src/HOL/Tools/inductive.ML
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```     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 get_monos: Proof.context -> thm list
```
```    31   val mono_add: attribute
```
```    32   val mono_del: attribute
```
```    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}
```
```    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     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     local_theory -> inductive_result * local_theory
```
```    85   val gen_ind_decl: add_ind_def -> bool -> (local_theory -> local_theory) parser
```
```    86 end;
```
```    87
```
```    88 structure Inductive: INDUCTIVE =
```
```    89 struct
```
```    90
```
```    91 (** theory context references **)
```
```    92
```
```    93 val inductive_forall_def = @{thm induct_forall_def};
```
```    94 val inductive_conj_name = "HOL.induct_conj";
```
```    95 val inductive_conj_def = @{thm induct_conj_def};
```
```    96 val inductive_conj = @{thms induct_conj};
```
```    97 val inductive_atomize = @{thms induct_atomize};
```
```    98 val inductive_rulify = @{thms induct_rulify};
```
```    99 val inductive_rulify_fallback = @{thms induct_rulify_fallback};
```
```   100
```
```   101 val simp_thms1 =
```
```   102   map mk_meta_eq
```
```   103     @{lemma "(~ True) = False" "(~ False) = True"
```
```   104         "(True --> P) = P" "(False --> P) = True"
```
```   105         "(P & True) = P" "(True & P) = P"
```
```   106       by (fact simp_thms)+};
```
```   107
```
```   108 val simp_thms2 =
```
```   109   map mk_meta_eq [@{thm inf_fun_def}, @{thm inf_bool_def}] @ simp_thms1;
```
```   110
```
```   111 val simp_thms3 =
```
```   112   map mk_meta_eq [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_def}, @{thm sup_bool_def}];
```
```   113
```
```   114
```
```   115
```
```   116 (** misc utilities **)
```
```   117
```
```   118 fun message quiet_mode s = if quiet_mode then () else writeln s;
```
```   119 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
```
```   120
```
```   121 fun coind_prefix true = "co"
```
```   122   | coind_prefix false = "";
```
```   123
```
```   124 fun log (b: int) m n = if m >= n then 0 else 1 + log b (b * m) n;
```
```   125
```
```   126 fun make_bool_args f g [] i = []
```
```   127   | make_bool_args f g (x :: xs) i =
```
```   128       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
```
```   129
```
```   130 fun make_bool_args' xs =
```
```   131   make_bool_args (K @{term False}) (K @{term True}) xs;
```
```   132
```
```   133 fun arg_types_of k c = drop k (binder_types (fastype_of c));
```
```   134
```
```   135 fun find_arg T x [] = raise Fail "find_arg"
```
```   136   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
```
```   137       apsnd (cons p) (find_arg T x ps)
```
```   138   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
```
```   139       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
```
```   140       else apsnd (cons p) (find_arg T x ps);
```
```   141
```
```   142 fun make_args Ts xs =
```
```   143   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
```
```   144     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
```
```   145
```
```   146 fun make_args' Ts xs Us =
```
```   147   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
```
```   148
```
```   149 fun dest_predicate cs params t =
```
```   150   let
```
```   151     val k = length params;
```
```   152     val (c, ts) = strip_comb t;
```
```   153     val (xs, ys) = chop k ts;
```
```   154     val i = find_index (fn c' => c' = c) cs;
```
```   155   in
```
```   156     if xs = params andalso i >= 0 then
```
```   157       SOME (c, i, ys, chop (length ys) (arg_types_of k c))
```
```   158     else NONE
```
```   159   end;
```
```   160
```
```   161 fun mk_names a 0 = []
```
```   162   | mk_names a 1 = [a]
```
```   163   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
```
```   164
```
```   165 fun select_disj 1 1 = []
```
```   166   | select_disj _ 1 = [rtac disjI1]
```
```   167   | select_disj n i = rtac disjI2 :: select_disj (n - 1) (i - 1);
```
```   168
```
```   169
```
```   170
```
```   171 (** context data **)
```
```   172
```
```   173 type inductive_result =
```
```   174   {preds: term list, elims: thm list, raw_induct: thm,
```
```   175    induct: thm, inducts: thm list, intrs: thm list, eqs: thm list};
```
```   176
```
```   177 fun transform_result phi {preds, elims, raw_induct: thm, induct, inducts, intrs, eqs} =
```
```   178   let
```
```   179     val term = Morphism.term phi;
```
```   180     val thm = Morphism.thm phi;
```
```   181     val fact = Morphism.fact phi;
```
```   182   in
```
```   183    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
```
```   184     induct = thm induct, inducts = fact inducts, intrs = fact intrs, eqs = fact eqs}
```
```   185   end;
```
```   186
```
```   187 type inductive_info = {names: string list, coind: bool} * inductive_result;
```
```   188
```
```   189 val empty_equations =
```
```   190   Item_Net.init Thm.eq_thm_prop
```
```   191     (single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
```
```   192
```
```   193 datatype data = Data of
```
```   194  {infos: inductive_info Symtab.table,
```
```   195   monos: thm list,
```
```   196   equations: thm Item_Net.T};
```
```   197
```
```   198 fun make_data (infos, monos, equations) =
```
```   199   Data {infos = infos, monos = monos, equations = equations};
```
```   200
```
```   201 structure Data = Generic_Data
```
```   202 (
```
```   203   type T = data;
```
```   204   val empty = make_data (Symtab.empty, [], empty_equations);
```
```   205   val extend = I;
```
```   206   fun merge (Data {infos = infos1, monos = monos1, equations = equations1},
```
```   207       Data {infos = infos2, monos = monos2, equations = equations2}) =
```
```   208     make_data (Symtab.merge (K true) (infos1, infos2),
```
```   209       Thm.merge_thms (monos1, monos2),
```
```   210       Item_Net.merge (equations1, equations2));
```
```   211 );
```
```   212
```
```   213 fun map_data f =
```
```   214   Data.map (fn Data {infos, monos, equations} => make_data (f (infos, monos, equations)));
```
```   215
```
```   216 fun rep_data ctxt = Data.get (Context.Proof ctxt) |> (fn Data rep => rep);
```
```   217
```
```   218 fun print_inductives ctxt =
```
```   219   let
```
```   220     val {infos, monos, ...} = rep_data ctxt;
```
```   221     val space = Consts.space_of (Proof_Context.consts_of ctxt);
```
```   222   in
```
```   223     [Pretty.block
```
```   224       (Pretty.breaks
```
```   225         (Pretty.str "(co)inductives:" ::
```
```   226           map (Pretty.mark_str o #1) (Name_Space.extern_table ctxt (space, infos)))),
