src/HOL/Tools/inductive.ML
 author wenzelm Sun Nov 27 14:20:31 2011 +0100 (2011-11-27) changeset 45647 96af0578571c parent 45592 8baa0b7f3f66 child 45648 7654f750fb43 permissions -rw-r--r--
misc tuning;
```     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, inducts: thm list, intrs: thm list, eqs: thm list}
```
```    26   val transform_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_def: thm
```
```    35   val rulify: thm -> thm
```
```    36   val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
```
```    37     thm list list * local_theory
```
```    38   val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
```
```    39     thm list list * local_theory
```
```    40   type inductive_flags =
```
```    41     {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
```
```    42       no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool}
```
```    43   val add_inductive_i:
```
```    44     inductive_flags -> ((binding * typ) * mixfix) list ->
```
```    45     (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
```
```    46     inductive_result * local_theory
```
```    47   val add_inductive: bool -> bool ->
```
```    48     (binding * string option * mixfix) list ->
```
```    49     (binding * string option * mixfix) list ->
```
```    50     (Attrib.binding * string) list ->
```
```    51     (Facts.ref * Attrib.src list) list ->
```
```    52     bool -> local_theory -> inductive_result * local_theory
```
```    53   val add_inductive_global: inductive_flags ->
```
```    54     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
```
```    55     thm list -> theory -> inductive_result * theory
```
```    56   val arities_of: thm -> (string * int) list
```
```    57   val params_of: thm -> term list
```
```    58   val partition_rules: thm -> thm list -> (string * thm list) list
```
```    59   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
```
```    60   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
```
```    61   val infer_intro_vars: thm -> int -> thm list -> term list list
```
```    62   val setup: theory -> theory
```
```    63 end;
```
```    64
```
```    65 signature INDUCTIVE =
```
```    66 sig
```
```    67   include BASIC_INDUCTIVE
```
```    68   type add_ind_def =
```
```    69     inductive_flags ->
```
```    70     term list -> (Attrib.binding * term) list -> thm list ->
```
```    71     term list -> (binding * mixfix) list ->
```
```    72     local_theory -> inductive_result * local_theory
```
```    73   val declare_rules: binding -> bool -> bool -> string list -> term list ->
```
```    74     thm list -> binding list -> Attrib.src list list -> (thm * string list * int) list ->
```
```    75     thm list -> thm -> local_theory -> thm list * thm list * thm list * thm * thm list * local_theory
```
```    76   val add_ind_def: add_ind_def
```
```    77   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
```
```    78     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
```
```    79     thm list -> local_theory -> inductive_result * local_theory
```
```    80   val gen_add_inductive: add_ind_def -> bool -> bool ->
```
```    81     (binding * string option * mixfix) list ->
```
```    82     (binding * string option * mixfix) list ->
```
```    83     (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
```
```    84     bool -> local_theory -> inductive_result * local_theory
```
```    85   val gen_ind_decl: add_ind_def -> bool -> (bool -> local_theory -> local_theory) parser
```
```    86 end;
```
```    87
```
```    88 structure Inductive: INDUCTIVE =
```
```    89 struct
```
```    90
```
```    91
```
```    92 (** theory context references **)
```
```    93
```
```    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
```
```   105 val simp_thms' = map mk_meta_eq
```
```   106   @{lemma "(~True) = False" "(~False) = True"
```
```   107       "(True --> P) = P" "(False --> P) = True"
```
```   108       "(P & True) = P" "(True & P) = P"
```
```   109     by (fact simp_thms)+};
```
```   110
```
```   111 val simp_thms'' = map mk_meta_eq [@{thm inf_fun_def}, @{thm inf_bool_def}] @ simp_thms';
```
```   112
```
```   113 val simp_thms''' = map mk_meta_eq
```
```   114   [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_def}, @{thm sup_bool_def}];
```
```   115
```
```   116
```
```   117
```
```   118 (** context data **)
```
```   119
```
```   120 type inductive_result =
```
```   121   {preds: term list, elims: thm list, raw_induct: thm,
```
```   122    induct: thm, inducts: thm list, intrs: thm list, eqs: thm list};
```
```   123
```
```   124 fun transform_result phi {preds, elims, raw_induct: thm, induct, inducts, intrs, eqs} =
```
```   125   let
```
```   126     val term = Morphism.term phi;
```
```   127     val thm = Morphism.thm phi;
```
```   128     val fact = Morphism.fact phi;
```
```   129   in
```
```   130    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
```
```   131     induct = thm induct, inducts = fact inducts, intrs = fact intrs, eqs = fact eqs}
```
```   132   end;
```
```   133
```
```   134 type inductive_info =
```
```   135   {names: string list, coind: bool} * inductive_result;
```
```   136
```
```   137 structure Data = Generic_Data
```
```   138 (
```
```   139   type T = inductive_info Symtab.table * thm list;
```
```   140   val empty = (Symtab.empty, []);
```
```   141   val extend = I;
```
```   142   fun merge ((tab1, monos1), (tab2, monos2)) : T =
```
```   143     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
```
```   144 );
```
```   145
```
```   146 val get_inductives = Data.get o Context.Proof;
```
```   147
```
```   148 fun print_inductives ctxt =
```
```   149   let
```
```   150     val (tab, monos) = get_inductives ctxt;
```
```   151     val space = Consts.space_of (Proof_Context.consts_of ctxt);
```
```   152   in
```
```   153     [Pretty.strs ("(co)inductives:" :: map #1 (Name_Space.extern_table ctxt (space, tab))),
```
```   154      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm ctxt) monos)]
```
```   155     |> Pretty.chunks |> Pretty.writeln
```
```   156   end;
```
```   157
```
```   158
```
```   159 (* get and put data *)
```
```   160
```
```   161 fun the_inductive ctxt name =
```
```   162   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
```
```   163     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
```
```   164   | SOME info => info);
```
```   165
```
```   166 fun put_inductives names info =
```
```   167   Data.map (apfst (fold (fn name => Symtab.update (name, info)) names));
```
```   168
```
```   169
```
```   170
```
```   171 (** monotonicity rules **)
```
```   172
```
```   173 val get_monos = #2 o get_inductives;
```
```   174 val map_monos = Data.map o apsnd;
```
```   175
```
```   176 fun mk_mono ctxt thm =
```
```   177   let
```
