => [([th],
   687            [Attrib.internal (K (ContextRules.intro_query NONE))])]) intrs) |>>
   688       map (hd o snd);
   689     val (((_, elims'), (_, [induct'])), ctxt2) =
   690       ctxt1 |>
   691       LocalTheory.note kind ((NameSpace.qualified rec_name "intros", []), intrs') ||>>
   692       fold_map (fn (name, (elim, cases)) =>
   693         LocalTheory.note kind ((NameSpace.qualified (Sign.base_name name) "cases",
   694           [Attrib.internal (K (RuleCases.case_names cases)),
   695            Attrib.internal (K (RuleCases.consumes 1)),
   696            Attrib.internal (K (Induct.cases_pred name)),
   697            Attrib.internal (K (ContextRules.elim_query NONE))]), [elim]) #>
   698         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   699       LocalTheory.note kind ((NameSpace.qualified rec_name (coind_prefix coind ^ "induct"),
   700         map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
   701 
   702     val ctxt3 = if no_ind orelse coind then ctxt2 else
   703       let val inducts = cnames ~~ ProjectRule.projects ctxt2 (1 upto length cnames) induct'
   704       in
   705         ctxt2 |>
   706         LocalTheory.notes kind [((NameSpace.qualified rec_name "inducts", []),
   707           inducts |> map (fn (name, th) => ([th],
   708             [Attrib.internal (K ind_case_names),
   709              Attrib.internal (K (RuleCases.consumes 1)),
   710              Attrib.internal (K (Induct.induct_pred name))])))] |> snd
   711       end
   712   in (intrs', elims', induct', ctxt3) end;
   713 
   714 type inductive_flags =
   715   {quiet_mode: bool, verbose: bool, kind: string, alt_name: bstring,
   716    coind: bool, no_elim: bool, no_ind: bool, skip_mono: bool}
   717 
   718 type add_ind_def =
   719   inductive_flags ->
   720   term list -> ((string * Attrib.src list) * term) list -> thm list ->
   721   term list -> (string * mixfix) list ->
   722   local_theory -> inductive_result * local_theory
   723 
   724 fun add_ind_def {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono}
   725     cs intros monos params cnames_syn ctxt =
   726   let
   727     val _ = null cnames_syn andalso error "No inductive predicates given";
   728     val _ = message (quiet_mode andalso not verbose)
   729       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^
   730         commas_quote (map fst cnames_syn));
   731 
   732     val cnames = map (Sign.full_name (ProofContext.theory_of ctxt) o #1) cnames_syn;  (* FIXME *)
   733     val ((intr_names, intr_atts), intr_ts) =
   734       apfst split_list (split_list (map (check_rule ctxt cs params) intros));
   735 
   736     val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   737       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts
   738         monos params cnames_syn ctxt;
   739 
   740     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
   741       params intr_ts rec_preds_defs ctxt1;
   742     val elims = if no_elim then [] else
   743       prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt1;
   744     val raw_induct = zero_var_indexes
   745       (if no_ind then Drule.asm_rl else
   746        if coind then
   747          singleton (ProofContext.export
   748            (snd (Variable.add_fixes (map (fst o dest_Free) params) ctxt1)) ctxt1)
   749            (rotate_prems ~1 (ObjectLogic.rulify (rule_by_tactic
   750              (rewrite_tac [le_fun_def, le_bool_def, sup_fun_eq, sup_bool_eq] THEN
   751                fold_tac rec_preds_defs) (mono RS (fp_def RS def_coinduct)))))
   752        else
   753          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
   754            rec_preds_defs ctxt1);
   755 
   756     val (intrs', elims', induct, ctxt2) = declare_rules kind rec_name coind no_ind
   757       cnames intrs intr_names intr_atts elims raw_induct ctxt1;
   758 
   759     val names = map #1 cnames_syn;
   760     val result =
   761       {preds = preds,
   762        intrs = intrs',
   763        elims = elims',
   764        raw_induct = rulify raw_induct,
   765        induct = induct};
   766 
   767     val ctxt3 = ctxt2
   768       |> LocalTheory.declaration (fn phi =>
   769         let val result' = morph_result phi result;
