src/HOL/Tools/inductive.ML
author Philipp Meyer
Wed Sep 30 11:33:59 2009 +0200 (2009-09-30)
changeset 32866 1238cbb7c08f
parent 32773 f6fd4ccd8eea
child 32952 aeb1e44fbc19
permissions -rw-r--r--
atp_minimal using chain_ths again
     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   val morph_result: morphism -> inductive_result -> inductive_result
    25   type inductive_info
    26   val the_inductive: Proof.context -> string -> inductive_info
    27   val print_inductives: Proof.context -> unit
    28   val mono_add: attribute
    29   val mono_del: attribute
    30   val get_monos: Proof.context -> thm list
    31   val mk_cases: Proof.context -> term -> thm
    32   val inductive_forall_name: string
    33   val inductive_forall_def: thm
    34   val rulify: thm -> thm
    35   val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
    36     thm list list * local_theory
    37   val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
    38     thm list list * local_theory
    39   type inductive_flags
    40   val add_inductive_i:
    41     inductive_flags -> ((binding * typ) * mixfix) list ->
    42     (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
    43     inductive_result * local_theory
    44   val add_inductive: bool -> bool ->
    45     (binding * string option * mixfix) list ->
    46     (binding * string option * mixfix) list ->
    47     (Attrib.binding * string) list ->
    48     (Facts.ref * Attrib.src list) list ->
    49     bool -> local_theory -> inductive_result * local_theory
    50   val add_inductive_global: string -> inductive_flags ->
    51     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    52     thm list -> theory -> inductive_result * theory
    53   val arities_of: thm -> (string * int) list
    54   val params_of: thm -> term list
    55   val partition_rules: thm -> thm list -> (string * thm list) list
    56   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
    57   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    58   val infer_intro_vars: thm -> int -> thm list -> term list list
    59   val setup: theory -> theory
    60 end;
    61 
    62 signature INDUCTIVE =
    63 sig
    64   include BASIC_INDUCTIVE
    65   type add_ind_def
    66   val declare_rules: string -> binding -> bool -> bool -> string list ->
    67     thm list -> binding list -> Attrib.src list list -> (thm * string list) list ->
    68     thm -> local_theory -> thm list * thm list * thm * local_theory
    69   val add_ind_def: add_ind_def
    70   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
    71     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    72     thm list -> local_theory -> inductive_result * local_theory
    73   val gen_add_inductive: add_ind_def -> bool -> bool ->
    74     (binding * string option * mixfix) list ->
    75     (binding * string option * mixfix) list ->
    76     (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
    77     bool -> local_theory -> inductive_result * local_theory
    78   val gen_ind_decl: add_ind_def -> bool ->
    79     OuterParse.token list -> (bool -> local_theory -> local_theory) * OuterParse.token list
    80 end;
    81 
    82 structure Inductive: INDUCTIVE =
    83 struct
    84 
    85 
    86 (** theory context references **)
    87 
    88 val inductive_forall_name = "HOL.induct_forall";
    89 val inductive_forall_def = @{thm induct_forall_def};
    90 val inductive_conj_name = "HOL.induct_conj";
    91 val inductive_conj_def = @{thm induct_conj_def};
    92 val inductive_conj = @{thms induct_conj};
    93 val inductive_atomize = @{thms induct_atomize};
    94 val inductive_rulify = @{thms induct_rulify};
    95 val inductive_rulify_fallback = @{thms induct_rulify_fallback};
    96 
    97 val notTrueE = TrueI RSN (2, notE);
    98 val notFalseI = Seq.hd (atac 1 notI);
    99 
   100 val simp_thms' = map mk_meta_eq
   101   @{lemma "(~True) = False" "(~False) = True"
   102       "(True --> P) = P" "(False --> P) = True"
   103       "(P & True) = P" "(True & P) = P"
   104     by (fact simp_thms)+};
   105 
   106 val simp_thms'' = map mk_meta_eq [@{thm inf_fun_eq}, @{thm inf_bool_eq}] @ simp_thms';
   107 
   108 val simp_thms''' = map mk_meta_eq
   109   [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_eq}, @{thm sup_bool_eq}];
   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 (Display.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
   161   (apfst (fold (fn name => Symtab.update (name, info)) names));
   162 
   163 
   164 
   165 (** monotonicity rules **)
   166 
   167 val get_monos = #2 o get_inductives;
   168 val map_monos = InductiveData.map o apsnd;
   169 
   170 fun mk_mono thm =
   171   let
   172     val concl = concl_of thm;
   173     fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @
   174       (case concl of
   175           (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
