src/HOL/Tools/inductive_package.ML
author berghofe
Tue Oct 17 09:51:04 2006 +0200 (2006-10-17)
changeset 21048 e57e91f72831
parent 21024 63ab84bb64d1
child 21350 6e58289b6685
permissions -rw-r--r--
Restructured and repaired code dealing with case names
in induction and elimination rules.
     1 (*  Title:      HOL/Tools/inductive_package.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Author:     Stefan Berghofer, TU Muenchen
     5     Author:     Markus Wenzel, TU Muenchen
     6 
     7 (Co)Inductive Definition module for HOL.
     8 
     9 Features:
    10   * least or greatest fixedpoints
    11   * mutually recursive definitions
    12   * definitions involving arbitrary monotone operators
    13   * automatically proves introduction and elimination rules
    14 
    15   Introduction rules have the form
    16   [| M Pj ti, ..., Q x, ... |] ==> Pk t
    17   where M is some monotone operator (usually the identity)
    18   Q x is any side condition on the free variables
    19   ti, t are any terms
    20   Pj, Pk are two of the predicates being defined in mutual recursion
    21 *)
    22 
    23 signature INDUCTIVE_PACKAGE =
    24 sig
    25   val quiet_mode: bool ref
    26   val trace: bool ref
    27   type inductive_result
    28   type inductive_info
    29   val get_inductive: Context.generic -> string -> inductive_info option
    30   val the_mk_cases: Context.generic -> string -> string -> thm
    31   val print_inductives: Context.generic -> unit
    32   val mono_add: attribute
    33   val mono_del: attribute
    34   val get_monos: Context.generic -> thm list
    35   val inductive_forall_name: string
    36   val inductive_forall_def: thm
    37   val rulify: thm -> thm
    38   val inductive_cases: ((bstring * Attrib.src list) * string list) list -> theory -> theory
    39   val inductive_cases_i: ((bstring * attribute list) * term list) list -> theory -> theory
    40   val add_inductive_i: bool -> bstring -> bool -> bool -> bool -> (string * typ option * mixfix) list ->
    41     (string * typ option) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
    42       local_theory -> local_theory * inductive_result
    43   val add_inductive: bool -> bool -> (string * string option * mixfix) list ->
    44     (string * string option * mixfix) list ->
    45     ((bstring * Attrib.src list) * string) list -> (thmref * Attrib.src list) list ->
    46     local_theory -> local_theory * inductive_result
    47   val setup: theory -> theory
    48 end;
    49 
    50 structure InductivePackage: INDUCTIVE_PACKAGE =
    51 struct
    52 
    53 
    54 (** theory context references **)
    55 
    56 val mono_name = "Orderings.mono";
    57 val gfp_name = "FixedPoint.gfp";
    58 val lfp_name = "FixedPoint.lfp";
    59 
    60 val inductive_forall_name = "HOL.induct_forall";
    61 val inductive_forall_def = thm "induct_forall_def";
    62 val inductive_conj_name = "HOL.induct_conj";
    63 val inductive_conj_def = thm "induct_conj_def";
    64 val inductive_conj = thms "induct_conj";
    65 val inductive_atomize = thms "induct_atomize";
    66 val inductive_rulify = thms "induct_rulify";
    67 val inductive_rulify_fallback = thms "induct_rulify_fallback";
    68 
    69 val notTrueE = TrueI RSN (2, notE);
    70 val notFalseI = Seq.hd (atac 1 notI);
    71 val simp_thms' = map (fn s => mk_meta_eq (the (find_first
    72   (equal (term_of (read_cterm HOL.thy (s, propT))) o prop_of) simp_thms)))
    73   ["(~True) = False", "(~False) = True",
    74    "(True --> ?P) = ?P", "(False --> ?P) = True",
    75    "(?P & True) = ?P", "(True & ?P) = ?P"];
    76 
    77 
    78 
    79 (** theory data **)
    80 
    81 type inductive_result =
    82   {preds: term list, defs: thm list, elims: thm list, raw_induct: thm,
    83    induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm};
    84 
    85 type inductive_info =
    86   {names: string list, coind: bool} * inductive_result;
    87 
    88 structure InductiveData = GenericDataFun
    89 (struct
    90   val name = "HOL/inductive2";
    91   type T = inductive_info Symtab.table * thm list;
    92 
    93   val empty = (Symtab.empty, []);
    94   val extend = I;
    95   fun merge _ ((tab1, monos1), (tab2, monos2)) =
    96     (Symtab.merge (K true) (tab1, tab2), Drule.merge_rules (monos1, monos2));
    97 
    98   fun print generic (tab, monos) =
    99     [Pretty.strs ("(co)inductives:" ::
