src/HOL/Tools/Nitpick/nitpick_hol.ML
author wenzelm
Fri Sep 24 15:53:13 2010 +0200 (2010-09-24)
changeset 39687 4e9b6ada3a21
parent 39557 fe5722fce758
child 40132 7ee65dbffa31
permissions -rw-r--r--
modernized structure Ord_List;
     1 (*  Title:      HOL/Tools/Nitpick/nitpick_hol.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2008, 2009, 2010
     4 
     5 Auxiliary HOL-related functions used by Nitpick.
     6 *)
     7 
     8 signature NITPICK_HOL =
     9 sig
    10   type styp = Nitpick_Util.styp
    11   type const_table = term list Symtab.table
    12   type special_fun = (styp * int list * term list) * styp
    13   type unrolled = styp * styp
    14   type wf_cache = (styp * (bool * bool)) list
    15 
    16   type hol_context =
    17     {thy: theory,
    18      ctxt: Proof.context,
    19      max_bisim_depth: int,
    20      boxes: (typ option * bool option) list,
    21      stds: (typ option * bool) list,
    22      wfs: (styp option * bool option) list,
    23      user_axioms: bool option,
    24      debug: bool,
    25      whacks: term list,
    26      binary_ints: bool option,
    27      destroy_constrs: bool,
    28      specialize: bool,
    29      star_linear_preds: bool,
    30      tac_timeout: Time.time option,
    31      evals: term list,
    32      case_names: (string * int) list,
    33      def_table: const_table,
    34      nondef_table: const_table,
    35      user_nondefs: term list,
    36      simp_table: const_table Unsynchronized.ref,
    37      psimp_table: const_table,
    38      choice_spec_table: const_table,
    39      intro_table: const_table,
    40      ground_thm_table: term list Inttab.table,
    41      ersatz_table: (string * string) list,
    42      skolems: (string * string list) list Unsynchronized.ref,
    43      special_funs: special_fun list Unsynchronized.ref,
    44      unrolled_preds: unrolled list Unsynchronized.ref,
    45      wf_cache: wf_cache Unsynchronized.ref,
    46      constr_cache: (typ * styp list) list Unsynchronized.ref}
    47 
    48   datatype fixpoint_kind = Lfp | Gfp | NoFp
    49   datatype boxability =
    50     InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
    51 
    52   val name_sep : string
    53   val numeral_prefix : string
    54   val base_prefix : string
    55   val step_prefix : string
    56   val unrolled_prefix : string
    57   val ubfp_prefix : string
    58   val lbfp_prefix : string
    59   val quot_normal_prefix : string
    60   val skolem_prefix : string
    61   val special_prefix : string
    62   val uncurry_prefix : string
    63   val eval_prefix : string
    64   val iter_var_prefix : string
    65   val strip_first_name_sep : string -> string * string
    66   val original_name : string -> string
    67   val abs_var : indexname * typ -> term -> term
    68   val s_let : string -> int -> typ -> typ -> (term -> term) -> term -> term
    69   val s_betapply : typ list -> term * term -> term
    70   val s_betapplys : typ list -> term * term list -> term
    71   val s_conj : term * term -> term
    72   val s_disj : term * term -> term
    73   val strip_any_connective : term -> term list * term
    74   val conjuncts_of : term -> term list
    75   val disjuncts_of : term -> term list
    76   val unarize_unbox_etc_type : typ -> typ
    77   val uniterize_unarize_unbox_etc_type : typ -> typ
    78   val string_for_type : Proof.context -> typ -> string
    79   val pretty_for_type : Proof.context -> typ -> Pretty.T
    80   val prefix_name : string -> string -> string
    81   val shortest_name : string -> string
    82   val short_name : string -> string
    83   val shorten_names_in_term : term -> term
    84   val strict_type_match : theory -> typ * typ -> bool
    85   val type_match : theory -> typ * typ -> bool
    86   val const_match : theory -> styp * styp -> bool
    87   val term_match : theory -> term * term -> bool
    88   val frac_from_term_pair : typ -> term -> term -> term
    89   val is_TFree : typ -> bool
    90   val is_higher_order_type : typ -> bool
    91   val is_fun_type : typ -> bool
    92   val is_set_type : typ -> bool
    93   val is_pair_type : typ -> bool
    94   val is_lfp_iterator_type : typ -> bool
    95   val is_gfp_iterator_type : typ -> bool
    96   val is_fp_iterator_type : typ -> bool
    97   val is_iterator_type : typ -> bool
    98   val is_boolean_type : typ -> bool
    99   val is_integer_type : typ -> bool
   100   val is_bit_type : typ -> bool
   101   val is_word_type : typ -> bool
   102   val is_integer_like_type : typ -> bool
   103   val is_record_type : typ -> bool
   104   val is_number_type : Proof.context -> typ -> bool
   105   val const_for_iterator_type : typ -> styp
   106   val strip_n_binders : int -> typ -> typ list * typ
   107   val nth_range_type : int -> typ -> typ
   108   val num_factors_in_type : typ -> int
   109   val num_binder_types : typ -> int
   110   val curried_binder_types : typ -> typ list
   111   val mk_flat_tuple : typ -> term list -> term
   112   val dest_n_tuple : int -> term -> term list
   113   val is_real_datatype : theory -> string -> bool
   114   val is_standard_datatype : theory -> (typ option * bool) list -> typ -> bool
   115   val is_codatatype : Proof.context -> typ -> bool
   116   val is_quot_type : Proof.context -> typ -> bool
   117   val is_pure_typedef : Proof.context -> typ -> bool
   118   val is_univ_typedef : Proof.context -> typ -> bool
   119   val is_datatype : Proof.context -> (typ option * bool) list -> typ -> bool
   120   val is_record_constr : styp -> bool
   121   val is_record_get : theory -> styp -> bool
   122   val is_record_update : theory -> styp -> bool
   123   val is_abs_fun : Proof.context -> styp -> bool
   124   val is_rep_fun : Proof.context -> styp -> bool
   125   val is_quot_abs_fun : Proof.context -> styp -> bool
   126   val is_quot_rep_fun : Proof.context -> styp -> bool
   127   val mate_of_rep_fun : Proof.context -> styp -> styp
   128   val is_constr_like : Proof.context -> styp -> bool
   129   val is_constr : Proof.context -> (typ option * bool) list -> styp -> bool
   130   val is_sel : string -> bool
   131   val is_sel_like_and_no_discr : string -> bool
   132   val box_type : hol_context -> boxability -> typ -> typ
   133   val binarize_nat_and_int_in_type : typ -> typ
   134   val binarize_nat_and_int_in_term : term -> term
   135   val discr_for_constr : styp -> styp
   136   val num_sels_for_constr_type : typ -> int
   137   val nth_sel_name_for_constr_name : string -> int -> string
   138   val nth_sel_for_constr : styp -> int -> styp
   139   val binarized_and_boxed_nth_sel_for_constr :
   140     hol_context -> bool -> styp -> int -> styp
   141   val sel_no_from_name : string -> int
   142   val close_form : term -> term
   143   val distinctness_formula : typ -> term list -> term
   144   val register_frac_type :
   145     string -> (string * string) list -> morphism -> Context.generic
   146     -> Context.generic
   147   val register_frac_type_global :
   148     string -> (string * string) list -> theory -> theory
   149   val unregister_frac_type :
   150     string -> morphism -> Context.generic -> Context.generic
   151   val unregister_frac_type_global : string -> theory -> theory
   152   val register_codatatype :
   153     typ -> string -> styp list -> morphism -> Context.generic -> Context.generic
   154   val register_codatatype_global :
   155     typ -> string -> styp list -> theory -> theory
   156   val unregister_codatatype :
   157     typ -> morphism -> Context.generic -> Context.generic
   158   val unregister_codatatype_global : typ -> theory -> theory
   159   val datatype_constrs : hol_context -> typ -> styp list
   160   val binarized_and_boxed_datatype_constrs :
   161     hol_context -> bool -> typ -> styp list
   162   val num_datatype_constrs : hol_context -> typ -> int
   163   val constr_name_for_sel_like : string -> string
   164   val binarized_and_boxed_constr_for_sel : hol_context -> bool -> styp -> styp
   165   val discriminate_value : hol_context -> styp -> term -> term
   166   val select_nth_constr_arg :
   167     Proof.context -> (typ option * bool) list -> styp -> term -> int -> typ
   168     -> term
   169   val construct_value :
   170     Proof.context -> (typ option * bool) list -> styp -> term list -> term
   171   val coerce_term : hol_context -> typ list -> typ -> typ -> term -> term
   172   val card_of_type : (typ * int) list -> typ -> int
   173   val bounded_card_of_type : int -> int -> (typ * int) list -> typ -> int
   174   val bounded_exact_card_of_type :
   175     hol_context -> typ list -> int -> int -> (typ * int) list -> typ -> int
   176   val is_finite_type : hol_context -> typ -> bool
   177   val is_small_finite_type : hol_context -> typ -> bool
   178   val special_bounds : term list -> (indexname * typ) list
   179   val is_funky_typedef : Proof.context -> typ -> bool
   180   val all_axioms_of :
   181     Proof.context -> (term * term) list -> term list * term list * term list
   182   val arity_of_built_in_const :
   183     theory -> (typ option * bool) list -> styp -> int option
   184   val is_built_in_const :
   185     theory -> (typ option * bool) list -> styp -> bool
   186   val term_under_def : term -> term
   187   val case_const_names :
   188     Proof.context -> (typ option * bool) list -> (string * int) list
   189   val unfold_defs_in_term : hol_context -> term -> term
   190   val const_def_table :
   191     Proof.context -> (term * term) list -> term list -> const_table
   192   val const_nondef_table : term list -> const_table
   193   val const_simp_table : Proof.context -> (term * term) list -> const_table
   194   val const_psimp_table : Proof.context -> (term * term) list -> const_table
   195   val const_choice_spec_table :
   196     Proof.context -> (term * term) list -> const_table
   197   val inductive_intro_table :
