src/HOL/Tools/inductive_package.ML
author wenzelm
Sat Sep 01 15:47:01 2007 +0200 (2007-09-01)
changeset 24509 23ee6b7788c2
parent 24133 75063f96618f
child 24516 c121834a8808
permissions -rw-r--r--
replaced ProofContext.read_term/prop by general Syntax.read_term/prop;
replaced ProofContext.cert_term/prop by general Syntax.check_term/prop (which also includes type-inference);
     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   val quiet_mode: bool ref
    25   type inductive_result
    26   val morph_result: morphism -> inductive_result -> inductive_result
    27   type inductive_info
    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: ((bstring * Attrib.src list) * string list) list ->
    38     Proof.context -> thm list list * local_theory
    39   val inductive_cases_i: ((bstring * Attrib.src list) * term list) list ->
    40     Proof.context -> thm list list * local_theory
    41   val add_inductive_i: bool -> bstring -> bool -> bool -> bool ->
    42     (string * typ option * mixfix) list ->
    43     (string * typ option) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
    44       local_theory -> inductive_result * local_theory
    45   val add_inductive: bool -> bool -> (string * string option * mixfix) list ->
    46     (string * string option * mixfix) list ->
    47     ((bstring * Attrib.src list) * string) list -> (thmref * Attrib.src list) list ->
    48     local_theory -> inductive_result * local_theory
    49   val add_inductive_global: bool -> bstring -> bool -> bool -> bool ->
    50     (string * typ option * mixfix) list -> (string * typ option) list ->
    51     ((bstring * Attrib.src list) * term) list -> thm list -> theory -> inductive_result * theory
    52   val arities_of: thm -> (string * int) list
    53   val params_of: thm -> term list
    54   val partition_rules: thm -> thm list -> (string * thm list) list
    55   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    56   val infer_intro_vars: thm -> int -> thm list -> term list list
    57   val setup: theory -> theory
    58 end;
    59 
    60 signature INDUCTIVE_PACKAGE =
    61 sig
    62   include BASIC_INDUCTIVE_PACKAGE
    63   type add_ind_def
    64   val declare_rules: bstring -> bool -> bool -> string list ->
    65     thm list -> bstring list -> Attrib.src list list -> (thm * string list) list ->
    66     thm -> local_theory -> thm list * thm list * thm * local_theory
    67   val add_ind_def: add_ind_def
    68   val gen_add_inductive_i: add_ind_def ->
    69     bool -> bstring -> bool -> bool -> bool ->
    70     (string * typ option * mixfix) list ->
    71     (string * typ option) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
    72       local_theory -> inductive_result * local_theory
    73   val gen_add_inductive: add_ind_def ->
    74     bool -> bool -> (string * string option * mixfix) list ->
    75     (string * string option * mixfix) list ->
    76     ((bstring * Attrib.src list) * string) list -> (thmref * Attrib.src list) list ->
    77     local_theory -> inductive_result * local_theory
    78   val gen_ind_decl: add_ind_def ->
    79     bool -> OuterParse.token list ->
    80     (Toplevel.transition -> Toplevel.transition) * OuterParse.token list
    81 end;
    82 
    83 structure InductivePackage: INDUCTIVE_PACKAGE =
    84 struct
    85 
    86 
    87 (** theory context references **)
    88 
    89 val mono_name = "Orderings.mono";
    90 val gfp_name = "FixedPoint.gfp";
    91 val lfp_name = "FixedPoint.lfp";
    92 
    93 val inductive_forall_name = "HOL.induct_forall";
    94 val inductive_forall_def = thm "induct_forall_def";
    95 val inductive_conj_name = "HOL.induct_conj";
    96 val inductive_conj_def = thm "induct_conj_def";
    97 val inductive_conj = thms "induct_conj";
    98 val inductive_atomize = thms "induct_atomize";
    99 val inductive_rulify = thms "induct_rulify";
   100 val inductive_rulify_fallback = thms "induct_rulify_fallback";
   101 
   102 val notTrueE = TrueI RSN (2, notE);
   103 val notFalseI = Seq.hd (atac 1 notI);
   104 val simp_thms' = map (fn s => mk_meta_eq (the (find_first
   105   (equal (Sign.read_prop HOL.thy s) o prop_of) simp_thms)))
   106   ["(~True) = False", "(~False) = True",
   107    "(True --> ?P) = ?P", "(False --> ?P) = True",
   108    "(?P & True) = ?P", "(True & ?P) = ?P"];
   109 
   110 
   111 
   112 (** context data **)
   113 
   114 type inductive_result =
   115   {preds: term list, elims: thm list, raw_induct: thm,
   116    induct: thm, intrs: thm list};
   117 
   118 fun morph_result phi {preds, elims, raw_induct: thm, induct, intrs} =
   119   let
   120     val term = Morphism.term phi;
   121     val thm = Morphism.thm phi;
   122     val fact = Morphism.fact phi;
   123   in
   124    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
   125     induct = thm induct, intrs = fact intrs}
   126   end;
   127 
   128 type inductive_info =
   129   {names: string list, coind: bool} * inductive_result;
   130 
   131 structure InductiveData = GenericDataFun
   132 (
   133   type T = inductive_info Symtab.table * thm list;
   134   val empty = (Symtab.empty, []);
   135   val extend = I;
   136   fun merge _ ((tab1, monos1), (tab2, monos2)) =
   137     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