   100       map #1 (NameSpace.extern_table
   101         (Sign.const_space (Context.theory_of generic), tab))),  (* FIXME? *)
   102      Pretty.big_list "monotonicity rules:"
   103         (map (ProofContext.pretty_thm (Context.proof_of generic)) monos)]
   104     |> Pretty.chunks |> Pretty.writeln;
   105 end);
   106 
   107 val print_inductives = InductiveData.print;
   108 
   109 
   110 (* get and put data *)
   111 
   112 val get_inductive = Symtab.lookup o #1 o InductiveData.get;
   113 
   114 fun the_inductive thy name =
   115   (case get_inductive thy name of
   116     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   117   | SOME info => info);
   118 
   119 val the_mk_cases = (#mk_cases o #2) oo the_inductive;
   120 
   121 fun put_inductives names info = InductiveData.map (apfst (fn tab =>
   122   fold (fn name => Symtab.update_new (name, info)) names tab
   123     handle Symtab.DUP dup => error ("Duplicate definition of (co)inductive predicate " ^ quote dup)));
   124 
   125 
   126 
   127 (** monotonicity rules **)
   128 
   129 val get_monos = #2 o InductiveData.get;
   130 val map_monos = InductiveData.map o Library.apsnd;
   131 
   132 fun mk_mono thm =
   133   let
   134     fun eq2mono thm' = [(*standard*) (thm' RS (thm' RS eq_to_mono))] @
   135       (case concl_of thm of
   136           (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
   137         | _ => [(*standard*) (thm' RS (thm' RS eq_to_mono2))]);
   138     val concl = concl_of thm
   139   in
   140     if can Logic.dest_equals concl then
   141       eq2mono (thm RS meta_eq_to_obj_eq)
   142     else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then
   143       eq2mono thm
   144     else [thm]
   145   end;
   146 
   147 
   148 (* attributes *)
   149 
   150 val mono_add = Thm.declaration_attribute (fn th =>
   151   map_monos (fold Drule.add_rule (mk_mono th)));
   152 
   153 val mono_del = Thm.declaration_attribute (fn th =>
   154   map_monos (fold Drule.del_rule (mk_mono th)));
   155 
   156 
   157 
   158 (** misc utilities **)
   159 
   160 val quiet_mode = ref false;
   161 val trace = ref false;  (*for debugging*)
   162 fun message s = if ! quiet_mode then () else writeln s;
   163 fun clean_message s = if ! quick_and_dirty then () else message s;
   164 
   165 fun coind_prefix true = "co"
   166   | coind_prefix false = "";
   167 
   168 fun log b m n = if m >= n then 0 else 1 + log b (b * m) n;
   169 
   170 fun make_bool_args f g [] i = []
   171   | make_bool_args f g (x :: xs) i =
   172       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   173 
   174 fun make_bool_args' xs =
   175   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
   176 
   177 fun find_arg T x [] = sys_error "find_arg"
   178   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   179       apsnd (cons p) (find_arg T x ps)
   180   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   181       if T = U then (y, (U, (SOME x, y)) :: ps)
   182       else apsnd (cons p) (find_arg T x ps);
   183 
   184 fun make_args Ts xs =
   185   map (fn (T, (NONE, ())) => Const ("arbitrary", T) | (_, (SOME t, ())) => t)
   186     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   187 
   188 fun make_args' Ts xs Us =
   189   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   190 
   191 fun dest_predicate cs params t =
   192   let
   193     val k = length params;
   194     val (c, ts) = strip_comb t;
   195     val (xs, ys) = chop k ts;
   196     val i = find_index_eq c cs;
   197   in
   198     if xs = params andalso i >= 0 then
   199       SOME (c, i, ys, chop (length ys)
   200         (List.drop (binder_types (fastype_of c), k)))
   201     else NONE
   202   end;
   203 
   204 fun mk_names a 0 = []
   205   | mk_names a 1 = [a]
   206   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   207 
   208 
   209 
   210 (** process rules **)
   211 
   212 local
   213 
   214 fun err_in_rule thy name t msg =
   215   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
   216     Sign.string_of_term thy t, msg]);
   217 
   218 fun err_in_prem thy name t p msg =
   219   error (cat_lines ["Ill-formed premise", Sign.string_of_term thy p,
   220     "in introduction rule " ^ quote name, Sign.string_of_term thy t, msg]);
   221 
   222 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   223 
   224 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   225 