```   178     fun eq2mono thm' = thm' RS (thm' RS eq_to_mono);
```
```   179     fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
```
```   180       handle THM _ => thm RS @{thm le_boolD}
```
```   181   in
```
```   182     (case concl_of thm of
```
```   183       Const ("==", _) \$ _ \$ _ => eq2mono (thm RS meta_eq_to_obj_eq)
```
```   184     | _ \$ (Const (@{const_name HOL.eq}, _) \$ _ \$ _) => eq2mono thm
```
```   185     | _ \$ (Const (@{const_name Orderings.less_eq}, _) \$ _ \$ _) =>
```
```   186       dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
```
```   187         (resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))
```
```   188     | _ => thm)
```
```   189   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm ctxt thm);
```
```   190
```
```   191 val mono_add =
```
```   192   Thm.declaration_attribute (fn thm => fn context =>
```
```   193     map_monos (Thm.add_thm (mk_mono (Context.proof_of context) thm)) context);
```
```   194
```
```   195 val mono_del =
```
```   196   Thm.declaration_attribute (fn thm => fn context =>
```
```   197     map_monos (Thm.del_thm (mk_mono (Context.proof_of context) thm)) context);
```
```   198
```
```   199
```
```   200
```
```   201 (** equations **)
```
```   202
```
```   203 structure Equation_Data = Generic_Data   (* FIXME just one data slot per module *)
```
```   204 (
```
```   205   type T = thm Item_Net.T;
```
```   206   val empty = Item_Net.init (op aconv o pairself Thm.prop_of)
```
```   207     (single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
```
```   208   val extend = I;
```
```   209   val merge = Item_Net.merge;
```
```   210 );
```
```   211
```
```   212 val add_equation = Thm.declaration_attribute (Equation_Data.map o Item_Net.update)
```
```   213
```
```   214
```
```   215
```
```   216 (** misc utilities **)
```
```   217
```
```   218 fun message quiet_mode s = if quiet_mode then () else writeln s;
```
```   219 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
```
```   220
```
```   221 fun coind_prefix true = "co"
```
```   222   | coind_prefix false = "";
```
```   223
```
```   224 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
```
```   225
```
```   226 fun make_bool_args f g [] i = []
```
```   227   | make_bool_args f g (x :: xs) i =
```
```   228       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
```
```   229
```
```   230 fun make_bool_args' xs =
```
```   231   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
```
```   232
```
```   233 fun arg_types_of k c = drop k (binder_types (fastype_of c));
```
```   234
```
```   235 fun find_arg T x [] = raise Fail "find_arg"
```
```   236   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
```
```   237       apsnd (cons p) (find_arg T x ps)
```
```   238   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
```
```   239       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
```
```   240       else apsnd (cons p) (find_arg T x ps);
```
```   241
```
```   242 fun make_args Ts xs =
```
```   243   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
```
```   244     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
```
```   245
```
```   246 fun make_args' Ts xs Us =
```
```   247   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
```
```   248
```
```   249 fun dest_predicate cs params t =
```
```   250   let
```
```   251     val k = length params;
```
```   252     val (c, ts) = strip_comb t;
```
```   253     val (xs, ys) = chop k ts;
```
```   254     val i = find_index (fn c' => c' = c) cs;
```
```   255   in
```
```   256     if xs = params andalso i >= 0 then
```
```   257       SOME (c, i, ys, chop (length ys) (arg_types_of k c))
```
```   258     else NONE
```
```   259   end;
```
```   260
```
```   261 fun mk_names a 0 = []
```
```   262   | mk_names a 1 = [a]
```
```   263   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
```
```   264
```
```   265 fun select_disj 1 1 = []
```
```   266   | select_disj _ 1 = [rtac disjI1]
```
```   267   | select_disj n i = rtac disjI2 :: select_disj (n - 1) (i - 1);
```
```   268
```
```   269
```
```   270
```
```   271 (** process rules **)
```
```   272
```
```   273 local
```
```   274
```
```   275 fun err_in_rule ctxt name t msg =
```
```   276   error (cat_lines ["Ill-formed introduction rule " ^ Binding.print name,
```
```   277     Syntax.string_of_term ctxt t, msg]);
```
```   278
```
```   279 fun err_in_prem ctxt name t p msg =
```
```   280   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
```
```   281     "in introduction rule " ^ Binding.print name, Syntax.string_of_term ctxt t, msg]);
```
```   282
```
```   283 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
```
```   284
```
```   285 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
```
```   286
```
```   287 val bad_app = "Inductive predicate must be applied to parameter(s) ";
```
```   288
```
```   289 fun atomize_term thy = Raw_Simplifier.rewrite_term thy inductive_atomize [];
```
```   290
```
```   291 in
```
```   292
```
```   293 fun check_rule ctxt cs params ((binding, att), rule) =
```
```   294   let
```
```   295     val params' = Term.variant_frees rule (Logic.strip_params rule);
```
```   296     val frees = rev (map Free params');
```
```   297     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
```
```   298     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
```
```   299     val rule' = Logic.list_implies (prems, concl);
```
```   300     val aprems = map (atomize_term (Proof_Context.theory_of ctxt)) prems;
```
```   301     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
```
```   302
```
```   303     fun check_ind err t =
```
```   304       (case dest_predicate cs params t of
```
```   305         NONE => err (bad_app ^
```
```   306           commas (map (Syntax.string_of_term ctxt) params))
```
```   307       | SOME (_, _, ys, _) =>
```
```   308           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
```
```   309           then err bad_ind_occ else ());
```
```   310
```
```   311     fun check_prem' prem t =
```
```   312       if member (op =) cs (head_of t) then
```
```   313         check_ind (err_in_prem ctxt binding rule prem) t
```
```   314       else
```
```   315         (case t of
```
```   316           Abs (_, _, t) => check_prem' prem t
```
```   317         | t \$ u => (check_prem' prem t; check_prem' prem u)
```
```   318         | _ => ());
```
```   319
```
```   320     fun check_prem (prem, aprem) =
```
```   321       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
```
```   322       else err_in_prem ctxt binding rule prem "Non-atomic premise";
```
```   323
```
```   324     val _ =
```
```   325       (case concl of
```
```   326         Const (@{const_name Trueprop}, _) \$ t =>
```
```   327           if member (op =) cs (head_of t) then
```