   138 );
   139 
   140 val get_inductives = InductiveData.get o Context.Proof;
   141 
   142 fun print_inductives ctxt =
   143   let
   144     val (tab, monos) = get_inductives ctxt;
   145     val space = Consts.space_of (ProofContext.consts_of ctxt);
   146   in
   147     [Pretty.strs ("(co)inductives:" :: map #1 (NameSpace.extern_table (space, tab))),
   148      Pretty.big_list "monotonicity rules:" (map (ProofContext.pretty_thm ctxt) monos)]
   149     |> Pretty.chunks |> Pretty.writeln
   150   end;
   151 
   152 
   153 (* get and put data *)
   154 
   155 fun the_inductive ctxt name =
   156   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
   157     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   158   | SOME info => info);
   159 
   160 fun put_inductives names info = InductiveData.map (apfst (fn tab =>
   161   fold (fn name => Symtab.update_new (name, info)) names tab
   162     handle Symtab.DUP d => error ("Duplicate definition of (co)inductive predicate " ^ quote d)));
   163 
   164 
   165 
   166 (** monotonicity rules **)
   167 
   168 val get_monos = #2 o get_inductives;
   169 val map_monos = InductiveData.map o apsnd;
   170 
   171 fun mk_mono thm =
   172   let
   173     val concl = concl_of thm;
   174     fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @
   175       (case concl of
   176           (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
   177         | _ => [thm' RS (thm' RS eq_to_mono2)]);
   178     fun dest_less_concl thm = dest_less_concl (thm RS le_funD)
   179       handle THM _ => thm RS le_boolD
   180   in
   181     case concl of
   182       Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq)
   183     | _ $ (Const ("op =", _) $ _ $ _) => eq2mono thm
   184     | _ $ (Const ("HOL.ord_class.less_eq", _) $ _ $ _) =>
   185       [dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   186          (resolve_tac [le_funI, le_boolI'])) thm))]
   187     | _ => [thm]
   188   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ string_of_thm thm);
   189 
   190 val mono_add = Thm.declaration_attribute (map_monos o fold Thm.add_thm o mk_mono);
   191 val mono_del = Thm.declaration_attribute (map_monos o fold Thm.del_thm o mk_mono);
   192 
   193 
   194 
   195 (** misc utilities **)
   196 
   197 val quiet_mode = ref false;
   198 fun message s = if ! quiet_mode then () else writeln s;
   199 fun clean_message s = if ! quick_and_dirty then () else message s;
   200 
   201 val note_theorems = LocalTheory.notes Thm.theoremK;
   202 val note_theorem = LocalTheory.note Thm.theoremK;
   203 
   204 fun coind_prefix true = "co"
   205   | coind_prefix false = "";
   206 
   207 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   208 
   209 fun make_bool_args f g [] i = []
   210   | make_bool_args f g (x :: xs) i =
   211       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   212 
   213 fun make_bool_args' xs =
   214   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
   215 
   216 fun find_arg T x [] = sys_error "find_arg"
   217   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   218       apsnd (cons p) (find_arg T x ps)
   219   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   220       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   221       else apsnd (cons p) (find_arg T x ps);
   222 
   223 fun make_args Ts xs =
   224   map (fn (T, (NONE, ())) => Const ("arbitrary", T) | (_, (SOME t, ())) => t)
   225     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   226 
   227 fun make_args' Ts xs Us =
   228   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   229 
   230 fun dest_predicate cs params t =
   231   let
   232     val k = length params;
   233     val (c, ts) = strip_comb t;
   234     val (xs, ys) = chop k ts;
   235     val i = find_index_eq c cs;
   236   in
   237     if xs = params andalso i >= 0 then
   238       SOME (c, i, ys, chop (length ys)
   239         (List.drop (binder_types (fastype_of c), k)))
   240     else NONE
   241   end;
   242 
   243 fun mk_names a 0 = []
   244   | mk_names a 1 = [a]
   245   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   246 
   247 
   248 
   249 (** process rules **)
   250 
   251 local
   252 
   253 fun err_in_rule ctxt name t msg =
   254   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
   255     ProofContext.string_of_term ctxt t, msg]);
   256 
   257 fun err_in_prem ctxt name t p msg =
   258   error (cat_lines ["Ill-formed premise", ProofContext.string_of_term ctxt p,
   259     "in introduction rule " ^ quote name, ProofContext.string_of_term ctxt t, msg]);
   260 
   261 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   262 
   263 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   264 
   265 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   266 
   267 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
   268 
   269 in
   270 
   271 fun check_rule ctxt cs params ((name, att), rule) =
   272   let
   273     val params' = Term.variant_frees rule (Logic.strip_params rule);
   274     val frees = rev (map Free params');
   275     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   276     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   277     val rule' = Logic.list_implies (prems, concl);