   770         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
   771   in (result, ctxt3) end;
   772 
   773 
   774 (* external interfaces *)
   775 
   776 fun gen_add_inductive_i mk_def
   777     (flags as {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono})
   778     cnames_syn pnames spec monos lthy =
   779   let
   780     val thy = ProofContext.theory_of lthy;
   781     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   782 
   783 
   784     (* abbrevs *)
   785 
   786     val (_, ctxt1) = Variable.add_fixes (map (fst o fst) cnames_syn) lthy;
   787 
   788     fun get_abbrev ((name, atts), t) =
   789       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
   790         let
   791           val _ = name = "" andalso null atts orelse
   792             error "Abbreviations may not have names or attributes";
   793           val ((x, T), rhs) = LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt1 t));
   794           val mx =
   795             (case find_first (fn ((c, _), _) => c = x) cnames_syn of
   796               NONE => error ("Undeclared head of abbreviation " ^ quote x)
   797             | SOME ((_, T'), mx) =>
   798                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
   799                 else mx);
   800         in SOME ((x, mx), rhs) end
   801       else NONE;
   802 
   803     val abbrevs = map_filter get_abbrev spec;
   804     val bs = map (fst o fst) abbrevs;
   805 
   806 
   807     (* predicates *)
   808 
   809     val pre_intros = filter_out (is_some o get_abbrev) spec;
   810     val cnames_syn' = filter_out (member (op =) bs o fst o fst) cnames_syn;
   811     val cs = map (Free o fst) cnames_syn';
   812     val ps = map Free pnames;
   813 
   814     val (_, ctxt2) = lthy |> Variable.add_fixes (map (fst o fst) cnames_syn');
   815     val _ = map (fn abbr => LocalDefs.fixed_abbrev abbr ctxt2) abbrevs;
   816     val ctxt3 = ctxt2 |> fold (snd oo LocalDefs.fixed_abbrev) abbrevs;
   817     val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy;
   818 
   819     fun close_rule r = list_all_free (rev (fold_aterms
   820       (fn t as Free (v as (s, _)) =>
   821           if Variable.is_fixed ctxt1 s orelse
   822             member (op =) ps t then I else insert (op =) v
   823         | _ => I) r []), r);
   824 
   825     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
   826     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
   827   in
   828     lthy
   829     |> mk_def flags cs intros monos ps preds
   830     ||> fold (snd oo LocalTheory.abbrev Syntax.mode_default) abbrevs
   831   end;
   832 
   833 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
   834   let
   835     val ((vars, specs), _) = lthy |> ProofContext.set_mode ProofContext.mode_abbrev
   836       |> Specification.read_specification
   837           (cnames_syn @ pnames_syn) (map (fn (a, s) => [(a, [s])]) intro_srcs);
   838     val (cs, ps) = chop (length cnames_syn) vars;
   839     val intrs = map (apsnd the_single) specs;
   840     val monos = Attrib.eval_thms lthy raw_monos;
   841     val flags = {quiet_mode = false, verbose = verbose, kind = Thm.theoremK, alt_name = "",
   842       coind = coind, no_elim = false, no_ind = false, skip_mono = false};
   843   in
   844     lthy
   845     |> LocalTheory.set_group (serial_string ())
   846     |> gen_add_inductive_i mk_def flags cs (map fst ps) intrs monos
   847   end;
   848 
   849 val add_inductive_i = gen_add_inductive_i add_ind_def;
   850 val add_inductive = gen_add_inductive add_ind_def;
   851 
   852 fun add_inductive_global group flags cnames_syn pnames pre_intros monos thy =
   853   let
   854     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
   855     val ctxt' = thy
   856       |> TheoryTarget.init NONE
   857       |> LocalTheory.set_group group
   858       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
   859       |> LocalTheory.exit;
   860     val info = #2 (the_inductive ctxt' name);
   861   in (info, ProofContext.theory_of ctxt') end;
   862 
   863 
   864 (* read off arities of inductive predicates from raw induction rule *)
   865 fun arities_of induct =
   866   map (fn (_ $ t $ u) =>