   176         | _ => [thm' RS (thm' RS eq_to_mono2)]);
   177     fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
   178       handle THM _ => thm RS @{thm le_boolD}
   179   in
   180     case concl of
   181       Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq)
   182     | _ $ (Const ("op =", _) $ _ $ _) => eq2mono thm
   183     | _ $ (Const (@{const_name HOL.less_eq}, _) $ _ $ _) =>
   184       [dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   185          (resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))]
   186     | _ => [thm]
   187   end handle THM _ =>
   188     error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm_without_context 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 fun message quiet_mode s = if quiet_mode then () else writeln s;
   198 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
   199 
   200 fun coind_prefix true = "co"
   201   | coind_prefix false = "";
   202 
   203 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   204 
   205 fun make_bool_args f g [] i = []
   206   | make_bool_args f g (x :: xs) i =
   207       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   208 
   209 fun make_bool_args' xs =
   210   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
   211 
   212 fun find_arg T x [] = sys_error "find_arg"
   213   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   214       apsnd (cons p) (find_arg T x ps)
   215   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   216       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   217       else apsnd (cons p) (find_arg T x ps);
   218 
   219 fun make_args Ts xs =
   220   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
   221     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   222 
   223 fun make_args' Ts xs Us =
   224   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   225 
   226 fun dest_predicate cs params t =
   227   let
   228     val k = length params;
   229     val (c, ts) = strip_comb t;
   230     val (xs, ys) = chop k ts;
   231     val i = find_index (fn c' => c' = c) cs;
   232   in
   233     if xs = params andalso i >= 0 then
   234       SOME (c, i, ys, chop (length ys)
   235         (List.drop (binder_types (fastype_of c), k)))
   236     else NONE
   237   end;
   238 
   239 fun mk_names a 0 = []
   240   | mk_names a 1 = [a]
   241   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   242 
   243 
   244 
   245 (** process rules **)
   246 
   247 local
   248 
   249 fun err_in_rule ctxt name t msg =
   250   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
   251     Syntax.string_of_term ctxt t, msg]);
   252 
   253 fun err_in_prem ctxt name t p msg =
   254   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
   255     "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
   256 
   257 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   258 
   259 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   260 
   261 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   262 
   263 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
   264 
   265 in
   266 
   267 fun check_rule ctxt cs params ((binding, att), rule) =
   268   let
   269     val err_name = Binding.str_of binding;
   270     val params' = Term.variant_frees rule (Logic.strip_params rule);
   271     val frees = rev (map Free params');
   272     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   273     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   274     val rule' = Logic.list_implies (prems, concl);
   275     val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
   276     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
   277 
   278     fun check_ind err t = case dest_predicate cs params t of
   279         NONE => err (bad_app ^
   280           commas (map (Syntax.string_of_term ctxt) params))
   281       | SOME (_, _, ys, _) =>
   282           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   283           then err bad_ind_occ else ();
   284 
   285     fun check_prem' prem t =
   286       if head_of t mem cs then
   287         check_ind (err_in_prem ctxt err_name rule prem) t
   288       else (case t of
   289           Abs (_, _, t) => check_prem' prem t
   290         | t $ u => (check_prem' prem t; check_prem' prem u)
   291         | _ => ());
   292 
   293     fun check_prem (prem, aprem) =
   294       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   295       else err_in_prem ctxt err_name rule prem "Non-atomic premise";
   296   in
   297     (case concl of
   298        Const ("Trueprop", _) $ t =>
   299          if head_of t mem cs then
   300            (check_ind (err_in_rule ctxt err_name rule') t;
   301             List.app check_prem (prems ~~ aprems))
   302          else err_in_rule ctxt err_name rule' bad_concl
   303      | _ => err_in_rule ctxt err_name rule' bad_concl);
