src/HOL/Tools/inductive.ML
author haftmann
Tue Nov 24 17:28:25 2009 +0100 (2009-11-24)
changeset 33955 fff6f11b1f09
parent 33766 c679f05600cd
child 33957 e9afca2118d4
permissions -rw-r--r--
curried take/drop
     1 (*  Title:      HOL/Tools/inductive.ML
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
     4 
     5 (Co)Inductive Definition module for HOL.
     6 
     7 Features:
     8   * least or greatest fixedpoints
     9   * mutually recursive definitions
    10   * definitions involving arbitrary monotone operators
    11   * automatically proves introduction and elimination rules
    12 
    13   Introduction rules have the form
    14   [| M Pj ti, ..., Q x, ... |] ==> Pk t
    15   where M is some monotone operator (usually the identity)
    16   Q x is any side condition on the free variables
    17   ti, t are any terms
    18   Pj, Pk are two of the predicates being defined in mutual recursion
    19 *)
    20 
    21 signature BASIC_INDUCTIVE =
    22 sig
    23   type inductive_result =
    24     {preds: term list, elims: thm list, raw_induct: thm,
    25      induct: thm, intrs: thm list}
    26   val morph_result: morphism -> inductive_result -> inductive_result
    27   type inductive_info = {names: string list, coind: bool} * inductive_result
    28   val the_inductive: Proof.context -> string -> inductive_info
    29   val print_inductives: Proof.context -> unit
    30   val mono_add: attribute
    31   val mono_del: attribute
    32   val get_monos: Proof.context -> thm list
    33   val mk_cases: Proof.context -> term -> thm
    34   val inductive_forall_name: string
    35   val inductive_forall_def: thm
    36   val rulify: thm -> thm
    37   val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
    38     thm list list * local_theory
    39   val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
    40     thm list list * local_theory
    41   type inductive_flags =
    42     {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
    43       no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool}
    44   val add_inductive_i:
    45     inductive_flags -> ((binding * typ) * mixfix) list ->
    46     (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
    47     inductive_result * local_theory
    48   val add_inductive: bool -> bool ->
    49     (binding * string option * mixfix) list ->
    50     (binding * string option * mixfix) list ->
    51     (Attrib.binding * string) list ->
    52     (Facts.ref * Attrib.src list) list ->
    53     bool -> local_theory -> inductive_result * local_theory
    54   val add_inductive_global: inductive_flags ->
    55     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    56     thm list -> theory -> inductive_result * theory
    57   val arities_of: thm -> (string * int) list
    58   val params_of: thm -> term list
    59   val partition_rules: thm -> thm list -> (string * thm list) list
    60   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
    61   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    62   val infer_intro_vars: thm -> int -> thm list -> term list list
    63   val setup: theory -> theory
    64 end;
    65 
    66 signature INDUCTIVE =
    67 sig
    68   include BASIC_INDUCTIVE
    69   type add_ind_def =
    70     inductive_flags ->
    71     term list -> (Attrib.binding * term) list -> thm list ->
    72     term list -> (binding * mixfix) list ->
    73     local_theory -> inductive_result * local_theory
    74   val declare_rules: binding -> bool -> bool -> string list ->
    75     thm list -> binding list -> Attrib.src list list -> (thm * string list) list ->
    76     thm -> local_theory -> thm list * thm list * thm * local_theory
    77   val add_ind_def: add_ind_def
    78   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
    79     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    80     thm list -> local_theory -> inductive_result * local_theory
    81   val gen_add_inductive: add_ind_def -> bool -> bool ->
    82     (binding * string option * mixfix) list ->
    83     (binding * string option * mixfix) list ->
    84     (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
    85     bool -> local_theory -> inductive_result * local_theory
    86   val gen_ind_decl: add_ind_def -> bool ->
    87     OuterParse.token list -> (bool -> local_theory -> local_theory) * OuterParse.token list
    88 end;
    89 
    90 structure Inductive: INDUCTIVE =
    91 struct
    92 
    93 
    94 (** theory context references **)
    95 
    96 val inductive_forall_name = "HOL.induct_forall";
    97 val inductive_forall_def = @{thm induct_forall_def};
    98 val inductive_conj_name = "HOL.induct_conj";
    99 val inductive_conj_def = @{thm induct_conj_def};
   100 val inductive_conj = @{thms induct_conj};
   101 val inductive_atomize = @{thms induct_atomize};
   102 val inductive_rulify = @{thms induct_rulify};
   103 val inductive_rulify_fallback = @{thms induct_rulify_fallback};
   104 
   105 val notTrueE = TrueI RSN (2, notE);
   106 val notFalseI = Seq.hd (atac 1 notI);
   107 
   108 val simp_thms' = map mk_meta_eq
   109   @{lemma "(~True) = False" "(~False) = True"
   110       "(True --> P) = P" "(False --> P) = True"
   111       "(P & True) = P" "(True & P) = P"
   112     by (fact simp_thms)+};
