src/HOL/Tools/inductive_package.ML
author wenzelm
Tue Sep 02 14:10:45 2008 +0200 (2008-09-02)
changeset 28083 103d9282a946
parent 27882 eaa9fef9f4c1
child 28084 a05ca48ef263
permissions -rw-r--r--
explicit type Name.binding for higher-specification elements;
     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: ((Name.binding * Attrib.src list) * string list) list ->
    37     Proof.context -> thm list list * local_theory
    38   val inductive_cases_i: ((Name.binding * 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 -> ((Name.binding * typ) * mixfix) list ->
    43     (string * typ) list -> ((Name.binding * Attrib.src list) * term) list -> thm list ->
    44       local_theory -> inductive_result * local_theory
    45   val add_inductive: bool -> bool ->
    46     (Name.binding * string option * mixfix) list ->
    47     (Name.binding * string option * mixfix) list ->
    48     ((Name.binding * Attrib.src list) * string) list ->
    49     (Facts.ref * Attrib.src list) list ->
    50     local_theory -> inductive_result * local_theory
    51   val add_inductive_global: string -> inductive_flags ->
    52     ((Name.binding * typ) * mixfix) list ->
    53     (string * typ) list ->
    54     ((Name.binding * Attrib.src list) * term) list -> thm list -> theory -> inductive_result * theory
    55   val arities_of: thm -> (string * int) list
    56   val params_of: thm -> term list
    57   val partition_rules: thm -> thm list -> (string * thm list) list
    58   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
    59   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    60   val infer_intro_vars: thm -> int -> thm list -> term list list
    61   val setup: theory -> theory
    62 end;
    63 
    64 signature INDUCTIVE_PACKAGE =
    65 sig
    66   include BASIC_INDUCTIVE_PACKAGE
    67   type add_ind_def
    68   val declare_rules: string -> Name.binding -> bool -> bool -> string list ->
    69     thm list -> Name.binding list -> Attrib.src list list -> (thm * string list) list ->
    70     thm -> local_theory -> thm list * thm list * thm * local_theory
    71   val add_ind_def: add_ind_def
    72   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
    73     ((Name.binding * typ) * mixfix) list ->
    74     (string * typ) list ->
    75     ((Name.binding * Attrib.src list) * term) list -> thm list ->
    76     local_theory -> inductive_result * local_theory
    77   val gen_add_inductive: add_ind_def -> bool -> bool ->
    78     (Name.binding * string option * mixfix) list ->
    79     (Name.binding * string option * mixfix) list ->
    80     ((Name.binding * Attrib.src list) * string) list -> (Facts.ref * Attrib.src list) list ->
    81     local_theory -> inductive_result * local_theory
    82   val gen_ind_decl: add_ind_def -> bool ->
    83     OuterParse.token list -> (local_theory -> local_theory) * OuterParse.token list
    84 end;
    85 
    86 structure InductivePackage: INDUCTIVE_PACKAGE =
    87 struct
    88 
    89 
    90 (** theory context references **)
    91 
    92 val inductive_forall_name = "HOL.induct_forall";
    93 val inductive_forall_def = thm "induct_forall_def";
    94 val inductive_conj_name = "HOL.induct_conj";
    95 val inductive_conj_def = thm "induct_conj_def";
    96 val inductive_conj = thms "induct_conj";
    97 val inductive_atomize = thms "induct_atomize";
    98 val inductive_rulify = thms "induct_rulify";
    99 val inductive_rulify_fallback = thms "induct_rulify_fallback";
   100 
   101 val notTrueE = TrueI RSN (2, notE);
   102 val notFalseI = Seq.hd (atac 1 notI);
   103 val simp_thms' = map (fn s => mk_meta_eq (the (find_first
   104   (equal (OldGoals.read_prop HOL.thy s) o prop_of) simp_thms)))
   105   ["(~True) = False", "(~False) = True",
   106    "(True --> ?P) = ?P", "(False --> ?P) = True",
   107    "(?P & True) = ?P", "(True & ?P) = ?P"];
   108 
   109 
   110 
   111 (** context data **)
   112 
   113 type inductive_result =
   114   {preds: term list, elims: thm list, raw_induct: thm,
   115    induct: thm, intrs: thm list};
   116 
   117 fun morph_result phi {preds, elims, raw_induct: thm, induct, intrs} =
   118   let
   119     val term = Morphism.term phi;
   120     val thm = Morphism.thm phi;
   121     val fact = Morphism.fact phi;
   122   in
   123    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
   124     induct = thm induct, intrs = fact intrs}
   125   end;
   126 
   127 type inductive_info =
   128   {names: string list, coind: bool} * inductive_result;
   129 
   130 structure InductiveData = GenericDataFun
   131 (
   132   type T = inductive_info Symtab.table * thm list;
   133   val empty = (Symtab.empty, []);
   134   val extend = I;
   135   fun merge _ ((tab1, monos1), (tab2, monos2)) =
   136     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
   137 );
   138 
   139 val get_inductives = InductiveData.get o Context.Proof;
   140 
   141 fun print_inductives ctxt =
   142   let
   143     val (tab, monos) = get_inductives ctxt;
   144     val space = Consts.space_of (ProofContext.consts_of ctxt);
   145   in