   278     val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
   279     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
   280 
   281     fun check_ind err t = case dest_predicate cs params t of
   282         NONE => err (bad_app ^
   283           commas (map (ProofContext.string_of_term ctxt) params))
   284       | SOME (_, _, ys, _) =>
   285           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   286           then err bad_ind_occ else ();
   287 
   288     fun check_prem' prem t =
   289       if head_of t mem cs then
   290         check_ind (err_in_prem ctxt name rule prem) t
   291       else (case t of
   292           Abs (_, _, t) => check_prem' prem t
   293         | t $ u => (check_prem' prem t; check_prem' prem u)
   294         | _ => ());
   295 
   296     fun check_prem (prem, aprem) =
   297       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   298       else err_in_prem ctxt name rule prem "Non-atomic premise";
   299   in
   300     (case concl of
   301        Const ("Trueprop", _) $ t =>
   302          if head_of t mem cs then
   303            (check_ind (err_in_rule ctxt name rule') t;
   304             List.app check_prem (prems ~~ aprems))
   305          else err_in_rule ctxt name rule' bad_concl
   306      | _ => err_in_rule ctxt name rule' bad_concl);
   307     ((name, att), arule)
   308   end;
   309 
   310 val rulify =  (* FIXME norm_hhf *)
   311   hol_simplify inductive_conj
   312   #> hol_simplify inductive_rulify
   313   #> hol_simplify inductive_rulify_fallback
   314   (*#> standard*);
   315 
   316 end;
   317 
   318 
   319 
   320 (** proofs for (co)inductive predicates **)
   321 
   322 (* prove monotonicity -- NOT subject to quick_and_dirty! *)
   323 
   324 fun prove_mono predT fp_fun monos ctxt =
   325  (message "  Proving monotonicity ...";
   326   Goal.prove ctxt [] []   (*NO quick_and_dirty here!*)
   327     (HOLogic.mk_Trueprop
   328       (Const (mono_name, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   329     (fn _ => EVERY [rtac monoI 1,
   330       REPEAT (resolve_tac [le_funI, le_boolI'] 1),
   331       REPEAT (FIRST
   332         [atac 1,
   333          resolve_tac (List.concat (map mk_mono monos) @ get_monos ctxt) 1,
   334          etac le_funE 1, dtac le_boolD 1])]));
   335 
   336 
   337 (* prove introduction rules *)
   338 
   339 fun prove_intrs coind mono fp_def k params intr_ts rec_preds_defs ctxt =
   340   let
   341     val _ = clean_message "  Proving the introduction rules ...";
   342 
   343     val unfold = funpow k (fn th => th RS fun_cong)
   344       (mono RS (fp_def RS
   345         (if coind then def_gfp_unfold else def_lfp_unfold)));
   346 
   347     fun select_disj 1 1 = []
   348       | select_disj _ 1 = [rtac disjI1]
   349       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   350 
   351     val rules = [refl, TrueI, notFalseI, exI, conjI];
   352 
   353     val intrs = map_index (fn (i, intr) => rulify
   354       (SkipProof.prove ctxt (map (fst o dest_Free) params) [] intr (fn _ => EVERY
   355        [rewrite_goals_tac rec_preds_defs,
   356         rtac (unfold RS iffD2) 1,
   357         EVERY1 (select_disj (length intr_ts) (i + 1)),
   358         (*Not ares_tac, since refl must be tried before any equality assumptions;
   359           backtracking may occur if the premises have extra variables!*)
   360         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts
   361 
   362   in (intrs, unfold) end;
   363 
   364 
   365 (* prove elimination rules *)
   366 
   367 fun prove_elims cs params intr_ts intr_names unfold rec_preds_defs ctxt =
   368   let
   369     val _ = clean_message "  Proving the elimination rules ...";
   370 
   371     val ([pname], ctxt') = ctxt |>
   372       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   373       Variable.variant_fixes ["P"];
   374     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   375 
   376     fun dest_intr r =
   377       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   378        Logic.strip_assums_hyp r, Logic.strip_params r);
   379 
   380     val intrs = map dest_intr intr_ts ~~ intr_names;
   381 
   382     val rules1 = [disjE, exE, FalseE];
   383     val rules2 = [conjE, FalseE, notTrueE];
   384 
   385     fun prove_elim c =
   386       let
   387         val Ts = List.drop (binder_types (fastype_of c), length params);
   388         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   389         val frees = map Free (anames ~~ Ts);
   390 
   391         fun mk_elim_prem ((_, _, us, _), ts, params') =
   392           list_all (params',
   393             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   394               (frees ~~ us) @ ts, P));
   395         val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs);
   396         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   397            map mk_elim_prem (map #1 c_intrs)
   398       in
   399         (SkipProof.prove ctxt'' [] prems P
   400           (fn {prems, ...} => EVERY
   401             [cut_facts_tac [hd prems] 1,
   402              rewrite_goals_tac rec_preds_defs,
   403              dtac (unfold RS iffD1) 1,