```   328            (check_ind (err_in_rule ctxt binding rule') t;
```
```   329             List.app check_prem (prems ~~ aprems))
```
```   330           else err_in_rule ctxt binding rule' bad_concl
```
```   331        | _ => err_in_rule ctxt binding rule' bad_concl);
```
```   332   in
```
```   333     ((binding, att), arule)
```
```   334   end;
```
```   335
```
```   336 val rulify =
```
```   337   hol_simplify inductive_conj
```
```   338   #> hol_simplify inductive_rulify
```
```   339   #> hol_simplify inductive_rulify_fallback
```
```   340   #> Simplifier.norm_hhf;
```
```   341
```
```   342 end;
```
```   343
```
```   344
```
```   345
```
```   346 (** proofs for (co)inductive predicates **)
```
```   347
```
```   348 (* prove monotonicity *)
```
```   349
```
```   350 fun prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt =
```
```   351  (message (quiet_mode orelse skip_mono andalso !quick_and_dirty orelse fork_mono)
```
```   352     "  Proving monotonicity ...";
```
```   353   (if skip_mono then Skip_Proof.prove else if fork_mono then Goal.prove_future else Goal.prove) ctxt
```
```   354     [] []
```
```   355     (HOLogic.mk_Trueprop
```
```   356       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) \$ fp_fun))
```
```   357     (fn _ => EVERY [rtac @{thm monoI} 1,
```
```   358       REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
```
```   359       REPEAT (FIRST
```
```   360         [atac 1,
```
```   361          resolve_tac (map (mk_mono ctxt) monos @ get_monos ctxt) 1,
```
```   362          etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
```
```   363
```
```   364
```
```   365 (* prove introduction rules *)
```
```   366
```
```   367 fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
```
```   368   let
```
```   369     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
```
```   370
```
```   371     val unfold = funpow k (fn th => th RS fun_cong)
```
```   372       (mono RS (fp_def RS
```
```   373         (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
```
```   374
```
```   375     val rules = [refl, TrueI, notFalseI, exI, conjI];
```
```   376
```
```   377     val intrs = map_index (fn (i, intr) =>
```
```   378       Skip_Proof.prove ctxt [] [] intr (fn _ => EVERY
```
```   379        [rewrite_goals_tac rec_preds_defs,
```
```   380         rtac (unfold RS iffD2) 1,
```
```   381         EVERY1 (select_disj (length intr_ts) (i + 1)),
```
```   382         (*Not ares_tac, since refl must be tried before any equality assumptions;
```
```   383           backtracking may occur if the premises have extra variables!*)
```
```   384         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)])
```
```   385        |> singleton (Proof_Context.export ctxt ctxt')) intr_ts
```
```   386
```
```   387   in (intrs, unfold) end;
```
```   388
```
```   389
```
```   390 (* prove elimination rules *)
```
```   391
```
```   392 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
```
```   393   let
```
```   394     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
```
```   395
```
```   396     val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
```
```   397     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
```
```   398
```
```   399     fun dest_intr r =
```
```   400       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
```
```   401        Logic.strip_assums_hyp r, Logic.strip_params r);
```
```   402
```
```   403     val intrs = map dest_intr intr_ts ~~ intr_names;
```
```   404
```
```   405     val rules1 = [disjE, exE, FalseE];
```
```   406     val rules2 = [conjE, FalseE, notTrueE];
```
```   407
```
```   408     fun prove_elim c =
```
```   409       let
```
```   410         val Ts = arg_types_of (length params) c;
```
```   411         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
```
```   412         val frees = map Free (anames ~~ Ts);
```
```   413
```
```   414         fun mk_elim_prem ((_, _, us, _), ts, params') =
```
```   415           list_all (params',
```
```   416             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
```
```   417               (frees ~~ us) @ ts, P));
```
```   418         val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
```
```   419         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
```
```   420            map mk_elim_prem (map #1 c_intrs)
```
```   421       in
```
```   422         (Skip_Proof.prove ctxt'' [] prems P
```
```   423           (fn {prems, ...} => EVERY
```
```   424             [cut_facts_tac [hd prems] 1,
```
```   425              rewrite_goals_tac rec_preds_defs,
```
```   426              dtac (unfold RS iffD1) 1,
```
```   427              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
```
```   428              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
```
```   429              EVERY (map (fn prem =>
```
```   430                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
```
```   431           |> singleton (Proof_Context.export ctxt'' ctxt'''),
```
```   432          map #2 c_intrs, length Ts)
```
```   433       end
```
```   434
```
```   435    in map prove_elim cs end;
```
```   436
```
```   437
```
```   438 (* prove simplification equations *)
```
```   439
```
```   440 fun prove_eqs quiet_mode cs params intr_ts intrs
```
```   441     (elims: (thm * bstring list * int) list) ctxt ctxt'' =  (* FIXME ctxt'' ?? *)
```
```   442   let
```
```   443     val _ = clean_message quiet_mode "  Proving the simplification rules ...";
```
```   444
```
```   445     fun dest_intr r =
```
```   446       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
```
```   447        Logic.strip_assums_hyp r, Logic.strip_params r);
```
```   448     val intr_ts' = map dest_intr intr_ts;
```
```   449
```
```   450     fun prove_eq c (elim: thm * 'a * 'b) =
```
```   451       let
```
```   452         val Ts = arg_types_of (length params) c;
```
```   453         val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
```
```   454         val frees = map Free (anames ~~ Ts);
```
```   455         val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
```
```   456         fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
```
```   457           let
```
```   458             fun list_ex ([], t) = t
```
```   459               | list_ex ((a, T) :: vars, t) =
```
```   460                   HOLogic.exists_const T \$ Abs (a, T, list_ex (vars, t));
```
```   461             val conjs = map2 (curry HOLogic.mk_eq) frees us @ (map HOLogic.dest_Trueprop ts);
```
```   462           in
```
```   463             list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
```
```   464           end;
```
```   465         val lhs = list_comb (c, params @ frees);
```
```   466         val rhs =
```
```   467           if null c_intrs then @{term False}
```