   146     [Pretty.strs ("(co)inductives:" :: map #1 (NameSpace.extern_table (space, tab))),
   147      Pretty.big_list "monotonicity rules:" (map (ProofContext.pretty_thm ctxt) monos)]
   148     |> Pretty.chunks |> Pretty.writeln
   149   end;
   150 
   151 
   152 (* get and put data *)
   153 
   154 fun the_inductive ctxt name =
   155   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
   156     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   157   | SOME info => info);
   158 
   159 fun put_inductives names info = InductiveData.map
   160   (apfst (fold (fn name => Symtab.update (name, info)) names));
   161 
   162 
   163 
   164 (** monotonicity rules **)
   165 
   166 val get_monos = #2 o get_inductives;
   167 val map_monos = InductiveData.map o apsnd;
   168 
   169 fun mk_mono thm =
   170   let
   171     val concl = concl_of thm;
   172     fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @
   173       (case concl of
   174           (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
   175         | _ => [thm' RS (thm' RS eq_to_mono2)]);
   176     fun dest_less_concl thm = dest_less_concl (thm RS le_funD)
   177       handle THM _ => thm RS le_boolD
   178   in
   179     case concl of
   180       Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq)
   181     | _ $ (Const ("op =", _) $ _ $ _) => eq2mono thm
   182     | _ $ (Const ("HOL.ord_class.less_eq", _) $ _ $ _) =>
   183       [dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   184          (resolve_tac [le_funI, le_boolI'])) thm))]
   185     | _ => [thm]
   186   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm thm);
   187 
   188 val mono_add = Thm.declaration_attribute (map_monos o fold Thm.add_thm o mk_mono);
   189 val mono_del = Thm.declaration_attribute (map_monos o fold Thm.del_thm o mk_mono);
   190 
   191 
   192 
   193 (** misc utilities **)
   194 
   195 fun message quiet_mode s = if quiet_mode then () else writeln s;
   196 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
   197 
   198 fun coind_prefix true = "co"
   199   | coind_prefix false = "";
   200 
   201 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   202 
   203 fun make_bool_args f g [] i = []
   204   | make_bool_args f g (x :: xs) i =
   205       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   206 
   207 fun make_bool_args' xs =
   208   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
   209 
   210 fun find_arg T x [] = sys_error "find_arg"
   211   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   212       apsnd (cons p) (find_arg T x ps)
   213   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   214       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   215       else apsnd (cons p) (find_arg T x ps);
   216 
   217 fun make_args Ts xs =
   218   map (fn (T, (NONE, ())) => Const ("arbitrary", T) | (_, (SOME t, ())) => t)
   219     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   220 
   221 fun make_args' Ts xs Us =
   222   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   223 
   224 fun dest_predicate cs params t =
   225   let
   226     val k = length params;
   227     val (c, ts) = strip_comb t;
   228     val (xs, ys) = chop k ts;
   229     val i = find_index_eq c cs;
   230   in
   231     if xs = params andalso i >= 0 then
   232       SOME (c, i, ys, chop (length ys)
   233         (List.drop (binder_types (fastype_of c), k)))
   234     else NONE
   235   end;
   236 
   237 fun mk_names a 0 = []
   238   | mk_names a 1 = [a]
   239   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   240 
   241 
   242 
   243 (** process rules **)
   244 
   245 local
   246 
   247 fun err_in_rule ctxt name t msg =
   248   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
   249     Syntax.string_of_term ctxt t, msg]);
   250 
   251 fun err_in_prem ctxt name t p msg =
   252   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
   253     "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
   254 
   255 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   256 
   257 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   258 
   259 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   260 
   261 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
   262 
   263 in
   264 
   265 fun check_rule ctxt cs params ((binding, att), rule) =
   266   let
   267     val name = Name.name_of binding;
   268     val params' = Term.variant_frees rule (Logic.strip_params rule);
   269     val frees = rev (map Free params');
   270     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   271     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   272     val rule' = Logic.list_implies (prems, concl);
   273     val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
   274     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
   275 
   276     fun check_ind err t = case dest_predicate cs params t of
   277         NONE => err (bad_app ^
   278           commas (map (Syntax.string_of_term ctxt) params))
   279       | SOME (_, _, ys, _) =>
   280           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   281           then err bad_ind_occ else ();