   404              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   405              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   406              EVERY (map (fn prem =>
   407                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
   408           |> rulify
   409           |> singleton (ProofContext.export ctxt'' ctxt),
   410          map #2 c_intrs)
   411       end
   412 
   413    in map prove_elim cs end;
   414 
   415 
   416 (* derivation of simplified elimination rules *)
   417 
   418 local
   419 
   420 (*delete needless equality assumptions*)
   421 val refl_thin = Goal.prove_global HOL.thy [] []
   422   (Sign.read_prop HOL.thy "!!P. a = a ==> P ==> P")
   423   (fn _ => assume_tac 1);
   424 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   425 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   426 
   427 fun simp_case_tac ss i =
   428   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
   429 
   430 in
   431 
   432 fun mk_cases ctxt prop =
   433   let
   434     val thy = ProofContext.theory_of ctxt;
   435     val ss = Simplifier.local_simpset_of ctxt;
   436 
   437     fun err msg =
   438       error (Pretty.string_of (Pretty.block
   439         [Pretty.str msg, Pretty.fbrk, ProofContext.pretty_term ctxt prop]));
   440 
   441     val elims = InductAttrib.find_casesS ctxt prop;
   442 
   443     val cprop = Thm.cterm_of thy prop;
   444     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
   445     fun mk_elim rl =
   446       Thm.implies_intr cprop (Tactic.rule_by_tactic tac (Thm.assume cprop RS rl))
   447       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   448   in
   449     (case get_first (try mk_elim) elims of
   450       SOME r => r
   451     | NONE => err "Proposition not an inductive predicate:")
   452   end;
   453 
   454 end;
   455 
   456 
   457 (* inductive_cases *)
   458 
   459 fun gen_inductive_cases prep_att prep_prop args lthy =
   460   let
   461     val thy = ProofContext.theory_of lthy;
   462     val facts = args |> map (fn ((a, atts), props) =>
   463       ((a, map (prep_att thy) atts),
   464         map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
   465   in lthy |> note_theorems facts |>> map snd end;
   466 
   467 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
   468 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   469 
   470 
   471 fun ind_cases src = Method.syntax (Scan.lift (Scan.repeat1 Args.name --
   472     Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.name) [])) src
   473   #> (fn ((raw_props, fixes), ctxt) =>
   474     let
   475       val (_, ctxt') = Variable.add_fixes fixes ctxt;
   476       val props = Syntax.read_props ctxt' raw_props;
   477       val ctxt'' = fold Variable.declare_term props ctxt';
   478       val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props)
   479     in Method.erule 0 rules end);
   480 
   481 
   482 
   483 (* prove induction rule *)
   484 
   485 fun prove_indrule cs argTs bs xs rec_const params intr_ts mono
   486     fp_def rec_preds_defs ctxt =
   487   let
   488     val _ = clean_message "  Proving the induction rule ...";
   489     val thy = ProofContext.theory_of ctxt;
   490 
   491     (* predicates for induction rule *)
   492 
   493     val (pnames, ctxt') = ctxt |>
   494       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   495       Variable.variant_fixes (mk_names "P" (length cs));
   496     val preds = map Free (pnames ~~
   497       map (fn c => List.drop (binder_types (fastype_of c), length params) --->
   498         HOLogic.boolT) cs);
   499 
   500     (* transform an introduction rule into a premise for induction rule *)
   501 
   502     fun mk_ind_prem r =
   503       let
   504         fun subst s = (case dest_predicate cs params s of
   505             SOME (_, i, ys, (_, Ts)) =>
   506               let
   507                 val k = length Ts;
   508                 val bs = map Bound (k - 1 downto 0);
   509                 val P = list_comb (List.nth (preds, i),
   510                   map (incr_boundvars k) ys @ bs);
   511                 val Q = list_abs (mk_names "x" k ~~ Ts,
   512                   HOLogic.mk_binop inductive_conj_name
   513                     (list_comb (incr_boundvars k s, bs), P))
   514               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   515           | NONE => (case s of
   516               (t $ u) => (fst (subst t) $ fst (subst u), NONE)
   517             | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
   518             | _ => (s, NONE)));
   519 
   520         fun mk_prem (s, prems) = (case subst s of
   521               (_, SOME (t, u)) => t :: u :: prems
   522             | (t, _) => t :: prems);
   523 
   524         val SOME (_, i, ys, _) = dest_predicate cs params
   525           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
   526 
   527       in list_all_free (Logic.strip_params r,
   528         Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
   529           [] (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r))),
   530             HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
   531       end;
   532 
   533     val ind_prems = map mk_ind_prem intr_ts;
   534 
   535 