   226 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   227 
   228 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
   229 
   230 in
   231 
   232 fun check_rule thy cs params ((name, att), rule) =
   233   let
   234     val params' = Term.variant_frees rule (Logic.strip_params rule);
   235     val frees = rev (map Free params');
   236     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   237     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   238     val aprems = map (atomize_term thy) prems;
   239     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
   240 
   241     fun check_ind err t = case dest_predicate cs params t of
   242         NONE => err (bad_app ^
   243           commas (map (Sign.string_of_term thy) params))
   244       | SOME (_, _, ys, _) =>
   245           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   246           then err bad_ind_occ else ();
   247 
   248     fun check_prem' prem t =
   249       if head_of t mem cs then
   250         check_ind (err_in_prem thy name rule prem) t
   251       else (case t of
   252           Abs (_, _, t) => check_prem' prem t
   253         | t $ u => (check_prem' prem t; check_prem' prem u)
   254         | _ => ());
   255 
   256     fun check_prem (prem, aprem) =
   257       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   258       else err_in_prem thy name rule prem "Non-atomic premise";
   259   in
   260     (case concl of
   261        Const ("Trueprop", _) $ t => 
   262          if head_of t mem cs then
   263            (check_ind (err_in_rule thy name rule) t;
   264             List.app check_prem (prems ~~ aprems))
   265          else err_in_rule thy name rule bad_concl
   266      | _ => err_in_rule thy name rule bad_concl);
   267     ((name, att), arule)
   268   end;
   269 
   270 val rulify =  (* FIXME norm_hhf *)
   271   hol_simplify inductive_conj
   272   #> hol_simplify inductive_rulify
   273   #> hol_simplify inductive_rulify_fallback
   274   (*#> standard*);
   275 
   276 end;
   277 
   278 
   279 
   280 (** proofs for (co)inductive predicates **)
   281 
   282 (* prove monotonicity -- NOT subject to quick_and_dirty! *)
   283 
   284 fun prove_mono predT fp_fun monos ctxt =
   285  (message "  Proving monotonicity ...";
   286   Goal.prove ctxt [] []   (*NO quick_and_dirty here!*)
   287     (HOLogic.mk_Trueprop
   288       (Const (mono_name, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   289     (fn _ => EVERY [rtac monoI 1,
   290       REPEAT (resolve_tac [le_funI, le_boolI'] 1),
   291       REPEAT (FIRST
   292         [atac 1,
   293          resolve_tac (List.concat (map mk_mono monos) @
   294            get_monos (Context.Proof ctxt)) 1,
   295          etac le_funE 1, dtac le_boolD 1])]));
   296 
   297 
   298 (* prove introduction rules *)
   299 
   300 fun prove_intrs coind mono fp_def k intr_ts rec_preds_defs ctxt =
   301   let
   302     val _ = clean_message "  Proving the introduction rules ...";
   303 
   304     val unfold = funpow k (fn th => th RS fun_cong)
   305       (mono RS (fp_def RS
   306         (if coind then def_gfp_unfold else def_lfp_unfold)));
   307 
   308     fun select_disj 1 1 = []
   309       | select_disj _ 1 = [rtac disjI1]
   310       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   311 
   312     val rules = [refl, TrueI, notFalseI, exI, conjI];
   313 
   314     val intrs = map_index (fn (i, intr) =>
   315       rulify (SkipProof.prove ctxt [] [] intr (fn _ => EVERY
   316        [rewrite_goals_tac rec_preds_defs,
   317         rtac (unfold RS iffD2) 1,
   318         EVERY1 (select_disj (length intr_ts) (i + 1)),
   319         (*Not ares_tac, since refl must be tried before any equality assumptions;
   320           backtracking may occur if the premises have extra variables!*)
   321         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts
   322 
   323   in (intrs, unfold) end;
   324 
   325 
   326 (* prove elimination rules *)
   327 
   328 fun prove_elims cs params intr_ts intr_names unfold rec_preds_defs ctxt =
   329   let
   330     val _ = clean_message "  Proving the elimination rules ...";
   331 
   332     val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
   333     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   334 
   335     fun dest_intr r =