   198     Proof.context -> (term * term) list -> const_table -> const_table
   199   val ground_theorem_table : theory -> term list Inttab.table
   200   val ersatz_table : Proof.context -> (string * string) list
   201   val add_simps : const_table Unsynchronized.ref -> string -> term list -> unit
   202   val inverse_axioms_for_rep_fun : Proof.context -> styp -> term list
   203   val optimized_typedef_axioms : Proof.context -> string * typ list -> term list
   204   val optimized_quot_type_axioms :
   205     Proof.context -> (typ option * bool) list -> string * typ list -> term list
   206   val def_of_const : theory -> const_table -> styp -> term option
   207   val fixpoint_kind_of_rhs : term -> fixpoint_kind
   208   val fixpoint_kind_of_const :
   209     theory -> const_table -> string * typ -> fixpoint_kind
   210   val is_real_inductive_pred : hol_context -> styp -> bool
   211   val is_constr_pattern_lhs : Proof.context -> term -> bool
   212   val is_constr_pattern_formula : Proof.context -> term -> bool
   213   val nondef_props_for_const :
   214     theory -> bool -> const_table -> styp -> term list
   215   val is_choice_spec_fun : hol_context -> styp -> bool
   216   val is_choice_spec_axiom : theory -> const_table -> term -> bool
   217   val is_real_equational_fun : hol_context -> styp -> bool
   218   val is_equational_fun_but_no_plain_def : hol_context -> styp -> bool
   219   val codatatype_bisim_axioms : hol_context -> typ -> term list
   220   val is_well_founded_inductive_pred : hol_context -> styp -> bool
   221   val unrolled_inductive_pred_const : hol_context -> bool -> styp -> term
   222   val equational_fun_axioms : hol_context -> styp -> term list
   223   val is_equational_fun_surely_complete : hol_context -> styp -> bool
   224   val merged_type_var_table_for_terms :
   225     theory -> term list -> (sort * string) list
   226   val merge_type_vars_in_term :
   227     theory -> bool -> (sort * string) list -> term -> term
   228   val ground_types_in_type : hol_context -> bool -> typ -> typ list
   229   val ground_types_in_terms : hol_context -> bool -> term list -> typ list
   230 end;
   231 
   232 structure Nitpick_HOL : NITPICK_HOL =
   233 struct
   234 
   235 open Nitpick_Util
   236 
   237 type const_table = term list Symtab.table
   238 type special_fun = (styp * int list * term list) * styp
   239 type unrolled = styp * styp
   240 type wf_cache = (styp * (bool * bool)) list
   241 
   242 type hol_context =
   243   {thy: theory,
   244    ctxt: Proof.context,
   245    max_bisim_depth: int,
   246    boxes: (typ option * bool option) list,
   247    stds: (typ option * bool) list,
   248    wfs: (styp option * bool option) list,
   249    user_axioms: bool option,
   250    debug: bool,
   251    whacks: term list,
   252    binary_ints: bool option,
   253    destroy_constrs: bool,
   254    specialize: bool,
   255    star_linear_preds: bool,
   256    tac_timeout: Time.time option,
   257    evals: term list,
   258    case_names: (string * int) list,
   259    def_table: const_table,
   260    nondef_table: const_table,
   261    user_nondefs: term list,
   262    simp_table: const_table Unsynchronized.ref,
   263    psimp_table: const_table,
   264    choice_spec_table: const_table,
   265    intro_table: const_table,
   266    ground_thm_table: term list Inttab.table,
   267    ersatz_table: (string * string) list,
   268    skolems: (string * string list) list Unsynchronized.ref,
   269    special_funs: special_fun list Unsynchronized.ref,
   270    unrolled_preds: unrolled list Unsynchronized.ref,
   271    wf_cache: wf_cache Unsynchronized.ref,
   272    constr_cache: (typ * styp list) list Unsynchronized.ref}
   273 
   274 datatype fixpoint_kind = Lfp | Gfp | NoFp
   275 datatype boxability =
   276   InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
   277 
   278 structure Data = Generic_Data(
   279   type T = {frac_types: (string * (string * string) list) list,
   280             codatatypes: (string * (string * styp list)) list}
   281   val empty = {frac_types = [], codatatypes = []}
   282   val extend = I
   283   fun merge ({frac_types = fs1, codatatypes = cs1},
   284              {frac_types = fs2, codatatypes = cs2}) : T =
   285     {frac_types = AList.merge (op =) (K true) (fs1, fs2),
   286      codatatypes = AList.merge (op =) (K true) (cs1, cs2)})
   287 
   288 val name_sep = "$"
   289 val numeral_prefix = nitpick_prefix ^ "num" ^ name_sep
   290 val sel_prefix = nitpick_prefix ^ "sel"
   291 val discr_prefix = nitpick_prefix ^ "is" ^ name_sep
   292 val set_prefix = nitpick_prefix ^ "set" ^ name_sep
   293 val lfp_iterator_prefix = nitpick_prefix ^ "lfpit" ^ name_sep
   294 val gfp_iterator_prefix = nitpick_prefix ^ "gfpit" ^ name_sep
   295 val unrolled_prefix = nitpick_prefix ^ "unroll" ^ name_sep
   296 val base_prefix = nitpick_prefix ^ "base" ^ name_sep
   297 val step_prefix = nitpick_prefix ^ "step" ^ name_sep
   298 val ubfp_prefix = nitpick_prefix ^ "ubfp" ^ name_sep
   299 val lbfp_prefix = nitpick_prefix ^ "lbfp" ^ name_sep
   300 val quot_normal_prefix = nitpick_prefix ^ "qn" ^ name_sep
   301 val skolem_prefix = nitpick_prefix ^ "sk"
   302 val special_prefix = nitpick_prefix ^ "sp"
   303 val uncurry_prefix = nitpick_prefix ^ "unc"
   304 val eval_prefix = nitpick_prefix ^ "eval"
   305 val iter_var_prefix = "i"
   306 
   307 (** Constant/type information and term/type manipulation **)
   308 
   309 fun sel_prefix_for j = sel_prefix ^ string_of_int j ^ name_sep
   310 fun quot_normal_name_for_type ctxt T =
   311   quot_normal_prefix ^ unyxml (Syntax.string_of_typ ctxt T)
   312 
   313 val strip_first_name_sep =
   314   Substring.full #> Substring.position name_sep ##> Substring.triml 1
   315   #> pairself Substring.string
   316 fun original_name s =
   317   if String.isPrefix nitpick_prefix s then
   318     case strip_first_name_sep s of (s1, "") => s1 | (_, s2) => original_name s2
   319   else
   320     s
   321 
   322 fun abs_var ((s, j), T) body = Abs (s, T, abstract_over (Var ((s, j), T), body))
   323 
   324 fun let_var s = (nitpick_prefix ^ s, 999)
   325 val let_inline_threshold = 20
   326 
   327 fun s_let s n abs_T body_T f t =
   328   if (n - 1) * (size_of_term t - 1) <= let_inline_threshold then
   329     f t
   330   else
   331     let val z = (let_var s, abs_T) in
   332       Const (@{const_name Let}, abs_T --> (abs_T --> body_T) --> body_T)
   333       $ t $ abs_var z (incr_boundvars 1 (f (Var z)))
   334     end
   335 
   336 fun loose_bvar1_count (Bound i, k) = if i = k then 1 else 0
   337   | loose_bvar1_count (t1 $ t2, k) =
   338     loose_bvar1_count (t1, k) + loose_bvar1_count (t2, k)
   339   | loose_bvar1_count (Abs (_, _, t), k) = loose_bvar1_count (t, k + 1)
   340   | loose_bvar1_count _ = 0
   341 
   342 fun s_betapply _ (Const (@{const_name If}, _) $ @{const True} $ t1', _) = t1'
   343   | s_betapply _ (Const (@{const_name If}, _) $ @{const False} $ _, t2) = t2
   344   | s_betapply Ts (Const (@{const_name Let},
   345                           Type (_, [bound_T, Type (_, [_, body_T])]))
   346                    $ t12 $ Abs (s, T, t13'), t2) =
   347     let val body_T' = range_type body_T in
   348       Const (@{const_name Let}, bound_T --> (bound_T --> body_T') --> body_T')
   349       $ t12 $ Abs (s, T, s_betapply (T :: Ts) (t13', incr_boundvars 1 t2))
   350     end
   351   | s_betapply Ts (t1 as Abs (s1, T1, t1'), t2) =
   352     (s_let s1 (loose_bvar1_count (t1', 0)) T1 (fastype_of1 (T1 :: Ts, t1'))
   353               (curry betapply t1) t2
   354      handle TERM _ => betapply (t1, t2)) (* FIXME: fix all uses *)
   355   | s_betapply _ (t1, t2) = t1 $ t2
   356 fun s_betapplys Ts = Library.foldl (s_betapply Ts)
   357 
   358 fun s_beta_norm Ts t =
   359   let
   360     fun aux _ (Var _) = raise Same.SAME
   361       | aux Ts (Abs (s, T, t')) = Abs (s, T, aux (T :: Ts) t')
   362       | aux Ts ((t1 as Abs _) $ t2) =
   363         Same.commit (aux Ts) (s_betapply Ts (t1, t2))
   364       | aux Ts (t1 $ t2) =
   365         ((case aux Ts t1 of
   366            t1 as Abs _ => Same.commit (aux Ts) (s_betapply Ts (t1, t2))
   367          | t1 => t1 $ Same.commit (aux Ts) t2)
   368         handle Same.SAME => t1 $ aux Ts t2)
   369       | aux _ _ = raise Same.SAME
   370   in aux Ts t handle Same.SAME => t end
   371 
   372 fun s_conj (t1, @{const True}) = t1
   373   | s_conj (@{const True}, t2) = t2
   374   | s_conj (t1, t2) =
   375     if t1 = @{const False} orelse t2 = @{const False} then @{const False}
   376     else HOLogic.mk_conj (t1, t2)
   377 fun s_disj (t1, @{const False}) = t1
   378   | s_disj (@{const False}, t2) = t2
   379   | s_disj (t1, t2) =
   380     if t1 = @{const True} orelse t2 = @{const True} then @{const True}
   381     else HOLogic.mk_disj (t1, t2)
   382 
   383 fun strip_connective conn_t (t as (t0 $ t1 $ t2)) =
   384     if t0 = conn_t then strip_connective t0 t2 @ strip_connective t0 t1 else [t]
   385   | strip_connective _ t = [t]
   386 fun strip_any_connective (t as (t0 $ _ $ _)) =
   387     if t0 = @{const HOL.conj} orelse t0 = @{const HOL.disj} then
   388       (strip_connective t0 t, t0)
   389     else
   390       ([t], @{const Not})
   391   | strip_any_connective t = ([t], @{const Not})
   392 val conjuncts_of = strip_connective @{const HOL.conj}
   393 val disjuncts_of = strip_connective @{const HOL.disj}
   394 
   395 (* When you add constants to these lists, make sure to handle them in
   396    "Nitpick_Nut.nut_from_term", and perhaps in "Nitpick_Mono.consider_term" as
   397    well. *)
   398 val built_in_consts =
   399   [(@{const_name all}, 1),
   400    (@{const_name "=="}, 2),
   401    (@{const_name "==>"}, 2),
   402    (@{const_name Pure.conjunction}, 2),
   403    (@{const_name Trueprop}, 1),
   404    (@{const_name Not}, 1),
   405    (@{const_name False}, 0),