   536     (* make conclusions for induction rules *)
   537 
   538     val Tss = map (binder_types o fastype_of) preds;
   539     val (xnames, ctxt'') =
   540       Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   541     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   542         (map (fn (((xnames, Ts), c), P) =>
   543            let val frees = map Free (xnames ~~ Ts)
   544            in HOLogic.mk_imp
   545              (list_comb (c, params @ frees), list_comb (P, frees))
   546            end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   547 
   548 
   549     (* make predicate for instantiation of abstract induction rule *)
   550 
   551     val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   552       (map_index (fn (i, P) => foldr HOLogic.mk_imp
   553          (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))
   554          (make_bool_args HOLogic.mk_not I bs i)) preds));
   555 
   556     val ind_concl = HOLogic.mk_Trueprop
   557       (HOLogic.mk_binrel "HOL.ord_class.less_eq" (rec_const, ind_pred));
   558 
   559     val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));
   560 
   561     val induct = SkipProof.prove ctxt'' [] ind_prems ind_concl
   562       (fn {prems, ...} => EVERY
   563         [rewrite_goals_tac [inductive_conj_def],
   564          DETERM (rtac raw_fp_induct 1),
   565          REPEAT (resolve_tac [le_funI, le_boolI] 1),
   566          rewrite_goals_tac (inf_fun_eq :: inf_bool_eq :: simp_thms'),
   567          (*This disjE separates out the introduction rules*)
   568          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
   569          (*Now break down the individual cases.  No disjE here in case
   570            some premise involves disjunction.*)
   571          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
   572          REPEAT (FIRSTGOAL
   573            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
   574          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
   575              (inductive_conj_def :: rec_preds_defs @ simp_thms') prem,
   576            conjI, refl] 1)) prems)]);
   577 
   578     val lemma = SkipProof.prove ctxt'' [] []
   579       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
   580         [rewrite_goals_tac rec_preds_defs,
   581          REPEAT (EVERY
   582            [REPEAT (resolve_tac [conjI, impI] 1),
   583             REPEAT (eresolve_tac [le_funE, le_boolE] 1),
   584             atac 1,
   585             rewrite_goals_tac simp_thms',
   586             atac 1])])
   587 
   588   in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end;
   589 
   590 
   591 
   592 (** specification of (co)inductive predicates **)
   593 
   594 fun mk_ind_def alt_name coind cs intr_ts monos
   595       params cnames_syn ctxt =
   596   let
   597     val fp_name = if coind then gfp_name else lfp_name;
   598 
   599     val argTs = fold (fn c => fn Ts => Ts @
   600       (List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs [];
   601     val k = log 2 1 (length cs);
   602     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   603     val p :: xs = map Free (Variable.variant_frees ctxt intr_ts
   604       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   605     val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts)
   606       (map (rpair HOLogic.boolT) (mk_names "b" k)));
   607 
   608     fun subst t = (case dest_predicate cs params t of
   609         SOME (_, i, ts, (Ts, Us)) =>
   610           let
   611             val l = length Us;
   612             val zs = map Bound (l - 1 downto 0)
   613           in
   614             list_abs (map (pair "z") Us, list_comb (p,
   615               make_bool_args' bs i @ make_args argTs
   616                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   617           end
   618       | NONE => (case t of
   619           t1 $ t2 => subst t1 $ subst t2
   620         | Abs (x, T, u) => Abs (x, T, subst u)
   621         | _ => t));
   622 
   623     (* transform an introduction rule into a conjunction  *)
   624     (*   [| p_i t; ... |] ==> p_j u                       *)
   625     (* is transformed into                                *)
   626     (*   b_j & x_j = u & p b_j t & ...                    *)
   627 
   628     fun transform_rule r =
   629       let
   630         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
   631           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   632         val ps = make_bool_args HOLogic.mk_not I bs i @
   633           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   634           map (subst o HOLogic.dest_Trueprop)
   635             (Logic.strip_assums_hyp r)
   636       in foldr (fn ((x, T), P) => HOLogic.exists_const T $ (Abs (x, T, P)))
   637         (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
   638         (Logic.strip_params r)
   639       end
   640 
   641     (* make a disjunction of all introduction rules *)
   642 
   643     val fp_fun = fold_rev lambda (p :: bs @ xs)
   644       (if null intr_ts then HOLogic.false_const
   645        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   646 
   647     (* add definiton of recursive predicates to theory *)
   648 
   649     val rec_name = if alt_name = "" then
