src/HOL/Tools/inductive.ML
 author bulwahn Fri Dec 03 08:40:47 2010 +0100 (2010-12-03) changeset 40902 7c652e4a924a parent 40316 665862241968 child 41075 4bed56dc95fb permissions -rw-r--r--
tuned
```     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 morph_result: morphism -> inductive_result -> inductive_result
```
```    27   type inductive_info = {names: string list, coind: bool} * inductive_result
```
```    28   val the_inductive: Proof.context -> string -> inductive_info
```
```    29   val print_inductives: Proof.context -> unit
```
```    30   val mono_add: attribute
```
```    31   val mono_del: attribute
```
```    32   val get_monos: Proof.context -> thm list
```
```    33   val mk_cases: Proof.context -> term -> thm
```
```    34   val inductive_forall_name: string
```
```    35   val inductive_forall_def: thm
```
```    36   val rulify: thm -> thm
```
```    37   val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
```
```    38     thm list list * local_theory
```
```    39   val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
```
```    40     thm list list * local_theory
```
```    41   type inductive_flags =
```
```    42     {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
```
```    43       no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool}
```
```    44   val add_inductive_i:
```
```    45     inductive_flags -> ((binding * typ) * mixfix) list ->
```
```    46     (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
```
```    47     inductive_result * local_theory
```
```    48   val add_inductive: bool -> bool ->
```
```    49     (binding * string option * mixfix) list ->
```
```    50     (binding * string option * mixfix) list ->
```
```    51     (Attrib.binding * string) list ->
```
```    52     (Facts.ref * Attrib.src list) list ->
```
```    53     bool -> local_theory -> inductive_result * local_theory
```
```    54   val add_inductive_global: inductive_flags ->
```
```    55     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
```
```    56     thm list -> theory -> inductive_result * theory
```
```    57   val arities_of: thm -> (string * int) list
```
```    58   val params_of: thm -> term list
```
```    59   val partition_rules: thm -> thm list -> (string * thm list) list
```
```    60   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
```
```    61   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
```
```    62   val infer_intro_vars: thm -> int -> thm list -> term list list
```
```    63   val setup: theory -> theory
```
```    64 end;
```
```    65
```
```    66 signature INDUCTIVE =
```
```    67 sig
```
```    68   include BASIC_INDUCTIVE
```
```    69   type add_ind_def =
```
```    70     inductive_flags ->
```
```    71     term list -> (Attrib.binding * term) list -> thm list ->
```
```    72     term list -> (binding * mixfix) list ->
```
```    73     local_theory -> inductive_result * local_theory
```
```    74   val declare_rules: binding -> bool -> bool -> string list -> term list ->
```
```    75     thm list -> binding list -> Attrib.src list list -> (thm * string list * int) list ->
```
```    76     thm list -> thm -> local_theory -> thm list * thm list * thm list * thm * thm list * local_theory
```
```    77   val add_ind_def: add_ind_def
```
```    78   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
```
```    79     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
```
```    80     thm list -> local_theory -> inductive_result * local_theory
```
```    81   val gen_add_inductive: add_ind_def -> bool -> bool ->
```
```    82     (binding * string option * mixfix) list ->
```
```    83     (binding * string option * mixfix) list ->
```
```    84     (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
```
```    85     bool -> local_theory -> inductive_result * local_theory
```
```    86   val gen_ind_decl: add_ind_def -> bool -> (bool -> local_theory -> local_theory) parser
```
```    87 end;
```
```    88
```
```    89 structure Inductive: INDUCTIVE =
```
```    90 struct
```
```    91
```
```    92
```
```    93 (** theory context references **)
```
```    94
```
```    95 val inductive_forall_name = "HOL.induct_forall";
```
```    96 val inductive_forall_def = @{thm induct_forall_def};
```
```    97 val inductive_conj_name = "HOL.induct_conj";
```
```    98 val inductive_conj_def = @{thm induct_conj_def};
```
```    99 val inductive_conj = @{thms induct_conj};
```
```   100 val inductive_atomize = @{thms induct_atomize};
```
```   101 val inductive_rulify = @{thms induct_rulify};
```
```   102 val inductive_rulify_fallback = @{thms induct_rulify_fallback};
```
```   103
```
```   104 val notTrueE = TrueI RSN (2, notE);
```
```   105 val notFalseI = Seq.hd (atac 1 notI);
```
```   106
```
```   107 val simp_thms' = map mk_meta_eq
```
```   108   @{lemma "(~True) = False" "(~False) = True"
```
```   109       "(True --> P) = P" "(False --> P) = True"
```
```   110       "(P & True) = P" "(True & P) = P"
```
```   111     by (fact simp_thms)+};
```
```   112
```
```   113 val simp_thms'' = map mk_meta_eq [@{thm inf_fun_eq}, @{thm inf_bool_eq}] @ simp_thms';
```
```   114
```
```   115 val simp_thms''' = map mk_meta_eq
```
```   116   [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_eq}, @{thm sup_bool_eq}];
```
```   117
```
```   118
```
```   119 (** context data **)
```
```   120
```
```   121 type inductive_result =
```
```   122   {preds: term list, elims: thm list, raw_induct: thm,
```
```   123    induct: thm, inducts: thm list, intrs: thm list, eqs: thm list};
```
```   124
```
```   125 fun morph_result phi {preds, elims, raw_induct: thm, induct, inducts, intrs, eqs} =
```
```   126   let
```
```   127     val term = Morphism.term phi;
```
```   128     val thm = Morphism.thm phi;
```
```   129     val fact = Morphism.fact phi;
```
```   130   in
```
```   131    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
```
```   132     induct = thm induct, inducts = fact inducts, intrs = fact intrs, eqs = fact eqs}
```
```   133   end;
```
```   134
```
```   135 type inductive_info =
```
```   136   {names: string list, coind: bool} * inductive_result;
```
```   137
```
```   138 structure InductiveData = Generic_Data
```
```   139 (
```
```   140   type T = inductive_info Symtab.table * thm list;
```
```   141   val empty = (Symtab.empty, []);
```
```   142   val extend = I;
```
```   143   fun merge ((tab1, monos1), (tab2, monos2)) : T =
```
```   144     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
```
```   145 );
```
```   146
```
```   147 val get_inductives = InductiveData.get o Context.Proof;
```
```   148
```
```   149 fun print_inductives ctxt =
```
```   150   let
```
```   151     val (tab, monos) = get_inductives ctxt;
```
```   152     val space = Consts.space_of (ProofContext.consts_of ctxt);
```
```   153   in
```