   336       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   337        Logic.strip_assums_hyp r, Logic.strip_params r);
   338 
   339     val intrs = map dest_intr intr_ts ~~ intr_names;
   340 
   341     val rules1 = [disjE, exE, FalseE];
   342     val rules2 = [conjE, FalseE, notTrueE];
   343 
   344     fun prove_elim c =
   345       let
   346         val Ts = List.drop (binder_types (fastype_of c), length params);
   347         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   348         val frees = map Free (anames ~~ Ts);
   349 
   350         fun mk_elim_prem ((_, _, us, _), ts, params') =
   351           list_all (params',
   352             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   353               (frees ~~ us) @ ts, P));
   354         val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs);
   355         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   356            map mk_elim_prem (map #1 c_intrs)
   357       in
   358         (SkipProof.prove ctxt'' [] prems P
   359           (fn {prems, ...} => EVERY
   360             [cut_facts_tac [hd prems] 1,
   361              rewrite_goals_tac rec_preds_defs,
   362              dtac (unfold RS iffD1) 1,
   363              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   364              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   365              EVERY (map (fn prem =>
   366                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
   367           |> rulify
   368           |> singleton (ProofContext.export ctxt'' ctxt),
   369          map #2 c_intrs)
   370       end
   371 
   372    in map prove_elim cs end;
   373 
   374 
   375 (* derivation of simplified elimination rules *)
   376 
   377 local
   378 
   379 (*cprop should have the form "Si t" where Si is an inductive predicate*)
   380 val mk_cases_err = "mk_cases: proposition not an inductive predicate";
   381 
   382 (*delete needless equality assumptions*)
   383 val refl_thin = prove_goal HOL.thy "!!P. a = a ==> P ==> P" (fn _ => [assume_tac 1]);
   384 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   385 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   386 
   387 fun simp_case_tac solved ss i =
   388   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i
   389   THEN_MAYBE (if solved then no_tac else all_tac);
   390 
   391 in
   392 
   393 fun mk_cases_i elims ss cprop =
   394   let
   395     val prem = Thm.assume cprop;
   396     val tac = ALLGOALS (simp_case_tac false ss) THEN prune_params_tac;
   397     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic tac (prem RS rl));
   398   in
   399     (case get_first (try mk_elim) elims of
   400       SOME r => r
   401     | NONE => error (Pretty.string_of (Pretty.block
   402         [Pretty.str mk_cases_err, Pretty.fbrk, Display.pretty_cterm cprop])))
   403   end;
   404 
   405 fun mk_cases elims s =
   406   mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.theory_of_thm (hd elims)) (s, propT));
   407 
   408 fun smart_mk_cases ctxt ss cprop =
   409   let
   410     val c = #1 (Term.dest_Const (Term.head_of (HOLogic.dest_Trueprop
   411       (Logic.strip_imp_concl (Thm.term_of cprop))))) handle TERM _ => error mk_cases_err;
   412     val (_, {elims, ...}) = the_inductive ctxt c;
   413   in mk_cases_i elims ss cprop end;
   414 
   415 end;
   416 
   417 
   418 (* inductive_cases(_i) *)
   419 
   420 fun gen_inductive_cases prep_att prep_prop args thy =
   421   let
   422     val cert_prop = Thm.cterm_of thy o prep_prop (ProofContext.init thy);
   423     val mk_cases = smart_mk_cases (Context.Theory thy) (Simplifier.simpset_of thy) o cert_prop;
   424 
   425     val facts = args |> map (fn ((a, atts), props) =>
   426      ((a, map (prep_att thy) atts), map (Thm.no_attributes o single o mk_cases) props));
   427   in thy |> PureThy.note_thmss_i "" facts |> snd end;
   428 
   429 val inductive_cases = gen_inductive_cases Attrib.attribute ProofContext.read_prop;
   430 val inductive_cases_i = gen_inductive_cases (K I) ProofContext.cert_prop;
   431 
   432 
   433 (* mk_cases_meth *)
   434 
   435 fun mk_cases_meth (ctxt, raw_props) =
   436   let
   437     val thy = ProofContext.theory_of ctxt;
   438     val ss = local_simpset_of ctxt;
   439     val cprops = map (Thm.cterm_of thy o ProofContext.read_prop ctxt) raw_props;
   440   in Method.erule 0 (map (smart_mk_cases (Context.Theory thy) ss) cprops) end;
   441 