```
```   227      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_item ctxt) monos)]
```
```   228   end |> Pretty.chunks |> Pretty.writeln;
```
```   229
```
```   230
```
```   231 (* inductive info *)
```
```   232
```
```   233 fun the_inductive ctxt name =
```
```   234   (case Symtab.lookup (#infos (rep_data ctxt)) name of
```
```   235     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
```
```   236   | SOME info => info);
```
```   237
```
```   238 fun put_inductives names info =
```
```   239   map_data (fn (infos, monos, equations) =>
```
```   240     (fold (fn name => Symtab.update (name, info)) names infos, monos, equations));
```
```   241
```
```   242
```
```   243 (* monotonicity rules *)
```
```   244
```
```   245 val get_monos = #monos o rep_data;
```
```   246
```
```   247 fun mk_mono ctxt thm =
```
```   248   let
```
```   249     fun eq_to_mono thm' = thm' RS (thm' RS @{thm eq_to_mono});
```
```   250     fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
```
```   251       handle THM _ => thm RS @{thm le_boolD}
```
```   252   in
```
```   253     (case concl_of thm of
```
```   254       Const ("==", _) \$ _ \$ _ => eq_to_mono (thm RS meta_eq_to_obj_eq)
```
```   255     | _ \$ (Const (@{const_name HOL.eq}, _) \$ _ \$ _) => eq_to_mono thm
```
```   256     | _ \$ (Const (@{const_name Orderings.less_eq}, _) \$ _ \$ _) =>
```
```   257       dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
```
```   258         (resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))
```
```   259     | _ => thm)
```
```   260   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm ctxt thm);
```
```   261
```
```   262 val mono_add =
```
```   263   Thm.declaration_attribute (fn thm => fn context =>
```
```   264     map_data (fn (infos, monos, equations) =>
```
```   265       (infos, Thm.add_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
```
```   266
```
```   267 val mono_del =
```
```   268   Thm.declaration_attribute (fn thm => fn context =>
```
```   269     map_data (fn (infos, monos, equations) =>
```
```   270       (infos, Thm.del_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
```
```   271
```
```   272
```
```   273 (* equations *)
```
```   274
```
```   275 val get_equations = #equations o rep_data;
```
```   276
```
```   277 val equation_add_permissive =
```
```   278   Thm.declaration_attribute (fn thm =>
```
```   279     map_data (fn (infos, monos, equations) =>
```
```   280       (infos, monos, perhaps (try (Item_Net.update thm)) equations)));
```
```   281
```
```   282
```
```   283
```
```   284 (** process rules **)
```
```   285
```
```   286 local
```
```   287
```
```   288 fun err_in_rule ctxt name t msg =
```
```   289   error (cat_lines ["Ill-formed introduction rule " ^ Binding.print name,
```
```   290     Syntax.string_of_term ctxt t, msg]);
```
```   291
```
```   292 fun err_in_prem ctxt name t p msg =
```
```   293   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
```
```   294     "in introduction rule " ^ Binding.print name, Syntax.string_of_term ctxt t, msg]);
```
```   295
```
```   296 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
```
```   297
```
```   298 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
```
```   299
```
```   300 val bad_app = "Inductive predicate must be applied to parameter(s) ";
```
```   301
```
```   302 fun atomize_term thy = Raw_Simplifier.rewrite_term thy inductive_atomize [];
```
```   303
```
```   304 in
```
```   305
```
```   306 fun check_rule ctxt cs params ((binding, att), rule) =
```
```   307   let
```
```   308     val params' = Term.variant_frees rule (Logic.strip_params rule);
```
```   309     val frees = rev (map Free params');
```
```   310     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
```
```   311     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
```
```   312     val rule' = Logic.list_implies (prems, concl);
```
```   313     val aprems = map (atomize_term (Proof_Context.theory_of ctxt)) prems;
```
```   314     val arule = fold_rev (Logic.all o Free) params' (Logic.list_implies (aprems, concl));
```
```   315
```
```   316     fun check_ind err t =
```
```   317       (case dest_predicate cs params t of
```
```   318         NONE => err (bad_app ^
```
```   319           commas (map (Syntax.string_of_term ctxt) params))
```
```   320       | SOME (_, _, ys, _) =>
```
```   321           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
```
```   322           then err bad_ind_occ else ());
```
```   323
```
```   324     fun check_prem' prem t =
```
```   325       if member (op =) cs (head_of t) then
```
```   326         check_ind (err_in_prem ctxt binding rule prem) t
```
```   327       else
```
```   328         (case t of
```
```   329           Abs (_, _, t) => check_prem' prem t
```
```   330         | t \$ u => (check_prem' prem t; check_prem' prem u)
```
```   331         | _ => ());
```
```   332
```
```   333     fun check_prem (prem, aprem) =
```
```   334       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
```
```   335       else err_in_prem ctxt binding rule prem "Non-atomic premise";
```
```   336
```
```   337     val _ =
```
```   338       (case concl of
```
```   339         Const (@{const_name Trueprop}, _) \$ t =>
```
```   340           if member (op =) cs (head_of t) then
```
```   341            (check_ind (err_in_rule ctxt binding rule') t;
```
```   342             List.app check_prem (prems ~~ aprems))
```
```   343           else err_in_rule ctxt binding rule' bad_concl
```
```   344        | _ => err_in_rule ctxt binding rule' bad_concl);
```
```   345   in
```
```   346     ((binding, att), arule)
```
```   347   end;
```
```   348
```
```   349 val rulify =
```
```   350   hol_simplify inductive_conj
```
```   351   #> hol_simplify inductive_rulify
```
```   352   #> hol_simplify inductive_rulify_fallback
```
```   353   #> Simplifier.norm_hhf;
```
```   354
```
```   355 end;
```
```   356
```
```   357
```
```   358
```
```   359 (** proofs for (co)inductive predicates **)
```
```   360
```
```   361 (* prove monotonicity *)
```
```   362
```
```   363 fun prove_mono quiet_mode skip_mono predT fp_fun monos ctxt =
```
```   364  (message (quiet_mode orelse skip_mono andalso ! quick_and_dirty)
```
```   365     "  Proving monotonicity ...";
```
```   366   (if skip_mono then Goal.prove_sorry else Goal.prove_future) ctxt
```
```   367     [] []
```
```   368     (HOLogic.mk_Trueprop
```
```   369       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) \$ fp_fun))
```
```   370     (fn _ => EVERY [rtac @{thm monoI} 1,
```
```   371       REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
```
```   372       REPEAT (FIRST
```
```   373         [atac 1,
```