```   154     [Pretty.strs ("(co)inductives:" :: map #1 (Name_Space.extern_table (space, tab))),
```
```   155      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm ctxt) monos)]
```
```   156     |> Pretty.chunks |> Pretty.writeln
```
```   157   end;
```
```   158
```
```   159
```
```   160 (* get and put data *)
```
```   161
```
```   162 fun the_inductive ctxt name =
```
```   163   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
```
```   164     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
```
```   165   | SOME info => info);
```
```   166
```
```   167 fun put_inductives names info = InductiveData.map
```
```   168   (apfst (fold (fn name => Symtab.update (name, info)) names));
```
```   169
```
```   170
```
```   171
```
```   172 (** monotonicity rules **)
```
```   173
```
```   174 val get_monos = #2 o get_inductives;
```
```   175 val map_monos = InductiveData.map o apsnd;
```
```   176
```
```   177 fun mk_mono thm =
```
```   178   let
```
```   179     fun eq2mono thm' = thm' RS (thm' RS eq_to_mono);
```
```   180     fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
```
```   181       handle THM _ => thm RS @{thm le_boolD}
```
```   182   in
```
```   183     case concl_of thm of
```
```   184       Const ("==", _) \$ _ \$ _ => eq2mono (thm RS meta_eq_to_obj_eq)
```
```   185     | _ \$ (Const (@{const_name HOL.eq}, _) \$ _ \$ _) => eq2mono thm
```
```   186     | _ \$ (Const (@{const_name Orderings.less_eq}, _) \$ _ \$ _) =>
```
```   187       dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
```
```   188         (resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))
```
```   189     | _ => thm
```
```   190   end handle THM _ =>
```
```   191     error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm_without_context thm);
```
```   192
```
```   193 val mono_add = Thm.declaration_attribute (map_monos o Thm.add_thm o mk_mono);
```
```   194 val mono_del = Thm.declaration_attribute (map_monos o Thm.del_thm o mk_mono);
```
```   195
```
```   196
```
```   197
```
```   198 (** equations **)
```
```   199
```
```   200 structure Equation_Data = Generic_Data
```
```   201 (
```
```   202   type T = thm Item_Net.T;
```
```   203   val empty = Item_Net.init (op aconv o pairself Thm.prop_of)
```
```   204     (single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
```
```   205   val extend = I;
```
```   206   val merge = Item_Net.merge;
```
```   207 );
```
```   208
```
```   209 val add_equation = Thm.declaration_attribute (Equation_Data.map o Item_Net.update)
```
```   210
```
```   211
```
```   212
```
```   213 (** misc utilities **)
```
```   214
```
```   215 fun message quiet_mode s = if quiet_mode then () else writeln s;
```
```   216 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
```
```   217
```
```   218 fun coind_prefix true = "co"
```
```   219   | coind_prefix false = "";
```
```   220
```
```   221 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
```
```   222
```
```   223 fun make_bool_args f g [] i = []
```
```   224   | make_bool_args f g (x :: xs) i =
```
```   225       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
```
```   226
```
```   227 fun make_bool_args' xs =
```
```   228   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
```
```   229
```
```   230 fun arg_types_of k c = drop k (binder_types (fastype_of c));
```
```   231
```
```   232 fun find_arg T x [] = raise Fail "find_arg"
```
```   233   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
```
```   234       apsnd (cons p) (find_arg T x ps)
```
```   235   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
```
```   236       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
```
```   237       else apsnd (cons p) (find_arg T x ps);
```
```   238
```
```   239 fun make_args Ts xs =
```
```   240   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
```
```   241     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
```
```   242
```
```   243 fun make_args' Ts xs Us =
```
```   244   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
```
```   245
```
```   246 fun dest_predicate cs params t =
```
```   247   let
```
```   248     val k = length params;
```
```   249     val (c, ts) = strip_comb t;
```
```   250     val (xs, ys) = chop k ts;
```
```   251     val i = find_index (fn c' => c' = c) cs;
```
```   252   in
```
```   253     if xs = params andalso i >= 0 then
```
```   254       SOME (c, i, ys, chop (length ys) (arg_types_of k c))
```
```   255     else NONE
```
```   256   end;
```
```   257
```
```   258 fun mk_names a 0 = []
```
```   259   | mk_names a 1 = [a]
```
```   260   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
```
```   261
```
```   262 fun select_disj 1 1 = []
```
```   263   | select_disj _ 1 = [rtac disjI1]
```
```   264   | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
```
```   265
```
```   266
```
```   267 (** process rules **)
```
```   268
```
```   269 local
```
```   270
```
```   271 fun err_in_rule ctxt name t msg =
```
```   272   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
```
```   273     Syntax.string_of_term ctxt t, msg]);
```
```   274
```
```   275 fun err_in_prem ctxt name t p msg =
```
```   276   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
```
```   277     "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
```
```   278
```
```   279 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
```
```   280
```
```   281 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
```
```   282
```
```   283 val bad_app = "Inductive predicate must be applied to parameter(s) ";
```
```   284
```
```   285 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
```
```   286
```
```   287 in
```
```   288
```
```   289 fun check_rule ctxt cs params ((binding, att), rule) =
```
```   290   let
```
```   291     val err_name = Binding.str_of binding;
```
```   292     val params' = Term.variant_frees rule (Logic.strip_params rule);
```
```   293     val frees = rev (map Free params');
```
```   294     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
```
```   295     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
```
```   296     val rule' = Logic.list_implies (prems, concl);
```
```   297     val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
```
```   298     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
```
```   299
```
```   300     fun check_ind err t = case dest_predicate cs params t of
```
```   301         NONE => err (bad_app ^
```
```   302           commas (map (Syntax.string_of_term ctxt) params))
```
```   303       | SOME (_, _, ys, _) =>
```
```   304           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
```
```   305           then err bad_ind_occ else ();
```
```   306
```
```   307     fun check_prem' prem t =
```
```   308       if member (op =) cs (head_of t) then
```
```   309         check_ind (err_in_prem ctxt err_name rule prem) t