   442 val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name));
   443 
   444 
   445 (* prove induction rule *)
   446 
   447 fun prove_indrule cs argTs bs xs rec_const params intr_ts mono
   448     fp_def rec_preds_defs ctxt =
   449   let
   450     val _ = clean_message "  Proving the induction rule ...";
   451     val thy = ProofContext.theory_of ctxt;
   452 
   453     (* predicates for induction rule *)
   454 
   455     val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
   456     val preds = map Free (pnames ~~
   457       map (fn c => List.drop (binder_types (fastype_of c), length params) --->
   458         HOLogic.boolT) cs);
   459 
   460     (* transform an introduction rule into a premise for induction rule *)
   461 
   462     fun mk_ind_prem r =
   463       let
   464         fun subst s = (case dest_predicate cs params s of
   465             SOME (_, i, ys, (_, Ts)) =>
   466               let
   467                 val k = length Ts;
   468                 val bs = map Bound (k - 1 downto 0);
   469                 val P = list_comb (List.nth (preds, i), ys @ bs);
   470                 val Q = list_abs (mk_names "x" k ~~ Ts,
   471                   HOLogic.mk_binop inductive_conj_name (list_comb (s, bs), P))
   472               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   473           | NONE => (case s of
   474               (t $ u) => (fst (subst t) $ fst (subst u), NONE)
   475             | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
   476             | _ => (s, NONE)));
   477 
   478         fun mk_prem (s, prems) = (case subst s of
   479               (_, SOME (t, u)) => t :: u :: prems
   480             | (t, _) => t :: prems);
   481 
   482         val SOME (_, i, ys, _) = dest_predicate cs params
   483           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
   484 
   485       in list_all_free (Logic.strip_params r,
   486         Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
   487           [] (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r))),
   488             HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
   489       end;
   490 
   491     val ind_prems = map mk_ind_prem intr_ts;
   492 
   493     (* make conclusions for induction rules *)
   494 
   495     val Tss = map (binder_types o fastype_of) preds;
   496     val (xnames, ctxt'') =
   497       Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   498     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   499         (map (fn (((xnames, Ts), c), P) =>
   500            let val frees = map Free (xnames ~~ Ts)
   501            in HOLogic.mk_imp
   502              (list_comb (c, params @ frees), list_comb (P, frees))
   503            end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   504 
   505     val dummy = if !trace then
   506                 (writeln "ind_prems = ";
   507                  List.app (writeln o Sign.string_of_term thy) ind_prems)
   508             else ();
   509 
   510     (* make predicate for instantiation of abstract induction rule *)
   511 
   512     val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   513       (map_index (fn (i, P) => foldr HOLogic.mk_imp
   514          (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))
   515          (make_bool_args HOLogic.mk_not I bs i)) preds));
   516 
   517     val ind_concl = HOLogic.mk_Trueprop
   518       (HOLogic.mk_binrel "Orderings.less_eq" (rec_const, ind_pred));
   519 
   520     val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));
   521 
   522     val dummy = if !trace then
   523                 (writeln "raw_fp_induct = "; print_thm raw_fp_induct)
   524             else ();
   525 
   526     val induct = SkipProof.prove ctxt'' [] ind_prems ind_concl
   527       (fn {prems, ...} => EVERY
   528         [rewrite_goals_tac [inductive_conj_def],
   529          DETERM (rtac raw_fp_induct 1),
   530          REPEAT (resolve_tac [le_funI, le_boolI] 1),
   531          rewrite_goals_tac (map mk_meta_eq [meet_fun_eq, meet_bool_eq] @ simp_thms'),
   532          (*This disjE separates out the introduction rules*)
   533          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
   534          (*Now break down the individual cases.  No disjE here in case
   535            some premise involves disjunction.*)
   536          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