   650       space_implode "_" (map fst cnames_syn) else alt_name;
   651 
   652     val ((rec_const, (_, fp_def)), ctxt') = ctxt |>
   653       LocalTheory.def Thm.internalK
   654         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   655          (("", []), fold_rev lambda params
   656            (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)));
   657     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
   658       (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params)));
   659     val specs = if length cs < 2 then [] else
   660       map_index (fn (i, (name_mx, c)) =>
   661         let
   662           val Ts = List.drop (binder_types (fastype_of c), length params);
   663           val xs = map Free (Variable.variant_frees ctxt intr_ts
   664             (mk_names "x" (length Ts) ~~ Ts))
   665         in
   666           (name_mx, (("", []), fold_rev lambda (params @ xs)
   667             (list_comb (rec_const, params @ make_bool_args' bs i @
   668               make_args argTs (xs ~~ Ts)))))
   669         end) (cnames_syn ~~ cs);
   670     val (consts_defs, ctxt'') = LocalTheory.defs Thm.internalK specs ctxt';
   671     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   672 
   673     val mono = prove_mono predT fp_fun monos ctxt''
   674 
   675   in (ctxt'', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
   676     list_comb (rec_const, params), preds, argTs, bs, xs)
   677   end;
   678 
   679 fun declare_rules rec_name coind no_ind cnames intrs intr_names intr_atts
   680       elims raw_induct ctxt =
   681   let
   682     val ind_case_names = RuleCases.case_names intr_names;
   683     val induct =
   684       if coind then
   685         (raw_induct, [RuleCases.case_names [rec_name],
   686           RuleCases.case_conclusion (rec_name, intr_names),
   687           RuleCases.consumes 1, InductAttrib.coinduct_set (hd cnames)])
   688       else if no_ind orelse length cnames > 1 then
   689         (raw_induct, [ind_case_names, RuleCases.consumes 0])
   690       else (raw_induct RSN (2, rev_mp), [ind_case_names, RuleCases.consumes 1]);
   691 
   692     val (intrs', ctxt1) =
   693       ctxt |>
   694       note_theorems
   695         (map (NameSpace.qualified rec_name) intr_names ~~
   696          intr_atts ~~ map (fn th => [([th],
   697            [Attrib.internal (K (ContextRules.intro_query NONE))])]) intrs) |>>
   698       map (hd o snd); (* FIXME? *)
   699     val (((_, elims'), (_, [induct'])), ctxt2) =
   700       ctxt1 |>
   701       note_theorem ((NameSpace.qualified rec_name "intros", []), intrs') ||>>
   702       fold_map (fn (name, (elim, cases)) =>
   703         note_theorem ((NameSpace.qualified (Sign.base_name name) "cases",
   704           [Attrib.internal (K (RuleCases.case_names cases)),
   705            Attrib.internal (K (RuleCases.consumes 1)),
   706            Attrib.internal (K (InductAttrib.cases_set name)),
   707            Attrib.internal (K (ContextRules.elim_query NONE))]), [elim]) #>
   708         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   709       note_theorem ((NameSpace.qualified rec_name (coind_prefix coind ^ "induct"),
   710         map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
   711 
   712     val ctxt3 = if no_ind orelse coind then ctxt2 else
   713       let val inducts = cnames ~~ ProjectRule.projects ctxt2 (1 upto length cnames) induct'
   714       in
   715         ctxt2 |>
   716         note_theorems [((NameSpace.qualified rec_name "inducts", []),
   717           inducts |> map (fn (name, th) => ([th],
   718             [Attrib.internal (K ind_case_names),
   719              Attrib.internal (K (RuleCases.consumes 1)),
   720              Attrib.internal (K (InductAttrib.induct_set name))])))] |> snd
   721       end
   722   in (intrs', elims', induct', ctxt3) end;
   723 
   724 type add_ind_def = bool -> bstring -> bool -> bool -> bool ->
   725   term list -> ((string * Attrib.src list) * term) list -> thm list ->
   726   term list -> (string * mixfix) list ->
   727   local_theory -> inductive_result * local_theory
   728 
   729 fun add_ind_def verbose alt_name coind no_elim no_ind cs
   730     intros monos params cnames_syn ctxt =
   731   let
   732     val _ =
   733       if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^
   734         commas_quote (map fst cnames_syn)) else ();
   735 
   736     val cnames = map (Sign.full_name (ProofContext.theory_of ctxt) o #1) cnames_syn;  (* FIXME *)
   737     val ((intr_names, intr_atts), intr_ts) =
   738       apfst split_list (split_list (map (check_rule ctxt cs params) intros));
   739 
   740     val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   741       argTs, bs, xs) = mk_ind_def alt_name coind cs intr_ts
   742         monos params cnames_syn ctxt;
   743 
   744     val (intrs, unfold) = prove_intrs coind mono fp_def (length bs + length xs)
   745       params intr_ts rec_preds_defs ctxt1;
   746     val elims = if no_elim then [] else
   747       prove_elims cs params intr_ts intr_names unfold rec_preds_defs ctxt1;
   748     val raw_induct = zero_var_indexes
   749       (if no_ind then Drule.asm_rl else
   750        if coind then
   751          singleton (ProofContext.export