   113 
   114 val simp_thms'' = map mk_meta_eq [@{thm inf_fun_eq}, @{thm inf_bool_eq}] @ simp_thms';
   115 
   116 val simp_thms''' = map mk_meta_eq
   117   [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_eq}, @{thm sup_bool_eq}];
   118 
   119 
   120 (** context data **)
   121 
   122 type inductive_result =
   123   {preds: term list, elims: thm list, raw_induct: thm,
   124    induct: thm, intrs: thm list};
   125 
   126 fun morph_result phi {preds, elims, raw_induct: thm, induct, intrs} =
   127   let
   128     val term = Morphism.term phi;
   129     val thm = Morphism.thm phi;
   130     val fact = Morphism.fact phi;
   131   in
   132    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
   133     induct = thm induct, intrs = fact intrs}
   134   end;
   135 
   136 type inductive_info =
   137   {names: string list, coind: bool} * inductive_result;
   138 
   139 structure InductiveData = Generic_Data
   140 (
   141   type T = inductive_info Symtab.table * thm list;
   142   val empty = (Symtab.empty, []);
   143   val extend = I;
   144   fun merge ((tab1, monos1), (tab2, monos2)) : T =
   145     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
   146 );
   147 
   148 val get_inductives = InductiveData.get o Context.Proof;
   149 
   150 fun print_inductives ctxt =
   151   let
   152     val (tab, monos) = get_inductives ctxt;
   153     val space = Consts.space_of (ProofContext.consts_of ctxt);
   154   in
   155     [Pretty.strs ("(co)inductives:" :: map #1 (Name_Space.extern_table (space, tab))),
   156      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm ctxt) monos)]
   157     |> Pretty.chunks |> Pretty.writeln
   158   end;
   159 
   160 
   161 (* get and put data *)
   162 
   163 fun the_inductive ctxt name =
   164   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
   165     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   166   | SOME info => info);
   167 
   168 fun put_inductives names info = InductiveData.map
   169   (apfst (fold (fn name => Symtab.update (name, info)) names));
   170 
   171 
   172 
   173 (** monotonicity rules **)
   174 
   175 val get_monos = #2 o get_inductives;
   176 val map_monos = InductiveData.map o apsnd;
   177 
   178 fun mk_mono thm =
   179   let
   180     val concl = concl_of thm;
   181     fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @
   182       (case concl of
   183           (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
   184         | _ => [thm' RS (thm' RS eq_to_mono2)]);
   185     fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
   186       handle THM _ => thm RS @{thm le_boolD}
   187   in
   188     case concl of
   189       Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq)
   190     | _ $ (Const ("op =", _) $ _ $ _) => eq2mono thm
   191     | _ $ (Const (@{const_name HOL.less_eq}, _) $ _ $ _) =>
   192       [dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   193          (resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))]
   194     | _ => [thm]
   195   end handle THM _ =>
   196     error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm_without_context thm);
   197 
   198 val mono_add = Thm.declaration_attribute (map_monos o fold Thm.add_thm o mk_mono);
   199 val mono_del = Thm.declaration_attribute (map_monos o fold Thm.del_thm o mk_mono);
   200 
   201 
   202 
   203 (** misc utilities **)
   204 
   205 fun message quiet_mode s = if quiet_mode then () else writeln s;
   206 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
   207 
   208 fun coind_prefix true = "co"
   209   | coind_prefix false = "";
   210 
   211 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   212 
   213 fun make_bool_args f g [] i = []
   214   | make_bool_args f g (x :: xs) i =
   215       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   216 
   217 fun make_bool_args' xs =
   218   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
   219 
   220 fun arg_types_of k c = (uncurry drop) (k, binder_types (fastype_of c));
   221 
   222 fun find_arg T x [] = sys_error "find_arg"
   223   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   224       apsnd (cons p) (find_arg T x ps)
   225   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   226       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   227       else apsnd (cons p) (find_arg T x ps);
   228 
   229 fun make_args Ts xs =
   230   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
   231     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   232 
   233 fun make_args' Ts xs Us =
   234   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   235 
   236 fun dest_predicate cs params t =
   237   let
   238     val k = length params;
   239     val (c, ts) = strip_comb t;
   240     val (xs, ys) = chop k ts;
   241     val i = find_index (fn c' => c' = c) cs;
   242   in
   243     if xs = params andalso i >= 0 then
   244       SOME (c, i, ys, chop (length ys) (arg_types_of k c))
   245     else NONE
   246   end;
   247 
   248 fun mk_names a 0 = []
   249   | mk_names a 1 = [a]
   250   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   251 
   252 
   253 
   254 (** process rules **)
   255 
   256 local
   257 
   258 fun err_in_rule ctxt name t msg =