   406    (@{const_name True}, 0),
   407    (@{const_name All}, 1),
   408    (@{const_name Ex}, 1),
   409    (@{const_name HOL.eq}, 1),
   410    (@{const_name HOL.conj}, 2),
   411    (@{const_name HOL.disj}, 2),
   412    (@{const_name HOL.implies}, 2),
   413    (@{const_name If}, 3),
   414    (@{const_name Let}, 2),
   415    (@{const_name Pair}, 2),
   416    (@{const_name fst}, 1),
   417    (@{const_name snd}, 1),
   418    (@{const_name Id}, 0),
   419    (@{const_name converse}, 1),
   420    (@{const_name trancl}, 1),
   421    (@{const_name rel_comp}, 2),
   422    (@{const_name image}, 2),
   423    (@{const_name finite}, 1),
   424    (@{const_name unknown}, 0),
   425    (@{const_name is_unknown}, 1),
   426    (@{const_name safe_The}, 1),
   427    (@{const_name Frac}, 0),
   428    (@{const_name norm_frac}, 0)]
   429 val built_in_nat_consts =
   430   [(@{const_name Suc}, 0),
   431    (@{const_name nat}, 0),
   432    (@{const_name nat_gcd}, 0),
   433    (@{const_name nat_lcm}, 0)]
   434 val built_in_typed_consts =
   435   [((@{const_name zero_class.zero}, int_T), 0),
   436    ((@{const_name one_class.one}, int_T), 0),
   437    ((@{const_name plus_class.plus}, int_T --> int_T --> int_T), 0),
   438    ((@{const_name minus_class.minus}, int_T --> int_T --> int_T), 0),
   439    ((@{const_name times_class.times}, int_T --> int_T --> int_T), 0),
   440    ((@{const_name div_class.div}, int_T --> int_T --> int_T), 0),
   441    ((@{const_name uminus_class.uminus}, int_T --> int_T), 0),
   442    ((@{const_name ord_class.less}, int_T --> int_T --> bool_T), 2),
   443    ((@{const_name ord_class.less_eq}, int_T --> int_T --> bool_T), 2)]
   444 val built_in_typed_nat_consts =
   445   [((@{const_name zero_class.zero}, nat_T), 0),
   446    ((@{const_name one_class.one}, nat_T), 0),
   447    ((@{const_name plus_class.plus}, nat_T --> nat_T --> nat_T), 0),
   448    ((@{const_name minus_class.minus}, nat_T --> nat_T --> nat_T), 0),
   449    ((@{const_name times_class.times}, nat_T --> nat_T --> nat_T), 0),
   450    ((@{const_name div_class.div}, nat_T --> nat_T --> nat_T), 0),
   451    ((@{const_name ord_class.less}, nat_T --> nat_T --> bool_T), 2),
   452    ((@{const_name ord_class.less_eq}, nat_T --> nat_T --> bool_T), 2),
   453    ((@{const_name of_nat}, nat_T --> int_T), 0)]
   454 val built_in_set_consts =
   455   [(@{const_name ord_class.less_eq}, 2)]
   456 
   457 fun unarize_type @{typ "unsigned_bit word"} = nat_T
   458   | unarize_type @{typ "signed_bit word"} = int_T
   459   | unarize_type (Type (s, Ts as _ :: _)) = Type (s, map unarize_type Ts)
   460   | unarize_type T = T
   461 fun unarize_unbox_etc_type (Type (@{type_name fin_fun}, Ts)) =
   462     unarize_unbox_etc_type (Type (@{type_name fun}, Ts))
   463   | unarize_unbox_etc_type (Type (@{type_name fun_box}, Ts)) =
   464     unarize_unbox_etc_type (Type (@{type_name fun}, Ts))
   465   | unarize_unbox_etc_type (Type (@{type_name pair_box}, Ts)) =
   466     Type (@{type_name prod}, map unarize_unbox_etc_type Ts)
   467   | unarize_unbox_etc_type @{typ "unsigned_bit word"} = nat_T
   468   | unarize_unbox_etc_type @{typ "signed_bit word"} = int_T
   469   | unarize_unbox_etc_type (Type (s, Ts as _ :: _)) =
   470     Type (s, map unarize_unbox_etc_type Ts)
   471   | unarize_unbox_etc_type T = T
   472 fun uniterize_type (Type (s, Ts as _ :: _)) = Type (s, map uniterize_type Ts)
   473   | uniterize_type @{typ bisim_iterator} = nat_T
   474   | uniterize_type T = T
   475 val uniterize_unarize_unbox_etc_type = uniterize_type o unarize_unbox_etc_type
   476 
   477 fun string_for_type ctxt = Syntax.string_of_typ ctxt o unarize_unbox_etc_type
   478 fun pretty_for_type ctxt = Syntax.pretty_typ ctxt o unarize_unbox_etc_type
   479 
   480 val prefix_name = Long_Name.qualify o Long_Name.base_name
   481 fun shortest_name s = List.last (space_explode "." s) handle List.Empty => ""
   482 val prefix_abs_vars = Term.map_abs_vars o prefix_name
   483 fun short_name s =
   484   case space_explode name_sep s of
   485     [_] => s |> String.isPrefix nitpick_prefix s ? unprefix nitpick_prefix
   486   | ss => map shortest_name ss |> space_implode "_"
   487 fun shorten_names_in_type (Type (s, Ts)) =
   488     Type (short_name s, map shorten_names_in_type Ts)
   489   | shorten_names_in_type T = T
   490 val shorten_names_in_term =
   491   map_aterms (fn Const (s, T) => Const (short_name s, T) | t => t)
   492   #> map_types shorten_names_in_type
   493 
   494 fun strict_type_match thy (T1, T2) =
   495   (Sign.typ_match thy (T2, T1) Vartab.empty; true)
   496   handle Type.TYPE_MATCH => false
   497 fun type_match thy = strict_type_match thy o pairself unarize_unbox_etc_type
   498 fun const_match thy ((s1, T1), (s2, T2)) =
   499   s1 = s2 andalso type_match thy (T1, T2)
   500 fun term_match thy (Const x1, Const x2) = const_match thy (x1, x2)
   501   | term_match thy (Free (s1, T1), Free (s2, T2)) =
   502     const_match thy ((shortest_name s1, T1), (shortest_name s2, T2))
   503   | term_match _ (t1, t2) = t1 aconv t2
   504 
   505 fun frac_from_term_pair T t1 t2 =
   506   case snd (HOLogic.dest_number t1) of
   507     0 => HOLogic.mk_number T 0
   508   | n1 => case snd (HOLogic.dest_number t2) of
   509             1 => HOLogic.mk_number T n1
   510           | n2 => Const (@{const_name divide}, T --> T --> T)
   511                   $ HOLogic.mk_number T n1 $ HOLogic.mk_number T n2
   512 
   513 fun is_TFree (TFree _) = true
   514   | is_TFree _ = false
   515 fun is_higher_order_type (Type (@{type_name fun}, _)) = true
   516   | is_higher_order_type (Type (_, Ts)) = exists is_higher_order_type Ts
   517   | is_higher_order_type _ = false
   518 fun is_fun_type (Type (@{type_name fun}, _)) = true
   519   | is_fun_type _ = false
   520 fun is_set_type (Type (@{type_name fun}, [_, @{typ bool}])) = true
   521   | is_set_type _ = false
   522 fun is_pair_type (Type (@{type_name prod}, _)) = true
   523   | is_pair_type _ = false
   524 fun is_lfp_iterator_type (Type (s, _)) = String.isPrefix lfp_iterator_prefix s
   525   | is_lfp_iterator_type _ = false
   526 fun is_gfp_iterator_type (Type (s, _)) = String.isPrefix gfp_iterator_prefix s
   527   | is_gfp_iterator_type _ = false
   528 val is_fp_iterator_type = is_lfp_iterator_type orf is_gfp_iterator_type
   529 fun is_iterator_type T =
   530   (T = @{typ bisim_iterator} orelse is_fp_iterator_type T)
   531 fun is_boolean_type T = (T = prop_T orelse T = bool_T)
   532 fun is_integer_type T = (T = nat_T orelse T = int_T)
   533 fun is_bit_type T = (T = @{typ unsigned_bit} orelse T = @{typ signed_bit})
   534 fun is_word_type (Type (@{type_name word}, _)) = true
   535   | is_word_type _ = false
   536 val is_integer_like_type = is_iterator_type orf is_integer_type orf is_word_type
   537 val is_record_type = not o null o Record.dest_recTs
   538 fun is_frac_type ctxt (Type (s, [])) =
   539     s |> AList.lookup (op =) (#frac_types (Data.get (Context.Proof ctxt)))
   540       |> these |> null |> not
   541   | is_frac_type _ _ = false
   542 fun is_number_type ctxt = is_integer_like_type orf is_frac_type ctxt
   543 
   544 fun iterator_type_for_const gfp (s, T) =
   545   Type ((if gfp then gfp_iterator_prefix else lfp_iterator_prefix) ^ s,
   546         binder_types T)
   547 fun const_for_iterator_type (Type (s, Ts)) =
   548     (strip_first_name_sep s |> snd, Ts ---> bool_T)
   549   | const_for_iterator_type T =
   550     raise TYPE ("Nitpick_HOL.const_for_iterator_type", [T], [])
   551 
   552 fun strip_n_binders 0 T = ([], T)
   553   | strip_n_binders n (Type (@{type_name fun}, [T1, T2])) =
   554     strip_n_binders (n - 1) T2 |>> cons T1
   555   | strip_n_binders n (Type (@{type_name fun_box}, Ts)) =
   556     strip_n_binders n (Type (@{type_name fun}, Ts))
   557   | strip_n_binders _ T = raise TYPE ("Nitpick_HOL.strip_n_binders", [T], [])
   558 val nth_range_type = snd oo strip_n_binders
   559 
   560 fun num_factors_in_type (Type (@{type_name prod}, [T1, T2])) =
   561     fold (Integer.add o num_factors_in_type) [T1, T2] 0
   562   | num_factors_in_type _ = 1
   563 fun num_binder_types (Type (@{type_name fun}, [_, T2])) =
   564     1 + num_binder_types T2
   565   | num_binder_types _ = 0
   566 val curried_binder_types = maps HOLogic.flatten_tupleT o binder_types
   567 fun maybe_curried_binder_types T =
   568   (if is_pair_type (body_type T) then binder_types else curried_binder_types) T
   569 
   570 fun mk_flat_tuple _ [t] = t
   571   | mk_flat_tuple (Type (@{type_name prod}, [T1, T2])) (t :: ts) =
   572     HOLogic.pair_const T1 T2 $ t $ (mk_flat_tuple T2 ts)
   573   | mk_flat_tuple T ts = raise TYPE ("Nitpick_HOL.mk_flat_tuple", [T], ts)
   574 fun dest_n_tuple 1 t = [t]
   575   | dest_n_tuple n t = HOLogic.dest_prod t ||> dest_n_tuple (n - 1) |> op ::
   576 
   577 type typedef_info =
   578   {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string,
   579    set_def: thm option, prop_of_Rep: thm, set_name: string,
   580    Abs_inverse: thm option, Rep_inverse: thm option}
   581 
   582 fun typedef_info ctxt s =
   583   if is_frac_type ctxt (Type (s, [])) then
   584     SOME {abs_type = Type (s, []), rep_type = @{typ "int * int"},
   585           Abs_name = @{const_name Abs_Frac}, Rep_name = @{const_name Rep_Frac},
   586           set_def = NONE, prop_of_Rep = @{prop "Rep_Frac x \<in> Frac"}
   587                           |> Logic.varify_global,
   588           set_name = @{const_name Frac}, Abs_inverse = NONE, Rep_inverse = NONE}
   589   else case Typedef.get_info ctxt s of
   590     (* When several entries are returned, it shouldn't matter much which one
   591        we take (according to Florian Haftmann). *)
   592     (* The "Logic.varifyT_global" calls are a temporary hack because these
   593        types's type variables sometimes clash with locally fixed type variables.