```   374          resolve_tac (map (mk_mono ctxt) monos @ get_monos ctxt) 1,
```
```   375          etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
```
```   376
```
```   377
```
```   378 (* prove introduction rules *)
```
```   379
```
```   380 fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
```
```   381   let
```
```   382     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
```
```   383
```
```   384     val unfold = funpow k (fn th => th RS fun_cong)
```
```   385       (mono RS (fp_def RS
```
```   386         (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
```
```   387
```
```   388     val rules = [refl, TrueI, @{lemma "~ False" by (rule notI)}, exI, conjI];
```
```   389
```
```   390     val intrs = map_index (fn (i, intr) =>
```
```   391       Goal.prove_sorry ctxt [] [] intr (fn _ => EVERY
```
```   392        [rewrite_goals_tac rec_preds_defs,
```
```   393         rtac (unfold RS iffD2) 1,
```
```   394         EVERY1 (select_disj (length intr_ts) (i + 1)),
```
```   395         (*Not ares_tac, since refl must be tried before any equality assumptions;
```
```   396           backtracking may occur if the premises have extra variables!*)
```
```   397         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)])
```
```   398        |> singleton (Proof_Context.export ctxt ctxt')) intr_ts
```
```   399
```
```   400   in (intrs, unfold) end;
```
```   401
```
```   402
```
```   403 (* prove elimination rules *)
```
```   404
```
```   405 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
```
```   406   let
```
```   407     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
```
```   408
```
```   409     val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
```
```   410     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
```
```   411
```
```   412     fun dest_intr r =
```
```   413       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
```
```   414        Logic.strip_assums_hyp r, Logic.strip_params r);
```
```   415
```
```   416     val intrs = map dest_intr intr_ts ~~ intr_names;
```
```   417
```
```   418     val rules1 = [disjE, exE, FalseE];
```
```   419     val rules2 = [conjE, FalseE, @{lemma "~ True ==> R" by (rule notE [OF _ TrueI])}];
```
```   420
```
```   421     fun prove_elim c =
```
```   422       let
```
```   423         val Ts = arg_types_of (length params) c;
```
```   424         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
```
```   425         val frees = map Free (anames ~~ Ts);
```
```   426
```
```   427         fun mk_elim_prem ((_, _, us, _), ts, params') =
```
```   428           Logic.list_all (params',
```
```   429             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
```
```   430               (frees ~~ us) @ ts, P));
```
```   431         val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
```
```   432         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
```
```   433            map mk_elim_prem (map #1 c_intrs)
```
```   434       in
```
```   435         (Goal.prove_sorry ctxt'' [] prems P
```
```   436           (fn {prems, ...} => EVERY
```
```   437             [cut_tac (hd prems) 1,
```
```   438              rewrite_goals_tac rec_preds_defs,
```
```   439              dtac (unfold RS iffD1) 1,
```
```   440              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
```
```   441              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
```
```   442              EVERY (map (fn prem =>
```
```   443                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
```
```   444           |> singleton (Proof_Context.export ctxt'' ctxt'''),
```
```   445          map #2 c_intrs, length Ts)
```
```   446       end
```
```   447
```
```   448    in map prove_elim cs end;
```
```   449
```
```   450
```
```   451 (* prove simplification equations *)
```
```   452
```
```   453 fun prove_eqs quiet_mode cs params intr_ts intrs
```
```   454     (elims: (thm * bstring list * int) list) ctxt ctxt'' =  (* FIXME ctxt'' ?? *)
```
```   455   let
```
```   456     val _ = clean_message quiet_mode "  Proving the simplification rules ...";
```
```   457
```
```   458     fun dest_intr r =
```
```   459       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
```
```   460        Logic.strip_assums_hyp r, Logic.strip_params r);
```
```   461     val intr_ts' = map dest_intr intr_ts;
```
```   462
```
```   463     fun prove_eq c (elim: thm * 'a * 'b) =
```
```   464       let
```
```   465         val Ts = arg_types_of (length params) c;
```
```   466         val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
```
```   467         val frees = map Free (anames ~~ Ts);
```
```   468         val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
```
```   469         fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
```
```   470           let
```
```   471             fun list_ex ([], t) = t
```
```   472               | list_ex ((a, T) :: vars, t) =
```
```   473                   HOLogic.exists_const T \$ Abs (a, T, list_ex (vars, t));
```
```   474             val conjs = map2 (curry HOLogic.mk_eq) frees us @ map HOLogic.dest_Trueprop ts;
```
```   475           in
```
```   476             list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
```
```   477           end;
```
```   478         val lhs = list_comb (c, params @ frees);
```
```   479         val rhs =
```
```   480           if null c_intrs then @{term False}
```
```   481           else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs);
```
```   482         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
```
```   483         fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
```
```   484             EVERY1 (select_disj (length c_intrs) (i + 1)) THEN
```
```   485             EVERY (replicate (length params) (rtac @{thm exI} 1)) THEN
```
```   486             (if null prems then rtac @{thm TrueI} 1
```
```   487              else
```
```   488               let
```
```   489                 val (prems', last_prem) = split_last prems;
```
```   490               in
```
```   491                 EVERY (map (fn prem => (rtac @{thm conjI} 1 THEN rtac prem 1)) prems') THEN
```
```   492                 rtac last_prem 1
```
```   493               end)) ctxt' 1;
```
```   494         fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
```
```   495           EVERY (replicate (length params') (etac @{thm exE} 1)) THEN
```
```   496           (if null ts andalso null us then rtac intr 1
```
```   497            else
```
```   498             EVERY (replicate (length ts + length us - 1) (etac @{thm conjE} 1)) THEN
```
```   499             Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
```
```   500               let
```