   282 
   283     fun check_prem' prem t =
   284       if head_of t mem cs then
   285         check_ind (err_in_prem ctxt name rule prem) t
   286       else (case t of
   287           Abs (_, _, t) => check_prem' prem t
   288         | t $ u => (check_prem' prem t; check_prem' prem u)
   289         | _ => ());
   290 
   291     fun check_prem (prem, aprem) =
   292       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   293       else err_in_prem ctxt name rule prem "Non-atomic premise";
   294   in
   295     (case concl of
   296        Const ("Trueprop", _) $ t =>
   297          if head_of t mem cs then
   298            (check_ind (err_in_rule ctxt name rule') t;
   299             List.app check_prem (prems ~~ aprems))
   300          else err_in_rule ctxt name rule' bad_concl
   301      | _ => err_in_rule ctxt name rule' bad_concl);
   302     ((binding, att), arule)
   303   end;
   304 
   305 val rulify =
   306   hol_simplify inductive_conj
   307   #> hol_simplify inductive_rulify
   308   #> hol_simplify inductive_rulify_fallback
   309   #> MetaSimplifier.norm_hhf;
   310 
   311 end;
   312 
   313 
   314 
   315 (** proofs for (co)inductive predicates **)
   316 
   317 (* prove monotonicity *)
   318 
   319 fun prove_mono quiet_mode skip_mono predT fp_fun monos ctxt =
   320  (message (quiet_mode orelse skip_mono andalso !quick_and_dirty)
   321     "  Proving monotonicity ...";
   322   (if skip_mono then SkipProof.prove else Goal.prove) ctxt [] []
   323     (HOLogic.mk_Trueprop
   324       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   325     (fn _ => EVERY [rtac @{thm monoI} 1,
   326       REPEAT (resolve_tac [le_funI, le_boolI'] 1),
   327       REPEAT (FIRST
   328         [atac 1,
   329          resolve_tac (List.concat (map mk_mono monos) @ get_monos ctxt) 1,
   330          etac le_funE 1, dtac le_boolD 1])]));
   331 
   332 
   333 (* prove introduction rules *)
   334 
   335 fun prove_intrs quiet_mode coind mono fp_def k params intr_ts rec_preds_defs ctxt =
   336   let
   337     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
   338 
   339     val unfold = funpow k (fn th => th RS fun_cong)
   340       (mono RS (fp_def RS
   341         (if coind then def_gfp_unfold else def_lfp_unfold)));
   342 
   343     fun select_disj 1 1 = []
   344       | select_disj _ 1 = [rtac disjI1]
   345       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   346 
   347     val rules = [refl, TrueI, notFalseI, exI, conjI];
   348 
   349     val intrs = map_index (fn (i, intr) => rulify
   350       (SkipProof.prove ctxt (map (fst o dest_Free) params) [] intr (fn _ => EVERY
   351        [rewrite_goals_tac rec_preds_defs,
   352         rtac (unfold RS iffD2) 1,
   353         EVERY1 (select_disj (length intr_ts) (i + 1)),
   354         (*Not ares_tac, since refl must be tried before any equality assumptions;
   355           backtracking may occur if the premises have extra variables!*)
   356         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts
   357 
   358   in (intrs, unfold) end;
   359 
   360 
   361 (* prove elimination rules *)
   362 
   363 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt =
   364   let
   365     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
   366 
   367     val ([pname], ctxt') = ctxt |>
   368       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   369       Variable.variant_fixes ["P"];
   370     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   371 
   372     fun dest_intr r =
   373       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   374        Logic.strip_assums_hyp r, Logic.strip_params r);
   375 
   376     val intrs = map dest_intr intr_ts ~~ intr_names;
   377 
   378     val rules1 = [disjE, exE, FalseE];
   379     val rules2 = [conjE, FalseE, notTrueE];
   380 
   381     fun prove_elim c =
   382       let
   383         val Ts = List.drop (binder_types (fastype_of c), length params);
   384         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   385         val frees = map Free (anames ~~ Ts);
   386 
   387         fun mk_elim_prem ((_, _, us, _), ts, params') =
   388           list_all (params',
   389             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   390               (frees ~~ us) @ ts, P));
   391         val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs);
   392         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   393            map mk_elim_prem (map #1 c_intrs)
   394       in
   395         (SkipProof.prove ctxt'' [] prems P
   396           (fn {prems, ...} => EVERY
   397             [cut_facts_tac [hd prems] 1,
   398              rewrite_goals_tac rec_preds_defs,
   399              dtac (unfold RS iffD1) 1,
   400              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   401              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   402              EVERY (map (fn prem =>
   403                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
   404           |> rulify
   405           |> singleton (ProofContext.export ctxt'' ctxt),
   406          map #2 c_intrs)