```   468           else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs);
```
```   469         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
```
```   470         fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
```
```   471             let
```
```   472               val (prems', last_prem) = split_last prems;
```
```   473             in
```
```   474               EVERY1 (select_disj (length c_intrs) (i + 1)) THEN
```
```   475               EVERY (replicate (length params) (rtac @{thm exI} 1)) THEN
```
```   476               EVERY (map (fn prem => (rtac @{thm conjI} 1 THEN rtac prem 1)) prems') THEN
```
```   477               rtac last_prem 1
```
```   478             end) ctxt' 1;
```
```   479         fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
```
```   480           EVERY (replicate (length params') (etac @{thm exE} 1)) THEN
```
```   481           EVERY (replicate (length ts + length us - 1) (etac @{thm conjE} 1)) THEN
```
```   482           Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
```
```   483             let
```
```   484               val (eqs, prems') = chop (length us) prems;
```
```   485               val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs;
```
```   486             in
```
```   487               rewrite_goal_tac rew_thms 1 THEN
```
```   488               rtac intr 1 THEN
```
```   489               EVERY (map (fn p => rtac p 1) prems')
```
```   490             end) ctxt' 1;
```
```   491       in
```
```   492         Skip_Proof.prove ctxt' [] [] eq (fn _ =>
```
```   493           rtac @{thm iffI} 1 THEN etac (#1 elim) 1 THEN
```
```   494           EVERY (map_index prove_intr1 c_intrs) THEN
```
```   495           (if null c_intrs then etac @{thm FalseE} 1
```
```   496            else
```
```   497             let val (c_intrs', last_c_intr) = split_last c_intrs in
```
```   498               EVERY (map (fn ci => etac @{thm disjE} 1 THEN prove_intr2 ci) c_intrs') THEN
```
```   499               prove_intr2 last_c_intr
```
```   500             end))
```
```   501         |> rulify
```
```   502         |> singleton (Proof_Context.export ctxt' ctxt'')
```
```   503       end;
```
```   504   in
```
```   505     map2 prove_eq cs elims
```
```   506   end;
```
```   507
```
```   508
```
```   509 (* derivation of simplified elimination rules *)
```
```   510
```
```   511 local
```
```   512
```
```   513 (*delete needless equality assumptions*)
```
```   514 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
```
```   515   (fn _ => assume_tac 1);
```
```   516 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
```
```   517 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
```
```   518
```
```   519 fun simp_case_tac ss i =
```
```   520   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
```
```   521
```
```   522 in
```
```   523
```
```   524 fun mk_cases ctxt prop =
```
```   525   let
```
```   526     val thy = Proof_Context.theory_of ctxt;
```
```   527     val ss = simpset_of ctxt;
```
```   528
```
```   529     fun err msg =
```
```   530       error (Pretty.string_of (Pretty.block
```
```   531         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
```
```   532
```
```   533     val elims = Induct.find_casesP ctxt prop;
```
```   534
```
```   535     val cprop = Thm.cterm_of thy prop;
```
```   536     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
```
```   537     fun mk_elim rl =
```
```   538       Thm.implies_intr cprop (Tactic.rule_by_tactic ctxt tac (Thm.assume cprop RS rl))
```
```   539       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
```
```   540   in
```
```   541     (case get_first (try mk_elim) elims of
```
```   542       SOME r => r
```
```   543     | NONE => err "Proposition not an inductive predicate:")
```
```   544   end;
```
```   545
```
```   546 end;
```
```   547
```
```   548
```
```   549 (* inductive_cases *)
```
```   550
```
```   551 fun gen_inductive_cases prep_att prep_prop args lthy =
```
```   552   let
```
```   553     val thy = Proof_Context.theory_of lthy;
```
```   554     val facts = args |> Par_List.map (fn ((a, atts), props) =>
```
```   555       ((a, map (prep_att thy) atts),
```
```   556         Par_List.map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
```
```   557   in lthy |> Local_Theory.notes facts |>> map snd end;
```
```   558
```
```   559 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
```
```   560 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
```
```   561
```
```   562
```
```   563 val ind_cases_setup =
```
```   564   Method.setup @{binding ind_cases}
```
```   565     (Scan.lift (Scan.repeat1 Args.name_source --
```
```   566       Scan.optional (Args.\$\$\$ "for" |-- Scan.repeat1 Args.binding) []) >>
```
```   567       (fn (raw_props, fixes) => fn ctxt =>
```
```   568         let
```
```   569           val (_, ctxt') = Variable.add_fixes_binding fixes ctxt;
```
```   570           val props = Syntax.read_props ctxt' raw_props;
```
```   571           val ctxt'' = fold Variable.declare_term props ctxt';
```
```   572           val rules = Proof_Context.export ctxt'' ctxt (map (mk_cases ctxt'') props)
```
```   573         in Method.erule 0 rules end))
```
```   574     "dynamic case analysis on predicates";
```
```   575
```
```   576
```
```   577 (* derivation of simplified equation *)
```
```   578
```
```   579 fun mk_simp_eq ctxt prop =
```
```   580   let
```
```   581     val thy = Proof_Context.theory_of ctxt;
```
```   582     val ctxt' = Variable.auto_fixes prop ctxt;
```
```   583     val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of;
```
```   584     val substs =
```
```   585       Item_Net.retrieve (Equation_Data.get (Context.Proof ctxt)) (HOLogic.dest_Trueprop prop)
```
```   586       |> map_filter
```
```   587         (fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
```
```   588             (Vartab.empty, Vartab.empty), eq)
```
```   589           handle Pattern.MATCH => NONE);
```
```   590     val (subst, eq) =
```
```   591       (case substs of
```
```   592         [s] => s
```
```   593       | _ => error
```
```   594         ("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique"));
```
```   595     val inst =
```
```   596       map (fn v => (cterm_of thy (Var v), cterm_of thy (Envir.subst_term subst (Var v))))
```
```   597         (Term.add_vars (lhs_of eq) []);
```
```   598   in
```
```   599     cterm_instantiate inst eq
```
```   600     |> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv (Simplifier.full_rewrite (simpset_of ctxt))))
```
```   601     |> singleton (Variable.export ctxt' ctxt)
```
```   602   end
```
```   603
```
```   604
```
```   605 (* inductive simps *)
```
```   606
```
```   607 fun gen_inductive_simps prep_att prep_prop args lthy =
```
```   608   let
```