```   501                 val (eqs, prems') = chop (length us) prems;
```
```   502                 val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs;
```
```   503               in
```
```   504                 rewrite_goal_tac rew_thms 1 THEN
```
```   505                 rtac intr 1 THEN
```
```   506                 EVERY (map (fn p => rtac p 1) prems')
```
```   507               end) ctxt' 1);
```
```   508       in
```
```   509         Goal.prove_sorry ctxt' [] [] eq (fn _ =>
```
```   510           rtac @{thm iffI} 1 THEN etac (#1 elim) 1 THEN
```
```   511           EVERY (map_index prove_intr1 c_intrs) THEN
```
```   512           (if null c_intrs then etac @{thm FalseE} 1
```
```   513            else
```
```   514             let val (c_intrs', last_c_intr) = split_last c_intrs in
```
```   515               EVERY (map (fn ci => etac @{thm disjE} 1 THEN prove_intr2 ci) c_intrs') THEN
```
```   516               prove_intr2 last_c_intr
```
```   517             end))
```
```   518         |> rulify
```
```   519         |> singleton (Proof_Context.export ctxt' ctxt'')
```
```   520       end;
```
```   521   in
```
```   522     map2 prove_eq cs elims
```
```   523   end;
```
```   524
```
```   525
```
```   526 (* derivation of simplified elimination rules *)
```
```   527
```
```   528 local
```
```   529
```
```   530 (*delete needless equality assumptions*)
```
```   531 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
```
```   532   (fn _ => assume_tac 1);
```
```   533 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
```
```   534 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
```
```   535
```
```   536 fun simp_case_tac ss i =
```
```   537   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
```
```   538
```
```   539 in
```
```   540
```
```   541 fun mk_cases ctxt prop =
```
```   542   let
```
```   543     val thy = Proof_Context.theory_of ctxt;
```
```   544     val ss = simpset_of ctxt;
```
```   545
```
```   546     fun err msg =
```
```   547       error (Pretty.string_of (Pretty.block
```
```   548         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
```
```   549
```
```   550     val elims = Induct.find_casesP ctxt prop;
```
```   551
```
```   552     val cprop = Thm.cterm_of thy prop;
```
```   553     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
```
```   554     fun mk_elim rl =
```
```   555       Thm.implies_intr cprop (Tactic.rule_by_tactic ctxt tac (Thm.assume cprop RS rl))
```
```   556       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
```
```   557   in
```
```   558     (case get_first (try mk_elim) elims of
```
```   559       SOME r => r
```
```   560     | NONE => err "Proposition not an inductive predicate:")
```
```   561   end;
```
```   562
```
```   563 end;
```
```   564
```
```   565
```
```   566 (* inductive_cases *)
```
```   567
```
```   568 fun gen_inductive_cases prep_att prep_prop args lthy =
```
```   569   let
```
```   570     val thy = Proof_Context.theory_of lthy;
```
```   571     val thmss =
```
```   572       map snd args
```
```   573       |> burrow (grouped 10 Par_List.map (mk_cases lthy o prep_prop lthy));
```
```   574     val facts =
```
```   575       map2 (fn ((a, atts), _) => fn thms => ((a, map (prep_att thy) atts), [(thms, [])]))
```
```   576         args thmss;
```
```   577   in lthy |> Local_Theory.notes facts |>> map snd end;
```
```   578
```
```   579 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
```
```   580 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
```
```   581
```
```   582
```
```   583 val ind_cases_setup =
```
```   584   Method.setup @{binding ind_cases}
```
```   585     (Scan.lift (Scan.repeat1 Args.name_source --
```
```   586       Scan.optional (Args.\$\$\$ "for" |-- Scan.repeat1 Args.binding) []) >>
```
```   587       (fn (raw_props, fixes) => fn ctxt =>
```
```   588         let
```
```   589           val (_, ctxt') = Variable.add_fixes_binding fixes ctxt;
```
```   590           val props = Syntax.read_props ctxt' raw_props;
```
```   591           val ctxt'' = fold Variable.declare_term props ctxt';
```
```   592           val rules = Proof_Context.export ctxt'' ctxt (map (mk_cases ctxt'') props)
```
```   593         in Method.erule 0 rules end))
```
```   594     "dynamic case analysis on predicates";
```
```   595
```
```   596
```
```   597 (* derivation of simplified equation *)
```
```   598
```
```   599 fun mk_simp_eq ctxt prop =
```
```   600   let
```
```   601     val thy = Proof_Context.theory_of ctxt;
```
```   602     val ctxt' = Variable.auto_fixes prop ctxt;
```
```   603     val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of;
```
```   604     val substs =
```
```   605       Item_Net.retrieve (get_equations ctxt) (HOLogic.dest_Trueprop prop)
```
```   606       |> map_filter
```
```   607         (fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
```
```   608             (Vartab.empty, Vartab.empty), eq)
```
```   609           handle Pattern.MATCH => NONE);
```
```   610     val (subst, eq) =
```
```   611       (case substs of
```
```   612         [s] => s
```
```   613       | _ => error
```
```   614         ("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique"));
```
```   615     val inst =
```
```   616       map (fn v => (cterm_of thy (Var v), cterm_of thy (Envir.subst_term subst (Var v))))
```
```   617         (Term.add_vars (lhs_of eq) []);
```
```   618   in
```
```   619     Drule.cterm_instantiate inst eq
```
```   620     |> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv (Simplifier.full_rewrite (simpset_of ctxt))))
```
```   621     |> singleton (Variable.export ctxt' ctxt)
```
```   622   end
```
```   623
```
```   624
```
```   625 (* inductive simps *)
```
```   626
```
```   627 fun gen_inductive_simps prep_att prep_prop args lthy =
```
```   628   let
```
```   629     val thy = Proof_Context.theory_of lthy;
```
```   630     val facts = args |> map (fn ((a, atts), props) =>
```
```   631       ((a, map (prep_att thy) atts),
```
```   632         map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
```
```   633   in lthy |> Local_Theory.notes facts |>> map snd end;
```
```   634
```
```   635 val inductive_simps = gen_inductive_simps Attrib.intern_src Syntax.read_prop;
```
```   636 val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
```
```   637
```
```   638
```
```   639 (* prove induction rule *)
```
```   640
```
```   641 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
```
```   642     fp_def rec_preds_defs ctxt ctxt''' =  (* FIXME ctxt''' ?? *)
```
```   643   let
```
```   644     val _ = clean_message quiet_mode "  Proving the induction rule ...";
```
```   645
```
```   646     (* predicates for induction rule *)
```
```   647
```