   407       end
   408 
   409    in map prove_elim cs end;
   410 
   411 
   412 (* derivation of simplified elimination rules *)
   413 
   414 local
   415 
   416 (*delete needless equality assumptions*)
   417 val refl_thin = Goal.prove_global HOL.thy [] [] @{prop "!!P. a = a ==> P ==> P"}
   418   (fn _ => assume_tac 1);
   419 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   420 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   421 
   422 fun simp_case_tac ss i =
   423   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
   424 
   425 in
   426 
   427 fun mk_cases ctxt prop =
   428   let
   429     val thy = ProofContext.theory_of ctxt;
   430     val ss = Simplifier.local_simpset_of ctxt;
   431 
   432     fun err msg =
   433       error (Pretty.string_of (Pretty.block
   434         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
   435 
   436     val elims = Induct.find_casesP ctxt prop;
   437 
   438     val cprop = Thm.cterm_of thy prop;
   439     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
   440     fun mk_elim rl =
   441       Thm.implies_intr cprop (Tactic.rule_by_tactic tac (Thm.assume cprop RS rl))
   442       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   443   in
   444     (case get_first (try mk_elim) elims of
   445       SOME r => r
   446     | NONE => err "Proposition not an inductive predicate:")
   447   end;
   448 
   449 end;
   450 
   451 
   452 (* inductive_cases *)
   453 
   454 fun gen_inductive_cases prep_att prep_prop args lthy =
   455   let
   456     val thy = ProofContext.theory_of lthy;
   457     val facts = args |> map (fn ((a, atts), props) =>
   458       ((a, map (prep_att thy) atts),
   459         map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
   460   in lthy |> LocalTheory.notes Thm.theoremK facts |>> map snd end;
   461 
   462 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
   463 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   464 
   465 
   466 fun ind_cases src = Method.syntax (Scan.lift (Scan.repeat1 Args.name_source --
   467     Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.name) [])) src
   468   #> (fn ((raw_props, fixes), ctxt) =>
   469     let
   470       val (_, ctxt') = Variable.add_fixes fixes ctxt;
   471       val props = Syntax.read_props ctxt' raw_props;
   472       val ctxt'' = fold Variable.declare_term props ctxt';
   473       val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props)
   474     in Method.erule 0 rules end);
   475 
   476 
   477 
   478 (* prove induction rule *)
   479 
   480 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
   481     fp_def rec_preds_defs ctxt =
   482   let
   483     val _ = clean_message quiet_mode "  Proving the induction rule ...";
   484     val thy = ProofContext.theory_of ctxt;
   485 
   486     (* predicates for induction rule *)
   487 
   488     val (pnames, ctxt') = ctxt |>
   489       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   490       Variable.variant_fixes (mk_names "P" (length cs));
   491     val preds = map Free (pnames ~~
   492       map (fn c => List.drop (binder_types (fastype_of c), length params) --->
   493         HOLogic.boolT) cs);
   494 
   495     (* transform an introduction rule into a premise for induction rule *)
   496 
   497     fun mk_ind_prem r =
   498       let
   499         fun subst s = (case dest_predicate cs params s of
   500             SOME (_, i, ys, (_, Ts)) =>
   501               let
   502                 val k = length Ts;
   503                 val bs = map Bound (k - 1 downto 0);
   504                 val P = list_comb (List.nth (preds, i),
   505                   map (incr_boundvars k) ys @ bs);
   506                 val Q = list_abs (mk_names "x" k ~~ Ts,
   507                   HOLogic.mk_binop inductive_conj_name
   508                     (list_comb (incr_boundvars k s, bs), P))
   509               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   510           | NONE => (case s of
   511               (t $ u) => (fst (subst t) $ fst (subst u), NONE)
   512             | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
   513             | _ => (s, NONE)));
   514 
   515         fun mk_prem (s, prems) = (case subst s of
   516               (_, SOME (t, u)) => t :: u :: prems
   517             | (t, _) => t :: prems);
   518 
   519         val SOME (_, i, ys, _) = dest_predicate cs params
   520           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
   521 
   522       in list_all_free (Logic.strip_params r,
   523         Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
   524           [] (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r))),
   525             HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
   526       end;
   527 
   528     val ind_prems = map mk_ind_prem intr_ts;
   529 
   530 
   531     (* make conclusions for induction rules *)
   532 
   533     val Tss = map (binder_types o fastype_of) preds;
   534     val (xnames, ctxt'') =