```   609     val thy = Proof_Context.theory_of lthy;
```
```   610     val facts = args |> map (fn ((a, atts), props) =>
```
```   611       ((a, map (prep_att thy) atts),
```
```   612         map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
```
```   613   in lthy |> Local_Theory.notes facts |>> map snd end;
```
```   614
```
```   615 val inductive_simps = gen_inductive_simps Attrib.intern_src Syntax.read_prop;
```
```   616 val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
```
```   617
```
```   618
```
```   619 (* prove induction rule *)
```
```   620
```
```   621 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
```
```   622     fp_def rec_preds_defs ctxt ctxt''' =  (* FIXME ctxt''' ?? *)
```
```   623   let
```
```   624     val _ = clean_message quiet_mode "  Proving the induction rule ...";
```
```   625
```
```   626     (* predicates for induction rule *)
```
```   627
```
```   628     val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
```
```   629     val preds =
```
```   630       map2 (curry Free) pnames
```
```   631         (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
```
```   632
```
```   633     (* transform an introduction rule into a premise for induction rule *)
```
```   634
```
```   635     fun mk_ind_prem r =
```
```   636       let
```
```   637         fun subst s =
```
```   638           (case dest_predicate cs params s of
```
```   639             SOME (_, i, ys, (_, Ts)) =>
```
```   640               let
```
```   641                 val k = length Ts;
```
```   642                 val bs = map Bound (k - 1 downto 0);
```
```   643                 val P = list_comb (nth preds i, map (incr_boundvars k) ys @ bs);
```
```   644                 val Q = list_abs (mk_names "x" k ~~ Ts,
```
```   645                   HOLogic.mk_binop inductive_conj_name
```
```   646                     (list_comb (incr_boundvars k s, bs), P));
```
```   647               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
```
```   648           | NONE =>
```
```   649               (case s of
```
```   650                 t \$ u => (fst (subst t) \$ fst (subst u), NONE)
```
```   651               | Abs (a, T, t) => (Abs (a, T, fst (subst t)), NONE)
```
```   652               | _ => (s, NONE)));
```
```   653
```
```   654         fun mk_prem s prems =
```
```   655           (case subst s of
```
```   656             (_, SOME (t, u)) => t :: u :: prems
```
```   657           | (t, _) => t :: prems);
```
```   658
```
```   659         val SOME (_, i, ys, _) =
```
```   660           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
```
```   661       in
```
```   662         list_all_free (Logic.strip_params r,
```
```   663           Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
```
```   664             (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
```
```   665               HOLogic.mk_Trueprop (list_comb (nth preds i, ys))))
```
```   666       end;
```
```   667
```
```   668     val ind_prems = map mk_ind_prem intr_ts;
```
```   669
```
```   670
```
```   671     (* make conclusions for induction rules *)
```
```   672
```
```   673     val Tss = map (binder_types o fastype_of) preds;
```
```   674     val (xnames, ctxt'') = Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
```
```   675     val mutual_ind_concl =
```
```   676       HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
```
```   677         (map (fn (((xnames, Ts), c), P) =>
```
```   678           let val frees = map Free (xnames ~~ Ts)
```
```   679           in HOLogic.mk_imp (list_comb (c, params @ frees), list_comb (P, frees)) end)
```
```   680         (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
```
```   681
```
```   682
```
```   683     (* make predicate for instantiation of abstract induction rule *)
```
```   684
```
```   685     val ind_pred =
```
```   686       fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
```
```   687         (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
```
```   688            (make_bool_args HOLogic.mk_not I bs i)
```
```   689            (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
```
```   690
```
```   691     val ind_concl =
```
```   692       HOLogic.mk_Trueprop
```
```   693         (HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
```
```   694
```
```   695     val raw_fp_induct = mono RS (fp_def RS @{thm def_lfp_induct});
```
```   696
```
```   697     val induct = Skip_Proof.prove ctxt'' [] ind_prems ind_concl
```
```   698       (fn {prems, ...} => EVERY
```
```   699         [rewrite_goals_tac [inductive_conj_def],
```
```   700          DETERM (rtac raw_fp_induct 1),
```
```   701          REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
```
```   702          rewrite_goals_tac simp_thms'',
```
```   703          (*This disjE separates out the introduction rules*)
```
```   704          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
```
```   705          (*Now break down the individual cases.  No disjE here in case
```
```   706            some premise involves disjunction.*)
```
```   707          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
```
```   708          REPEAT (FIRSTGOAL
```
```   709            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
```
```   710          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
```
```   711              (inductive_conj_def :: rec_preds_defs @ simp_thms'') prem,
```
```   712            conjI, refl] 1)) prems)]);
```
```   713
```
```   714     val lemma = Skip_Proof.prove ctxt'' [] []
```
```   715       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
```
```   716         [rewrite_goals_tac rec_preds_defs,
```
```   717          REPEAT (EVERY
```
```   718            [REPEAT (resolve_tac [conjI, impI] 1),
```
```   719             REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
```
```   720             atac 1,
```
```   721             rewrite_goals_tac simp_thms',
```
```   722             atac 1])]);
```
```   723
```
```   724   in singleton (Proof_Context.export ctxt'' ctxt''') (induct RS lemma) end;
```
```   725
```
```   726
```
```   727
```
```   728 (** specification of (co)inductive predicates **)
```
```   729
```
```   730 fun mk_ind_def quiet_mode skip_mono fork_mono alt_name coind
```
```   731     cs intr_ts monos params cnames_syn lthy =
```
```   732   let
```
```   733     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
```
```   734
```
```   735     val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
```
```   736     val k = log 2 1 (length cs);
```
```   737     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
```
```   738     val p :: xs =
```
```   739       map Free (Variable.variant_frees lthy intr_ts
```
```   740         (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
```