```   648     val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
```
```   649     val preds =
```
```   650       map2 (curry Free) pnames
```
```   651         (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
```
```   652
```
```   653     (* transform an introduction rule into a premise for induction rule *)
```
```   654
```
```   655     fun mk_ind_prem r =
```
```   656       let
```
```   657         fun subst s =
```
```   658           (case dest_predicate cs params s of
```
```   659             SOME (_, i, ys, (_, Ts)) =>
```
```   660               let
```
```   661                 val k = length Ts;
```
```   662                 val bs = map Bound (k - 1 downto 0);
```
```   663                 val P = list_comb (nth preds i, map (incr_boundvars k) ys @ bs);
```
```   664                 val Q =
```
```   665                   fold_rev Term.abs (mk_names "x" k ~~ Ts)
```
```   666                     (HOLogic.mk_binop inductive_conj_name
```
```   667                       (list_comb (incr_boundvars k s, bs), P));
```
```   668               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
```
```   669           | NONE =>
```
```   670               (case s of
```
```   671                 t \$ u => (fst (subst t) \$ fst (subst u), NONE)
```
```   672               | Abs (a, T, t) => (Abs (a, T, fst (subst t)), NONE)
```
```   673               | _ => (s, NONE)));
```
```   674
```
```   675         fun mk_prem s prems =
```
```   676           (case subst s of
```
```   677             (_, SOME (t, u)) => t :: u :: prems
```
```   678           | (t, _) => t :: prems);
```
```   679
```
```   680         val SOME (_, i, ys, _) =
```
```   681           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
```
```   682       in
```
```   683         fold_rev (Logic.all o Free) (Logic.strip_params r)
```
```   684           (Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
```
```   685             (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
```
```   686               HOLogic.mk_Trueprop (list_comb (nth preds i, ys))))
```
```   687       end;
```
```   688
```
```   689     val ind_prems = map mk_ind_prem intr_ts;
```
```   690
```
```   691
```
```   692     (* make conclusions for induction rules *)
```
```   693
```
```   694     val Tss = map (binder_types o fastype_of) preds;
```
```   695     val (xnames, ctxt'') = Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
```
```   696     val mutual_ind_concl =
```
```   697       HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
```
```   698         (map (fn (((xnames, Ts), c), P) =>
```
```   699           let val frees = map Free (xnames ~~ Ts)
```
```   700           in HOLogic.mk_imp (list_comb (c, params @ frees), list_comb (P, frees)) end)
```
```   701         (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
```
```   702
```
```   703
```
```   704     (* make predicate for instantiation of abstract induction rule *)
```
```   705
```
```   706     val ind_pred =
```
```   707       fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
```
```   708         (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
```
```   709            (make_bool_args HOLogic.mk_not I bs i)
```
```   710            (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
```
```   711
```
```   712     val ind_concl =
```
```   713       HOLogic.mk_Trueprop
```
```   714         (HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
```
```   715
```
```   716     val raw_fp_induct = mono RS (fp_def RS @{thm def_lfp_induct});
```
```   717
```
```   718     val induct = Goal.prove_sorry ctxt'' [] ind_prems ind_concl
```
```   719       (fn {prems, ...} => EVERY
```
```   720         [rewrite_goals_tac [inductive_conj_def],
```
```   721          DETERM (rtac raw_fp_induct 1),
```
```   722          REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
```
```   723          rewrite_goals_tac simp_thms2,
```
```   724          (*This disjE separates out the introduction rules*)
```
```   725          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
```
```   726          (*Now break down the individual cases.  No disjE here in case
```
```   727            some premise involves disjunction.*)
```
```   728          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
```
```   729          REPEAT (FIRSTGOAL
```
```   730            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
```
```   731          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
```
```   732              (inductive_conj_def :: rec_preds_defs @ simp_thms2) prem,
```
```   733            conjI, refl] 1)) prems)]);
```
```   734
```
```   735     val lemma = Goal.prove_sorry ctxt'' [] []
```
```   736       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
```
```   737         [rewrite_goals_tac rec_preds_defs,
```
```   738          REPEAT (EVERY
```
```   739            [REPEAT (resolve_tac [conjI, impI] 1),
```
```   740             REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
```
```   741             atac 1,
```
```   742             rewrite_goals_tac simp_thms1,
```
```   743             atac 1])]);
```
```   744
```
```   745   in singleton (Proof_Context.export ctxt'' ctxt''') (induct RS lemma) end;
```
```   746
```
```   747
```
```   748
```
```   749 (** specification of (co)inductive predicates **)
```
```   750
```
```   751 fun mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts monos params cnames_syn lthy =
```
```   752   let
```
```   753     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
```
```   754
```
```   755     val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
```
```   756     val k = log 2 1 (length cs);
```
```   757     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
```
```   758     val p :: xs =
```
```   759       map Free (Variable.variant_frees lthy intr_ts
```
```   760         (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
```
```   761     val bs =
```
```   762       map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
```
```   763         (map (rpair HOLogic.boolT) (mk_names "b" k)));
```
```   764
```
```   765     fun subst t =
```
```   766       (case dest_predicate cs params t of
```
```   767         SOME (_, i, ts, (Ts, Us)) =>
```
```   768           let
```
```   769             val l = length Us;
```
```   770             val zs = map Bound (l - 1 downto 0);
```
```   771           in
```
```   772             fold_rev (Term.abs o pair "z") Us
```
```   773               (list_comb (p,
```
```   774                 make_bool_args' bs i @ make_args argTs
```
```   775                   ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
```
```   776           end
```
```   777       | NONE =>
```
```   778           (case t of