   535       Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   536     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   537         (map (fn (((xnames, Ts), c), P) =>
   538            let val frees = map Free (xnames ~~ Ts)
   539            in HOLogic.mk_imp
   540              (list_comb (c, params @ frees), list_comb (P, frees))
   541            end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   542 
   543 
   544     (* make predicate for instantiation of abstract induction rule *)
   545 
   546     val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   547       (map_index (fn (i, P) => foldr HOLogic.mk_imp
   548          (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))
   549          (make_bool_args HOLogic.mk_not I bs i)) preds));
   550 
   551     val ind_concl = HOLogic.mk_Trueprop
   552       (HOLogic.mk_binrel "HOL.ord_class.less_eq" (rec_const, ind_pred));
   553 
   554     val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));
   555 
   556     val induct = SkipProof.prove ctxt'' [] ind_prems ind_concl
   557       (fn {prems, ...} => EVERY
   558         [rewrite_goals_tac [inductive_conj_def],
   559          DETERM (rtac raw_fp_induct 1),
   560          REPEAT (resolve_tac [le_funI, le_boolI] 1),
   561          rewrite_goals_tac (inf_fun_eq :: inf_bool_eq :: simp_thms'),
   562          (*This disjE separates out the introduction rules*)
   563          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
   564          (*Now break down the individual cases.  No disjE here in case
   565            some premise involves disjunction.*)
   566          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
   567          REPEAT (FIRSTGOAL
   568            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
   569          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
   570              (inductive_conj_def :: rec_preds_defs @ simp_thms') prem,
   571            conjI, refl] 1)) prems)]);
   572 
   573     val lemma = SkipProof.prove ctxt'' [] []
   574       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
   575         [rewrite_goals_tac rec_preds_defs,
   576          REPEAT (EVERY
   577            [REPEAT (resolve_tac [conjI, impI] 1),
   578             REPEAT (eresolve_tac [le_funE, le_boolE] 1),
   579             atac 1,
   580             rewrite_goals_tac simp_thms',
   581             atac 1])])
   582 
   583   in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end;
   584 
   585 
   586 
   587 (** specification of (co)inductive predicates **)
   588 
   589 fun mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts monos params cnames_syn ctxt =
   590   let
   591     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
   592 
   593     val argTs = fold (fn c => fn Ts => Ts @
   594       (List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs [];
   595     val k = log 2 1 (length cs);
   596     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   597     val p :: xs = map Free (Variable.variant_frees ctxt intr_ts
   598       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   599     val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts)
   600       (map (rpair HOLogic.boolT) (mk_names "b" k)));
   601 
   602     fun subst t = (case dest_predicate cs params t of
   603         SOME (_, i, ts, (Ts, Us)) =>
   604           let
   605             val l = length Us;
   606             val zs = map Bound (l - 1 downto 0)
   607           in
   608             list_abs (map (pair "z") Us, list_comb (p,
   609               make_bool_args' bs i @ make_args argTs
   610                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   611           end
   612       | NONE => (case t of
   613           t1 $ t2 => subst t1 $ subst t2
   614         | Abs (x, T, u) => Abs (x, T, subst u)
   615         | _ => t));
   616 
   617     (* transform an introduction rule into a conjunction  *)
   618     (*   [| p_i t; ... |] ==> p_j u                       *)
   619     (* is transformed into                                *)
   620     (*   b_j & x_j = u & p b_j t & ...                    *)
   621 
   622     fun transform_rule r =
   623       let
   624         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
   625           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   626         val ps = make_bool_args HOLogic.mk_not I bs i @
   627           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   628           map (subst o HOLogic.dest_Trueprop)
   629             (Logic.strip_assums_hyp r)
   630       in foldr (fn ((x, T), P) => HOLogic.exists_const T $ (Abs (x, T, P)))
   631         (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
   632         (Logic.strip_params r)
   633       end
   634 
   635     (* make a disjunction of all introduction rules *)
   636 
   637     val fp_fun = fold_rev lambda (p :: bs @ xs)
   638       (if null intr_ts then HOLogic.false_const
   639        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   640 
   641     (* add definiton of recursive predicates to theory *)
   642 
   643     val rec_name =
   644       if Name.name_of alt_name = "" then