   752            (snd (Variable.add_fixes (map (fst o dest_Free) params) ctxt1)) ctxt1)
   753            (rotate_prems ~1 (ObjectLogic.rulify (rule_by_tactic
   754              (rewrite_tac [le_fun_def, le_bool_def, sup_fun_eq, sup_bool_eq] THEN
   755                fold_tac rec_preds_defs) (mono RS (fp_def RS def_coinduct)))))
   756        else
   757          prove_indrule cs argTs bs xs rec_const params intr_ts mono fp_def
   758            rec_preds_defs ctxt1);
   759 
   760     val (intrs', elims', induct, ctxt2) = declare_rules rec_name coind no_ind
   761       cnames intrs intr_names intr_atts elims raw_induct ctxt1;
   762 
   763     val names = map #1 cnames_syn;
   764     val result =
   765       {preds = preds,
   766        intrs = intrs',
   767        elims = elims',
   768        raw_induct = rulify raw_induct,
   769        induct = induct};
   770 
   771     val ctxt3 = ctxt2
   772       |> Context.proof_map (put_inductives names ({names = names, coind = coind}, result))
   773       |> LocalTheory.declaration (fn phi =>
   774         let
   775           val names' = map (LocalTheory.target_name ctxt2 o Morphism.name phi) names;
   776           val result' = morph_result phi result;
   777         in put_inductives names' ({names = names', coind = coind}, result') end);
   778   in (result, ctxt3) end;
   779 
   780 
   781 (* external interfaces *)
   782 
   783 fun gen_add_inductive_i mk_def verbose alt_name coind no_elim no_ind
   784     cnames_syn pnames pre_intros monos ctxt =
   785   let
   786     val thy = ProofContext.theory_of ctxt;
   787     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   788 
   789     val frees = fold (Term.add_frees o snd) pre_intros [];
   790     fun type_of s = (case AList.lookup op = frees s of
   791       NONE => error ("No such variable: " ^ s) | SOME T => T);
   792 
   793     fun is_abbrev ((name, atts), t) =
   794       can (Logic.strip_assums_concl #> Logic.dest_equals) t andalso
   795       (name = "" andalso null atts orelse
   796        error "Abbreviations may not have names or attributes");
   797 
   798     fun expand_atom tab (t as Free xT) =
   799           the_default t (AList.lookup op = tab xT)
   800       | expand_atom tab t = t;
   801     fun expand [] r = r
   802       | expand tab r = Envir.beta_norm (Term.map_aterms (expand_atom tab) r);
   803 
   804     val (_, ctxt') = Variable.add_fixes (map #1 cnames_syn) ctxt;
   805 
   806     fun prep_abbrevs [] abbrevs' abbrevs'' = (rev abbrevs', rev abbrevs'')
   807       | prep_abbrevs ((_, abbrev) :: abbrevs) abbrevs' abbrevs'' =
   808           let val ((s, T), t) =
   809             LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt' abbrev))
   810           in case find_first (equal s o #1) cnames_syn of
   811               NONE => error ("Head of abbreviation " ^ quote s ^ " undeclared")
   812             | SOME (_, _, mx) => prep_abbrevs abbrevs
   813                 (((s, T), expand abbrevs' t) :: abbrevs')
   814                 (((s, mx), expand abbrevs' t) :: abbrevs'') (* FIXME: do not expand *)
   815           end;
   816 
   817     val (abbrevs, pre_intros') = List.partition is_abbrev pre_intros;
   818     val (abbrevs', abbrevs'') = prep_abbrevs abbrevs [] [];
   819     val _ = (case gen_inter (op = o apsnd fst)
   820       (fold (Term.add_frees o snd) abbrevs' [], abbrevs') of
   821         [] => ()
   822       | xs => error ("Bad abbreviation(s): " ^ commas (map fst xs)));
   823 
   824     val params = map
   825       (fn (s, SOME T) => Free (s, T) | (s, NONE) => Free (s, type_of s)) pnames;
   826     val cnames_syn' = filter_out (fn (s, _, _) =>
   827       exists (equal s o fst o fst) abbrevs') cnames_syn;
   828     val cs = map
   829       (fn (s, SOME T, _) => Free (s, T) | (s, NONE, _) => Free (s, type_of s)) cnames_syn';
   830     val cnames_syn'' = map (fn (s, _, mx) => (s, mx)) cnames_syn';
   831 
   832     fun close_rule (x, r) = (x, list_all_free (rev (fold_aterms
   833       (fn t as Free (v as (s, _)) =>
   834             if Variable.is_fixed ctxt s orelse member op = cs t orelse
   835               member op = params t then I else insert op = v
   836         | _ => I) r []), r));
   837 
   838     val intros = map (apsnd (expand abbrevs') #> close_rule) pre_intros';
   839   in
   840     ctxt |>
   841     mk_def verbose alt_name coind no_elim no_ind cs intros monos
   842       params cnames_syn'' ||>
   843     fold (snd oo LocalTheory.abbrev Syntax.default_mode) abbrevs''
   844   end;
   845 
   846 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos ctxt =
   847   let
   848     val (_, ctxt') = Specification.read_specification (cnames_syn @ pnames_syn) [] ctxt;
   849     val intrs = map (fn ((name, att), s) => apsnd hd (hd (snd (fst
   850       (Specification.read_specification [] [((name, att), [s])] ctxt'))))
   851       handle ERROR msg =>
   852         cat_error msg ("The error(s) above occurred for\n" ^
   853           (if name = "" then "" else name ^ ": ") ^ s)) intro_srcs;
   854     val pnames = map (fn (s, _, _) =>
   855       (s, SOME (ProofContext.infer_type ctxt' s))) pnames_syn;
   856     val cnames_syn' = map (fn (s, _, mx) =>
   857       (s, SOME (ProofContext.infer_type ctxt' s), mx)) cnames_syn;