   259   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
   260     Syntax.string_of_term ctxt t, msg]);
   261 
   262 fun err_in_prem ctxt name t p msg =
   263   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
   264     "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
   265 
   266 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   267 
   268 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   269 
   270 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   271 
   272 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
   273 
   274 in
   275 
   276 fun check_rule ctxt cs params ((binding, att), rule) =
   277   let
   278     val err_name = Binding.str_of binding;
   279     val params' = Term.variant_frees rule (Logic.strip_params rule);
   280     val frees = rev (map Free params');
   281     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   282     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   283     val rule' = Logic.list_implies (prems, concl);
   284     val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
   285     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
   286 
   287     fun check_ind err t = case dest_predicate cs params t of
   288         NONE => err (bad_app ^
   289           commas (map (Syntax.string_of_term ctxt) params))
   290       | SOME (_, _, ys, _) =>
   291           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   292           then err bad_ind_occ else ();
   293 
   294     fun check_prem' prem t =
   295       if head_of t mem cs then
   296         check_ind (err_in_prem ctxt err_name rule prem) t
   297       else (case t of
   298           Abs (_, _, t) => check_prem' prem t
   299         | t $ u => (check_prem' prem t; check_prem' prem u)
   300         | _ => ());
   301 
   302     fun check_prem (prem, aprem) =
   303       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   304       else err_in_prem ctxt err_name rule prem "Non-atomic premise";
   305   in
   306     (case concl of
   307        Const ("Trueprop", _) $ t =>
   308          if head_of t mem cs then
   309            (check_ind (err_in_rule ctxt err_name rule') t;
   310             List.app check_prem (prems ~~ aprems))
   311          else err_in_rule ctxt err_name rule' bad_concl
   312      | _ => err_in_rule ctxt err_name rule' bad_concl);
   313     ((binding, att), arule)
   314   end;
   315 
   316 val rulify =
   317   hol_simplify inductive_conj
   318   #> hol_simplify inductive_rulify
   319   #> hol_simplify inductive_rulify_fallback
   320   #> Simplifier.norm_hhf;
   321 
   322 end;
   323 
   324 
   325 
   326 (** proofs for (co)inductive predicates **)
   327 
   328 (* prove monotonicity *)
   329 
   330 fun prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt =
   331  (message (quiet_mode orelse skip_mono andalso !quick_and_dirty orelse fork_mono)
   332     "  Proving monotonicity ...";
   333   (if skip_mono then Skip_Proof.prove else if fork_mono then Goal.prove_future else Goal.prove) ctxt
   334     [] []
   335     (HOLogic.mk_Trueprop
   336       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   337     (fn _ => EVERY [rtac @{thm monoI} 1,
   338       REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
   339       REPEAT (FIRST
   340         [atac 1,
   341          resolve_tac (maps mk_mono monos @ get_monos ctxt) 1,
   342          etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
   343 
   344 
   345 (* prove introduction rules *)
   346 
   347 fun prove_intrs quiet_mode coind mono fp_def k params intr_ts rec_preds_defs ctxt =
   348   let
   349     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
   350 
   351     val unfold = funpow k (fn th => th RS fun_cong)
   352       (mono RS (fp_def RS
   353         (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
   354 
   355     fun select_disj 1 1 = []
   356       | select_disj _ 1 = [rtac disjI1]
   357       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   358 
   359     val rules = [refl, TrueI, notFalseI, exI, conjI];
   360 
   361     val intrs = map_index (fn (i, intr) => rulify
   362       (Skip_Proof.prove ctxt (map (fst o dest_Free) params) [] intr (fn _ => EVERY
   363        [rewrite_goals_tac rec_preds_defs,
   364         rtac (unfold RS iffD2) 1,
   365         EVERY1 (select_disj (length intr_ts) (i + 1)),
   366         (*Not ares_tac, since refl must be tried before any equality assumptions;
   367           backtracking may occur if the premises have extra variables!*)
   368         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts
   369 
   370   in (intrs, unfold) end;
   371 
   372 
   373 (* prove elimination rules *)
   374 
   375 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt =
   376   let
   377     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
   378 
   379     val ([pname], ctxt') = ctxt |>
   380       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   381       Variable.variant_fixes ["P"];
   382     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   383 
   384     fun dest_intr r =
   385       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   386        Logic.strip_assums_hyp r, Logic.strip_params r);
   387 