```
```   310       else (case t of
```
```   311           Abs (_, _, t) => check_prem' prem t
```
```   312         | t \$ u => (check_prem' prem t; check_prem' prem u)
```
```   313         | _ => ());
```
```   314
```
```   315     fun check_prem (prem, aprem) =
```
```   316       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
```
```   317       else err_in_prem ctxt err_name rule prem "Non-atomic premise";
```
```   318   in
```
```   319     (case concl of
```
```   320        Const (@{const_name Trueprop}, _) \$ t =>
```
```   321          if member (op =) cs (head_of t) then
```
```   322            (check_ind (err_in_rule ctxt err_name rule') t;
```
```   323             List.app check_prem (prems ~~ aprems))
```
```   324          else err_in_rule ctxt err_name rule' bad_concl
```
```   325      | _ => err_in_rule ctxt err_name rule' bad_concl);
```
```   326     ((binding, att), arule)
```
```   327   end;
```
```   328
```
```   329 val rulify =
```
```   330   hol_simplify inductive_conj
```
```   331   #> hol_simplify inductive_rulify
```
```   332   #> hol_simplify inductive_rulify_fallback
```
```   333   #> Simplifier.norm_hhf;
```
```   334
```
```   335 end;
```
```   336
```
```   337
```
```   338
```
```   339 (** proofs for (co)inductive predicates **)
```
```   340
```
```   341 (* prove monotonicity *)
```
```   342
```
```   343 fun prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt =
```
```   344  (message (quiet_mode orelse skip_mono andalso !quick_and_dirty orelse fork_mono)
```
```   345     "  Proving monotonicity ...";
```
```   346   (if skip_mono then Skip_Proof.prove else if fork_mono then Goal.prove_future else Goal.prove) ctxt
```
```   347     [] []
```
```   348     (HOLogic.mk_Trueprop
```
```   349       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) \$ fp_fun))
```
```   350     (fn _ => EVERY [rtac @{thm monoI} 1,
```
```   351       REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
```
```   352       REPEAT (FIRST
```
```   353         [atac 1,
```
```   354          resolve_tac (map mk_mono monos @ get_monos ctxt) 1,
```
```   355          etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
```
```   356
```
```   357
```
```   358 (* prove introduction rules *)
```
```   359
```
```   360 fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
```
```   361   let
```
```   362     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
```
```   363
```
```   364     val unfold = funpow k (fn th => th RS fun_cong)
```
```   365       (mono RS (fp_def RS
```
```   366         (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
```
```   367
```
```   368     val rules = [refl, TrueI, notFalseI, exI, conjI];
```
```   369
```
```   370     val intrs = map_index (fn (i, intr) =>
```
```   371       Skip_Proof.prove ctxt [] [] intr (fn _ => EVERY
```
```   372        [rewrite_goals_tac rec_preds_defs,
```
```   373         rtac (unfold RS iffD2) 1,
```
```   374         EVERY1 (select_disj (length intr_ts) (i + 1)),
```
```   375         (*Not ares_tac, since refl must be tried before any equality assumptions;
```
```   376           backtracking may occur if the premises have extra variables!*)
```
```   377         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)])
```
```   378        |> singleton (ProofContext.export ctxt ctxt')) intr_ts
```
```   379
```
```   380   in (intrs, unfold) end;
```
```   381
```
```   382
```
```   383 (* prove elimination rules *)
```
```   384
```
```   385 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
```
```   386   let
```
```   387     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
```
```   388
```
```   389     val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
```
```   390     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
```
```   391
```
```   392     fun dest_intr r =
```
```   393       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
```
```   394        Logic.strip_assums_hyp r, Logic.strip_params r);
```
```   395
```
```   396     val intrs = map dest_intr intr_ts ~~ intr_names;
```
```   397
```
```   398     val rules1 = [disjE, exE, FalseE];
```
```   399     val rules2 = [conjE, FalseE, notTrueE];
```
```   400
```
```   401     fun prove_elim c =
```
```   402       let
```
```   403         val Ts = arg_types_of (length params) c;
```
```   404         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
```
```   405         val frees = map Free (anames ~~ Ts);
```
```   406
```
```   407         fun mk_elim_prem ((_, _, us, _), ts, params') =
```
```   408           list_all (params',
```
```   409             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
```
```   410               (frees ~~ us) @ ts, P));
```
```   411         val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
```
```   412         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
```
```   413            map mk_elim_prem (map #1 c_intrs)
```
```   414       in
```
```   415         (Skip_Proof.prove ctxt'' [] prems P
```
```   416           (fn {prems, ...} => EVERY
```
```   417             [cut_facts_tac [hd prems] 1,
```
```   418              rewrite_goals_tac rec_preds_defs,
```
```   419              dtac (unfold RS iffD1) 1,
```
```   420              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
```
```   421              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
```
```   422              EVERY (map (fn prem =>
```
```   423                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
```
```   424           |> singleton (ProofContext.export ctxt'' ctxt'''),
```
```   425          map #2 c_intrs, length Ts)
```
```   426       end
```
```   427
```
```   428    in map prove_elim cs end;
```
```   429
```
```   430 (* prove simplification equations *)
```
```   431
```
```   432 fun prove_eqs quiet_mode cs params intr_ts intrs (elims: (thm * bstring list * int) list) ctxt ctxt'' =
```
```   433   let
```
```   434     val _ = clean_message quiet_mode "  Proving the simplification rules ...";
```
```   435
```
```   436     fun dest_intr r =
```
```   437       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
```
```   438        Logic.strip_assums_hyp r, Logic.strip_params r);
```
```   439     val intr_ts' = map dest_intr intr_ts;
```
```   440     fun prove_eq c (elim: thm * 'a * 'b) =
```
```   441       let
```
```   442         val Ts = arg_types_of (length params) c;
```
```   443         val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
```
```   444         val frees = map Free (anames ~~ Ts);
```
```   445         val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
```
```   446         fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
```
```   447           let
```
```   448             fun list_ex ([], t) = t
```
```   449               | list_ex ((a,T)::vars, t) =