   645         Name.binding (space_implode "_" (map (Name.name_of o fst) cnames_syn))
   646       else alt_name;
   647 
   648     val ((rec_const, (_, fp_def)), ctxt') = ctxt |>
   649       LocalTheory.define Thm.internalK
   650         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   651          ((Name.no_binding, []), fold_rev lambda params
   652            (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)));
   653     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
   654       (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params)));
   655     val specs = if length cs < 2 then [] else
   656       map_index (fn (i, (name_mx, c)) =>
   657         let
   658           val Ts = List.drop (binder_types (fastype_of c), length params);
   659           val xs = map Free (Variable.variant_frees ctxt intr_ts
   660             (mk_names "x" (length Ts) ~~ Ts))
   661         in
   662           (name_mx, ((Name.no_binding, []), fold_rev lambda (params @ xs)
   663             (list_comb (rec_const, params @ make_bool_args' bs i @
   664               make_args argTs (xs ~~ Ts)))))
   665         end) (cnames_syn ~~ cs);
   666     val (consts_defs, ctxt'') = fold_map (LocalTheory.define Thm.internalK) specs ctxt';
   667     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   668 
   669     val mono = prove_mono quiet_mode skip_mono predT fp_fun monos ctxt''
   670 
   671   in (ctxt'', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
   672     list_comb (rec_const, params), preds, argTs, bs, xs)
   673   end;
   674 
   675 fun declare_rules kind rec_binding coind no_ind cnames intrs intr_bindings intr_atts
   676       elims raw_induct ctxt =
   677   let
   678     val rec_name = Name.name_of rec_binding;
   679     val rec_qualified = NameSpace.qualified rec_name;
   680     val intr_names = map Name.name_of intr_bindings;
   681     val ind_case_names = RuleCases.case_names intr_names;
   682     val induct =
   683       if coind then
   684         (raw_induct, [RuleCases.case_names [rec_name],
   685           RuleCases.case_conclusion (rec_name, intr_names),
   686           RuleCases.consumes 1, Induct.coinduct_pred (hd cnames)])
   687       else if no_ind orelse length cnames > 1 then
   688         (raw_induct, [ind_case_names, RuleCases.consumes 0])
   689       else (raw_induct RSN (2, rev_mp), [ind_case_names, RuleCases.consumes 1]);
   690 
   691     val (intrs', ctxt1) =
   692       ctxt |>
   693       LocalTheory.notes kind
   694         (map (Name.map_name rec_qualified) intr_bindings ~~
   695          intr_atts ~~ map (fn th => [([th],
   696            [Attrib.internal (K (ContextRules.intro_query NONE))])]) intrs) |>>
   697       map (hd o snd);
   698     val (((_, elims'), (_, [induct'])), ctxt2) =
   699       ctxt1 |>
   700       LocalTheory.note kind ((Name.binding (rec_qualified "intros"), []), intrs') ||>>
   701       fold_map (fn (name, (elim, cases)) =>
   702         LocalTheory.note kind ((Name.binding (NameSpace.qualified (Sign.base_name name) "cases"),
   703           [Attrib.internal (K (RuleCases.case_names cases)),
   704            Attrib.internal (K (RuleCases.consumes 1)),
   705            Attrib.internal (K (Induct.cases_pred name)),
   706            Attrib.internal (K (ContextRules.elim_query NONE))]), [elim]) #>
   707         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   708       LocalTheory.note kind ((Name.binding (rec_qualified (coind_prefix coind ^ "induct")),
   709         map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
   710 
   711     val ctxt3 = if no_ind orelse coind then ctxt2 else
   712       let val inducts = cnames ~~ ProjectRule.projects ctxt2 (1 upto length cnames) induct'
   713       in
   714         ctxt2 |>
   715         LocalTheory.notes kind [((Name.binding (rec_qualified "inducts"), []),
   716           inducts |> map (fn (name, th) => ([th],
   717             [Attrib.internal (K ind_case_names),
   718              Attrib.internal (K (RuleCases.consumes 1)),
   719              Attrib.internal (K (Induct.induct_pred name))])))] |> snd
   720       end
   721   in (intrs', elims', induct', ctxt3) end;
   722 
   723 type inductive_flags =
   724   {quiet_mode: bool, verbose: bool, kind: string, alt_name: Name.binding,
   725    coind: bool, no_elim: bool, no_ind: bool, skip_mono: bool}
   726 
   727 type add_ind_def =
   728   inductive_flags ->
   729   term list -> ((Name.binding * Attrib.src list) * term) list -> thm list ->
   730   term list -> (Name.binding * mixfix) list ->
   731   local_theory -> inductive_result * local_theory
   732 
   733 fun (add_ind_def: add_ind_def)
   734     {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono}
   735     cs intros monos params cnames_syn ctxt =
   736   let
   737     val _ = null cnames_syn andalso error "No inductive predicates given";
   738     val names = map (Name.name_of o fst) cnames_syn;
   739     val _ = message (quiet_mode andalso not verbose)
   740       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
   741 
   742     val cnames = map (Sign.full_name (ProofContext.theory_of ctxt) o Name.name_of o #1) cnames_syn;  (* FIXME *)