   594        Remove these calls once "Typedef" is fully localized. *)
   595     ({abs_type, rep_type, Abs_name, Rep_name, ...},
   596      {set_def, Rep, Abs_inverse, Rep_inverse, ...}) :: _ =>
   597     SOME {abs_type = Logic.varifyT_global abs_type,
   598           rep_type = Logic.varifyT_global rep_type, Abs_name = Abs_name,
   599           Rep_name = Rep_name, set_def = set_def, prop_of_Rep = prop_of Rep,
   600           set_name = set_prefix ^ s, Abs_inverse = SOME Abs_inverse,
   601           Rep_inverse = SOME Rep_inverse}
   602   | _ => NONE
   603 
   604 val is_typedef = is_some oo typedef_info
   605 val is_real_datatype = is_some oo Datatype.get_info
   606 fun is_standard_datatype thy = the oo triple_lookup (type_match thy)
   607 
   608 (* FIXME: Use antiquotation for "code_numeral" below or detect "rep_datatype",
   609    e.g., by adding a field to "Datatype_Aux.info". *)
   610 fun is_basic_datatype thy stds s =
   611   member (op =) [@{type_name prod}, @{type_name bool}, @{type_name int},
   612                  "Code_Numeral.code_numeral"] s orelse
   613   (s = @{type_name nat} andalso is_standard_datatype thy stds nat_T)
   614 
   615 (* TODO: use "Term_Subst.instantiateT" instead? *)
   616 fun instantiate_type thy T1 T1' T2 =
   617   Same.commit (Envir.subst_type_same
   618                    (Sign.typ_match thy (T1, T1') Vartab.empty)) T2
   619   handle Type.TYPE_MATCH =>
   620          raise TYPE ("Nitpick_HOL.instantiate_type", [T1, T1'], [])
   621 fun varify_and_instantiate_type ctxt T1 T1' T2 =
   622   let val thy = ProofContext.theory_of ctxt in
   623     instantiate_type thy (varify_type ctxt T1) T1' (varify_type ctxt T2)
   624   end
   625 
   626 fun repair_constr_type ctxt body_T' T =
   627   varify_and_instantiate_type ctxt (body_type T) body_T' T
   628 
   629 fun register_frac_type_generic frac_s ersaetze generic =
   630   let
   631     val {frac_types, codatatypes} = Data.get generic
   632     val frac_types = AList.update (op =) (frac_s, ersaetze) frac_types
   633   in Data.put {frac_types = frac_types, codatatypes = codatatypes} generic end
   634 (* TODO: Consider morphism. *)
   635 fun register_frac_type frac_s ersaetze (_ : morphism) =
   636   register_frac_type_generic frac_s ersaetze
   637 val register_frac_type_global = Context.theory_map oo register_frac_type_generic
   638 
   639 fun unregister_frac_type_generic frac_s = register_frac_type_generic frac_s []
   640 (* TODO: Consider morphism. *)
   641 fun unregister_frac_type frac_s (_ : morphism) =
   642   unregister_frac_type_generic frac_s
   643 val unregister_frac_type_global =
   644   Context.theory_map o unregister_frac_type_generic
   645 
   646 fun register_codatatype_generic co_T case_name constr_xs generic =
   647   let
   648     val ctxt = Context.proof_of generic
   649     val thy = Context.theory_of generic
   650     val {frac_types, codatatypes} = Data.get generic
   651     val constr_xs = map (apsnd (repair_constr_type ctxt co_T)) constr_xs
   652     val (co_s, co_Ts) = dest_Type co_T
   653     val _ =
   654       if forall is_TFree co_Ts andalso not (has_duplicates (op =) co_Ts) andalso
   655          co_s <> @{type_name fun} andalso
   656          not (is_basic_datatype thy [(NONE, true)] co_s) then
   657         ()
   658       else
   659         raise TYPE ("Nitpick_HOL.register_codatatype_generic", [co_T], [])
   660     val codatatypes = AList.update (op =) (co_s, (case_name, constr_xs))
   661                                    codatatypes
   662   in Data.put {frac_types = frac_types, codatatypes = codatatypes} generic end
   663 (* TODO: Consider morphism. *)
   664 fun register_codatatype co_T case_name constr_xs (_ : morphism) =
   665   register_codatatype_generic co_T case_name constr_xs
   666 val register_codatatype_global =
   667   Context.theory_map ooo register_codatatype_generic
   668 
   669 fun unregister_codatatype_generic co_T = register_codatatype_generic co_T "" []
   670 (* TODO: Consider morphism. *)
   671 fun unregister_codatatype co_T (_ : morphism) =
   672   unregister_codatatype_generic co_T
   673 val unregister_codatatype_global =
   674   Context.theory_map o unregister_codatatype_generic
   675 
   676 fun is_codatatype ctxt (Type (s, _)) =
   677     s |> AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   678       |> Option.map snd |> these |> null |> not
   679   | is_codatatype _ _ = false
   680 fun is_real_quot_type thy (Type (s, _)) =
   681     is_some (Quotient_Info.quotdata_lookup_raw thy s)
   682   | is_real_quot_type _ _ = false
   683 fun is_quot_type ctxt T =
   684   let val thy = ProofContext.theory_of ctxt in
   685     is_real_quot_type thy T andalso not (is_codatatype ctxt T)
   686   end
   687 fun is_pure_typedef ctxt (T as Type (s, _)) =
   688     let val thy = ProofContext.theory_of ctxt in
   689       is_typedef ctxt s andalso
   690       not (is_real_datatype thy s orelse is_real_quot_type thy T orelse
   691            is_codatatype ctxt T orelse is_record_type T orelse
   692            is_integer_like_type T)
   693     end
   694   | is_pure_typedef _ _ = false
   695 fun is_univ_typedef ctxt (Type (s, _)) =
   696     (case typedef_info ctxt s of
   697        SOME {set_def, prop_of_Rep, ...} =>
   698        let
   699          val t_opt =
   700            case set_def of
   701              SOME thm => try (snd o Logic.dest_equals o prop_of) thm
   702            | NONE => try (snd o HOLogic.dest_mem o HOLogic.dest_Trueprop)
   703                          prop_of_Rep
   704        in
   705          case t_opt of
   706            SOME (Const (@{const_name top}, _)) => true
   707            (* "Multiset.multiset" *)
   708          | SOME (Const (@{const_name Collect}, _)
   709                  $ Abs (_, _, Const (@{const_name finite}, _) $ _)) => true
   710            (* "FinFun.finfun" *)
   711          | SOME (Const (@{const_name Collect}, _) $ Abs (_, _,
   712                      Const (@{const_name Ex}, _) $ Abs (_, _,
   713                          Const (@{const_name finite}, _) $ _))) => true
   714          | _ => false
   715        end
   716      | NONE => false)
   717   | is_univ_typedef _ _ = false
   718 fun is_datatype ctxt stds (T as Type (s, _)) =
   719     let val thy = ProofContext.theory_of ctxt in
   720       (is_typedef ctxt s orelse is_codatatype ctxt T orelse
   721        T = @{typ ind} orelse is_real_quot_type thy T) andalso
   722       not (is_basic_datatype thy stds s)
   723     end
   724   | is_datatype _ _ _ = false
   725 
   726 fun all_record_fields thy T =
   727   let val (recs, more) = Record.get_extT_fields thy T in
   728     recs @ more :: all_record_fields thy (snd more)
   729   end
   730   handle TYPE _ => []
   731 fun is_record_constr (s, T) =
   732   String.isSuffix Record.extN s andalso
   733   let val dataT = body_type T in
   734     is_record_type dataT andalso
   735     s = unsuffix Record.ext_typeN (fst (dest_Type dataT)) ^ Record.extN
   736   end
   737 val num_record_fields = Integer.add 1 o length o fst oo Record.get_extT_fields
   738 fun no_of_record_field thy s T1 =
   739   find_index (curry (op =) s o fst)
   740              (Record.get_extT_fields thy T1 ||> single |> op @)
   741 fun is_record_get thy (s, Type (@{type_name fun}, [T1, _])) =
   742     exists (curry (op =) s o fst) (all_record_fields thy T1)
   743   | is_record_get _ _ = false
   744 fun is_record_update thy (s, T) =
   745   String.isSuffix Record.updateN s andalso
   746   exists (curry (op =) (unsuffix Record.updateN s) o fst)
   747          (all_record_fields thy (body_type T))
   748   handle TYPE _ => false
   749 fun is_abs_fun ctxt (s, Type (@{type_name fun}, [_, Type (s', _)])) =
   750     (case typedef_info ctxt s' of
   751        SOME {Abs_name, ...} => s = Abs_name
   752      | NONE => false)
   753   | is_abs_fun _ _ = false
   754 fun is_rep_fun ctxt (s, Type (@{type_name fun}, [Type (s', _), _])) =
   755     (case typedef_info ctxt s' of
   756        SOME {Rep_name, ...} => s = Rep_name
   757      | NONE => false)
   758   | is_rep_fun _ _ = false
   759 fun is_quot_abs_fun ctxt (x as (_, Type (@{type_name fun},
   760                                          [_, abs_T as Type (s', _)]))) =
   761     try (Quotient_Term.absrep_const_chk Quotient_Term.AbsF ctxt) s'
   762     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   763   | is_quot_abs_fun _ _ = false
   764 fun is_quot_rep_fun ctxt (x as (_, Type (@{type_name fun},
   765                                          [abs_T as Type (s', _), _]))) =
   766     try (Quotient_Term.absrep_const_chk Quotient_Term.RepF ctxt) s'
   767     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   768   | is_quot_rep_fun _ _ = false
   769 
   770 fun mate_of_rep_fun ctxt (x as (_, Type (@{type_name fun},
   771                                          [T1 as Type (s', _), T2]))) =
   772     (case typedef_info ctxt s' of
   773        SOME {Abs_name, ...} => (Abs_name, Type (@{type_name fun}, [T2, T1]))
   774      | NONE => raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x]))
   775   | mate_of_rep_fun _ x = raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x])
   776 fun rep_type_for_quot_type thy (T as Type (s, _)) =
   777     let val {qtyp, rtyp, ...} = Quotient_Info.quotdata_lookup thy s in
   778       instantiate_type thy qtyp T rtyp
   779     end
   780   | rep_type_for_quot_type _ T =
   781     raise TYPE ("Nitpick_HOL.rep_type_for_quot_type", [T], [])
   782 fun equiv_relation_for_quot_type thy (Type (s, Ts)) =
   783     let
   784       val {qtyp, equiv_rel, equiv_thm, ...} =
   785         Quotient_Info.quotdata_lookup thy s
   786       val partial =
   787         case prop_of equiv_thm of
   788           @{const Trueprop} $ (Const (@{const_name equivp}, _) $ _) => false
   789         | @{const Trueprop} $ (Const (@{const_name part_equivp}, _) $ _) => true
   790         | _ => raise NOT_SUPPORTED "Ill-formed quotient type equivalence \
   791                                    \relation theorem"
   792       val Ts' = qtyp |> dest_Type |> snd
   793     in (subst_atomic_types (Ts' ~~ Ts) equiv_rel, partial) end
   794   | equiv_relation_for_quot_type _ T =
   795     raise TYPE ("Nitpick_HOL.equiv_relation_for_quot_type", [T], [])
   796 
   797 fun is_coconstr ctxt (s, T) =
   798   case body_type T of
   799     co_T as Type (co_s, _) =>
   800     let val {codatatypes, ...} = Data.get (Context.Proof ctxt) in