   867       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
   868     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
   869 
   870 (* read off parameters of inductive predicate from raw induction rule *)
   871 fun params_of induct =
   872   let
   873     val (_ $ t $ u :: _) =
   874       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
   875     val (_, ts) = strip_comb t;
   876     val (_, us) = strip_comb u
   877   in
   878     List.take (ts, length ts - length us)
   879   end;
   880 
   881 val pname_of_intr =
   882   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
   883 
   884 (* partition introduction rules according to predicate name *)
   885 fun gen_partition_rules f induct intros =
   886   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
   887     (map (rpair [] o fst) (arities_of induct));
   888 
   889 val partition_rules = gen_partition_rules I;
   890 fun partition_rules' induct = gen_partition_rules fst induct;
   891 
   892 fun unpartition_rules intros xs =
   893   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
   894     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
   895 
   896 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
   897 fun infer_intro_vars elim arity intros =
   898   let
   899     val thy = theory_of_thm elim;
   900     val _ :: cases = prems_of elim;
   901     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
   902     fun mtch (t, u) =
   903       let
   904         val params = Logic.strip_params t;
   905         val vars = map (Var o apfst (rpair 0))
   906           (Name.variant_list used (map fst params) ~~ map snd params);
   907         val ts = map (curry subst_bounds (rev vars))
   908           (List.drop (Logic.strip_assums_hyp t, arity));
   909         val us = Logic.strip_imp_prems u;
   910         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
   911           (Vartab.empty, Vartab.empty);
   912       in
   913         map (Envir.subst_vars tab) vars
   914       end
   915   in
   916     map (mtch o apsnd prop_of) (cases ~~ intros)
   917   end;
   918 
   919 
   920 
   921 (** package setup **)
   922 
   923 (* setup theory *)
   924 
   925 val setup =
   926   Method.add_methods [("ind_cases", ind_cases,
   927     "dynamic case analysis on predicates")] #>
   928   Attrib.add_attributes [("mono", Attrib.add_del_args mono_add mono_del,
   929     "declaration of monotonicity rule")];
   930 
   931 
   932 (* outer syntax *)
   933 
   934 local structure P = OuterParse and K = OuterKeyword in
   935 
   936 val _ = OuterSyntax.keywords ["monos"];
   937 
   938 fun flatten_specification specs = specs |> maps
   939   (fn (a, (concl, [])) => concl |> map
   940         (fn ((b, atts), [B]) =>
   941               if a = "" then ((b, atts), B)
   942               else if b = "" then ((a, atts), B)
   943               else error ("Illegal nested case names " ^ quote (NameSpace.append a b))
   944           | ((b, _), _) => error ("Illegal simultaneous specification " ^ quote b))
   945     | (a, _) => error ("Illegal local specification parameters for " ^ quote a));
   946 
   947 fun gen_ind_decl mk_def coind =
   948   P.opt_target --
   949   P.fixes -- P.for_fixes --
   950   Scan.optional (P.$$$ "where" |-- P.!!! SpecParse.specification) [] --
   951   Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.xthms1) []
   952   >> (fn ((((loc, preds), params), specs), monos) =>
   953     Toplevel.local_theory loc
   954       (fn lthy => lthy |> gen_add_inductive mk_def true coind preds params
   955          (flatten_specification specs) monos |> snd));
   956 
   957 val ind_decl = gen_ind_decl add_ind_def;
   958 
   959 val _ = OuterSyntax.command "inductive" "define inductive predicates" K.thy_decl (ind_decl false);
   960 val _ = OuterSyntax.command "coinductive" "define coinductive predicates" K.thy_decl (ind_decl true);
   961 
   962 val _ =
   963   OuterSyntax.command "inductive_cases"
   964     "create simplified instances of elimination rules (improper)" K.thy_script
   965     (P.opt_target -- P.and_list1 SpecParse.spec
   966       >> (fn (loc, specs) => Toplevel.local_theory loc (snd o inductive_cases specs)));
   967 
   968 end;
   969 
   970 end;