   388     val intrs = map dest_intr intr_ts ~~ intr_names;
   389 
   390     val rules1 = [disjE, exE, FalseE];
   391     val rules2 = [conjE, FalseE, notTrueE];
   392 
   393     fun prove_elim c =
   394       let
   395         val Ts = arg_types_of (length params) c;
   396         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   397         val frees = map Free (anames ~~ Ts);
   398 
   399         fun mk_elim_prem ((_, _, us, _), ts, params') =
   400           list_all (params',
   401             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   402               (frees ~~ us) @ ts, P));
   403         val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
   404         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   405            map mk_elim_prem (map #1 c_intrs)
   406       in
   407         (Skip_Proof.prove ctxt'' [] prems P
   408           (fn {prems, ...} => EVERY
   409             [cut_facts_tac [hd prems] 1,
   410              rewrite_goals_tac rec_preds_defs,
   411              dtac (unfold RS iffD1) 1,
   412              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   413              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   414              EVERY (map (fn prem =>
   415                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
   416           |> rulify
   417           |> singleton (ProofContext.export ctxt'' ctxt),
   418          map #2 c_intrs)
   419       end
   420 
   421    in map prove_elim cs end;
   422 
   423 
   424 (* derivation of simplified elimination rules *)
   425 
   426 local
   427 
   428 (*delete needless equality assumptions*)
   429 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
   430   (fn _ => assume_tac 1);
   431 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   432 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   433 
   434 fun simp_case_tac ss i =
   435   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
   436 
   437 in
   438 
   439 fun mk_cases ctxt prop =
   440   let
   441     val thy = ProofContext.theory_of ctxt;
   442     val ss = simpset_of ctxt;
   443 
   444     fun err msg =
   445       error (Pretty.string_of (Pretty.block
   446         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
   447 
   448     val elims = Induct.find_casesP ctxt prop;
   449 
   450     val cprop = Thm.cterm_of thy prop;
   451     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
   452     fun mk_elim rl =
   453       Thm.implies_intr cprop (Tactic.rule_by_tactic tac (Thm.assume cprop RS rl))
   454       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   455   in
   456     (case get_first (try mk_elim) elims of
   457       SOME r => r
   458     | NONE => err "Proposition not an inductive predicate:")
   459   end;
   460 
   461 end;
   462 
   463 
   464 (* inductive_cases *)
   465 
   466 fun gen_inductive_cases prep_att prep_prop args lthy =
   467   let
   468     val thy = ProofContext.theory_of lthy;
   469     val facts = args |> map (fn ((a, atts), props) =>
   470       ((a, map (prep_att thy) atts),
   471         map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
   472   in lthy |> Local_Theory.notes facts |>> map snd end;
   473 
   474 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
   475 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   476 
   477 
   478 val ind_cases_setup =
   479   Method.setup @{binding ind_cases}
   480     (Scan.lift (Scan.repeat1 Args.name_source --
   481       Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.name) []) >>
   482       (fn (raw_props, fixes) => fn ctxt =>
   483         let
   484           val (_, ctxt') = Variable.add_fixes fixes ctxt;
   485           val props = Syntax.read_props ctxt' raw_props;
   486           val ctxt'' = fold Variable.declare_term props ctxt';
   487           val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props)
   488         in Method.erule 0 rules end))
   489     "dynamic case analysis on predicates";
   490 
   491 
   492 (* prove induction rule *)
   493 
   494 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
   495     fp_def rec_preds_defs ctxt =
   496   let
   497     val _ = clean_message quiet_mode "  Proving the induction rule ...";
   498     val thy = ProofContext.theory_of ctxt;
   499 
   500     (* predicates for induction rule *)
   501 
   502     val (pnames, ctxt') = ctxt |>
   503       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   504       Variable.variant_fixes (mk_names "P" (length cs));
   505     val preds = map2 (curry Free) pnames
   506       (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
   507 
   508     (* transform an introduction rule into a premise for induction rule *)
   509 
   510     fun mk_ind_prem r =
   511       let
   512         fun subst s =
   513           (case dest_predicate cs params s of
   514             SOME (_, i, ys, (_, Ts)) =>
   515               let
   516                 val k = length Ts;
   517                 val bs = map Bound (k - 1 downto 0);
   518                 val P = list_comb (List.nth (preds, i),
   519                   map (incr_boundvars k) ys @ bs);
   520                 val Q = list_abs (mk_names "x" k ~~ Ts,
   521                   HOLogic.mk_binop inductive_conj_name