   801       exists (fn (s', T') => s = s' andalso repair_constr_type ctxt co_T T' = T)
   802              (AList.lookup (op =) codatatypes co_s |> Option.map snd |> these)
   803     end
   804   | _ => false
   805 fun is_constr_like ctxt (s, T) =
   806   member (op =) [@{const_name FinFun}, @{const_name FunBox},
   807                  @{const_name PairBox}, @{const_name Quot},
   808                  @{const_name Zero_Rep}, @{const_name Suc_Rep}] s orelse
   809   let
   810     val thy = ProofContext.theory_of ctxt
   811     val (x as (_, T)) = (s, unarize_unbox_etc_type T)
   812   in
   813     is_real_constr thy x orelse is_record_constr x orelse
   814     (is_abs_fun ctxt x andalso is_pure_typedef ctxt (range_type T)) orelse
   815     is_coconstr ctxt x
   816   end
   817 fun is_stale_constr ctxt (x as (_, T)) =
   818   is_codatatype ctxt (body_type T) andalso is_constr_like ctxt x andalso
   819   not (is_coconstr ctxt x)
   820 fun is_constr ctxt stds (x as (_, T)) =
   821   let val thy = ProofContext.theory_of ctxt in
   822     is_constr_like ctxt x andalso
   823     not (is_basic_datatype thy stds
   824                          (fst (dest_Type (unarize_type (body_type T))))) andalso
   825     not (is_stale_constr ctxt x)
   826   end
   827 val is_sel = String.isPrefix discr_prefix orf String.isPrefix sel_prefix
   828 val is_sel_like_and_no_discr =
   829   String.isPrefix sel_prefix orf
   830   (member (op =) [@{const_name fst}, @{const_name snd}])
   831 
   832 fun in_fun_lhs_for InConstr = InSel
   833   | in_fun_lhs_for _ = InFunLHS
   834 fun in_fun_rhs_for InConstr = InConstr
   835   | in_fun_rhs_for InSel = InSel
   836   | in_fun_rhs_for InFunRHS1 = InFunRHS2
   837   | in_fun_rhs_for _ = InFunRHS1
   838 
   839 fun is_boxing_worth_it (hol_ctxt : hol_context) boxy T =
   840   case T of
   841     Type (@{type_name fun}, _) =>
   842     (boxy = InPair orelse boxy = InFunLHS) andalso
   843     not (is_boolean_type (body_type T))
   844   | Type (@{type_name prod}, Ts) =>
   845     boxy = InPair orelse boxy = InFunRHS1 orelse boxy = InFunRHS2 orelse
   846     ((boxy = InExpr orelse boxy = InFunLHS) andalso
   847      exists (is_boxing_worth_it hol_ctxt InPair)
   848             (map (box_type hol_ctxt InPair) Ts))
   849   | _ => false
   850 and should_box_type (hol_ctxt as {thy, boxes, ...}) boxy z =
   851   case triple_lookup (type_match thy) boxes (Type z) of
   852     SOME (SOME box_me) => box_me
   853   | _ => is_boxing_worth_it hol_ctxt boxy (Type z)
   854 and box_type hol_ctxt boxy T =
   855   case T of
   856     Type (z as (@{type_name fun}, [T1, T2])) =>
   857     if boxy <> InConstr andalso boxy <> InSel andalso
   858        should_box_type hol_ctxt boxy z then
   859       Type (@{type_name fun_box},
   860             [box_type hol_ctxt InFunLHS T1, box_type hol_ctxt InFunRHS1 T2])
   861     else
   862       box_type hol_ctxt (in_fun_lhs_for boxy) T1
   863       --> box_type hol_ctxt (in_fun_rhs_for boxy) T2
   864   | Type (z as (@{type_name prod}, Ts)) =>
   865     if boxy <> InConstr andalso boxy <> InSel
   866        andalso should_box_type hol_ctxt boxy z then
   867       Type (@{type_name pair_box}, map (box_type hol_ctxt InSel) Ts)
   868     else
   869       Type (@{type_name prod},
   870             map (box_type hol_ctxt
   871                           (if boxy = InConstr orelse boxy = InSel then boxy
   872                            else InPair)) Ts)
   873   | _ => T
   874 
   875 fun binarize_nat_and_int_in_type @{typ nat} = @{typ "unsigned_bit word"}
   876   | binarize_nat_and_int_in_type @{typ int} = @{typ "signed_bit word"}
   877   | binarize_nat_and_int_in_type (Type (s, Ts)) =
   878     Type (s, map binarize_nat_and_int_in_type Ts)
   879   | binarize_nat_and_int_in_type T = T
   880 val binarize_nat_and_int_in_term = map_types binarize_nat_and_int_in_type
   881 
   882 fun discr_for_constr (s, T) = (discr_prefix ^ s, body_type T --> bool_T)
   883 
   884 fun num_sels_for_constr_type T = length (maybe_curried_binder_types T)
   885 fun nth_sel_name_for_constr_name s n =
   886   if s = @{const_name Pair} then
   887     if n = 0 then @{const_name fst} else @{const_name snd}
   888   else
   889     sel_prefix_for n ^ s
   890 fun nth_sel_for_constr x ~1 = discr_for_constr x
   891   | nth_sel_for_constr (s, T) n =
   892     (nth_sel_name_for_constr_name s n,
   893      body_type T --> nth (maybe_curried_binder_types T) n)
   894 fun binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize =
   895   apsnd ((binarize ? binarize_nat_and_int_in_type) o box_type hol_ctxt InSel)
   896   oo nth_sel_for_constr
   897 
   898 fun sel_no_from_name s =
   899   if String.isPrefix discr_prefix s then
   900     ~1
   901   else if String.isPrefix sel_prefix s then
   902     s |> unprefix sel_prefix |> Int.fromString |> the
   903   else if s = @{const_name snd} then
   904     1
   905   else
   906     0
   907 
   908 val close_form =
   909   let
   910     fun close_up zs zs' =
   911       fold (fn (z as ((s, _), T)) => fn t' =>
   912                Term.all T $ Abs (s, T, abstract_over (Var z, t')))
   913            (take (length zs' - length zs) zs')
   914     fun aux zs (@{const "==>"} $ t1 $ t2) =
   915         let val zs' = Term.add_vars t1 zs in
   916           close_up zs zs' (Logic.mk_implies (t1, aux zs' t2))
   917         end
   918       | aux zs t = close_up zs (Term.add_vars t zs) t
   919   in aux [] end
   920 
   921 fun distinctness_formula T =
   922   all_distinct_unordered_pairs_of
   923   #> map (fn (t1, t2) => @{const Not} $ (HOLogic.eq_const T $ t1 $ t2))
   924   #> List.foldr (s_conj o swap) @{const True}
   925 
   926 fun zero_const T = Const (@{const_name zero_class.zero}, T)
   927 fun suc_const T = Const (@{const_name Suc}, T --> T)
   928 
   929 fun uncached_datatype_constrs ({thy, ctxt, stds, ...} : hol_context)
   930                               (T as Type (s, Ts)) =
   931     (case AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   932                        s of
   933        SOME (_, xs' as (_ :: _)) => map (apsnd (repair_constr_type ctxt T)) xs'
   934      | _ =>
   935        if is_datatype ctxt stds T then
   936          case Datatype.get_info thy s of
   937            SOME {index, descr, ...} =>
   938            let
   939              val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the
   940            in
   941              map (apsnd (fn Us =>
   942                             map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))
   943                  constrs
   944            end
   945          | NONE =>
   946            if is_record_type T then
   947              let
   948                val s' = unsuffix Record.ext_typeN s ^ Record.extN
   949                val T' = (Record.get_extT_fields thy T
   950                         |> apsnd single |> uncurry append |> map snd) ---> T
   951              in [(s', T')] end
   952            else if is_real_quot_type thy T then
   953              [(@{const_name Quot}, rep_type_for_quot_type thy T --> T)]
   954            else case typedef_info ctxt s of
   955              SOME {abs_type, rep_type, Abs_name, ...} =>
   956              [(Abs_name,
   957                varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
   958            | NONE =>
   959              if T = @{typ ind} then
   960                [dest_Const @{const Zero_Rep}, dest_Const @{const Suc_Rep}]
   961              else
   962                []
   963        else
   964          [])
   965   | uncached_datatype_constrs _ _ = []
   966 fun datatype_constrs (hol_ctxt as {constr_cache, ...}) T =
   967   case AList.lookup (op =) (!constr_cache) T of
   968     SOME xs => xs
   969   | NONE =>
   970     let val xs = uncached_datatype_constrs hol_ctxt T in
   971       (Unsynchronized.change constr_cache (cons (T, xs)); xs)
   972     end
   973 fun binarized_and_boxed_datatype_constrs hol_ctxt binarize =
   974   map (apsnd ((binarize ? binarize_nat_and_int_in_type)
   975               o box_type hol_ctxt InConstr)) o datatype_constrs hol_ctxt
   976 val num_datatype_constrs = length oo datatype_constrs
   977 
   978 fun constr_name_for_sel_like @{const_name fst} = @{const_name Pair}
   979   | constr_name_for_sel_like @{const_name snd} = @{const_name Pair}
   980   | constr_name_for_sel_like s' = original_name s'
   981 fun binarized_and_boxed_constr_for_sel hol_ctxt binarize (s', T') =
   982   let val s = constr_name_for_sel_like s' in
   983     AList.lookup (op =)
   984         (binarized_and_boxed_datatype_constrs hol_ctxt binarize (domain_type T'))
   985         s
   986     |> the |> pair s
   987   end
   988 
   989 fun discr_term_for_constr hol_ctxt (x as (s, T)) =
   990   let val dataT = body_type T in
   991     if s = @{const_name Suc} then
   992       Abs (Name.uu, dataT,
   993            @{const Not} $ HOLogic.mk_eq (zero_const dataT, Bound 0))
   994     else if num_datatype_constrs hol_ctxt dataT >= 2 then
   995       Const (discr_for_constr x)
   996     else
   997       Abs (Name.uu, dataT, @{const True})
   998   end
   999 fun discriminate_value (hol_ctxt as {ctxt, ...}) x t =
  1000   case head_of t of
  1001     Const x' =>
  1002     if x = x' then @{const True}
  1003     else if is_constr_like ctxt x' then @{const False}
  1004     else s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1005   | _ => s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1006 
  1007 fun nth_arg_sel_term_for_constr thy stds (x as (s, T)) n =
  1008   let val (arg_Ts, dataT) = strip_type T in
  1009     if dataT = nat_T andalso is_standard_datatype thy stds nat_T then
  1010       @{term "%n::nat. n - 1"}
  1011     else if is_pair_type dataT then
  1012       Const (nth_sel_for_constr x n)
  1013     else
  1014       let
  1015         fun aux m (Type (@{type_name prod}, [T1, T2])) =
  1016             let
  1017               val (m, t1) = aux m T1
  1018               val (m, t2) = aux m T2
  1019             in (m, HOLogic.mk_prod (t1, t2)) end
  1020           | aux m T =
  1021             (m + 1, Const (nth_sel_name_for_constr_name s m, dataT --> T)
  1022                     $ Bound 0)
  1023         val m = fold (Integer.add o num_factors_in_type)
  1024                      (List.take (arg_Ts, n)) 0
  1025       in Abs ("x", dataT, aux m (nth arg_Ts n) |> snd) end
  1026   end
  1027 fun select_nth_constr_arg ctxt stds x t n res_T =
  1028   let val thy = ProofContext.theory_of ctxt in