   304     ((binding, att), arule)
   305   end;
   306 
   307 val rulify =
   308   hol_simplify inductive_conj
   309   #> hol_simplify inductive_rulify
   310   #> hol_simplify inductive_rulify_fallback
   311   #> Simplifier.norm_hhf;
   312 
   313 end;
   314 
   315 
   316 
   317 (** proofs for (co)inductive predicates **)
   318 
   319 (* prove monotonicity *)
   320 
   321 fun prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt =
   322  (message (quiet_mode orelse skip_mono andalso !quick_and_dirty orelse fork_mono)
   323     "  Proving monotonicity ...";
   324   (if skip_mono then SkipProof.prove else if fork_mono then Goal.prove_future else Goal.prove) ctxt
   325     [] []
   326     (HOLogic.mk_Trueprop
   327       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   328     (fn _ => EVERY [rtac @{thm monoI} 1,
   329       REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
   330       REPEAT (FIRST
   331         [atac 1,
   332          resolve_tac (List.concat (map mk_mono monos) @ get_monos ctxt) 1,
   333          etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
   334 
   335 
   336 (* prove introduction rules *)
   337 
   338 fun prove_intrs quiet_mode coind mono fp_def k params intr_ts rec_preds_defs ctxt =
   339   let
   340     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
   341 
   342     val unfold = funpow k (fn th => th RS fun_cong)
   343       (mono RS (fp_def RS
   344         (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
   345 
   346     fun select_disj 1 1 = []
   347       | select_disj _ 1 = [rtac disjI1]
   348       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   349 
   350     val rules = [refl, TrueI, notFalseI, exI, conjI];
   351 
   352     val intrs = map_index (fn (i, intr) => rulify
   353       (SkipProof.prove ctxt (map (fst o dest_Free) params) [] intr (fn _ => EVERY
   354        [rewrite_goals_tac rec_preds_defs,
   355         rtac (unfold RS iffD2) 1,
   356         EVERY1 (select_disj (length intr_ts) (i + 1)),
   357         (*Not ares_tac, since refl must be tried before any equality assumptions;
   358           backtracking may occur if the premises have extra variables!*)
   359         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts
   360 
   361   in (intrs, unfold) end;
   362 
   363 
   364 (* prove elimination rules *)
   365 
   366 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt =
   367   let
   368     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
   369 
   370     val ([pname], ctxt') = ctxt |>
   371       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   372       Variable.variant_fixes ["P"];
   373     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   374 
   375     fun dest_intr r =
   376       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   377        Logic.strip_assums_hyp r, Logic.strip_params r);
   378 
   379     val intrs = map dest_intr intr_ts ~~ intr_names;
   380 
   381     val rules1 = [disjE, exE, FalseE];
   382     val rules2 = [conjE, FalseE, notTrueE];
   383 
   384     fun prove_elim c =
   385       let
   386         val Ts = List.drop (binder_types (fastype_of c), length params);
   387         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   388         val frees = map Free (anames ~~ Ts);
   389 
   390         fun mk_elim_prem ((_, _, us, _), ts, params') =
   391           list_all (params',
   392             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   393               (frees ~~ us) @ ts, P));
   394         val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs);
   395         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   396            map mk_elim_prem (map #1 c_intrs)
   397       in
   398         (SkipProof.prove ctxt'' [] prems P
   399           (fn {prems, ...} => EVERY
   400             [cut_facts_tac [hd prems] 1,
   401              rewrite_goals_tac rec_preds_defs,
   402              dtac (unfold RS iffD1) 1,
   403              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   404              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   405              EVERY (map (fn prem =>
   406                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
   407           |> rulify
   408           |> singleton (ProofContext.export ctxt'' ctxt),
   409          map #2 c_intrs)
   410       end
   411 
   412    in map prove_elim cs end;
   413 
   414 
   415 (* derivation of simplified elimination rules *)
   416 
   417 local
   418 
   419 (*delete needless equality assumptions*)
   420 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
   421   (fn _ => assume_tac 1);
   422 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   423 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   424 
   425 fun simp_case_tac ss i =
   426   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