   522                     (list_comb (incr_boundvars k s, bs), P))
   523               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   524           | NONE =>
   525               (case s of
   526                 (t $ u) => (fst (subst t) $ fst (subst u), NONE)
   527               | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
   528               | _ => (s, NONE)));
   529 
   530         fun mk_prem s prems =
   531           (case subst s of
   532             (_, SOME (t, u)) => t :: u :: prems
   533           | (t, _) => t :: prems);
   534 
   535         val SOME (_, i, ys, _) = dest_predicate cs params
   536           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
   537 
   538       in list_all_free (Logic.strip_params r,
   539         Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
   540           (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
   541             HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
   542       end;
   543 
   544     val ind_prems = map mk_ind_prem intr_ts;
   545 
   546 
   547     (* make conclusions for induction rules *)
   548 
   549     val Tss = map (binder_types o fastype_of) preds;
   550     val (xnames, ctxt'') =
   551       Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   552     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   553         (map (fn (((xnames, Ts), c), P) =>
   554            let val frees = map Free (xnames ~~ Ts)
   555            in HOLogic.mk_imp
   556              (list_comb (c, params @ frees), list_comb (P, frees))
   557            end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   558 
   559 
   560     (* make predicate for instantiation of abstract induction rule *)
   561 
   562     val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   563       (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
   564          (make_bool_args HOLogic.mk_not I bs i)
   565          (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
   566 
   567     val ind_concl = HOLogic.mk_Trueprop
   568       (HOLogic.mk_binrel "HOL.ord_class.less_eq" (rec_const, ind_pred));
   569 
   570     val raw_fp_induct = (mono RS (fp_def RS @{thm def_lfp_induct}));
   571 
   572     val induct = Skip_Proof.prove ctxt'' [] ind_prems ind_concl
   573       (fn {prems, ...} => EVERY
   574         [rewrite_goals_tac [inductive_conj_def],
   575          DETERM (rtac raw_fp_induct 1),
   576          REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
   577          rewrite_goals_tac simp_thms'',
   578          (*This disjE separates out the introduction rules*)
   579          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
   580          (*Now break down the individual cases.  No disjE here in case
   581            some premise involves disjunction.*)
   582          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
   583          REPEAT (FIRSTGOAL
   584            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
   585          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
   586              (inductive_conj_def :: rec_preds_defs @ simp_thms'') prem,
   587            conjI, refl] 1)) prems)]);
   588 
   589     val lemma = Skip_Proof.prove ctxt'' [] []
   590       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
   591         [rewrite_goals_tac rec_preds_defs,
   592          REPEAT (EVERY
   593            [REPEAT (resolve_tac [conjI, impI] 1),
   594             REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
   595             atac 1,
   596             rewrite_goals_tac simp_thms',
   597             atac 1])])
   598 
   599   in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end;
   600 
   601 
   602 
   603 (** specification of (co)inductive predicates **)
   604 
   605 fun mk_ind_def quiet_mode skip_mono fork_mono alt_name coind
   606     cs intr_ts monos params cnames_syn lthy =
   607   let
   608     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
   609 
   610     val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
   611     val k = log 2 1 (length cs);
   612     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   613     val p :: xs = map Free (Variable.variant_frees lthy intr_ts
   614       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   615     val bs = map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
   616       (map (rpair HOLogic.boolT) (mk_names "b" k)));
   617 
   618     fun subst t =
   619       (case dest_predicate cs params t of
   620         SOME (_, i, ts, (Ts, Us)) =>
   621           let
   622             val l = length Us;
   623             val zs = map Bound (l - 1 downto 0);
   624           in
   625             list_abs (map (pair "z") Us, list_comb (p,
   626               make_bool_args' bs i @ make_args argTs
   627                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   628           end
   629       | NONE =>
   630           (case t of
   631             t1 $ t2 => subst t1 $ subst t2
   632           | Abs (x, T, u) => Abs (x, T, subst u)
   633           | _ => t));
   634 
   635     (* transform an introduction rule into a conjunction  *)
   636     (*   [| p_i t; ... |] ==> p_j u                       *)