   537          REPEAT (FIRSTGOAL
   538            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
   539          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
   540            (inductive_conj_def :: rec_preds_defs) prem, conjI, refl] 1)) prems)]);
   541 
   542     val lemma = SkipProof.prove ctxt'' [] []
   543       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
   544         [rewrite_goals_tac rec_preds_defs,
   545          REPEAT (EVERY
   546            [REPEAT (resolve_tac [conjI, impI] 1),
   547             REPEAT (eresolve_tac [le_funE, le_boolE] 1),
   548             atac 1,
   549             rewrite_goals_tac simp_thms',
   550             atac 1])])
   551 
   552   in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end;
   553 
   554 
   555 
   556 (** specification of (co)inductive predicates **)
   557 
   558 fun mk_ind_def alt_name coind cs intr_ts monos
   559       params cnames_syn ctxt =
   560   let
   561     val fp_name = if coind then gfp_name else lfp_name;
   562 
   563     val argTs = fold (fn c => fn Ts => Ts @
   564       (List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs [];
   565     val k = log 2 1 (length cs);
   566     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   567     val p :: xs = map Free (Variable.variant_frees ctxt intr_ts
   568       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   569     val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts)
   570       (map (rpair HOLogic.boolT) (mk_names "b" k)));
   571 
   572     fun subst t = (case dest_predicate cs params t of
   573         SOME (_, i, ts, (Ts, Us)) =>
   574           let val zs = map Bound (length Us - 1 downto 0)
   575           in
   576             list_abs (map (pair "z") Us, list_comb (p,
   577               make_bool_args' bs i @ make_args argTs ((ts ~~ Ts) @ (zs ~~ Us))))
   578           end
   579       | NONE => (case t of
   580           t1 $ t2 => subst t1 $ subst t2
   581         | Abs (x, T, u) => Abs (x, T, subst u)
   582         | _ => t));
   583 
   584     (* transform an introduction rule into a conjunction  *)
   585     (*   [| p_i t; ... |] ==> p_j u                       *)
   586     (* is transformed into                                *)
   587     (*   b_j & x_j = u & p b_j t & ...                    *)
   588 
   589     fun transform_rule r =
   590       let
   591         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
   592           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   593         val ps = make_bool_args HOLogic.mk_not I bs i @
   594           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   595           map (subst o HOLogic.dest_Trueprop)
   596             (Logic.strip_assums_hyp r)
   597       in foldr (fn ((x, T), P) => HOLogic.exists_const T $ (Abs (x, T, P)))
   598         (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
   599         (Logic.strip_params r)
   600       end
   601 
   602     (* make a disjunction of all introduction rules *)
   603 
   604     val fp_fun = fold_rev lambda (p :: bs @ xs)
   605       (if null intr_ts then HOLogic.false_const
   606        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   607 
   608     (* add definiton of recursive predicates to theory *)
   609 
   610     val rec_name = if alt_name = "" then
   611       space_implode "_" (map fst cnames_syn) else alt_name;
   612 
   613     val ((rec_const, (_, fp_def)), ctxt') = ctxt |>
   614       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   615       fold Variable.declare_term intr_ts |>
   616       LocalTheory.def
   617         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   618          (("", []), fold_rev lambda params
   619            (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)));
   620     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
   621       (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params)));
   622     val specs = if length cs < 2 then [] else
   623       map_index (fn (i, (name_mx, c)) =>
   624         let
   625           val Ts = List.drop (binder_types (fastype_of c), length params);
   626           val xs = map Free (Variable.variant_frees ctxt intr_ts
   627             (mk_names "x" (length Ts) ~~ Ts))
   628         in
   629           (name_mx, (("", []), fold_rev lambda (params @ xs)
   630             (list_comb (rec_const, params @ make_bool_args' bs i @