```   741     val bs =
```
```   742       map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
```
```   743         (map (rpair HOLogic.boolT) (mk_names "b" k)));
```
```   744
```
```   745     fun subst t =
```
```   746       (case dest_predicate cs params t of
```
```   747         SOME (_, i, ts, (Ts, Us)) =>
```
```   748           let
```
```   749             val l = length Us;
```
```   750             val zs = map Bound (l - 1 downto 0);
```
```   751           in
```
```   752             list_abs (map (pair "z") Us, list_comb (p,
```
```   753               make_bool_args' bs i @ make_args argTs
```
```   754                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
```
```   755           end
```
```   756       | NONE =>
```
```   757           (case t of
```
```   758             t1 \$ t2 => subst t1 \$ subst t2
```
```   759           | Abs (x, T, u) => Abs (x, T, subst u)
```
```   760           | _ => t));
```
```   761
```
```   762     (* transform an introduction rule into a conjunction  *)
```
```   763     (*   [| p_i t; ... |] ==> p_j u                       *)
```
```   764     (* is transformed into                                *)
```
```   765     (*   b_j & x_j = u & p b_j t & ...                    *)
```
```   766
```
```   767     fun transform_rule r =
```
```   768       let
```
```   769         val SOME (_, i, ts, (Ts, _)) =
```
```   770           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
```
```   771         val ps =
```
```   772           make_bool_args HOLogic.mk_not I bs i @
```
```   773           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
```
```   774           map (subst o HOLogic.dest_Trueprop) (Logic.strip_assums_hyp r);
```
```   775       in
```
```   776         fold_rev (fn (x, T) => fn P => HOLogic.exists_const T \$ Abs (x, T, P))
```
```   777           (Logic.strip_params r)
```
```   778           (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
```
```   779       end;
```
```   780
```
```   781     (* make a disjunction of all introduction rules *)
```
```   782
```
```   783     val fp_fun =
```
```   784       fold_rev lambda (p :: bs @ xs)
```
```   785         (if null intr_ts then HOLogic.false_const
```
```   786          else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
```
```   787
```
```   788     (* add definiton of recursive predicates to theory *)
```
```   789
```
```   790     val rec_name =
```
```   791       if Binding.is_empty alt_name then
```
```   792         Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
```
```   793       else alt_name;
```
```   794
```
```   795     val ((rec_const, (_, fp_def)), lthy') = lthy
```
```   796       |> Local_Theory.conceal
```
```   797       |> Local_Theory.define
```
```   798         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
```
```   799          ((Binding.empty, @{attributes [nitpick_unfold]}),
```
```   800            fold_rev lambda params
```
```   801              (Const (fp_name, (predT --> predT) --> predT) \$ fp_fun)))
```
```   802       ||> Local_Theory.restore_naming lthy;
```
```   803     val fp_def' =
```
```   804       Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
```
```   805         (cterm_of (Proof_Context.theory_of lthy') (list_comb (rec_const, params)));
```
```   806     val specs =
```
```   807       if length cs < 2 then []
```
```   808       else
```
```   809         map_index (fn (i, (name_mx, c)) =>
```
```   810           let
```
```   811             val Ts = arg_types_of (length params) c;
```
```   812             val xs =
```
```   813               map Free (Variable.variant_frees lthy intr_ts (mk_names "x" (length Ts) ~~ Ts));
```
```   814           in
```
```   815             (name_mx, (apfst Binding.conceal Attrib.empty_binding, fold_rev lambda (params @ xs)
```
```   816               (list_comb (rec_const, params @ make_bool_args' bs i @
```
```   817                 make_args argTs (xs ~~ Ts)))))
```
```   818           end) (cnames_syn ~~ cs);
```
```   819     val (consts_defs, lthy'') = lthy'
```
```   820       |> fold_map Local_Theory.define specs;
```
```   821     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
```
```   822
```
```   823     val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
```
```   824     val mono = prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos lthy''';
```
```   825     val (_, lthy'''') =
```
```   826       Local_Theory.note (apfst Binding.conceal Attrib.empty_binding,
```
```   827         Proof_Context.export lthy''' lthy'' [mono]) lthy'';
```
```   828
```
```   829   in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
```
```   830     list_comb (rec_const, params), preds, argTs, bs, xs)
```
```   831   end;
```
```   832
```
```   833 fun declare_rules rec_binding coind no_ind cnames
```
```   834     preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
```
```   835   let
```
```   836     val rec_name = Binding.name_of rec_binding;
```
```   837     fun rec_qualified qualified = Binding.qualify qualified rec_name;
```
```   838     val intr_names = map Binding.name_of intr_bindings;
```
```   839     val ind_case_names = Rule_Cases.case_names intr_names;
```
```   840     val induct =
```
```   841       if coind then
```
```   842         (raw_induct, [Rule_Cases.case_names [rec_name],
```
```   843           Rule_Cases.case_conclusion (rec_name, intr_names),
```
```   844           Rule_Cases.consumes 1, Induct.coinduct_pred (hd cnames)])
```
```   845       else if no_ind orelse length cnames > 1 then
```
```   846         (raw_induct, [ind_case_names, Rule_Cases.consumes 0])
```
```   847       else (raw_induct RSN (2, rev_mp), [ind_case_names, Rule_Cases.consumes 1]);
```
```   848
```
```   849     val (intrs', lthy1) =
```
```   850       lthy |>
```
```   851       Spec_Rules.add
```
```   852         (if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
```
```   853       Local_Theory.notes
```
```   854         (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
```
```   855           map (fn th => [([th],
```
```   856            [Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
```
```   857       map (hd o snd);
```
```   858     val (((_, elims'), (_, [induct'])), lthy2) =
```
```   859       lthy1 |>
```
```   860       Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
```
```   861       fold_map (fn (name, (elim, cases, k)) =>
```
```   862         Local_Theory.note
```
```   863           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
```
```   864             [Attrib.internal (K (Rule_Cases.case_names cases)),
```
```   865              Attrib.internal (K (Rule_Cases.consumes 1)),
```
```   866              Attrib.internal (K (Rule_Cases.constraints k)),
```
```   867              Attrib.internal (K (Induct.cases_pred name)),