   743     val ((intr_names, intr_atts), intr_ts) =
   744       apfst split_list (split_list (map (check_rule ctxt cs params) intros));
   745 
   746     val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   747       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts
   748         monos params cnames_syn ctxt;
   749 
   750     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
   751       params intr_ts rec_preds_defs ctxt1;
   752     val elims = if no_elim then [] else
   753       prove_elims quiet_mode cs params intr_ts (map Name.name_of intr_names)
   754         unfold rec_preds_defs ctxt1;
   755     val raw_induct = zero_var_indexes
   756       (if no_ind then Drule.asm_rl else
   757        if coind then
   758          singleton (ProofContext.export
   759            (snd (Variable.add_fixes (map (fst o dest_Free) params) ctxt1)) ctxt1)
   760            (rotate_prems ~1 (ObjectLogic.rulify (rule_by_tactic
   761              (rewrite_tac [le_fun_def, le_bool_def, sup_fun_eq, sup_bool_eq] THEN
   762                fold_tac rec_preds_defs) (mono RS (fp_def RS def_coinduct)))))
   763        else
   764          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
   765            rec_preds_defs ctxt1);
   766 
   767     val (intrs', elims', induct, ctxt2) = declare_rules kind rec_name coind no_ind
   768       cnames intrs intr_names intr_atts elims raw_induct ctxt1;
   769 
   770     val result =
   771       {preds = preds,
   772        intrs = intrs',
   773        elims = elims',
   774        raw_induct = rulify raw_induct,
   775        induct = induct};
   776 
   777     val ctxt3 = ctxt2
   778       |> LocalTheory.declaration (fn phi =>
   779         let val result' = morph_result phi result;
   780         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
   781   in (result, ctxt3) end;
   782 
   783 
   784 (* external interfaces *)
   785 
   786 fun gen_add_inductive_i mk_def
   787     (flags as {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono})
   788     cnames_syn pnames spec monos lthy =
   789   let
   790     val thy = ProofContext.theory_of lthy;
   791     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   792 
   793 
   794     (* abbrevs *)
   795 
   796     val (_, ctxt1) = Variable.add_fixes (map (Name.name_of o fst o fst) cnames_syn) lthy;
   797 
   798     fun get_abbrev ((name, atts), t) =
   799       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
   800         let
   801           val _ = Name.name_of name = "" andalso null atts orelse
   802             error "Abbreviations may not have names or attributes";
   803           val ((x, T), rhs) = LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt1 t));
   804           val var =
   805             (case find_first (fn ((c, _), _) => Name.name_of c = x) cnames_syn of
   806               NONE => error ("Undeclared head of abbreviation " ^ quote x)
   807             | SOME ((b, T'), mx) =>
   808                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
   809                 else (b, mx));
   810         in SOME (var, rhs) end
   811       else NONE;
   812 
   813     val abbrevs = map_filter get_abbrev spec;
   814     val bs = map (Name.name_of o fst o fst) abbrevs;
   815 
   816 
   817     (* predicates *)
   818 
   819     val pre_intros = filter_out (is_some o get_abbrev) spec;
   820     val cnames_syn' = filter_out (member (op =) bs o Name.name_of o fst o fst) cnames_syn;
   821     val cs = map (Free o apfst Name.name_of o fst) cnames_syn';
   822     val ps = map Free pnames;
   823 
   824     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Name.name_of o fst o fst) cnames_syn');
   825     val _ = map (fn abbr => LocalDefs.fixed_abbrev abbr ctxt2) abbrevs;
   826     val ctxt3 = ctxt2 |> fold (snd oo LocalDefs.fixed_abbrev) abbrevs;
   827     val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy;
   828 
   829     fun close_rule r = list_all_free (rev (fold_aterms
   830       (fn t as Free (v as (s, _)) =>
   831           if Variable.is_fixed ctxt1 s orelse
   832             member (op =) ps t then I else insert (op =) v
   833         | _ => I) r []), r);
   834 
   835     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
   836     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
   837   in
   838     lthy
   839     |> mk_def flags cs intros monos ps preds
   840     ||> fold (snd oo LocalTheory.abbrev Syntax.mode_default) abbrevs
   841   end;
   842 
   843 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
   844   let
   845     val ((vars, specs), _) = lthy |> ProofContext.set_mode ProofContext.mode_abbrev
   846       |> Specification.read_specification
   847           (cnames_syn @ pnames_syn) (map (fn (a, s) => [(a, [s])]) intro_srcs);
   848     val (cs, ps) = chop (length cnames_syn) vars;
   849     val intrs = map (apsnd the_single) specs;
   850     val monos = Attrib.eval_thms lthy raw_monos;
   851     val flags = {quiet_mode = false, verbose = verbose, kind = Thm.theoremK,
   852       alt_name = Name.no_binding, coind = coind, no_elim = false, no_ind = false, skip_mono = false};