```
```   779             t1 \$ t2 => subst t1 \$ subst t2
```
```   780           | Abs (x, T, u) => Abs (x, T, subst u)
```
```   781           | _ => t));
```
```   782
```
```   783     (* transform an introduction rule into a conjunction  *)
```
```   784     (*   [| p_i t; ... |] ==> p_j u                       *)
```
```   785     (* is transformed into                                *)
```
```   786     (*   b_j & x_j = u & p b_j t & ...                    *)
```
```   787
```
```   788     fun transform_rule r =
```
```   789       let
```
```   790         val SOME (_, i, ts, (Ts, _)) =
```
```   791           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
```
```   792         val ps =
```
```   793           make_bool_args HOLogic.mk_not I bs i @
```
```   794           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
```
```   795           map (subst o HOLogic.dest_Trueprop) (Logic.strip_assums_hyp r);
```
```   796       in
```
```   797         fold_rev (fn (x, T) => fn P => HOLogic.exists_const T \$ Abs (x, T, P))
```
```   798           (Logic.strip_params r)
```
```   799           (if null ps then @{term True} else foldr1 HOLogic.mk_conj ps)
```
```   800       end;
```
```   801
```
```   802     (* make a disjunction of all introduction rules *)
```
```   803
```
```   804     val fp_fun =
```
```   805       fold_rev lambda (p :: bs @ xs)
```
```   806         (if null intr_ts then @{term False}
```
```   807          else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
```
```   808
```
```   809     (* add definiton of recursive predicates to theory *)
```
```   810
```
```   811     val rec_name =
```
```   812       if Binding.is_empty alt_name then
```
```   813         Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
```
```   814       else alt_name;
```
```   815
```
```   816     val ((rec_const, (_, fp_def)), lthy') = lthy
```
```   817       |> Local_Theory.conceal
```
```   818       |> Local_Theory.define
```
```   819         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
```
```   820          ((Thm.def_binding rec_name, @{attributes [nitpick_unfold]}),
```
```   821            fold_rev lambda params
```
```   822              (Const (fp_name, (predT --> predT) --> predT) \$ fp_fun)))
```
```   823       ||> Local_Theory.restore_naming lthy;
```
```   824     val fp_def' =
```
```   825       Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
```
```   826         (cterm_of (Proof_Context.theory_of lthy') (list_comb (rec_const, params)));
```
```   827     val specs =
```
```   828       if length cs < 2 then []
```
```   829       else
```
```   830         map_index (fn (i, (name_mx, c)) =>
```
```   831           let
```
```   832             val Ts = arg_types_of (length params) c;
```
```   833             val xs =
```
```   834               map Free (Variable.variant_frees lthy intr_ts (mk_names "x" (length Ts) ~~ Ts));
```
```   835           in
```
```   836             (name_mx, (apfst Binding.conceal Attrib.empty_binding, fold_rev lambda (params @ xs)
```
```   837               (list_comb (rec_const, params @ make_bool_args' bs i @
```
```   838                 make_args argTs (xs ~~ Ts)))))
```
```   839           end) (cnames_syn ~~ cs);
```
```   840     val (consts_defs, lthy'') = lthy'
```
```   841       |> fold_map Local_Theory.define specs;
```
```   842     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
```
```   843
```
```   844     val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
```
```   845     val mono = prove_mono quiet_mode skip_mono predT fp_fun monos lthy''';
```
```   846     val (_, lthy'''') =
```
```   847       Local_Theory.note (apfst Binding.conceal Attrib.empty_binding,
```
```   848         Proof_Context.export lthy''' lthy'' [mono]) lthy'';
```
```   849
```
```   850   in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
```
```   851     list_comb (rec_const, params), preds, argTs, bs, xs)
```
```   852   end;
```
```   853
```
```   854 fun declare_rules rec_binding coind no_ind cnames
```
```   855     preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
```
```   856   let
```
```   857     val rec_name = Binding.name_of rec_binding;
```
```   858     fun rec_qualified qualified = Binding.qualify qualified rec_name;
```
```   859     val intr_names = map Binding.name_of intr_bindings;
```
```   860     val ind_case_names = Rule_Cases.case_names intr_names;
```
```   861     val induct =
```
```   862       if coind then
```
```   863         (raw_induct,
```
```   864          [Rule_Cases.case_names [rec_name],
```
```   865           Rule_Cases.case_conclusion (rec_name, intr_names),
```
```   866           Rule_Cases.consumes (1 - Thm.nprems_of raw_induct),
```
```   867           Induct.coinduct_pred (hd cnames)])
```
```   868       else if no_ind orelse length cnames > 1 then
```
```   869         (raw_induct,
```
```   870           [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of raw_induct))])
```
```   871       else
```
```   872         (raw_induct RSN (2, rev_mp),
```
```   873           [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of raw_induct))]);
```
```   874
```
```   875     val (intrs', lthy1) =
```
```   876       lthy |>
```
```   877       Spec_Rules.add
```
```   878         (if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
```
```   879       Local_Theory.notes
```
```   880         (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
```
```   881           map (fn th => [([th],
```
```   882            [Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
```
```   883       map (hd o snd);
```
```   884     val (((_, elims'), (_, [induct'])), lthy2) =
```
```   885       lthy1 |>
```
```   886       Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
```
```   887       fold_map (fn (name, (elim, cases, k)) =>
```
```   888         Local_Theory.note
```
```   889           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
```
```   890             [Attrib.internal (K (Rule_Cases.case_names cases)),
```
```   891              Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of elim))),
```
```   892              Attrib.internal (K (Rule_Cases.constraints k)),
```
```   893              Attrib.internal (K (Induct.cases_pred name)),
```
```   894              Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
```
```   895         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
```
```   896       Local_Theory.note
```
```   897         ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
```
```   898           map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
```
```   899
```
```   900     val (eqs', lthy3) = lthy2 |>
```
```   901       fold_map (fn (name, eq) => Local_Theory.note