   427 
   428 in
   429 
   430 fun mk_cases ctxt prop =
   431   let
   432     val thy = ProofContext.theory_of ctxt;
   433     val ss = simpset_of ctxt;
   434 
   435     fun err msg =
   436       error (Pretty.string_of (Pretty.block
   437         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
   438 
   439     val elims = Induct.find_casesP ctxt prop;
   440 
   441     val cprop = Thm.cterm_of thy prop;
   442     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
   443     fun mk_elim rl =
   444       Thm.implies_intr cprop (Tactic.rule_by_tactic tac (Thm.assume cprop RS rl))
   445       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   446   in
   447     (case get_first (try mk_elim) elims of
   448       SOME r => r
   449     | NONE => err "Proposition not an inductive predicate:")
   450   end;
   451 
   452 end;
   453 
   454 
   455 (* inductive_cases *)
   456 
   457 fun gen_inductive_cases prep_att prep_prop args lthy =
   458   let
   459     val thy = ProofContext.theory_of lthy;
   460     val facts = args |> map (fn ((a, atts), props) =>
   461       ((a, map (prep_att thy) atts),
   462         map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
   463   in lthy |> LocalTheory.notes Thm.generatedK facts |>> map snd end;
   464 
   465 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
   466 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   467 
   468 
   469 val ind_cases_setup =
   470   Method.setup @{binding ind_cases}
   471     (Scan.lift (Scan.repeat1 Args.name_source --
   472       Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.name) []) >>
   473       (fn (raw_props, fixes) => fn 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     "dynamic case analysis on predicates";
   481 
   482 
   483 (* prove induction rule *)
   484 
   485 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
   486     fp_def rec_preds_defs ctxt =
   487   let
   488     val _ = clean_message quiet_mode "  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 (List.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) => List.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 @{thm 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 [@{thm le_funI}, @{thm le_boolI}] 1),
   566          rewrite_goals_tac 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 [@{thm le_funE}, @{thm 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 quiet_mode skip_mono fork_mono alt_name coind cs intr_ts monos params cnames_syn ctxt =
   595   let
   596     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
   597 
   598     val argTs = fold (fn c => fn Ts => Ts @
   599       (List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs [];
   600     val k = log 2 1 (length cs);
   601     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   602     val p :: xs = map Free (Variable.variant_frees ctxt intr_ts
   603       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   604     val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts)
   605       (map (rpair HOLogic.boolT) (mk_names "b" k)));
   606 
   607     fun subst t = (case dest_predicate cs params t of
   608         SOME (_, i, ts, (Ts, Us)) =>
   609           let
   610             val l = length Us;
   611             val zs = map Bound (l - 1 downto 0)
   612           in
   613             list_abs (map (pair "z") Us, list_comb (p,
   614               make_bool_args' bs i @ make_args argTs
   615                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   616           end
   617       | NONE => (case t of
   618           t1 $ t2 => subst t1 $ subst t2
   619         | Abs (x, T, u) => Abs (x, T, subst u)
   620         | _ => t));
   621 
   622     (* transform an introduction rule into a conjunction  *)
   623     (*   [| p_i t; ... |] ==> p_j u                       *)
   624     (* is transformed into                                *)
   625     (*   b_j & x_j = u & p b_j t & ...                    *)
   626 
   627     fun transform_rule r =
   628       let
   629         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
   630           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   631         val ps = make_bool_args HOLogic.mk_not I bs i @
   632           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   633           map (subst o HOLogic.dest_Trueprop)
   634             (Logic.strip_assums_hyp r)
   635       in List.foldr (fn ((x, T), P) => HOLogic.exists_const T $ (Abs (x, T, P)))
   636         (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
   637         (Logic.strip_params r)
   638       end
   639 
   640     (* make a disjunction of all introduction rules *)
   641 
   642     val fp_fun = fold_rev lambda (p :: bs @ xs)
   643       (if null intr_ts then HOLogic.false_const
   644        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   645 