```
```   450                  (HOLogic.exists_const T) \$ (Abs(a, T, list_ex(vars,t)));
```
```   451             val conjs = map2 (curry HOLogic.mk_eq) frees us @ (map HOLogic.dest_Trueprop ts)
```
```   452           in
```
```   453             list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
```
```   454           end;
```
```   455         val lhs = list_comb (c, params @ frees)
```
```   456         val rhs =
```
```   457           if null c_intrs then @{term False} else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs)
```
```   458         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
```
```   459         fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
```
```   460             let
```
```   461               val (prems', last_prem) = split_last prems
```
```   462             in
```
```   463               EVERY1 (select_disj (length c_intrs) (i + 1))
```
```   464               THEN EVERY (replicate (length params) (rtac @{thm exI} 1))
```
```   465               THEN EVERY (map (fn prem => (rtac @{thm conjI} 1 THEN rtac prem 1)) prems')
```
```   466               THEN rtac last_prem 1
```
```   467             end) ctxt' 1
```
```   468         fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
```
```   469           EVERY (replicate (length params') (etac @{thm exE} 1))
```
```   470           THEN EVERY (replicate (length ts + length us - 1) (etac @{thm conjE} 1))
```
```   471           THEN Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
```
```   472             let
```
```   473               val (eqs, prems') = chop (length us) prems
```
```   474               val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs
```
```   475             in
```
```   476               rewrite_goal_tac rew_thms 1
```
```   477               THEN rtac intr 1
```
```   478               THEN (EVERY (map (fn p => rtac p 1) prems'))
```
```   479             end) ctxt' 1
```
```   480       in
```
```   481         Skip_Proof.prove ctxt' [] [] eq (fn {...} =>
```
```   482           rtac @{thm iffI} 1 THEN etac (#1 elim) 1
```
```   483           THEN EVERY (map_index prove_intr1 c_intrs)
```
```   484           THEN (if null c_intrs then etac @{thm FalseE} 1 else
```
```   485             let val (c_intrs', last_c_intr) = split_last c_intrs in
```
```   486               EVERY (map (fn ci => etac @{thm disjE} 1 THEN prove_intr2 ci) c_intrs')
```
```   487               THEN prove_intr2 last_c_intr
```
```   488             end))
```
```   489         |> rulify
```
```   490         |> singleton (ProofContext.export ctxt' ctxt'')
```
```   491       end;
```
```   492   in
```
```   493     map2 prove_eq cs elims
```
```   494   end;
```
```   495
```
```   496 (* derivation of simplified elimination rules *)
```
```   497
```
```   498 local
```
```   499
```
```   500 (*delete needless equality assumptions*)
```
```   501 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
```
```   502   (fn _ => assume_tac 1);
```
```   503 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
```
```   504 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
```
```   505
```
```   506 fun simp_case_tac ss i =
```
```   507   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
```
```   508
```
```   509 in
```
```   510
```
```   511 fun mk_cases ctxt prop =
```
```   512   let
```
```   513     val thy = ProofContext.theory_of ctxt;
```
```   514     val ss = simpset_of ctxt;
```
```   515
```
```   516     fun err msg =
```
```   517       error (Pretty.string_of (Pretty.block
```
```   518         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
```
```   519
```
```   520     val elims = Induct.find_casesP ctxt prop;
```
```   521
```
```   522     val cprop = Thm.cterm_of thy prop;
```
```   523     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
```
```   524     fun mk_elim rl =
```
```   525       Thm.implies_intr cprop (Tactic.rule_by_tactic ctxt tac (Thm.assume cprop RS rl))
```
```   526       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
```
```   527   in
```
```   528     (case get_first (try mk_elim) elims of
```
```   529       SOME r => r
```
```   530     | NONE => err "Proposition not an inductive predicate:")
```
```   531   end;
```
```   532
```
```   533 end;
```
```   534
```
```   535 (* inductive_cases *)
```
```   536
```
```   537 fun gen_inductive_cases prep_att prep_prop args lthy =
```
```   538   let
```
```   539     val thy = ProofContext.theory_of lthy;
```
```   540     val facts = args |> Par_List.map (fn ((a, atts), props) =>
```
```   541       ((a, map (prep_att thy) atts),
```
```   542         Par_List.map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
```
```   543   in lthy |> Local_Theory.notes facts |>> map snd end;
```
```   544
```
```   545 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
```
```   546 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
```
```   547
```
```   548
```
```   549 val ind_cases_setup =
```
```   550   Method.setup @{binding ind_cases}
```
```   551     (Scan.lift (Scan.repeat1 Args.name_source --
```
```   552       Scan.optional (Args.\$\$\$ "for" |-- Scan.repeat1 Args.name) []) >>
```
```   553       (fn (raw_props, fixes) => fn ctxt =>
```
```   554         let
```
```   555           val (_, ctxt') = Variable.add_fixes fixes ctxt;
```
```   556           val props = Syntax.read_props ctxt' raw_props;
```
```   557           val ctxt'' = fold Variable.declare_term props ctxt';
```
```   558           val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props)
```
```   559         in Method.erule 0 rules end))
```
```   560     "dynamic case analysis on predicates";
```
```   561
```
```   562 (* derivation of simplified equation *)
```
```   563
```
```   564 fun mk_simp_eq ctxt prop =
```
```   565   let
```
```   566     val thy = ProofContext.theory_of ctxt
```
```   567     val ctxt' = Variable.auto_fixes prop ctxt
```
```   568     val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of
```
```   569     val substs = Item_Net.retrieve (Equation_Data.get (Context.Proof ctxt)) (HOLogic.dest_Trueprop prop)
```
```   570       |> map_filter
```
```   571         (fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
```
```   572             (Vartab.empty, Vartab.empty), eq)
```
```   573           handle Pattern.MATCH => NONE)
```
```   574     val (subst, eq) = case substs of
```
```   575         [s] => s
```
```   576       | _ => error
```
```   577         ("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique")
```
```   578     val inst = map (fn v => (cterm_of thy (Var v), cterm_of thy (Envir.subst_term subst (Var v))))
```
```   579       (Term.add_vars (lhs_of eq) [])
```
```   580    in
```
```   581     cterm_instantiate inst eq
```
```   582     |> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv
```
```   583       (Simplifier.full_rewrite (simpset_of ctxt))))
```
```   584     |> singleton (Variable.export ctxt' ctxt)
```
```   585   end
```
```   586
```
```   587 (* inductive simps *)
```
```   588
```
```   589 fun gen_inductive_simps prep_att prep_prop args lthy =
```
```   590   let
```
```   591     val thy = ProofContext.theory_of lthy;
```
```   592     val facts = args |> map (fn ((a, atts), props) =>
```
```   593       ((a, map (prep_att thy) atts),
```
```   594         map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
```
```   595   in lthy |> Local_Theory.notes facts |>> map snd end;
```
```   596
```
```   597 val inductive_simps = gen_inductive_simps Attrib.intern_src Syntax.read_prop;
```
```   598 val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
```
```   599
```
```   600 (* prove induction rule *)
```
```   601
```
```   602 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
```
```   603     fp_def rec_preds_defs ctxt ctxt''' =
```
```   604   let
```
```   605     val _ = clean_message quiet_mode "  Proving the induction rule ...";
```
```   606     val thy = ProofContext.theory_of ctxt;
```
```   607
```
```   608     (* predicates for induction rule *)
```
```   609
```
```   610     val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
```
```   611     val preds = map2 (curry Free) pnames
```
```   612       (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
```
```   613
```
```   614     (* transform an introduction rule into a premise for induction rule *)
```
```   615
```
```   616     fun mk_ind_prem r =
```
```   617       let
```
```   618         fun subst s =
```
```   619           (case dest_predicate cs params s of
```
```   620             SOME (_, i, ys, (_, Ts)) =>
```
```   621               let
```
```   622                 val k = length Ts;
```
```   623                 val bs = map Bound (k - 1 downto 0);
```
```   624                 val P = list_comb (List.nth (preds, i),
```
```   625                   map (incr_boundvars k) ys @ bs);
```
```   626                 val Q = list_abs (mk_names "x" k ~~ Ts,
```
```   627                   HOLogic.mk_binop inductive_conj_name
```
```   628                     (list_comb (incr_boundvars k s, bs), P))
```
```   629               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
```
```   630           | NONE =>
```
```   631               (case s of
```
```   632                 (t \$ u) => (fst (subst t) \$ fst (subst u), NONE)
```
```   633               | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
```
```   634               | _ => (s, NONE)));
```
```   635
```
```   636         fun mk_prem s prems =
```
```   637           (case subst s of
```
```   638             (_, SOME (t, u)) => t :: u :: prems
```
```   639           | (t, _) => t :: prems);
```
```   640
```
```   641         val SOME (_, i, ys, _) = dest_predicate cs params
```
```   642           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
```
```   643
```
```   644       in list_all_free (Logic.strip_params r,
```
```   645         Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
```
```   646           (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
```
```   647             HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
```
```   648       end;
```
```   649
```
```   650     val ind_prems = map mk_ind_prem intr_ts;
```
```   651
```
```   652
```
```   653     (* make conclusions for induction rules *)
```
```   654
```
```   655     val Tss = map (binder_types o fastype_of) preds;
```
```   656     val (xnames, ctxt'') =
```
```   657       Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
```
```   658     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
```
```   659         (map (fn (((xnames, Ts), c), P) =>
```
```   660            let val frees = map Free (xnames ~~ Ts)
```
```   661            in HOLogic.mk_imp
```
```   662              (list_comb (c, params @ frees), list_comb (P, frees))
```
```   663            end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
```
```   664
```
```   665
```
```   666     (* make predicate for instantiation of abstract induction rule *)
```
```   667
```
```   668     val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
```
```   669       (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
```
```   670          (make_bool_args HOLogic.mk_not I bs i)
```
```   671          (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
```
```   672
```
```   673     val ind_concl = HOLogic.mk_Trueprop
```
```   674       (HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
```
```   675
```
```   676     val raw_fp_induct = (mono RS (fp_def RS @{thm def_lfp_induct}));
```
```   677
```
```   678     val induct = Skip_Proof.prove ctxt'' [] ind_prems ind_concl
```
```   679       (fn {prems, ...} => EVERY
```
```   680         [rewrite_goals_tac [inductive_conj_def],
```
```   681          DETERM (rtac raw_fp_induct 1),
```
```   682          REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
```
```   683          rewrite_goals_tac simp_thms'',
```
```   684          (*This disjE separates out the introduction rules*)
```
```   685          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
```
```   686          (*Now break down the individual cases.  No disjE here in case
```
```   687            some premise involves disjunction.*)
```
```   688          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
```
```   689          REPEAT (FIRSTGOAL
```
```   690            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
```
```   691          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
```
```   692              (inductive_conj_def :: rec_preds_defs @ simp_thms'') prem,
```
```   693            conjI, refl] 1)) prems)]);
```
```   694
```
```   695     val lemma = Skip_Proof.prove ctxt'' [] []
```
```   696       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
```
```   697         [rewrite_goals_tac rec_preds_defs,
```
```   698          REPEAT (EVERY
```
```   699            [REPEAT (resolve_tac [conjI, impI] 1),
```
```   700             REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
```
```   701             atac 1,
```
```   702             rewrite_goals_tac simp_thms',
```
```   703             atac 1])])
```
```   704
```
```   705   in singleton (ProofContext.export ctxt'' ctxt''') (induct RS lemma) end;
```
```   706
```
```   707
```
```   708
```
```   709 (** specification of (co)inductive predicates **)
```
```   710
```
```   711 fun mk_ind_def quiet_mode skip_mono fork_mono alt_name coind
```
```   712     cs intr_ts monos params cnames_syn lthy =
```
```   713   let
```
```   714     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
```
```   715
```
```   716     val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
```