  1029     (case strip_comb t of
  1030        (Const x', args) =>
  1031        if x = x' then nth args n
  1032        else if is_constr_like ctxt x' then Const (@{const_name unknown}, res_T)
  1033        else raise SAME ()
  1034      | _ => raise SAME())
  1035     handle SAME () =>
  1036            s_betapply [] (nth_arg_sel_term_for_constr thy stds x n, t)
  1037   end
  1038 
  1039 fun construct_value _ _ x [] = Const x
  1040   | construct_value ctxt stds (x as (s, _)) args =
  1041     let val args = map Envir.eta_contract args in
  1042       case hd args of
  1043         Const (s', _) $ t =>
  1044         if is_sel_like_and_no_discr s' andalso
  1045            constr_name_for_sel_like s' = s andalso
  1046            forall (fn (n, t') =>
  1047                       select_nth_constr_arg ctxt stds x t n dummyT = t')
  1048                   (index_seq 0 (length args) ~~ args) then
  1049           t
  1050         else
  1051           list_comb (Const x, args)
  1052       | _ => list_comb (Const x, args)
  1053     end
  1054 
  1055 fun constr_expand (hol_ctxt as {ctxt, stds, ...}) T t =
  1056   (case head_of t of
  1057      Const x => if is_constr_like ctxt x then t else raise SAME ()
  1058    | _ => raise SAME ())
  1059   handle SAME () =>
  1060          let
  1061            val x' as (_, T') =
  1062              if is_pair_type T then
  1063                let val (T1, T2) = HOLogic.dest_prodT T in
  1064                  (@{const_name Pair}, T1 --> T2 --> T)
  1065                end
  1066              else
  1067                datatype_constrs hol_ctxt T |> hd
  1068            val arg_Ts = binder_types T'
  1069          in
  1070            list_comb (Const x', map2 (select_nth_constr_arg ctxt stds x' t)
  1071                                      (index_seq 0 (length arg_Ts)) arg_Ts)
  1072          end
  1073 
  1074 fun coerce_bound_no f j t =
  1075   case t of
  1076     t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2
  1077   | Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')
  1078   | Bound j' => if j' = j then f t else t
  1079   | _ => t
  1080 fun coerce_bound_0_in_term hol_ctxt new_T old_T =
  1081   old_T <> new_T ? coerce_bound_no (coerce_term hol_ctxt [new_T] old_T new_T) 0
  1082 and coerce_term (hol_ctxt as {ctxt, stds, ...}) Ts new_T old_T t =
  1083   if old_T = new_T then
  1084     t
  1085   else
  1086     case (new_T, old_T) of
  1087       (Type (new_s, new_Ts as [new_T1, new_T2]),
  1088        Type (@{type_name fun}, [old_T1, old_T2])) =>
  1089       (case eta_expand Ts t 1 of
  1090          Abs (s, _, t') =>
  1091          Abs (s, new_T1,
  1092               t' |> coerce_bound_0_in_term hol_ctxt new_T1 old_T1
  1093                  |> coerce_term hol_ctxt (new_T1 :: Ts) new_T2 old_T2)
  1094          |> Envir.eta_contract
  1095          |> new_s <> @{type_name fun}
  1096             ? construct_value ctxt stds
  1097                   (if new_s = @{type_name fin_fun} then @{const_name FinFun}
  1098                    else @{const_name FunBox},
  1099                    Type (@{type_name fun}, new_Ts) --> new_T)
  1100               o single
  1101        | t' => raise TERM ("Nitpick_HOL.coerce_term", [t']))
  1102     | (Type (new_s, new_Ts as [new_T1, new_T2]),
  1103        Type (old_s, old_Ts as [old_T1, old_T2])) =>
  1104       if old_s = @{type_name fin_fun} orelse old_s = @{type_name fun_box} orelse
  1105          old_s = @{type_name pair_box} orelse old_s = @{type_name prod} then
  1106         case constr_expand hol_ctxt old_T t of
  1107           Const (old_s, _) $ t1 =>
  1108           if new_s = @{type_name fun} then
  1109             coerce_term hol_ctxt Ts new_T (Type (@{type_name fun}, old_Ts)) t1
  1110           else
  1111             construct_value ctxt stds
  1112                 (old_s, Type (@{type_name fun}, new_Ts) --> new_T)
  1113                 [coerce_term hol_ctxt Ts (Type (@{type_name fun}, new_Ts))
  1114                              (Type (@{type_name fun}, old_Ts)) t1]
  1115         | Const _ $ t1 $ t2 =>
  1116           construct_value ctxt stds
  1117               (if new_s = @{type_name prod} then @{const_name Pair}
  1118                else @{const_name PairBox}, new_Ts ---> new_T)
  1119               (map3 (coerce_term hol_ctxt Ts) [new_T1, new_T2] [old_T1, old_T2]
  1120                     [t1, t2])
  1121         | t' => raise TERM ("Nitpick_HOL.coerce_term", [t'])
  1122       else
  1123         raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1124     | _ => raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1125 
  1126 fun card_of_type assigns (Type (@{type_name fun}, [T1, T2])) =
  1127     reasonable_power (card_of_type assigns T2) (card_of_type assigns T1)
  1128   | card_of_type assigns (Type (@{type_name prod}, [T1, T2])) =
  1129     card_of_type assigns T1 * card_of_type assigns T2
  1130   | card_of_type _ (Type (@{type_name itself}, _)) = 1
  1131   | card_of_type _ @{typ prop} = 2
  1132   | card_of_type _ @{typ bool} = 2
  1133   | card_of_type assigns T =
  1134     case AList.lookup (op =) assigns T of
  1135       SOME k => k
  1136     | NONE => if T = @{typ bisim_iterator} then 0
  1137               else raise TYPE ("Nitpick_HOL.card_of_type", [T], [])
  1138 fun bounded_card_of_type max default_card assigns
  1139                          (Type (@{type_name fun}, [T1, T2])) =
  1140     let
  1141       val k1 = bounded_card_of_type max default_card assigns T1
  1142       val k2 = bounded_card_of_type max default_card assigns T2
  1143     in
  1144       if k1 = max orelse k2 = max then max
  1145       else Int.min (max, reasonable_power k2 k1)
  1146     end
  1147   | bounded_card_of_type max default_card assigns
  1148                          (Type (@{type_name prod}, [T1, T2])) =
  1149     let
  1150       val k1 = bounded_card_of_type max default_card assigns T1
  1151       val k2 = bounded_card_of_type max default_card assigns T2
  1152     in if k1 = max orelse k2 = max then max else Int.min (max, k1 * k2) end
  1153   | bounded_card_of_type max default_card assigns T =
  1154     Int.min (max, if default_card = ~1 then
  1155                     card_of_type assigns T
  1156                   else
  1157                     card_of_type assigns T
  1158                     handle TYPE ("Nitpick_HOL.card_of_type", _, _) =>
  1159                            default_card)
  1160 fun bounded_exact_card_of_type hol_ctxt finitizable_dataTs max default_card
  1161                                assigns T =
  1162   let
  1163     fun aux avoid T =
  1164       (if member (op =) avoid T then
  1165          0
  1166        else if member (op =) finitizable_dataTs T then
  1167          raise SAME ()
  1168        else case T of
  1169          Type (@{type_name fun}, [T1, T2]) =>
  1170          let
  1171            val k1 = aux avoid T1
  1172            val k2 = aux avoid T2
  1173          in
  1174            if k1 = 0 orelse k2 = 0 then 0
  1175            else if k1 >= max orelse k2 >= max then max
  1176            else Int.min (max, reasonable_power k2 k1)
  1177          end
  1178        | Type (@{type_name prod}, [T1, T2]) =>
  1179          let
  1180            val k1 = aux avoid T1
  1181            val k2 = aux avoid T2
  1182          in
  1183            if k1 = 0 orelse k2 = 0 then 0
  1184            else if k1 >= max orelse k2 >= max then max
  1185            else Int.min (max, k1 * k2)
  1186          end
  1187        | Type (@{type_name itself}, _) => 1
  1188        | @{typ prop} => 2
  1189        | @{typ bool} => 2
  1190        | Type _ =>
  1191          (case datatype_constrs hol_ctxt T of
  1192             [] => if is_integer_type T orelse is_bit_type T then 0
  1193                   else raise SAME ()
  1194           | constrs =>
  1195             let
  1196               val constr_cards =
  1197                 map (Integer.prod o map (aux (T :: avoid)) o binder_types o snd)
  1198                     constrs
  1199             in
  1200               if exists (curry (op =) 0) constr_cards then 0
  1201               else Integer.sum constr_cards
  1202             end)
  1203        | _ => raise SAME ())
  1204       handle SAME () =>
  1205              AList.lookup (op =) assigns T |> the_default default_card
  1206   in Int.min (max, aux [] T) end
  1207 
  1208 val small_type_max_card = 5
  1209 
  1210 fun is_finite_type hol_ctxt T =
  1211   bounded_exact_card_of_type hol_ctxt [] 1 2 [] T > 0
  1212 fun is_small_finite_type hol_ctxt T =
  1213   let val n = bounded_exact_card_of_type hol_ctxt [] 1 2 [] T in
  1214     n > 0 andalso n <= small_type_max_card
  1215   end
  1216 
  1217 fun is_ground_term (t1 $ t2) = is_ground_term t1 andalso is_ground_term t2
  1218   | is_ground_term (Const _) = true
  1219   | is_ground_term _ = false
  1220 
  1221 fun hashw_term (t1 $ t2) = hashw (hashw_term t1, hashw_term t2)
  1222   | hashw_term (Const (s, _)) = hashw_string (s, 0w0)
  1223   | hashw_term _ = 0w0
  1224 val hash_term = Word.toInt o hashw_term
  1225 
  1226 fun special_bounds ts =
  1227   fold Term.add_vars ts [] |> sort (Term_Ord.fast_indexname_ord o pairself fst)
  1228 
  1229 (* FIXME: detect "rep_datatype"? *)
  1230 fun is_funky_typedef_name ctxt s =
  1231   member (op =) [@{type_name unit}, @{type_name prod},
  1232                  @{type_name Sum_Type.sum}, @{type_name int}] s orelse
  1233   is_frac_type ctxt (Type (s, []))
  1234 fun is_funky_typedef ctxt (Type (s, _)) = is_funky_typedef_name ctxt s
  1235   | is_funky_typedef _ _ = false
  1236 fun is_arity_type_axiom (Const (@{const_name HOL.type_class}, _)
  1237                          $ Const (@{const_name TYPE}, _)) = true
  1238   | is_arity_type_axiom _ = false
  1239 fun is_typedef_axiom ctxt boring (@{const "==>"} $ _ $ t2) =
  1240     is_typedef_axiom ctxt boring t2
  1241   | is_typedef_axiom ctxt boring
  1242         (@{const Trueprop} $ (Const (@{const_name Typedef.type_definition}, _)
  1243          $ Const (_, Type (@{type_name fun}, [Type (s, _), _]))
  1244          $ Const _ $ _)) =
  1245     boring <> is_funky_typedef_name ctxt s andalso is_typedef ctxt s
  1246   | is_typedef_axiom _ _ _ = false
  1247 val is_class_axiom =
  1248   Logic.strip_horn #> swap #> op :: #> forall (can Logic.dest_of_class)
  1249 
  1250 (* Distinguishes between (1) constant definition axioms, (2) type arity and
  1251    typedef axioms, and (3) other axioms, and returns the pair ((1), (3)).