   631               make_args argTs (xs ~~ Ts)))))
   632         end) (cnames_syn ~~ cs);
   633     val (consts_defs, ctxt'') = fold_map LocalTheory.def specs ctxt';
   634     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   635 
   636     val mono = prove_mono predT fp_fun monos ctxt''
   637 
   638   in (ctxt'', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
   639     list_comb (rec_const, params), preds, argTs, bs, xs)
   640   end;
   641 
   642 fun add_ind_def verbose alt_name coind no_elim no_ind cs
   643     intros monos params cnames_syn ctxt =
   644   let
   645     val _ =
   646       if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^
   647         commas_quote (map fst cnames_syn)) else ();
   648 
   649     val cnames = map (Sign.full_name (ProofContext.theory_of ctxt) o #1) cnames_syn;
   650     val ((intr_names, intr_atts), intr_ts) = apfst split_list (split_list intros);
   651 
   652     val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   653       argTs, bs, xs) = mk_ind_def alt_name coind cs intr_ts
   654         monos params cnames_syn ctxt;
   655 
   656     val (intrs, unfold) = prove_intrs coind mono fp_def (length bs + length xs)
   657       intr_ts rec_preds_defs ctxt1;
   658     val elims = if no_elim then [] else
   659       cnames ~~ map (apfst (singleton (ProofContext.export ctxt1 ctxt)))
   660         (prove_elims cs params intr_ts intr_names unfold rec_preds_defs ctxt1);
   661     val raw_induct = singleton (ProofContext.export ctxt1 ctxt)
   662       (if no_ind then Drule.asm_rl else
   663        if coind then ObjectLogic.rulify (rule_by_tactic
   664          (rewrite_tac [le_fun_def, le_bool_def] THEN
   665            fold_tac rec_preds_defs) (mono RS (fp_def RS def_coinduct)))
   666        else
   667          prove_indrule cs argTs bs xs rec_const params intr_ts mono fp_def
   668            rec_preds_defs ctxt1);
   669     val induct_cases = map (#1 o #1) intros;
   670     val ind_case_names = RuleCases.case_names induct_cases;
   671     val induct =
   672       if coind then
   673         (raw_induct, [RuleCases.case_names [rec_name],
   674           RuleCases.case_conclusion (rec_name, induct_cases),
   675           RuleCases.consumes 1])
   676       else if no_ind orelse length cs > 1 then
   677         (raw_induct, [ind_case_names, RuleCases.consumes 0])
   678       else (raw_induct RSN (2, rev_mp), [ind_case_names, RuleCases.consumes 1]);
   679 
   680     val (intrs', ctxt2) =
   681       ctxt1 |>
   682       LocalTheory.notes
   683         (map (fn "" => "" | name => NameSpace.append rec_name name) intr_names ~~
   684          intr_atts ~~
   685          map (single o rpair [] o single) (ProofContext.export ctxt1 ctxt intrs)) |>>
   686       map (hd o snd); (* FIXME? *)
   687     val (((_, elims'), (_, [induct'])), ctxt3) =
   688       ctxt2 |>
   689       LocalTheory.note ((NameSpace.append rec_name "intros", []), intrs') ||>>
   690       fold_map (fn (name, (elim, cases)) =>
   691         LocalTheory.note ((NameSpace.append (Sign.base_name name) "cases",
   692           [Attrib.internal (RuleCases.case_names cases),
   693            Attrib.internal (RuleCases.consumes 1),
   694            Attrib.internal (InductAttrib.cases_set name)]), [elim]) #>
   695         apfst (hd o snd)) elims ||>>
   696       LocalTheory.note ((NameSpace.append rec_name (coind_prefix coind ^ "induct"),
   697         map Attrib.internal (#2 induct)), [rulify (#1 induct)]);
   698 
   699     val induct_att = if coind then InductAttrib.coinduct_set else InductAttrib.induct_set;
   700     val ctxt4 = if no_ind then ctxt3 else
   701       let val inducts = cnames ~~ ProjectRule.projects ctxt (1 upto length cnames) induct'
   702       in
   703         ctxt3 |>
   704         LocalTheory.notes (inducts |> map (fn (name, th) => (("",
   705           [Attrib.internal ind_case_names,
   706            Attrib.internal (RuleCases.consumes 1),
   707            Attrib.internal (induct_att name)]), [([th], [])]))) |> snd |>
   708         LocalTheory.note ((NameSpace.append rec_name (coind_prefix coind ^ "inducts"),
   709           [Attrib.internal ind_case_names,
   710            Attrib.internal (RuleCases.consumes 1)]), map snd inducts) |> snd
   711       end;
   712 
   713     val result =
   714       {preds = preds,
   715        defs = fp_def :: rec_preds_defs,