   646     (* add definiton of recursive predicates to theory *)
   647 
   648     val rec_name =
   649       if Binding.is_empty alt_name then
   650         Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
   651       else alt_name;
   652 
   653     val ((rec_const, (_, fp_def)), ctxt') = ctxt |>
   654       LocalTheory.define Thm.internalK
   655         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   656          (Attrib.empty_binding, fold_rev lambda params
   657            (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)));
   658     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
   659       (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params)));
   660     val specs = if length cs < 2 then [] else
   661       map_index (fn (i, (name_mx, c)) =>
   662         let
   663           val Ts = List.drop (binder_types (fastype_of c), length params);
   664           val xs = map Free (Variable.variant_frees ctxt intr_ts
   665             (mk_names "x" (length Ts) ~~ Ts))
   666         in
   667           (name_mx, (Attrib.empty_binding, fold_rev lambda (params @ xs)
   668             (list_comb (rec_const, params @ make_bool_args' bs i @
   669               make_args argTs (xs ~~ Ts)))))
   670         end) (cnames_syn ~~ cs);
   671     val (consts_defs, ctxt'') = fold_map (LocalTheory.define Thm.internalK) specs ctxt';
   672     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   673 
   674     val mono = prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt'';
   675     val ((_, [mono']), ctxt''') =
   676       LocalTheory.note Thm.internalK (Attrib.empty_binding, [mono]) ctxt'';
   677 
   678   in (ctxt''', rec_name, mono', fp_def', map (#2 o #2) consts_defs,
   679     list_comb (rec_const, params), preds, argTs, bs, xs)
   680   end;
   681 
   682 fun declare_rules kind rec_binding coind no_ind cnames intrs intr_bindings intr_atts
   683       elims raw_induct ctxt =
   684   let
   685     val rec_name = Binding.name_of rec_binding;
   686     fun rec_qualified qualified = Binding.qualify qualified rec_name;
   687     val intr_names = map Binding.name_of intr_bindings;
   688     val ind_case_names = RuleCases.case_names intr_names;
   689     val induct =
   690       if coind then
   691         (raw_induct, [RuleCases.case_names [rec_name],
   692           RuleCases.case_conclusion (rec_name, intr_names),
   693           RuleCases.consumes 1, Induct.coinduct_pred (hd cnames)])
   694       else if no_ind orelse length cnames > 1 then
   695         (raw_induct, [ind_case_names, RuleCases.consumes 0])
   696       else (raw_induct RSN (2, rev_mp), [ind_case_names, RuleCases.consumes 1]);
   697 
   698     val (intrs', ctxt1) =
   699       ctxt |>
   700       LocalTheory.notes kind
   701         (map (rec_qualified false) intr_bindings ~~ intr_atts ~~ map (fn th => [([th],
   702            [Attrib.internal (K (ContextRules.intro_query NONE)),
   703             Attrib.internal (K Nitpick_Ind_Intros.add)])]) intrs) |>>
   704       map (hd o snd);
   705     val (((_, elims'), (_, [induct'])), ctxt2) =
   706       ctxt1 |>
   707       LocalTheory.note kind ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
   708       fold_map (fn (name, (elim, cases)) =>
   709         LocalTheory.note kind ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
   710           [Attrib.internal (K (RuleCases.case_names cases)),
   711            Attrib.internal (K (RuleCases.consumes 1)),
   712            Attrib.internal (K (Induct.cases_pred name)),
   713            Attrib.internal (K (ContextRules.elim_query NONE))]), [elim]) #>
   714         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   715       LocalTheory.note kind
   716         ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
   717           map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
   718 
   719     val ctxt3 = if no_ind orelse coind then ctxt2 else
   720       let val inducts = cnames ~~ Project_Rule.projects ctxt2 (1 upto length cnames) induct'
   721       in
   722         ctxt2 |>
   723         LocalTheory.notes kind [((rec_qualified true (Binding.name "inducts"), []),
   724           inducts |> map (fn (name, th) => ([th],
   725             [Attrib.internal (K ind_case_names),
   726              Attrib.internal (K (RuleCases.consumes 1)),
   727              Attrib.internal (K (Induct.induct_pred name))])))] |> snd
   728       end
   729   in (intrs', elims', induct', ctxt3) end;
   730 
   731 type inductive_flags =
   732   {quiet_mode: bool, verbose: bool, kind: string, alt_name: binding,
   733    coind: bool, no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool}
   734 
   735 type add_ind_def =
   736   inductive_flags ->
   737   term list -> (Attrib.binding * term) list -> thm list ->
   738   term list -> (binding * mixfix) list ->
   739   local_theory -> inductive_result * local_theory