   637     (* is transformed into                                *)
   638     (*   b_j & x_j = u & p b_j t & ...                    *)
   639 
   640     fun transform_rule r =
   641       let
   642         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
   643           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   644         val ps = make_bool_args HOLogic.mk_not I bs i @
   645           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   646           map (subst o HOLogic.dest_Trueprop)
   647             (Logic.strip_assums_hyp r)
   648       in
   649         fold_rev (fn (x, T) => fn P => HOLogic.exists_const T $ Abs (x, T, P))
   650           (Logic.strip_params r)
   651           (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
   652       end
   653 
   654     (* make a disjunction of all introduction rules *)
   655 
   656     val fp_fun = fold_rev lambda (p :: bs @ xs)
   657       (if null intr_ts then HOLogic.false_const
   658        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   659 
   660     (* add definiton of recursive predicates to theory *)
   661 
   662     val rec_name =
   663       if Binding.is_empty alt_name then
   664         Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
   665       else alt_name;
   666 
   667     val ((rec_const, (_, fp_def)), lthy') = lthy
   668       |> Local_Theory.conceal
   669       |> Local_Theory.define
   670         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   671          ((Binding.empty, [Attrib.internal (K Nitpick_Defs.add)]),
   672          fold_rev lambda params
   673            (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)))
   674       ||> Local_Theory.restore_naming lthy;
   675     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
   676       (cterm_of (ProofContext.theory_of lthy') (list_comb (rec_const, params)));
   677     val specs =
   678       if length cs < 2 then []
   679       else
   680         map_index (fn (i, (name_mx, c)) =>
   681           let
   682             val Ts = arg_types_of (length params) c;
   683             val xs = map Free (Variable.variant_frees lthy intr_ts
   684               (mk_names "x" (length Ts) ~~ Ts))
   685           in
   686             (name_mx, (Attrib.empty_binding, fold_rev lambda (params @ xs)
   687               (list_comb (rec_const, params @ make_bool_args' bs i @
   688                 make_args argTs (xs ~~ Ts)))))
   689           end) (cnames_syn ~~ cs);
   690     val (consts_defs, lthy'') = lthy'
   691       |> Local_Theory.conceal
   692       |> fold_map Local_Theory.define specs
   693       ||> Local_Theory.restore_naming lthy';
   694     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   695 
   696     val mono = prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos lthy'';
   697     val ((_, [mono']), lthy''') =
   698       Local_Theory.note (apfst Binding.conceal Attrib.empty_binding, [mono]) lthy'';
   699 
   700   in (lthy''', rec_name, mono', fp_def', map (#2 o #2) consts_defs,
   701     list_comb (rec_const, params), preds, argTs, bs, xs)
   702   end;
   703 
   704 fun declare_rules rec_binding coind no_ind cnames
   705     intrs intr_bindings intr_atts elims raw_induct lthy =
   706   let
   707     val rec_name = Binding.name_of rec_binding;
   708     fun rec_qualified qualified = Binding.qualify qualified rec_name;
   709     val intr_names = map Binding.name_of intr_bindings;
   710     val ind_case_names = Rule_Cases.case_names intr_names;
   711     val induct =
   712       if coind then
   713         (raw_induct, [Rule_Cases.case_names [rec_name],
   714           Rule_Cases.case_conclusion (rec_name, intr_names),
   715           Rule_Cases.consumes 1, Induct.coinduct_pred (hd cnames)])
   716       else if no_ind orelse length cnames > 1 then
   717         (raw_induct, [ind_case_names, Rule_Cases.consumes 0])
   718       else (raw_induct RSN (2, rev_mp), [ind_case_names, Rule_Cases.consumes 1]);
   719 
   720     val (intrs', lthy1) =
   721       lthy |>
   722       Local_Theory.notes
   723         (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
   724           map (fn th => [([th],
   725            [Attrib.internal (K (Context_Rules.intro_query NONE)),
   726             Attrib.internal (K Nitpick_Intros.add)])]) intrs) |>>
   727       map (hd o snd);
   728     val (((_, elims'), (_, [induct'])), lthy2) =
   729       lthy1 |>
   730       Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
   731       fold_map (fn (name, (elim, cases)) =>
   732         Local_Theory.note
   733           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
   734             [Attrib.internal (K (Rule_Cases.case_names cases)),
   735              Attrib.internal (K (Rule_Cases.consumes 1)),
   736              Attrib.internal (K (Induct.cases_pred name)),
   737              Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
   738         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   739       Local_Theory.note
   740         ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
   741           map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
   742 
   743     val lthy3 =
   744       if no_ind orelse coind then lthy2
   745       else