```
```   868              Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
```
```   869         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
```
```   870       Local_Theory.note
```
```   871         ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
```
```   872           map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
```
```   873
```
```   874     val (eqs', lthy3) = lthy2 |>
```
```   875       fold_map (fn (name, eq) => Local_Theory.note
```
```   876           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
```
```   877             [Attrib.internal (K add_equation)]), [eq])
```
```   878           #> apfst (hd o snd))
```
```   879         (if null eqs then [] else (cnames ~~ eqs))
```
```   880     val (inducts, lthy4) =
```
```   881       if no_ind orelse coind then ([], lthy3)
```
```   882       else
```
```   883         let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
```
```   884           lthy3 |>
```
```   885           Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
```
```   886             inducts |> map (fn (name, th) => ([th],
```
```   887               [Attrib.internal (K ind_case_names),
```
```   888                Attrib.internal (K (Rule_Cases.consumes 1)),
```
```   889                Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
```
```   890         end;
```
```   891   in (intrs', elims', eqs', induct', inducts, lthy4) end;
```
```   892
```
```   893 type inductive_flags =
```
```   894   {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
```
```   895     no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool};
```
```   896
```
```   897 type add_ind_def =
```
```   898   inductive_flags ->
```
```   899   term list -> (Attrib.binding * term) list -> thm list ->
```
```   900   term list -> (binding * mixfix) list ->
```
```   901   local_theory -> inductive_result * local_theory;
```
```   902
```
```   903 fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}
```
```   904     cs intros monos params cnames_syn lthy =
```
```   905   let
```
```   906     val _ = null cnames_syn andalso error "No inductive predicates given";
```
```   907     val names = map (Binding.name_of o fst) cnames_syn;
```
```   908     val _ = message (quiet_mode andalso not verbose)
```
```   909       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
```
```   910
```
```   911     val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
```
```   912     val ((intr_names, intr_atts), intr_ts) =
```
```   913       apfst split_list (split_list (map (check_rule lthy cs params) intros));
```
```   914
```
```   915     val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
```
```   916       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts
```
```   917         monos params cnames_syn lthy;
```
```   918
```
```   919     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
```
```   920       intr_ts rec_preds_defs lthy2 lthy1;
```
```   921     val elims =
```
```   922       if no_elim then []
```
```   923       else
```
```   924         prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
```
```   925           unfold rec_preds_defs lthy2 lthy1;
```
```   926     val raw_induct = zero_var_indexes
```
```   927       (if no_ind then Drule.asm_rl
```
```   928        else if coind then
```
```   929          singleton (Proof_Context.export lthy2 lthy1)
```
```   930            (rotate_prems ~1 (Object_Logic.rulify
```
```   931              (fold_rule rec_preds_defs
```
```   932                (rewrite_rule simp_thms'''
```
```   933                 (mono RS (fp_def RS @{thm def_coinduct}))))))
```
```   934        else
```
```   935          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
```
```   936            rec_preds_defs lthy2 lthy1);
```
```   937     val eqs =
```
```   938       if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1;
```
```   939
```
```   940     val elims' = map (fn (th, ns, i) => (rulify th, ns, i)) elims;
```
```   941     val intrs' = map rulify intrs;
```
```   942
```
```   943     val (intrs'', elims'', eqs', induct, inducts, lthy3) =
```
```   944       declare_rules rec_name coind no_ind
```
```   945         cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
```
```   946
```
```   947     val result =
```
```   948       {preds = preds,
```
```   949        intrs = intrs'',
```
```   950        elims = elims'',
```
```   951        raw_induct = rulify raw_induct,
```
```   952        induct = induct,
```
```   953        inducts = inducts,
```
```   954        eqs = eqs'};
```
```   955
```
```   956     val lthy4 = lthy3
```
```   957       |> Local_Theory.declaration {syntax = false, pervasive = false} (fn phi =>
```
```   958         let val result' = transform_result phi result;
```
```   959         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
```
```   960   in (result, lthy4) end;
```
```   961
```
```   962
```
```   963 (* external interfaces *)
```
```   964
```
```   965 fun gen_add_inductive_i mk_def
```
```   966     (flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono})
```
```   967     cnames_syn pnames spec monos lthy =
```
```   968   let
```
```   969     val thy = Proof_Context.theory_of lthy;
```
```   970     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
```
```   971
```
```   972
```
```   973     (* abbrevs *)
```
```   974
```
```   975     val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
```
```   976
```
```   977     fun get_abbrev ((name, atts), t) =
```
```   978       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
```
```   979         let
```
```   980           val _ = Binding.is_empty name andalso null atts orelse
```
```   981             error "Abbreviations may not have names or attributes";
```
```   982           val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
```
```   983           val var =
```
```   984             (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
```
```   985               NONE => error ("Undeclared head of abbreviation " ^ quote x)
```
```   986             | SOME ((b, T'), mx) =>
```
```   987                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
```
```   988                 else (b, mx));
```
```   989         in SOME (var, rhs) end
```
```   990       else NONE;
```
```   991
```
```   992     val abbrevs = map_filter get_abbrev spec;
```
```   993     val bs = map (Binding.name_of o fst o fst) abbrevs;
```
```   994
```
```   995
```
```   996     (* predicates *)
```
```   997
```
```   998     val pre_intros = filter_out (is_some o get_abbrev) spec;
```
```   999     val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
```
```  1000     val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
```
```  1001     val ps = map Free pnames;
```
```  1002
```