   853   in
   854     lthy
   855     |> LocalTheory.set_group (serial_string ())
   856     |> gen_add_inductive_i mk_def flags cs (map (apfst Name.name_of o fst) ps) intrs monos
   857   end;
   858 
   859 val add_inductive_i = gen_add_inductive_i add_ind_def;
   860 val add_inductive = gen_add_inductive add_ind_def;
   861 
   862 fun add_inductive_global group flags cnames_syn pnames pre_intros monos thy =
   863   let
   864     val name = Sign.full_name thy (Name.name_of (fst (fst (hd cnames_syn))));
   865     val ctxt' = thy
   866       |> TheoryTarget.init NONE
   867       |> LocalTheory.set_group group
   868       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
   869       |> LocalTheory.exit;
   870     val info = #2 (the_inductive ctxt' name);
   871   in (info, ProofContext.theory_of ctxt') end;
   872 
   873 
   874 (* read off arities of inductive predicates from raw induction rule *)
   875 fun arities_of induct =
   876   map (fn (_ $ t $ u) =>
   877       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
   878     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
   879 
   880 (* read off parameters of inductive predicate from raw induction rule *)
   881 fun params_of induct =
   882   let
   883     val (_ $ t $ u :: _) =
   884       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
   885     val (_, ts) = strip_comb t;
   886     val (_, us) = strip_comb u
   887   in
   888     List.take (ts, length ts - length us)
   889   end;
   890 
   891 val pname_of_intr =
   892   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
   893 
   894 (* partition introduction rules according to predicate name *)
   895 fun gen_partition_rules f induct intros =
   896   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
   897     (map (rpair [] o fst) (arities_of induct));
   898 
   899 val partition_rules = gen_partition_rules I;
   900 fun partition_rules' induct = gen_partition_rules fst induct;
   901 
   902 fun unpartition_rules intros xs =
   903   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
   904     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
   905 
   906 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
   907 fun infer_intro_vars elim arity intros =
   908   let
   909     val thy = theory_of_thm elim;
   910     val _ :: cases = prems_of elim;
   911     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
   912     fun mtch (t, u) =
   913       let
   914         val params = Logic.strip_params t;
   915         val vars = map (Var o apfst (rpair 0))
   916           (Name.variant_list used (map fst params) ~~ map snd params);
   917         val ts = map (curry subst_bounds (rev vars))
   918           (List.drop (Logic.strip_assums_hyp t, arity));
   919         val us = Logic.strip_imp_prems u;
   920         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
   921           (Vartab.empty, Vartab.empty);
   922       in
   923         map (Envir.subst_vars tab) vars
   924       end
   925   in
   926     map (mtch o apsnd prop_of) (cases ~~ intros)
   927   end;
   928 
   929 
   930 
   931 (** package setup **)
   932 
   933 (* setup theory *)
   934 
   935 val setup =
   936   Method.add_methods [("ind_cases", ind_cases,
   937     "dynamic case analysis on predicates")] #>
   938   Attrib.add_attributes [("mono", Attrib.add_del_args mono_add mono_del,
   939     "declaration of monotonicity rule")];
   940 
   941 
   942 (* outer syntax *)
   943 
   944 local structure P = OuterParse and K = OuterKeyword in
   945 
   946 val _ = OuterKeyword.keyword "monos";
   947 
   948 (* FIXME eliminate *)
   949 fun flatten_specification specs = specs |> maps
   950   (fn (a, (concl, [])) => concl |> map
   951         (fn ((b, atts), [B]) =>
   952               if Name.name_of a = "" then ((b, atts), B)
   953               else if Name.name_of b = "" then ((a, atts), B)
   954               else error "Illegal nested case names"
   955           | ((b, _), _) => error "Illegal simultaneous specification")
   956     | (a, _) => error ("Illegal local specification parameters for " ^ quote (Name.name_of a)));
   957 
   958 fun gen_ind_decl mk_def coind =
   959   P.fixes -- P.for_fixes --
   960   Scan.optional (P.$$$ "where" |-- P.!!! SpecParse.specification) [] --
   961   Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.xthms1) []
   962   >> (fn (((preds, params), specs), monos) =>
   963       (snd o gen_add_inductive mk_def true coind preds params (flatten_specification specs) monos));
   964 
   965 val ind_decl = gen_ind_decl add_ind_def;
   966 
   967 val _ = OuterSyntax.local_theory "inductive" "define inductive predicates" K.thy_decl (ind_decl false);
   968 val _ = OuterSyntax.local_theory "coinductive" "define coinductive predicates" K.thy_decl (ind_decl true);
   969 
   970 val _ =
   971   OuterSyntax.local_theory "inductive_cases"
   972     "create simplified instances of elimination rules (improper)" K.thy_script
   973     (P.and_list1 SpecParse.spec >> (fn specs => snd o inductive_cases specs));
   974 
   975 end;
   976 
   977 end;