```
```   902           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
```
```   903             [Attrib.internal (K equation_add_permissive)]), [eq])
```
```   904           #> apfst (hd o snd))
```
```   905         (if null eqs then [] else (cnames ~~ eqs))
```
```   906     val (inducts, lthy4) =
```
```   907       if no_ind orelse coind then ([], lthy3)
```
```   908       else
```
```   909         let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
```
```   910           lthy3 |>
```
```   911           Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
```
```   912             inducts |> map (fn (name, th) => ([th],
```
```   913               [Attrib.internal (K ind_case_names),
```
```   914                Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of th))),
```
```   915                Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
```
```   916         end;
```
```   917   in (intrs', elims', eqs', induct', inducts, lthy4) end;
```
```   918
```
```   919 type inductive_flags =
```
```   920   {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
```
```   921     no_elim: bool, no_ind: bool, skip_mono: bool};
```
```   922
```
```   923 type add_ind_def =
```
```   924   inductive_flags ->
```
```   925   term list -> (Attrib.binding * term) list -> thm list ->
```
```   926   term list -> (binding * mixfix) list ->
```
```   927   local_theory -> inductive_result * local_theory;
```
```   928
```
```   929 fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono}
```
```   930     cs intros monos params cnames_syn lthy =
```
```   931   let
```
```   932     val _ = null cnames_syn andalso error "No inductive predicates given";
```
```   933     val names = map (Binding.name_of o fst) cnames_syn;
```
```   934     val _ = message (quiet_mode andalso not verbose)
```
```   935       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
```
```   936
```
```   937     val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
```
```   938     val ((intr_names, intr_atts), intr_ts) =
```
```   939       apfst split_list (split_list (map (check_rule lthy cs params) intros));
```
```   940
```
```   941     val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
```
```   942       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts
```
```   943         monos params cnames_syn lthy;
```
```   944
```
```   945     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
```
```   946       intr_ts rec_preds_defs lthy2 lthy1;
```
```   947     val elims =
```
```   948       if no_elim then []
```
```   949       else
```
```   950         prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
```
```   951           unfold rec_preds_defs lthy2 lthy1;
```
```   952     val raw_induct = zero_var_indexes
```
```   953       (if no_ind then Drule.asm_rl
```
```   954        else if coind then
```
```   955          singleton (Proof_Context.export lthy2 lthy1)
```
```   956            (rotate_prems ~1 (Object_Logic.rulify
```
```   957              (fold_rule rec_preds_defs
```
```   958                (rewrite_rule simp_thms3
```
```   959                 (mono RS (fp_def RS @{thm def_coinduct}))))))
```
```   960        else
```
```   961          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
```
```   962            rec_preds_defs lthy2 lthy1);
```
```   963     val eqs =
```
```   964       if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1;
```
```   965
```
```   966     val elims' = map (fn (th, ns, i) => (rulify th, ns, i)) elims;
```
```   967     val intrs' = map rulify intrs;
```
```   968
```
```   969     val (intrs'', elims'', eqs', induct, inducts, lthy3) =
```
```   970       declare_rules rec_name coind no_ind
```
```   971         cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
```
```   972
```
```   973     val result =
```
```   974       {preds = preds,
```
```   975        intrs = intrs'',
```
```   976        elims = elims'',
```
```   977        raw_induct = rulify raw_induct,
```
```   978        induct = induct,
```
```   979        inducts = inducts,
```
```   980        eqs = eqs'};
```
```   981
```
```   982     val lthy4 = lthy3
```
```   983       |> Local_Theory.declaration {syntax = false, pervasive = false} (fn phi =>
```
```   984         let val result' = transform_result phi result;
```
```   985         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
```
```   986   in (result, lthy4) end;
```
```   987
```
```   988
```
```   989 (* external interfaces *)
```
```   990
```
```   991 fun gen_add_inductive_i mk_def
```
```   992     (flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono})
```
```   993     cnames_syn pnames spec monos lthy =
```
```   994   let
```
```   995     val thy = Proof_Context.theory_of lthy;
```
```   996     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
```
```   997
```
```   998
```
```   999     (* abbrevs *)
```
```  1000
```
```  1001     val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
```
```  1002
```
```  1003     fun get_abbrev ((name, atts), t) =
```
```  1004       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
```
```  1005         let
```
```  1006           val _ = Binding.is_empty name andalso null atts orelse
```
```  1007             error "Abbreviations may not have names or attributes";
```
```  1008           val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
```
```  1009           val var =
```
```  1010             (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
```
```  1011               NONE => error ("Undeclared head of abbreviation " ^ quote x)
```
```  1012             | SOME ((b, T'), mx) =>
```
```  1013                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
```
```  1014                 else (b, mx));
```
```  1015         in SOME (var, rhs) end
```
```  1016       else NONE;
```
```  1017
```
```  1018     val abbrevs = map_filter get_abbrev spec;
```
```  1019     val bs = map (Binding.name_of o fst o fst) abbrevs;
```
```  1020
```
```  1021
```
```  1022     (* predicates *)
```
```  1023
```
```  1024     val pre_intros = filter_out (is_some o get_abbrev) spec;
```
```  1025     val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
```
```  1026     val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
```
```  1027     val ps = map Free pnames;
```
```  1028
```
```  1029     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
```
```  1030     val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
```
```  1031     val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
```
```  1032     val expand = Assumption.export_term ctxt3 lthy #> Proof_Context.cert_term lthy;
```
```  1033
```
```  1034     fun close_rule r =
```
```  1035       fold (Logic.all o Free) (fold_aterms