  1252    Typedef axioms are uninteresting to Nitpick, because it can retrieve them
  1253    using "typedef_info". *)
  1254 fun partition_axioms_by_definitionality ctxt axioms def_names =
  1255   let
  1256     val axioms = sort (fast_string_ord o pairself fst) axioms
  1257     val defs = Ord_List.inter (fast_string_ord o apsnd fst) def_names axioms
  1258     val nondefs =
  1259       Ord_List.subtract (fast_string_ord o apsnd fst) def_names axioms
  1260       |> filter_out ((is_arity_type_axiom orf is_typedef_axiom ctxt true) o snd)
  1261   in pairself (map snd) (defs, nondefs) end
  1262 
  1263 (* Ideally we would check against "Complex_Main", not "Refute", but any theory
  1264    will do as long as it contains all the "axioms" and "axiomatization"
  1265    commands. *)
  1266 fun is_built_in_theory thy = Theory.subthy (thy, @{theory Refute})
  1267 
  1268 val is_trivial_definition =
  1269   the_default false o try (op aconv o Logic.dest_equals)
  1270 val is_plain_definition =
  1271   let
  1272     fun do_lhs t1 =
  1273       case strip_comb t1 of
  1274         (Const _, args) =>
  1275         forall is_Var args andalso not (has_duplicates (op =) args)
  1276       | _ => false
  1277     fun do_eq (Const (@{const_name "=="}, _) $ t1 $ _) = do_lhs t1
  1278       | do_eq (@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)) =
  1279         do_lhs t1
  1280       | do_eq _ = false
  1281   in do_eq end
  1282 
  1283 fun all_axioms_of ctxt subst =
  1284   let
  1285     val thy = ProofContext.theory_of ctxt
  1286     val axioms_of_thys =
  1287       maps Thm.axioms_of
  1288       #> map (apsnd (subst_atomic subst o prop_of))
  1289       #> filter_out (is_class_axiom o snd)
  1290     val specs = Defs.all_specifications_of (Theory.defs_of thy)
  1291     val def_names = specs |> maps snd |> map_filter #def
  1292                     |> Ord_List.make fast_string_ord
  1293     val thys = thy :: Theory.ancestors_of thy
  1294     val (built_in_thys, user_thys) = List.partition is_built_in_theory thys
  1295     val built_in_axioms = axioms_of_thys built_in_thys
  1296     val user_axioms = axioms_of_thys user_thys
  1297     val (built_in_defs, built_in_nondefs) =
  1298       partition_axioms_by_definitionality ctxt built_in_axioms def_names
  1299       ||> filter (is_typedef_axiom ctxt false)
  1300     val (user_defs, user_nondefs) =
  1301       partition_axioms_by_definitionality ctxt user_axioms def_names
  1302     val (built_in_nondefs, user_nondefs) =
  1303       List.partition (is_typedef_axiom ctxt false) user_nondefs
  1304       |>> append built_in_nondefs
  1305     val defs =
  1306       (thy |> Global_Theory.all_thms_of
  1307            |> filter (curry (op =) Thm.definitionK o Thm.get_kind o snd)
  1308            |> map (prop_of o snd)
  1309            |> filter_out is_trivial_definition
  1310            |> filter is_plain_definition) @
  1311       user_defs @ built_in_defs
  1312   in (defs, built_in_nondefs, user_nondefs) end
  1313 
  1314 fun arity_of_built_in_const thy stds (s, T) =
  1315   if s = @{const_name If} then
  1316     if nth_range_type 3 T = @{typ bool} then NONE else SOME 3
  1317   else
  1318     let val std_nats = is_standard_datatype thy stds nat_T in
  1319       case AList.lookup (op =)
  1320                     (built_in_consts
  1321                      |> std_nats ? append built_in_nat_consts) s of
  1322         SOME n => SOME n
  1323       | NONE =>
  1324         case AList.lookup (op =)
  1325                  (built_in_typed_consts
  1326                   |> std_nats ? append built_in_typed_nat_consts)
  1327                  (s, unarize_type T) of
  1328           SOME n => SOME n
  1329         | NONE =>
  1330           case s of
  1331             @{const_name zero_class.zero} =>
  1332             if is_iterator_type T then SOME 0 else NONE
  1333           | @{const_name Suc} =>
  1334             if is_iterator_type (domain_type T) then SOME 0 else NONE
  1335           | _ => if is_fun_type T andalso is_set_type (domain_type T) then
  1336                    AList.lookup (op =) built_in_set_consts s
  1337                  else
  1338                    NONE
  1339     end
  1340 val is_built_in_const = is_some ooo arity_of_built_in_const
  1341 
  1342 (* This function is designed to work for both real definition axioms and
  1343    simplification rules (equational specifications). *)
  1344 fun term_under_def t =
  1345   case t of
  1346     @{const "==>"} $ _ $ t2 => term_under_def t2
  1347   | Const (@{const_name "=="}, _) $ t1 $ _ => term_under_def t1
  1348   | @{const Trueprop} $ t1 => term_under_def t1
  1349   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => term_under_def t1
  1350   | Abs (_, _, t') => term_under_def t'
  1351   | t1 $ _ => term_under_def t1
  1352   | _ => t
  1353 
  1354 (* Here we crucially rely on "specialize_type" performing a preorder traversal
  1355    of the term, without which the wrong occurrence of a constant could be
  1356    matched in the face of overloading. *)
  1357 fun def_props_for_const thy stds table (x as (s, _)) =
  1358   if is_built_in_const thy stds x then
  1359     []
  1360   else
  1361     these (Symtab.lookup table s)
  1362     |> map_filter (try (specialize_type thy x))
  1363     |> filter (curry (op =) (Const x) o term_under_def)
  1364 
  1365 fun normalized_rhs_of t =
  1366   let
  1367     fun aux (v as Var _) (SOME t) = SOME (lambda v t)
  1368       | aux (c as Const (@{const_name TYPE}, _)) (SOME t) = SOME (lambda c t)
  1369       | aux _ _ = NONE
  1370     val (lhs, rhs) =
  1371       case t of
  1372         Const (@{const_name "=="}, _) $ t1 $ t2 => (t1, t2)
  1373       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ t2) =>
  1374         (t1, t2)
  1375       | _ => raise TERM ("Nitpick_HOL.normalized_rhs_of", [t])
  1376     val args = strip_comb lhs |> snd
  1377   in fold_rev aux args (SOME rhs) end
  1378 
  1379 fun def_of_const thy table (x as (s, _)) =
  1380   if is_built_in_const thy [(NONE, false)] x orelse
  1381      original_name s <> s then
  1382     NONE
  1383   else
  1384     x |> def_props_for_const thy [(NONE, false)] table |> List.last
  1385       |> normalized_rhs_of |> Option.map (prefix_abs_vars s)
  1386     handle List.Empty => NONE
  1387 
  1388 fun fixpoint_kind_of_rhs (Abs (_, _, t)) = fixpoint_kind_of_rhs t
  1389   | fixpoint_kind_of_rhs (Const (@{const_name lfp}, _) $ Abs _) = Lfp
  1390   | fixpoint_kind_of_rhs (Const (@{const_name gfp}, _) $ Abs _) = Gfp
  1391   | fixpoint_kind_of_rhs _ = NoFp
  1392 
  1393 fun is_mutually_inductive_pred_def thy table t =
  1394   let
  1395     fun is_good_arg (Bound _) = true
  1396       | is_good_arg (Const (s, _)) =
  1397         s = @{const_name True} orelse s = @{const_name False} orelse
  1398         s = @{const_name undefined}
  1399       | is_good_arg _ = false
  1400   in
  1401     case t |> strip_abs_body |> strip_comb of
  1402       (Const x, ts as (_ :: _)) =>
  1403       (case def_of_const thy table x of
  1404          SOME t' => fixpoint_kind_of_rhs t' <> NoFp andalso
  1405                     forall is_good_arg ts
  1406        | NONE => false)
  1407     | _ => false
  1408   end
  1409 fun unfold_mutually_inductive_preds thy table =
  1410   map_aterms (fn t as Const x =>
  1411                  (case def_of_const thy table x of
  1412                     SOME t' =>
  1413                     let val t' = Envir.eta_contract t' in
  1414                       if is_mutually_inductive_pred_def thy table t' then t'
  1415                       else t
  1416                     end
  1417                  | NONE => t)
  1418                | t => t)
  1419 
  1420 fun case_const_names ctxt stds =
  1421   let val thy = ProofContext.theory_of ctxt in
  1422     Symtab.fold (fn (dtype_s, {index, descr, case_name, ...}) =>
  1423                     if is_basic_datatype thy stds dtype_s then
  1424                       I
  1425                     else
  1426                       cons (case_name, AList.lookup (op =) descr index
  1427                                        |> the |> #3 |> length))
  1428                 (Datatype.get_all thy) [] @
  1429     map (apsnd length o snd) (#codatatypes (Data.get (Context.Proof ctxt)))
  1430   end
  1431 
  1432 fun fixpoint_kind_of_const thy table x =
  1433   if is_built_in_const thy [(NONE, false)] x then NoFp
  1434   else fixpoint_kind_of_rhs (the (def_of_const thy table x))
  1435   handle Option.Option => NoFp
  1436 
  1437 fun is_real_inductive_pred ({thy, stds, def_table, intro_table, ...}
  1438                             : hol_context) x =
  1439   fixpoint_kind_of_const thy def_table x <> NoFp andalso
  1440   not (null (def_props_for_const thy stds intro_table x))
  1441 fun is_inductive_pred hol_ctxt (x as (s, _)) =
  1442   is_real_inductive_pred hol_ctxt x orelse String.isPrefix ubfp_prefix s orelse
  1443   String.isPrefix lbfp_prefix s
  1444 
  1445 fun lhs_of_equation t =
  1446   case t of
  1447     Const (@{const_name all}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1448   | Const (@{const_name "=="}, _) $ t1 $ _ => SOME t1
  1449   | @{const "==>"} $ _ $ t2 => lhs_of_equation t2
  1450   | @{const Trueprop} $ t1 => lhs_of_equation t1
  1451   | Const (@{const_name All}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1452   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => SOME t1
  1453   | @{const HOL.implies} $ _ $ t2 => lhs_of_equation t2
  1454   | _ => NONE
  1455 fun is_constr_pattern _ (Bound _) = true
  1456   | is_constr_pattern _ (Var _) = true
  1457   | is_constr_pattern ctxt t =
  1458     case strip_comb t of