   858     val (monos, ctxt'') = LocalTheory.theory_result (IsarCmd.apply_theorems raw_monos) ctxt;
   859   in
   860     gen_add_inductive_i mk_def verbose "" coind false false cnames_syn' pnames intrs monos ctxt''
   861   end;
   862 
   863 val add_inductive_i = gen_add_inductive_i add_ind_def;
   864 val add_inductive = gen_add_inductive add_ind_def;
   865 
   866 fun add_inductive_global verbose alt_name coind no_elim no_ind cnames_syn pnames pre_intros monos =
   867   TheoryTarget.init NONE #>
   868   add_inductive_i verbose alt_name coind no_elim no_ind cnames_syn pnames pre_intros monos #>
   869   (fn (_, lthy) =>
   870     (#2 (the_inductive (LocalTheory.target_of lthy)
   871       (LocalTheory.target_name lthy (#1 (hd cnames_syn)))),
   872     ProofContext.theory_of (LocalTheory.exit lthy)));
   873 
   874 
   875 (* read off arities of inductive predicates from raw induction rule *)
   876 fun arities_of induct =
   877   map (fn (_ $ t $ u) =>
   878       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
   879     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
   880 
   881 (* read off parameters of inductive predicate from raw induction rule *)
   882 fun params_of induct =
   883   let
   884     val (_ $ t $ u :: _) =
   885       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
   886     val (_, ts) = strip_comb t;
   887     val (_, us) = strip_comb u
   888   in
   889     List.take (ts, length ts - length us)
   890   end;
   891 
   892 val pname_of_intr =
   893   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
   894 
   895 (* partition introduction rules according to predicate name *)
   896 fun partition_rules induct intros =
   897   fold_rev (fn r => AList.map_entry op = (pname_of_intr r) (cons r)) intros
   898     (map (rpair [] o fst) (arities_of induct));
   899 
   900 fun unpartition_rules intros xs =
   901   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
   902     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
   903 
   904 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
   905 fun infer_intro_vars elim arity intros =
   906   let
   907     val thy = theory_of_thm elim;
   908     val _ :: cases = prems_of elim;
   909     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
   910     fun mtch (t, u) =
   911       let
   912         val params = Logic.strip_params t;
   913         val vars = map (Var o apfst (rpair 0))
   914           (Name.variant_list used (map fst params) ~~ map snd params);
   915         val ts = map (curry subst_bounds (rev vars))
   916           (List.drop (Logic.strip_assums_hyp t, arity));
   917         val us = Logic.strip_imp_prems u;
   918         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
   919           (Vartab.empty, Vartab.empty);
   920       in
   921         map (Envir.subst_vars tab) vars
   922       end
   923   in
   924     map (mtch o apsnd prop_of) (cases ~~ intros)
   925   end;
   926 
   927 
   928 (** package setup **)
   929 
   930 (* setup theory *)
   931 
   932 val setup =
   933   Method.add_methods [("ind_cases", ind_cases,
   934     "dynamic case analysis on predicates")] #>
   935   Attrib.add_attributes [("mono", Attrib.add_del_args mono_add mono_del,
   936     "declaration of monotonicity rule")];
   937 
   938 
   939 (* outer syntax *)
   940 
   941 local structure P = OuterParse and K = OuterKeyword in
   942 
   943 (* FIXME tmp *)
   944 fun flatten_specification specs = specs |> maps
   945   (fn (a, (concl, [])) => concl |> map
   946         (fn ((b, atts), [B]) =>
   947               if a = "" then ((b, atts), B)
   948               else if b = "" then ((a, atts), B)
   949               else error ("Illegal nested case names " ^ quote (NameSpace.append a b))
   950           | ((b, _), _) => error ("Illegal simultaneous specification " ^ quote b))
   951     | (a, _) => error ("Illegal local specification parameters for " ^ quote a));
   952 
   953 fun gen_ind_decl mk_def coind =
   954   P.opt_target --
   955   P.fixes -- P.for_fixes --
   956   Scan.optional (P.$$$ "where" |-- P.!!! SpecParse.specification) [] --
   957   Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.xthms1) []
   958   >> (fn ((((loc, preds), params), specs), monos) =>
   959     Toplevel.local_theory loc
   960       (fn lthy => lthy |> gen_add_inductive mk_def true coind preds params
   961          (flatten_specification specs) monos |> snd));
   962 
   963 val ind_decl = gen_ind_decl add_ind_def;
   964 
   965 val inductiveP =
   966   OuterSyntax.command "inductive" "define inductive predicates" K.thy_decl (ind_decl false);
   967 
   968 val coinductiveP =
   969   OuterSyntax.command "coinductive" "define coinductive predicates" K.thy_decl (ind_decl true);
   970 
   971 
   972 val inductive_casesP =
   973   OuterSyntax.command "inductive_cases"
   974     "create simplified instances of elimination rules (improper)" K.thy_script
   975     (P.opt_target -- P.and_list1 SpecParse.spec
   976       >> (fn (loc, specs) => Toplevel.local_theory loc (snd o inductive_cases specs)));
   977 
   978 val _ = OuterSyntax.add_keywords ["monos"];
   979 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
   980 
   981 end;
   982 
   983 end;