   740 
   741 fun add_ind_def {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}
   742     cs intros monos params cnames_syn ctxt =
   743   let
   744     val _ = null cnames_syn andalso error "No inductive predicates given";
   745     val names = map (Binding.name_of o fst) cnames_syn;
   746     val _ = message (quiet_mode andalso not verbose)
   747       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
   748 
   749     val cnames = map (LocalTheory.full_name ctxt o #1) cnames_syn;  (* FIXME *)
   750     val ((intr_names, intr_atts), intr_ts) =
   751       apfst split_list (split_list (map (check_rule ctxt cs params) intros));
   752 
   753     val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   754       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts
   755         monos params cnames_syn ctxt;
   756 
   757     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
   758       params intr_ts rec_preds_defs ctxt1;
   759     val elims = if no_elim then [] else
   760       prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
   761         unfold rec_preds_defs ctxt1;
   762     val raw_induct = zero_var_indexes
   763       (if no_ind then Drule.asm_rl else
   764        if coind then
   765          singleton (ProofContext.export
   766            (snd (Variable.add_fixes (map (fst o dest_Free) params) ctxt1)) ctxt1)
   767            (rotate_prems ~1 (ObjectLogic.rulify
   768              (fold_rule rec_preds_defs
   769                (rewrite_rule simp_thms'''
   770                 (mono RS (fp_def RS @{thm def_coinduct}))))))
   771        else
   772          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
   773            rec_preds_defs ctxt1);
   774 
   775     val (intrs', elims', induct, ctxt2) = declare_rules kind rec_name coind no_ind
   776       cnames intrs intr_names intr_atts elims raw_induct ctxt1;
   777 
   778     val result =
   779       {preds = preds,
   780        intrs = intrs',
   781        elims = elims',
   782        raw_induct = rulify raw_induct,
   783        induct = induct};
   784 
   785     val ctxt3 = ctxt2
   786       |> LocalTheory.declaration (fn phi =>
   787         let val result' = morph_result phi result;
   788         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
   789   in (result, ctxt3) end;
   790 
   791 
   792 (* external interfaces *)
   793 
   794 fun gen_add_inductive_i mk_def
   795     (flags as {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono})
   796     cnames_syn pnames spec monos lthy =
   797   let
   798     val thy = ProofContext.theory_of lthy;
   799     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   800 
   801 
   802     (* abbrevs *)
   803 
   804     val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
   805 
   806     fun get_abbrev ((name, atts), t) =
   807       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
   808         let
   809           val _ = Binding.is_empty name andalso null atts orelse
   810             error "Abbreviations may not have names or attributes";
   811           val ((x, T), rhs) = LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt1 t));
   812           val var =
   813             (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
   814               NONE => error ("Undeclared head of abbreviation " ^ quote x)
   815             | SOME ((b, T'), mx) =>
   816                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
   817                 else (b, mx));
   818         in SOME (var, rhs) end
   819       else NONE;
   820 
   821     val abbrevs = map_filter get_abbrev spec;
   822     val bs = map (Binding.name_of o fst o fst) abbrevs;
   823 
   824 
   825     (* predicates *)
   826 
   827     val pre_intros = filter_out (is_some o get_abbrev) spec;
   828     val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
   829     val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
   830     val ps = map Free pnames;
   831 
   832     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
   833     val _ = map (fn abbr => LocalDefs.fixed_abbrev abbr ctxt2) abbrevs;
   834     val ctxt3 = ctxt2 |> fold (snd oo LocalDefs.fixed_abbrev) abbrevs;
   835     val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy;
   836 
   837     fun close_rule r = list_all_free (rev (fold_aterms
   838       (fn t as Free (v as (s, _)) =>
   839           if Variable.is_fixed ctxt1 s orelse
   840             member (op =) ps t then I else insert (op =) v
   841         | _ => I) r []), r);
   842 
   843     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
   844     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
   845   in
   846     lthy
   847     |> mk_def flags cs intros monos ps preds
   848     ||> fold (snd oo LocalTheory.abbrev Syntax.mode_default) abbrevs
   849   end;
   850 