   746         let val inducts = cnames ~~ Project_Rule.projects lthy2 (1 upto length cnames) induct' in
   747           lthy2 |>
   748           Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
   749             inducts |> map (fn (name, th) => ([th],
   750               [Attrib.internal (K ind_case_names),
   751                Attrib.internal (K (Rule_Cases.consumes 1)),
   752                Attrib.internal (K (Induct.induct_pred name))])))] |> snd
   753         end;
   754   in (intrs', elims', induct', lthy3) end;
   755 
   756 type inductive_flags =
   757   {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
   758     no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool};
   759 
   760 type add_ind_def =
   761   inductive_flags ->
   762   term list -> (Attrib.binding * term) list -> thm list ->
   763   term list -> (binding * mixfix) list ->
   764   local_theory -> inductive_result * local_theory;
   765 
   766 fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}
   767     cs intros monos params cnames_syn lthy =
   768   let
   769     val _ = null cnames_syn andalso error "No inductive predicates given";
   770     val names = map (Binding.name_of o fst) cnames_syn;
   771     val _ = message (quiet_mode andalso not verbose)
   772       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
   773 
   774     val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
   775     val ((intr_names, intr_atts), intr_ts) =
   776       apfst split_list (split_list (map (check_rule lthy cs params) intros));
   777 
   778     val (lthy1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   779       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts
   780         monos params cnames_syn lthy;
   781 
   782     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
   783       params intr_ts rec_preds_defs lthy1;
   784     val elims =
   785       if no_elim then []
   786       else
   787         prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
   788           unfold rec_preds_defs lthy1;
   789     val raw_induct = zero_var_indexes
   790       (if no_ind then Drule.asm_rl
   791        else if coind then
   792          singleton (ProofContext.export
   793            (snd (Variable.add_fixes (map (fst o dest_Free) params) lthy1)) lthy1)
   794            (rotate_prems ~1 (ObjectLogic.rulify
   795              (fold_rule rec_preds_defs
   796                (rewrite_rule simp_thms'''
   797                 (mono RS (fp_def RS @{thm def_coinduct}))))))
   798        else
   799          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
   800            rec_preds_defs lthy1);
   801 
   802     val (intrs', elims', induct, lthy2) = declare_rules rec_name coind no_ind
   803       cnames intrs intr_names intr_atts elims raw_induct lthy1;
   804 
   805     val result =
   806       {preds = preds,
   807        intrs = intrs',
   808        elims = elims',
   809        raw_induct = rulify raw_induct,
   810        induct = induct};
   811 
   812     val lthy3 = lthy2
   813       |> Local_Theory.declaration false (fn phi =>
   814         let val result' = morph_result phi result;
   815         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
   816   in (result, lthy3) end;
   817 
   818 
   819 (* external interfaces *)
   820 
   821 fun gen_add_inductive_i mk_def
   822     (flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono})
   823     cnames_syn pnames spec monos lthy =
   824   let
   825     val thy = ProofContext.theory_of lthy;
   826     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   827 
   828 
   829     (* abbrevs *)
   830 
   831     val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
   832 
   833     fun get_abbrev ((name, atts), t) =
   834       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
   835         let
   836           val _ = Binding.is_empty name andalso null atts orelse
   837             error "Abbreviations may not have names or attributes";
   838           val ((x, T), rhs) = LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt1 t));
   839           val var =
   840             (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
   841               NONE => error ("Undeclared head of abbreviation " ^ quote x)
   842             | SOME ((b, T'), mx) =>
   843                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
   844                 else (b, mx));
   845         in SOME (var, rhs) end
   846       else NONE;
   847 
   848     val abbrevs = map_filter get_abbrev spec;
   849     val bs = map (Binding.name_of o fst o fst) abbrevs;
   850 
   851 
   852     (* predicates *)
   853 
   854     val pre_intros = filter_out (is_some o get_abbrev) spec;
   855     val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
   856     val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
   857     val ps = map Free pnames;
   858 
   859     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
   860     val _ = map (fn abbr => LocalDefs.fixed_abbrev abbr ctxt2) abbrevs;
   861     val ctxt3 = ctxt2 |> fold (snd oo LocalDefs.fixed_abbrev) abbrevs;
   862     val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy;
   863 
   864     fun close_rule r = list_all_free (rev (fold_aterms
   865       (fn t as Free (v as (s, _)) =>