```
```  1036         (fn t as Free (v as (s, _)) =>
```
```  1037             if Variable.is_fixed ctxt1 s orelse
```
```  1038               member (op =) ps t then I else insert (op =) v
```
```  1039           | _ => I) r []) r;
```
```  1040
```
```  1041     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
```
```  1042     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
```
```  1043   in
```
```  1044     lthy
```
```  1045     |> mk_def flags cs intros monos ps preds
```
```  1046     ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
```
```  1047   end;
```
```  1048
```
```  1049 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
```
```  1050   let
```
```  1051     val ((vars, intrs), _) = lthy
```
```  1052       |> Proof_Context.set_mode Proof_Context.mode_abbrev
```
```  1053       |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
```
```  1054     val (cs, ps) = chop (length cnames_syn) vars;
```
```  1055     val monos = Attrib.eval_thms lthy raw_monos;
```
```  1056     val flags =
```
```  1057      {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
```
```  1058       coind = coind, no_elim = false, no_ind = false, skip_mono = false};
```
```  1059   in
```
```  1060     lthy
```
```  1061     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
```
```  1062   end;
```
```  1063
```
```  1064 val add_inductive_i = gen_add_inductive_i add_ind_def;
```
```  1065 val add_inductive = gen_add_inductive add_ind_def;
```
```  1066
```
```  1067 fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
```
```  1068   let
```
```  1069     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
```
```  1070     val ctxt' = thy
```
```  1071       |> Named_Target.theory_init
```
```  1072       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
```
```  1073       |> Local_Theory.exit;
```
```  1074     val info = #2 (the_inductive ctxt' name);
```
```  1075   in (info, Proof_Context.theory_of ctxt') end;
```
```  1076
```
```  1077
```
```  1078 (* read off arities of inductive predicates from raw induction rule *)
```
```  1079 fun arities_of induct =
```
```  1080   map (fn (_ \$ t \$ u) =>
```
```  1081       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
```
```  1082     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
```
```  1083
```
```  1084 (* read off parameters of inductive predicate from raw induction rule *)
```
```  1085 fun params_of induct =
```
```  1086   let
```
```  1087     val (_ \$ t \$ u :: _) = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
```
```  1088     val (_, ts) = strip_comb t;
```
```  1089     val (_, us) = strip_comb u;
```
```  1090   in
```
```  1091     List.take (ts, length ts - length us)
```
```  1092   end;
```
```  1093
```
```  1094 val pname_of_intr =
```
```  1095   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
```
```  1096
```
```  1097 (* partition introduction rules according to predicate name *)
```
```  1098 fun gen_partition_rules f induct intros =
```
```  1099   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
```
```  1100     (map (rpair [] o fst) (arities_of induct));
```
```  1101
```
```  1102 val partition_rules = gen_partition_rules I;
```
```  1103 fun partition_rules' induct = gen_partition_rules fst induct;
```
```  1104
```
```  1105 fun unpartition_rules intros xs =
```
```  1106   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
```
```  1107     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
```
```  1108
```
```  1109 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
```
```  1110 fun infer_intro_vars elim arity intros =
```
```  1111   let
```
```  1112     val thy = theory_of_thm elim;
```
```  1113     val _ :: cases = prems_of elim;
```
```  1114     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
```
```  1115     fun mtch (t, u) =
```
```  1116       let
```
```  1117         val params = Logic.strip_params t;
```
```  1118         val vars =
```
```  1119           map (Var o apfst (rpair 0))
```
```  1120             (Name.variant_list used (map fst params) ~~ map snd params);
```
```  1121         val ts =
```
```  1122           map (curry subst_bounds (rev vars))
```
```  1123             (List.drop (Logic.strip_assums_hyp t, arity));
```
```  1124         val us = Logic.strip_imp_prems u;
```
```  1125         val tab =
```
```  1126           fold (Pattern.first_order_match thy) (ts ~~ us) (Vartab.empty, Vartab.empty);
```
```  1127       in
```
```  1128         map (Envir.subst_term tab) vars
```
```  1129       end
```
```  1130   in
```
```  1131     map (mtch o apsnd prop_of) (cases ~~ intros)
```
```  1132   end;
```
```  1133
```
```  1134
```
```  1135
```
```  1136 (** package setup **)
```
```  1137
```
```  1138 (* setup theory *)
```
```  1139
```
```  1140 val setup =
```
```  1141   ind_cases_setup #>
```
```  1142   Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
```
```  1143     "declaration of monotonicity rule";
```
```  1144
```
```  1145
```
```  1146 (* outer syntax *)
```
```  1147
```
```  1148 fun gen_ind_decl mk_def coind =
```
```  1149   Parse.fixes -- Parse.for_fixes --
```
```  1150   Scan.optional Parse_Spec.where_alt_specs [] --
```
```  1151   Scan.optional (@{keyword "monos"} |-- Parse.!!! Parse_Spec.xthms1) []
```
```  1152   >> (fn (((preds, params), specs), monos) =>
```
```  1153       (snd o gen_add_inductive mk_def true coind preds params specs monos));
```
```  1154
```
```  1155 val ind_decl = gen_ind_decl add_ind_def;
```
```  1156
```
```  1157 val _ =
```
```  1158   Outer_Syntax.local_theory @{command_spec "inductive"} "define inductive predicates"
```
```  1159     (ind_decl false);
```
```  1160
```
```  1161 val _ =
```
```  1162   Outer_Syntax.local_theory @{command_spec "coinductive"} "define coinductive predicates"
```
```  1163     (ind_decl true);
```
```  1164
```
```  1165 val _ =
```
```  1166   Outer_Syntax.local_theory @{command_spec "inductive_cases"}
```
```  1167     "create simplified instances of elimination rules"
```
```  1168     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
```
```  1169
```
```  1170 val _ =
```
```  1171   Outer_Syntax.local_theory @{command_spec "inductive_simps"}
```
```  1172     "create simplification rules for inductive predicates"
```
```  1173     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
```
```  1174
```
```  1175 val _ =
```
```  1176   Outer_Syntax.improper_command @{command_spec "print_inductives"}
```
```  1177     "print (co)inductive definitions and monotonicity rules"
```
```  1178     (Scan.succeed
```
```  1179       (Toplevel.no_timing o Toplevel.unknown_context o
```
```  1180         Toplevel.keep (print_inductives o Toplevel.context_of)));
```
```  1181
```
```  1182 end;
```