   716        mono = singleton (ProofContext.export ctxt1 ctxt) mono,
   717        unfold = singleton (ProofContext.export ctxt1 ctxt) unfold,
   718        intrs = intrs',
   719        elims = elims',
   720        mk_cases = mk_cases elims',
   721        raw_induct = rulify raw_induct,
   722        induct = induct'}
   723       
   724   in
   725     (LocalTheory.declaration
   726        (put_inductives cnames ({names = cnames, coind = coind}, result)) ctxt4,
   727      result)
   728   end;
   729 
   730 
   731 (* external interfaces *)
   732 
   733 fun add_inductive_i verbose alt_name coind no_elim no_ind cnames_syn pnames pre_intros monos ctxt =
   734   let
   735     val thy = ProofContext.theory_of ctxt;
   736     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   737 
   738     val frees = fold (Term.add_frees o snd) pre_intros [];
   739     fun type_of s = (case AList.lookup op = frees s of
   740       NONE => error ("No such variable: " ^ s) | SOME T => T);
   741 
   742     val params = map
   743       (fn (s, SOME T) => Free (s, T) | (s, NONE) => Free (s, type_of s)) pnames;
   744     val cs = map
   745       (fn (s, SOME T, _) => Free (s, T) | (s, NONE, _) => Free (s, type_of s)) cnames_syn;
   746     val cnames_syn' = map (fn (s, _, mx) => (s, mx)) cnames_syn;
   747 
   748     fun close_rule (x, r) = (x, list_all_free (rev (fold_aterms
   749       (fn t as Free (v as (s, _)) =>
   750             if Variable.is_fixed ctxt s orelse member op = cs t orelse
   751               member op = params t then I else insert op = v
   752         | _ => I) r []), r));
   753 
   754     val intros = map (close_rule o check_rule thy cs params) pre_intros;
   755   in
   756     add_ind_def verbose alt_name coind no_elim no_ind cs intros monos
   757       params cnames_syn' ctxt
   758   end;
   759 
   760 fun add_inductive verbose coind cnames_syn pnames_syn intro_srcs raw_monos ctxt =
   761   let
   762     val (_, ctxt') = Specification.read_specification (cnames_syn @ pnames_syn) [] ctxt;
   763     val intrs = map (fn spec => apsnd hd (hd (snd (fst
   764       (Specification.read_specification [] [apsnd single spec] ctxt'))))) intro_srcs;
   765     val pnames = map (fn (s, _, _) =>
   766       (s, SOME (ProofContext.infer_type ctxt' s))) pnames_syn;
   767     val cnames_syn' = map (fn (s, _, mx) =>
   768       (s, SOME (ProofContext.infer_type ctxt' s), mx)) cnames_syn;
   769     val (monos, ctxt'') = LocalTheory.theory_result (IsarThy.apply_theorems raw_monos) ctxt;
   770   in
   771     add_inductive_i verbose "" coind false false cnames_syn' pnames intrs monos ctxt''
   772   end;
   773 
   774 
   775 
   776 (** package setup **)
   777 
   778 (* setup theory *)
   779 
   780 val setup =
   781   InductiveData.init #>
   782   Method.add_methods [("ind_cases2", mk_cases_meth oo mk_cases_args,
   783     "dynamic case analysis on predicates")] #>
   784   Attrib.add_attributes [("mono2", Attrib.add_del_args mono_add mono_del,
   785     "declaration of monotonicity rule")];
   786 
   787 
   788 (* outer syntax *)
   789 
   790 local structure P = OuterParse and K = OuterKeyword in
   791 
   792 fun mk_ind coind ((((loc, preds), params), intrs), monos) =
   793   Toplevel.local_theory loc
   794     (#1 o add_inductive true coind preds params intrs monos);
   795 
   796 fun ind_decl coind =
   797   P.opt_locale_target --
   798   P.fixes -- Scan.optional (P.$$$ "for" |-- P.fixes) [] --
   799   (P.$$$ "intros" |--
   800     P.!!! (Scan.repeat (P.opt_thm_name ":" -- P.prop))) --
   801   Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1) []
   802   >> mk_ind coind;
   803 
   804 val inductiveP =
   805   OuterSyntax.command "inductive2" "define inductive predicates" K.thy_decl (ind_decl false);
   806 
   807 val coinductiveP =
   808   OuterSyntax.command "coinductive2" "define coinductive predicates" K.thy_decl (ind_decl true);
   809 
   810 
   811 val ind_cases =
   812   P.and_list1 (P.opt_thm_name ":" -- Scan.repeat1 P.prop)
   813   >> (Toplevel.theory o inductive_cases);
   814 
   815 val inductive_casesP =
   816   OuterSyntax.command "inductive_cases2"
   817     "create simplified instances of elimination rules (improper)" K.thy_script ind_cases;
   818 
   819 val _ = OuterSyntax.add_keywords ["intros", "monos"];
   820 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
   821 
   822 end;
   823 
   824 end;
   825