   866           if Variable.is_fixed ctxt1 s orelse
   867             member (op =) ps t then I else insert (op =) v
   868         | _ => I) r []), r);
   869 
   870     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
   871     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
   872   in
   873     lthy
   874     |> mk_def flags cs intros monos ps preds
   875     ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
   876   end;
   877 
   878 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos int lthy =
   879   let
   880     val ((vars, intrs), _) = lthy
   881       |> ProofContext.set_mode ProofContext.mode_abbrev
   882       |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
   883     val (cs, ps) = chop (length cnames_syn) vars;
   884     val monos = Attrib.eval_thms lthy raw_monos;
   885     val flags = {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
   886       coind = coind, no_elim = false, no_ind = false, skip_mono = false, fork_mono = not int};
   887   in
   888     lthy
   889     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
   890   end;
   891 
   892 val add_inductive_i = gen_add_inductive_i add_ind_def;
   893 val add_inductive = gen_add_inductive add_ind_def;
   894 
   895 fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
   896   let
   897     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
   898     val ctxt' = thy
   899       |> Theory_Target.init NONE
   900       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
   901       |> Local_Theory.exit;
   902     val info = #2 (the_inductive ctxt' name);
   903   in (info, ProofContext.theory_of ctxt') end;
   904 
   905 
   906 (* read off arities of inductive predicates from raw induction rule *)
   907 fun arities_of induct =
   908   map (fn (_ $ t $ u) =>
   909       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
   910     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
   911 
   912 (* read off parameters of inductive predicate from raw induction rule *)
   913 fun params_of induct =
   914   let
   915     val (_ $ t $ u :: _) =
   916       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
   917     val (_, ts) = strip_comb t;
   918     val (_, us) = strip_comb u
   919   in
   920     List.take (ts, length ts - length us)
   921   end;
   922 
   923 val pname_of_intr =
   924   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
   925 
   926 (* partition introduction rules according to predicate name *)
   927 fun gen_partition_rules f induct intros =
   928   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
   929     (map (rpair [] o fst) (arities_of induct));
   930 
   931 val partition_rules = gen_partition_rules I;
   932 fun partition_rules' induct = gen_partition_rules fst induct;
   933 
   934 fun unpartition_rules intros xs =
   935   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
   936     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
   937 
   938 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
   939 fun infer_intro_vars elim arity intros =
   940   let
   941     val thy = theory_of_thm elim;
   942     val _ :: cases = prems_of elim;
   943     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
   944     fun mtch (t, u) =
   945       let
   946         val params = Logic.strip_params t;
   947         val vars = map (Var o apfst (rpair 0))
   948           (Name.variant_list used (map fst params) ~~ map snd params);
   949         val ts = map (curry subst_bounds (rev vars))
   950           (List.drop (Logic.strip_assums_hyp t, arity));
   951         val us = Logic.strip_imp_prems u;
   952         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
   953           (Vartab.empty, Vartab.empty);
   954       in
   955         map (Envir.subst_term tab) vars
   956       end
   957   in
   958     map (mtch o apsnd prop_of) (cases ~~ intros)
   959   end;
   960 
   961 
   962 
   963 (** package setup **)
   964 
   965 (* setup theory *)
   966 
   967 val setup =
   968   ind_cases_setup #>
   969   Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
   970     "declaration of monotonicity rule";
   971 
   972 
   973 (* outer syntax *)
   974 
   975 local structure P = OuterParse and K = OuterKeyword in
   976 
   977 val _ = OuterKeyword.keyword "monos";
   978 
   979 fun gen_ind_decl mk_def coind =
   980   P.fixes -- P.for_fixes --
   981   Scan.optional SpecParse.where_alt_specs [] --
   982   Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.xthms1) []
   983   >> (fn (((preds, params), specs), monos) =>
   984       (snd oo gen_add_inductive mk_def true coind preds params specs monos));
   985 
   986 val ind_decl = gen_ind_decl add_ind_def;
   987 
   988 val _ =
   989   OuterSyntax.local_theory' "inductive" "define inductive predicates" K.thy_decl
   990     (ind_decl false);
   991 
   992 val _ =
   993   OuterSyntax.local_theory' "coinductive" "define coinductive predicates" K.thy_decl
   994     (ind_decl true);
   995 
   996 val _ =
   997   OuterSyntax.local_theory "inductive_cases"
   998     "create simplified instances of elimination rules (improper)" K.thy_script
   999     (P.and_list1 SpecParse.specs >> (snd oo inductive_cases));
  1000 
  1001 end;
  1002 
  1003 end;