```   717     val k = log 2 1 (length cs);
```
```   718     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
```
```   719     val p :: xs = map Free (Variable.variant_frees lthy intr_ts
```
```   720       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
```
```   721     val bs = map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
```
```   722       (map (rpair HOLogic.boolT) (mk_names "b" k)));
```
```   723
```
```   724     fun subst t =
```
```   725       (case dest_predicate cs params t of
```
```   726         SOME (_, i, ts, (Ts, Us)) =>
```
```   727           let
```
```   728             val l = length Us;
```
```   729             val zs = map Bound (l - 1 downto 0);
```
```   730           in
```
```   731             list_abs (map (pair "z") Us, list_comb (p,
```
```   732               make_bool_args' bs i @ make_args argTs
```
```   733                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
```
```   734           end
```
```   735       | NONE =>
```
```   736           (case t of
```
```   737             t1 \$ t2 => subst t1 \$ subst t2
```
```   738           | Abs (x, T, u) => Abs (x, T, subst u)
```
```   739           | _ => t));
```
```   740
```
```   741     (* transform an introduction rule into a conjunction  *)
```
```   742     (*   [| p_i t; ... |] ==> p_j u                       *)
```
```   743     (* is transformed into                                *)
```
```   744     (*   b_j & x_j = u & p b_j t & ...                    *)
```
```   745
```
```   746     fun transform_rule r =
```
```   747       let
```
```   748         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
```
```   749           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
```
```   750         val ps = make_bool_args HOLogic.mk_not I bs i @
```
```   751           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
```
```   752           map (subst o HOLogic.dest_Trueprop)
```
```   753             (Logic.strip_assums_hyp r)
```
```   754       in
```
```   755         fold_rev (fn (x, T) => fn P => HOLogic.exists_const T \$ Abs (x, T, P))
```
```   756           (Logic.strip_params r)
```
```   757           (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
```
```   758       end
```
```   759
```
```   760     (* make a disjunction of all introduction rules *)
```
```   761
```
```   762     val fp_fun = fold_rev lambda (p :: bs @ xs)
```
```   763       (if null intr_ts then HOLogic.false_const
```
```   764        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
```
```   765
```
```   766     (* add definiton of recursive predicates to theory *)
```
```   767
```
```   768     val rec_name =
```
```   769       if Binding.is_empty alt_name then
```
```   770         Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
```
```   771       else alt_name;
```
```   772
```
```   773     val ((rec_const, (_, fp_def)), lthy') = lthy
```
```   774       |> Local_Theory.conceal
```
```   775       |> Local_Theory.define
```
```   776         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
```
```   777          ((Binding.empty, [Attrib.internal (K Nitpick_Defs.add)]),
```
```   778          fold_rev lambda params
```
```   779            (Const (fp_name, (predT --> predT) --> predT) \$ fp_fun)))
```
```   780       ||> Local_Theory.restore_naming lthy;
```
```   781     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
```
```   782       (cterm_of (ProofContext.theory_of lthy') (list_comb (rec_const, params)));
```
```   783     val specs =
```
```   784       if length cs < 2 then []
```
```   785       else
```
```   786         map_index (fn (i, (name_mx, c)) =>
```
```   787           let
```
```   788             val Ts = arg_types_of (length params) c;
```
```   789             val xs = map Free (Variable.variant_frees lthy intr_ts
```
```   790               (mk_names "x" (length Ts) ~~ Ts))
```
```   791           in
```
```   792             (name_mx, (apfst Binding.conceal Attrib.empty_binding, fold_rev lambda (params @ xs)
```
```   793               (list_comb (rec_const, params @ make_bool_args' bs i @
```
```   794                 make_args argTs (xs ~~ Ts)))))
```
```   795           end) (cnames_syn ~~ cs);
```
```   796     val (consts_defs, lthy'') = lthy'
```
```   797       |> fold_map Local_Theory.define specs;
```
```   798     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
```
```   799
```
```   800     val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
```
```   801     val mono = prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos lthy''';
```
```   802     val (_, lthy'''') =
```
```   803       Local_Theory.note (apfst Binding.conceal Attrib.empty_binding,
```
```   804         ProofContext.export lthy''' lthy'' [mono]) lthy'';
```
```   805
```
```   806   in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
```
```   807     list_comb (rec_const, params), preds, argTs, bs, xs)
```
```   808   end;
```
```   809
```
```   810 fun declare_rules rec_binding coind no_ind cnames
```
```   811     preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
```
```   812   let
```
```   813     val rec_name = Binding.name_of rec_binding;
```
```   814     fun rec_qualified qualified = Binding.qualify qualified rec_name;
```
```   815     val intr_names = map Binding.name_of intr_bindings;
```
```   816     val ind_case_names = Rule_Cases.case_names intr_names;
```
```   817     val induct =
```
```   818       if coind then
```
```   819         (raw_induct, [Rule_Cases.case_names [rec_name],
```
```   820           Rule_Cases.case_conclusion (rec_name, intr_names),
```
```   821           Rule_Cases.consumes 1, Induct.coinduct_pred (hd cnames)])
```
```   822       else if no_ind orelse length cnames > 1 then
```
```   823         (raw_induct, [ind_case_names, Rule_Cases.consumes 0])
```
```   824       else (raw_induct RSN (2, rev_mp), [ind_case_names, Rule_Cases.consumes 1]);
```
```   825
```
```   826     val (intrs', lthy1) =
```
```   827       lthy |>
```
```   828       Spec_Rules.add
```
```   829         (if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
```
```   830       Local_Theory.notes
```
```   831         (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
```
```   832           map (fn th => [([th],
```
```   833            [Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
```
```   834       map (hd o snd);
```
```   835     val (((_, elims'), (_, [induct'])), lthy2) =
```
```   836       lthy1 |>
```
```   837       Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
```
```   838       fold_map (fn (name, (elim, cases, k)) =>
```
```   839         Local_Theory.note
```
```   840           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
```
```   841             [Attrib.internal (K (Rule_Cases.case_names cases)),
```
```   842              Attrib.internal (K (Rule_Cases.consumes 1)),