   851 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos int lthy =
   852   let
   853     val ((vars, intrs), _) = lthy
   854       |> ProofContext.set_mode ProofContext.mode_abbrev
   855       |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
   856     val (cs, ps) = chop (length cnames_syn) vars;
   857     val monos = Attrib.eval_thms lthy raw_monos;
   858     val flags = {quiet_mode = false, verbose = verbose, kind = Thm.generatedK,
   859       alt_name = Binding.empty, coind = coind, no_elim = false, no_ind = false,
   860       skip_mono = false, fork_mono = not int};
   861   in
   862     lthy
   863     |> LocalTheory.set_group (serial_string ())
   864     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
   865   end;
   866 
   867 val add_inductive_i = gen_add_inductive_i add_ind_def;
   868 val add_inductive = gen_add_inductive add_ind_def;
   869 
   870 fun add_inductive_global group flags cnames_syn pnames pre_intros monos thy =
   871   let
   872     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
   873     val ctxt' = thy
   874       |> TheoryTarget.init NONE
   875       |> LocalTheory.set_group group
   876       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
   877       |> LocalTheory.exit;
   878     val info = #2 (the_inductive ctxt' name);
   879   in (info, ProofContext.theory_of ctxt') end;
   880 
   881 
   882 (* read off arities of inductive predicates from raw induction rule *)
   883 fun arities_of induct =
   884   map (fn (_ $ t $ u) =>
   885       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
   886     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
   887 
   888 (* read off parameters of inductive predicate from raw induction rule *)
   889 fun params_of induct =
   890   let
   891     val (_ $ t $ u :: _) =
   892       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
   893     val (_, ts) = strip_comb t;
   894     val (_, us) = strip_comb u
   895   in
   896     List.take (ts, length ts - length us)
   897   end;
   898 
   899 val pname_of_intr =
   900   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
   901 
   902 (* partition introduction rules according to predicate name *)
   903 fun gen_partition_rules f induct intros =
   904   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
   905     (map (rpair [] o fst) (arities_of induct));
   906 
   907 val partition_rules = gen_partition_rules I;
   908 fun partition_rules' induct = gen_partition_rules fst induct;
   909 
   910 fun unpartition_rules intros xs =
   911   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
   912     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
   913 
   914 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
   915 fun infer_intro_vars elim arity intros =
   916   let
   917     val thy = theory_of_thm elim;
   918     val _ :: cases = prems_of elim;
   919     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
   920     fun mtch (t, u) =
   921       let
   922         val params = Logic.strip_params t;
   923         val vars = map (Var o apfst (rpair 0))
   924           (Name.variant_list used (map fst params) ~~ map snd params);
   925         val ts = map (curry subst_bounds (rev vars))
   926           (List.drop (Logic.strip_assums_hyp t, arity));
   927         val us = Logic.strip_imp_prems u;
   928         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
   929           (Vartab.empty, Vartab.empty);
   930       in
   931         map (Envir.subst_term tab) vars
   932       end
   933   in
   934     map (mtch o apsnd prop_of) (cases ~~ intros)
   935   end;
   936 
   937 
   938 
   939 (** package setup **)
   940 
   941 (* setup theory *)
   942 
   943 val setup =
   944   ind_cases_setup #>
   945   Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
   946     "declaration of monotonicity rule";
   947 
   948 
   949 (* outer syntax *)
   950 
   951 local structure P = OuterParse and K = OuterKeyword in
   952 
   953 val _ = OuterKeyword.keyword "monos";
   954 
   955 fun gen_ind_decl mk_def coind =
   956   P.fixes -- P.for_fixes --
   957   Scan.optional SpecParse.where_alt_specs [] --
   958   Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.xthms1) []
   959   >> (fn (((preds, params), specs), monos) =>
   960       (snd oo gen_add_inductive mk_def true coind preds params specs monos));
   961 
   962 val ind_decl = gen_ind_decl add_ind_def;
   963 
   964 val _ = OuterSyntax.local_theory' "inductive" "define inductive predicates" K.thy_decl (ind_decl false);
   965 val _ = OuterSyntax.local_theory' "coinductive" "define coinductive predicates" K.thy_decl (ind_decl true);
   966 
   967 val _ =
   968   OuterSyntax.local_theory "inductive_cases"
   969     "create simplified instances of elimination rules (improper)" K.thy_script
   970     (P.and_list1 SpecParse.specs >> (snd oo inductive_cases));
   971 
   972 end;
   973 
   974 end;