```  1003     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
```
```  1004     val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
```
```  1005     val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
```
```  1006     val expand = Assumption.export_term ctxt3 lthy #> Proof_Context.cert_term lthy;
```
```  1007
```
```  1008     fun close_rule r = list_all_free (rev (fold_aterms
```
```  1009       (fn t as Free (v as (s, _)) =>
```
```  1010           if Variable.is_fixed ctxt1 s orelse
```
```  1011             member (op =) ps t then I else insert (op =) v
```
```  1012         | _ => I) r []), r);
```
```  1013
```
```  1014     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
```
```  1015     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
```
```  1016   in
```
```  1017     lthy
```
```  1018     |> mk_def flags cs intros monos ps preds
```
```  1019     ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
```
```  1020   end;
```
```  1021
```
```  1022 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos int lthy =
```
```  1023   let
```
```  1024     val ((vars, intrs), _) = lthy
```
```  1025       |> Proof_Context.set_mode Proof_Context.mode_abbrev
```
```  1026       |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
```
```  1027     val (cs, ps) = chop (length cnames_syn) vars;
```
```  1028     val monos = Attrib.eval_thms lthy raw_monos;
```
```  1029     val flags = {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
```
```  1030       coind = coind, no_elim = false, no_ind = false, skip_mono = false, fork_mono = not int};
```
```  1031   in
```
```  1032     lthy
```
```  1033     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
```
```  1034   end;
```
```  1035
```
```  1036 val add_inductive_i = gen_add_inductive_i add_ind_def;
```
```  1037 val add_inductive = gen_add_inductive add_ind_def;
```
```  1038
```
```  1039 fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
```
```  1040   let
```
```  1041     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
```
```  1042     val ctxt' = thy
```
```  1043       |> Named_Target.theory_init
```
```  1044       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
```
```  1045       |> Local_Theory.exit;
```
```  1046     val info = #2 (the_inductive ctxt' name);
```
```  1047   in (info, Proof_Context.theory_of ctxt') end;
```
```  1048
```
```  1049
```
```  1050 (* read off arities of inductive predicates from raw induction rule *)
```
```  1051 fun arities_of induct =
```
```  1052   map (fn (_ \$ t \$ u) =>
```
```  1053       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
```
```  1054     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
```
```  1055
```
```  1056 (* read off parameters of inductive predicate from raw induction rule *)
```
```  1057 fun params_of induct =
```
```  1058   let
```
```  1059     val (_ \$ t \$ u :: _) = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
```
```  1060     val (_, ts) = strip_comb t;
```
```  1061     val (_, us) = strip_comb u;
```
```  1062   in
```
```  1063     List.take (ts, length ts - length us)
```
```  1064   end;
```
```  1065
```
```  1066 val pname_of_intr =
```
```  1067   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
```
```  1068
```
```  1069 (* partition introduction rules according to predicate name *)
```
```  1070 fun gen_partition_rules f induct intros =
```
```  1071   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
```
```  1072     (map (rpair [] o fst) (arities_of induct));
```
```  1073
```
```  1074 val partition_rules = gen_partition_rules I;
```
```  1075 fun partition_rules' induct = gen_partition_rules fst induct;
```
```  1076
```
```  1077 fun unpartition_rules intros xs =
```
```  1078   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
```
```  1079     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
```
```  1080
```
```  1081 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
```
```  1082 fun infer_intro_vars elim arity intros =
```
```  1083   let
```
```  1084     val thy = theory_of_thm elim;
```
```  1085     val _ :: cases = prems_of elim;
```
```  1086     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
```
```  1087     fun mtch (t, u) =
```
```  1088       let
```
```  1089         val params = Logic.strip_params t;
```
```  1090         val vars =
```
```  1091           map (Var o apfst (rpair 0))
```
```  1092             (Name.variant_list used (map fst params) ~~ map snd params);
```
```  1093         val ts =
```
```  1094           map (curry subst_bounds (rev vars))
```
```  1095             (List.drop (Logic.strip_assums_hyp t, arity));
```
```  1096         val us = Logic.strip_imp_prems u;
```
```  1097         val tab =
```
```  1098           fold (Pattern.first_order_match thy) (ts ~~ us) (Vartab.empty, Vartab.empty);
```
```  1099       in
```
```  1100         map (Envir.subst_term tab) vars
```
```  1101       end
```
```  1102   in
```
```  1103     map (mtch o apsnd prop_of) (cases ~~ intros)
```
```  1104   end;
```
```  1105
```
```  1106
```
```  1107
```
```  1108 (** package setup **)
```
```  1109
```
```  1110 (* setup theory *)
```
```  1111
```
```  1112 val setup =
```
```  1113   ind_cases_setup #>
```
```  1114   Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
```
```  1115     "declaration of monotonicity rule";
```
```  1116
```
```  1117
```
```  1118 (* outer syntax *)
```
```  1119
```
```  1120 val _ = Keyword.keyword "monos";
```
```  1121
```
```  1122 fun gen_ind_decl mk_def coind =
```
```  1123   Parse.fixes -- Parse.for_fixes --
```
```  1124   Scan.optional Parse_Spec.where_alt_specs [] --
```
```  1125   Scan.optional (Parse.\$\$\$ "monos" |-- Parse.!!! Parse_Spec.xthms1) []
```
```  1126   >> (fn (((preds, params), specs), monos) =>
```
```  1127       (snd oo gen_add_inductive mk_def true coind preds params specs monos));
```
```  1128
```
```  1129 val ind_decl = gen_ind_decl add_ind_def;
```
```  1130
```
```  1131 val _ =
```
```  1132   Outer_Syntax.local_theory' "inductive" "define inductive predicates" Keyword.thy_decl
```
```  1133     (ind_decl false);
```
```  1134
```
```  1135 val _ =
```
```  1136   Outer_Syntax.local_theory' "coinductive" "define coinductive predicates" Keyword.thy_decl
```
```  1137     (ind_decl true);
```
```  1138
```
```  1139 val _ =
```
```  1140   Outer_Syntax.local_theory "inductive_cases"
```
```  1141     "create simplified instances of elimination rules (improper)" Keyword.thy_script
```
```  1142     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
```
```  1143
```
```  1144 val _ =
```
```  1145   Outer_Syntax.local_theory "inductive_simps"
```
```  1146     "create simplification rules for inductive predicates" Keyword.thy_script
```
```  1147     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
```
```  1148
```
```  1149 end;
```