```
```   843              Attrib.internal (K (Rule_Cases.constraints k)),
```
```   844              Attrib.internal (K (Induct.cases_pred name)),
```
```   845              Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
```
```   846         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
```
```   847       Local_Theory.note
```
```   848         ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
```
```   849           map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
```
```   850
```
```   851     val (eqs', lthy3) = lthy2 |>
```
```   852       fold_map (fn (name, eq) => Local_Theory.note
```
```   853           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
```
```   854             [Attrib.internal (K add_equation)]), [eq])
```
```   855           #> apfst (hd o snd))
```
```   856         (if null eqs then [] else (cnames ~~ eqs))
```
```   857     val (inducts, lthy4) =
```
```   858       if no_ind orelse coind then ([], lthy3)
```
```   859       else
```
```   860         let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
```
```   861           lthy3 |>
```
```   862           Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
```
```   863             inducts |> map (fn (name, th) => ([th],
```
```   864               [Attrib.internal (K ind_case_names),
```
```   865                Attrib.internal (K (Rule_Cases.consumes 1)),
```
```   866                Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
```
```   867         end;
```
```   868   in (intrs', elims', eqs', induct', inducts, lthy4) end;
```
```   869
```
```   870 type inductive_flags =
```
```   871   {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
```
```   872     no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool};
```
```   873
```
```   874 type add_ind_def =
```
```   875   inductive_flags ->
```
```   876   term list -> (Attrib.binding * term) list -> thm list ->
```
```   877   term list -> (binding * mixfix) list ->
```
```   878   local_theory -> inductive_result * local_theory;
```
```   879
```
```   880 fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}
```
```   881     cs intros monos params cnames_syn lthy =
```
```   882   let
```
```   883     val _ = null cnames_syn andalso error "No inductive predicates given";
```
```   884     val names = map (Binding.name_of o fst) cnames_syn;
```
```   885     val _ = message (quiet_mode andalso not verbose)
```
```   886       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
```
```   887
```
```   888     val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
```
```   889     val ((intr_names, intr_atts), intr_ts) =
```
```   890       apfst split_list (split_list (map (check_rule lthy cs params) intros));
```
```   891
```
```   892     val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
```
```   893       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts
```
```   894         monos params cnames_syn lthy;
```
```   895
```
```   896     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
```
```   897       intr_ts rec_preds_defs lthy2 lthy1;
```
```   898     val elims =
```
```   899       if no_elim then []
```
```   900       else
```
```   901         prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
```
```   902           unfold rec_preds_defs lthy2 lthy1;
```
```   903     val raw_induct = zero_var_indexes
```
```   904       (if no_ind then Drule.asm_rl
```
```   905        else if coind then
```
```   906          singleton (ProofContext.export lthy2 lthy1)
```
```   907            (rotate_prems ~1 (Object_Logic.rulify
```
```   908              (fold_rule rec_preds_defs
```
```   909                (rewrite_rule simp_thms'''
```
```   910                 (mono RS (fp_def RS @{thm def_coinduct}))))))
```
```   911        else
```
```   912          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
```
```   913            rec_preds_defs lthy2 lthy1);
```
```   914     val eqs =
```
```   915       if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1
```
```   916
```
```   917     val elims' = map (fn (th, ns, i) => (rulify th, ns, i)) elims
```
```   918     val intrs' = map rulify intrs
```
```   919
```
```   920     val (intrs'', elims'', eqs', induct, inducts, lthy3) = declare_rules rec_name coind no_ind
```
```   921       cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
```
```   922
```
```   923     val result =
```
```   924       {preds = preds,
```
```   925        intrs = intrs'',
```
```   926        elims = elims'',
```
```   927        raw_induct = rulify raw_induct,
```
```   928        induct = induct,
```
```   929        inducts = inducts,
```
```   930        eqs = eqs'};
```
```   931
```
```   932     val lthy4 = lthy3
```
```   933       |> Local_Theory.declaration false (fn phi =>
```
```   934         let val result' = morph_result phi result;
```
```   935         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
```
```   936   in (result, lthy4) end;
```
```   937
```
```   938
```
```   939 (* external interfaces *)
```
```   940
```
```   941 fun gen_add_inductive_i mk_def
```
```   942     (flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono})
```
```   943     cnames_syn pnames spec monos lthy =
```
```   944   let
```
```   945     val thy = ProofContext.theory_of lthy;
```
```   946     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
```
```   947
```
```   948
```
```   949     (* abbrevs *)
```
```   950
```
```   951     val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
```
```   952
```
```   953     fun get_abbrev ((name, atts), t) =
```
```   954       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
```
```   955         let
```
```   956           val _ = Binding.is_empty name andalso null atts orelse
```
```   957             error "Abbreviations may not have names or attributes";
```
```   958           val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
```
```   959           val var =
```
```   960             (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
```
```   961               NONE => error ("Undeclared head of abbreviation " ^ quote x)
```
```   962             | SOME ((b, T'), mx) =>
```
```   963                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
```
```   964                 else (b, mx));
```
```   965         in SOME (var, rhs) end
```
```   966       else NONE;
```
```   967
```
```   968     val abbrevs = map_filter get_abbrev spec;
```
```   969     val bs = map (Binding.name_of o fst o fst) abbrevs;
```
```   970
```
```   971
```
```   972     (* predicates *)
```
```   973
```
```   974     val pre_intros = filter_out (is_some o get_abbrev) spec;
```
```   975     val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
```
```   976     val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
```
```   977     val ps = map Free pnames;
```
```   978
```
```   979     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
```
```   980     val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
```
```   981     val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
```
```   982     val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy;
```
```   983
```
```   984     fun close_rule r = list_all_free (rev (fold_aterms
```
```   985       (fn t as Free (v as (s, _)) =>
```
```   986           if Variable.is_fixed ctxt1 s orelse
```
```   987             member (op =) ps t then I else insert (op =) v
```
```   988         | _ => I) r []), r);
```
```   989
```
```   990     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
```
```   991     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
```
```   992   in
```
```   993     lthy
```
```   994     |> mk_def flags cs intros monos ps preds
```
```   995     ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
```
```   996   end;
```
```   997
```
```   998 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos int lthy =
```
```   999   let
```
```  1000     val ((vars, intrs), _) = lthy
```
```  1001       |> ProofContext.set_mode ProofContext.mode_abbrev
```
```  1002       |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
```
```  1003     val (cs, ps) = chop (length cnames_syn) vars;
```
```  1004     val monos = Attrib.eval_thms lthy raw_monos;
```
```  1005     val flags = {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
```
```  1006       coind = coind, no_elim = false, no_ind = false, skip_mono = false, fork_mono = not int};
```
```  1007   in
```
```  1008     lthy
```
```  1009     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
```
```  1010   end;
```
```  1011
```
```  1012 val add_inductive_i = gen_add_inductive_i add_ind_def;
```
```  1013 val add_inductive = gen_add_inductive add_ind_def;
```
```  1014
```
```  1015 fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
```
```  1016   let
```
```  1017     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
```
```  1018     val ctxt' = thy
```
```  1019       |> Named_Target.theory_init
```
```  1020       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
```
```  1021       |> Local_Theory.exit;
```
```  1022     val info = #2 (the_inductive ctxt' name);
```
```  1023   in (info, ProofContext.theory_of ctxt') end;
```
```  1024
```
```  1025
```
```  1026 (* read off arities of inductive predicates from raw induction rule *)
```
```  1027 fun arities_of induct =
```
```  1028   map (fn (_ \$ t \$ u) =>
```
```  1029       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
```
```  1030     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
```
```  1031
```
```  1032 (* read off parameters of inductive predicate from raw induction rule *)
```
```  1033 fun params_of induct =
```
```  1034   let
```
```  1035     val (_ \$ t \$ u :: _) =
```
```  1036       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
```
```  1037     val (_, ts) = strip_comb t;
```
```  1038     val (_, us) = strip_comb u
```
```  1039   in
```
```  1040     List.take (ts, length ts - length us)
```
```  1041   end;
```
```  1042
```
```  1043 val pname_of_intr =
```
```  1044   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
```
```  1045
```
```  1046 (* partition introduction rules according to predicate name *)
```
```  1047 fun gen_partition_rules f induct intros =
```
```  1048   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
```
```  1049     (map (rpair [] o fst) (arities_of induct));
```
```  1050
```
```  1051 val partition_rules = gen_partition_rules I;
```
```  1052 fun partition_rules' induct = gen_partition_rules fst induct;
```
```  1053
```
```  1054 fun unpartition_rules intros xs =
```
```  1055   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
```
```  1056     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
```
```  1057
```
```  1058 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
```
```  1059 fun infer_intro_vars elim arity intros =
```
```  1060   let
```
```  1061     val thy = theory_of_thm elim;
```
```  1062     val _ :: cases = prems_of elim;
```
```  1063     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
```
```  1064     fun mtch (t, u) =
```
```  1065       let
```
```  1066         val params = Logic.strip_params t;
```
```  1067         val vars = map (Var o apfst (rpair 0))
```
```  1068           (Name.variant_list used (map fst params) ~~ map snd params);
```
```  1069         val ts = map (curry subst_bounds (rev vars))
```
```  1070           (List.drop (Logic.strip_assums_hyp t, arity));
```
```  1071         val us = Logic.strip_imp_prems u;
```
```  1072         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
```
```  1073           (Vartab.empty, Vartab.empty);
```
```  1074       in
```
```  1075         map (Envir.subst_term tab) vars
```
```  1076       end
```
```  1077   in
```
```  1078     map (mtch o apsnd prop_of) (cases ~~ intros)
```
```  1079   end;
```
```  1080
```
```  1081
```
```  1082
```
```  1083 (** package setup **)
```
```  1084
```
```  1085 (* setup theory *)
```
```  1086
```
```  1087 val setup =
```
```  1088   ind_cases_setup #>
```
```  1089   Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
```
```  1090     "declaration of monotonicity rule";
```
```  1091
```
```  1092
```
```  1093 (* outer syntax *)
```
```  1094
```
```  1095 val _ = Keyword.keyword "monos";
```
```  1096
```
```  1097 fun gen_ind_decl mk_def coind =
```
```  1098   Parse.fixes -- Parse.for_fixes --
```
```  1099   Scan.optional Parse_Spec.where_alt_specs [] --
```
```  1100   Scan.optional (Parse.\$\$\$ "monos" |-- Parse.!!! Parse_Spec.xthms1) []
```
```  1101   >> (fn (((preds, params), specs), monos) =>
```
```  1102       (snd oo gen_add_inductive mk_def true coind preds params specs monos));
```
```  1103
```
```  1104 val ind_decl = gen_ind_decl add_ind_def;
```
```  1105
```
```  1106 val _ =
```
```  1107   Outer_Syntax.local_theory' "inductive" "define inductive predicates" Keyword.thy_decl
```
```  1108     (ind_decl false);
```
```  1109
```
```  1110 val _ =
```
```  1111   Outer_Syntax.local_theory' "coinductive" "define coinductive predicates" Keyword.thy_decl
```
```  1112     (ind_decl true);
```
```  1113
```
```  1114 val _ =
```
```  1115   Outer_Syntax.local_theory "inductive_cases"
```
```  1116     "create simplified instances of elimination rules (improper)" Keyword.thy_script
```
```  1117     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
```
```  1118
```
```  1119 val _ =
```
```  1120   Outer_Syntax.local_theory "inductive_simps"
```
```  1121     "create simplification rules for inductive predicates" Keyword.thy_script
```
```  1122     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
```
```  1123
```
```  1124 end;
```