  1459       (Const x, args) =>
  1460       is_constr_like ctxt x andalso forall (is_constr_pattern ctxt) args
  1461     | _ => false
  1462 fun is_constr_pattern_lhs ctxt t =
  1463   forall (is_constr_pattern ctxt) (snd (strip_comb t))
  1464 fun is_constr_pattern_formula ctxt t =
  1465   case lhs_of_equation t of
  1466     SOME t' => is_constr_pattern_lhs ctxt t'
  1467   | NONE => false
  1468 
  1469 (* Similar to "specialize_type" but returns all matches rather than only the
  1470    first (preorder) match. *)
  1471 fun multi_specialize_type thy slack (s, T) t =
  1472   let
  1473     fun aux (Const (s', T')) ys =
  1474         if s = s' then
  1475           ys |> (if AList.defined (op =) ys T' then
  1476                    I
  1477                  else
  1478                    cons (T', monomorphic_term (Sign.typ_match thy (T', T)
  1479                                                               Vartab.empty) t)
  1480                    handle Type.TYPE_MATCH => I
  1481                         | TERM _ =>
  1482                           if slack then
  1483                             I
  1484                           else
  1485                             raise NOT_SUPPORTED
  1486                                       ("too much polymorphism in axiom \"" ^
  1487                                        Syntax.string_of_term_global thy t ^
  1488                                        "\" involving " ^ quote s))
  1489         else
  1490           ys
  1491       | aux _ ys = ys
  1492   in map snd (fold_aterms aux t []) end
  1493 fun nondef_props_for_const thy slack table (x as (s, _)) =
  1494   these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)
  1495 
  1496 fun unvarify_term (t1 $ t2) = unvarify_term t1 $ unvarify_term t2
  1497   | unvarify_term (Var ((s, 0), T)) = Free (s, T)
  1498   | unvarify_term (Abs (s, T, t')) = Abs (s, T, unvarify_term t')
  1499   | unvarify_term t = t
  1500 fun axiom_for_choice_spec thy =
  1501   unvarify_term
  1502   #> Object_Logic.atomize_term thy
  1503   #> Choice_Specification.close_form
  1504   #> HOLogic.mk_Trueprop
  1505 fun is_choice_spec_fun ({thy, def_table, nondef_table, choice_spec_table, ...}
  1506                         : hol_context) x =
  1507   case nondef_props_for_const thy true choice_spec_table x of
  1508     [] => false
  1509   | ts => case def_of_const thy def_table x of
  1510             SOME (Const (@{const_name Eps}, _) $ _) => true
  1511           | SOME _ => false
  1512           | NONE =>
  1513             let val ts' = nondef_props_for_const thy true nondef_table x in
  1514               length ts' = length ts andalso
  1515               forall (fn t =>
  1516                          exists (curry (op aconv) (axiom_for_choice_spec thy t))
  1517                                 ts') ts
  1518             end
  1519 
  1520 fun is_choice_spec_axiom thy choice_spec_table t =
  1521   Symtab.exists (fn (_, ts) =>
  1522                     exists (curry (op aconv) t o axiom_for_choice_spec thy) ts)
  1523                 choice_spec_table
  1524 
  1525 fun is_real_equational_fun ({thy, stds, simp_table, psimp_table, ...}
  1526                             : hol_context) x =
  1527   exists (fn table => not (null (def_props_for_const thy stds table x)))
  1528          [!simp_table, psimp_table]
  1529 fun is_equational_fun_but_no_plain_def hol_ctxt =
  1530   is_real_equational_fun hol_ctxt orf is_inductive_pred hol_ctxt
  1531 
  1532 (** Constant unfolding **)
  1533 
  1534 fun constr_case_body ctxt stds (func_t, (x as (_, T))) =
  1535   let val arg_Ts = binder_types T in
  1536     s_betapplys [] (func_t, map2 (select_nth_constr_arg ctxt stds x (Bound 0))
  1537                                  (index_seq 0 (length arg_Ts)) arg_Ts)
  1538   end
  1539 fun add_constr_case res_T (body_t, guard_t) res_t =
  1540   if res_T = bool_T then
  1541     s_conj (HOLogic.mk_imp (guard_t, body_t), res_t)
  1542   else
  1543     Const (@{const_name If}, bool_T --> res_T --> res_T --> res_T)
  1544     $ guard_t $ body_t $ res_t
  1545 fun optimized_case_def (hol_ctxt as {ctxt, stds, ...}) dataT res_T func_ts =
  1546   let
  1547     val xs = datatype_constrs hol_ctxt dataT
  1548     val cases =
  1549       func_ts ~~ xs
  1550       |> map (fn (func_t, x) =>
  1551                  (constr_case_body ctxt stds (incr_boundvars 1 func_t, x),
  1552                   discriminate_value hol_ctxt x (Bound 0)))
  1553       |> AList.group (op aconv)
  1554       |> map (apsnd (List.foldl s_disj @{const False}))
  1555       |> sort (int_ord o pairself (size_of_term o snd))
  1556       |> rev
  1557   in
  1558     if res_T = bool_T then
  1559       if forall (member (op =) [@{const False}, @{const True}] o fst) cases then
  1560         case cases of
  1561           [(body_t, _)] => body_t
  1562         | [_, (@{const True}, head_t2)] => head_t2
  1563         | [_, (@{const False}, head_t2)] => @{const Not} $ head_t2
  1564         | _ => raise BAD ("Nitpick_HOL.optimized_case_def", "impossible cases")
  1565       else
  1566         @{const True} |> fold_rev (add_constr_case res_T) cases
  1567     else
  1568       fst (hd cases) |> fold_rev (add_constr_case res_T) (tl cases)
  1569   end
  1570   |> curry absdummy dataT
  1571 
  1572 fun optimized_record_get (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T res_T t =
  1573   let val constr_x = hd (datatype_constrs hol_ctxt rec_T) in
  1574     case no_of_record_field thy s rec_T of
  1575       ~1 => (case rec_T of
  1576                Type (_, Ts as _ :: _) =>
  1577                let
  1578                  val rec_T' = List.last Ts
  1579                  val j = num_record_fields thy rec_T - 1
  1580                in
  1581                  select_nth_constr_arg ctxt stds constr_x t j res_T
  1582                  |> optimized_record_get hol_ctxt s rec_T' res_T
  1583                end
  1584              | _ => raise TYPE ("Nitpick_HOL.optimized_record_get", [rec_T],
  1585                                 []))
  1586     | j => select_nth_constr_arg ctxt stds constr_x t j res_T
  1587   end
  1588 fun optimized_record_update (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T fun_t
  1589                             rec_t =
  1590   let
  1591     val constr_x as (_, constr_T) = hd (datatype_constrs hol_ctxt rec_T)
  1592     val Ts = binder_types constr_T
  1593     val n = length Ts
  1594     val special_j = no_of_record_field thy s rec_T
  1595     val ts =
  1596       map2 (fn j => fn T =>
  1597                let val t = select_nth_constr_arg ctxt stds constr_x rec_t j T in
  1598                  if j = special_j then
  1599                    s_betapply [] (fun_t, t)
  1600                  else if j = n - 1 andalso special_j = ~1 then
  1601                    optimized_record_update hol_ctxt s
  1602                        (rec_T |> dest_Type |> snd |> List.last) fun_t t
  1603                  else
  1604                    t
  1605                end) (index_seq 0 n) Ts
  1606   in list_comb (Const constr_x, ts) end
  1607 
  1608 (* Prevents divergence in case of cyclic or infinite definition dependencies. *)
  1609 val unfold_max_depth = 255
  1610 
  1611 (* Inline definitions or define as an equational constant? Booleans tend to
  1612    benefit more from inlining, due to the polarity analysis. *)
  1613 val def_inline_threshold_for_booleans = 50
  1614 val def_inline_threshold_for_non_booleans = 20
  1615 
  1616 fun unfold_defs_in_term
  1617         (hol_ctxt as {thy, ctxt, stds, whacks, case_names, def_table,
  1618                       ground_thm_table, ersatz_table, ...}) =
  1619   let
  1620     fun do_term depth Ts t =
  1621       case t of
  1622         (t0 as Const (@{const_name Int.number_class.number_of},
  1623                       Type (@{type_name fun}, [_, ran_T]))) $ t1 =>
  1624         ((if is_number_type ctxt ran_T then
  1625             let
  1626               val j = t1 |> HOLogic.dest_numeral
  1627                          |> ran_T = nat_T ? Integer.max 0
  1628               val s = numeral_prefix ^ signed_string_of_int j
  1629             in
  1630               if is_integer_like_type ran_T then
  1631                 if is_standard_datatype thy stds ran_T then
  1632                   Const (s, ran_T)
  1633                 else
  1634                   funpow j (curry (op $) (suc_const ran_T)) (zero_const ran_T)
  1635               else
  1636                 do_term depth Ts (Const (@{const_name of_int}, int_T --> ran_T)
  1637                                   $ Const (s, int_T))
  1638             end
  1639             handle TERM _ => raise SAME ()
  1640           else
  1641             raise SAME ())
  1642          handle SAME () =>
  1643                 s_betapply [] (do_term depth Ts t0, do_term depth Ts t1))
  1644       | Const (@{const_name refl_on}, T) $ Const (@{const_name top}, _) $ t2 =>
  1645         do_const depth Ts t (@{const_name refl'}, range_type T) [t2]
  1646       | (t0 as Const (@{const_name Sigma}, _)) $ t1 $ (t2 as Abs (_, _, t2')) =>
  1647         s_betapplys Ts (t0 |> loose_bvar1 (t2', 0) ? do_term depth Ts,
  1648                         map (do_term depth Ts) [t1, t2])
  1649       | Const (x as (@{const_name distinct},
  1650                Type (@{type_name fun}, [Type (@{type_name list}, [T']), _])))
  1651         $ (t1 as _ $ _) =>
  1652         (t1 |> HOLogic.dest_list |> distinctness_formula T'
  1653          handle TERM _ => do_const depth Ts t x [t1])
  1654       | Const (x as (@{const_name If}, _)) $ t1 $ t2 $ t3 =>
  1655         if is_ground_term t1 andalso
  1656            exists (Pattern.matches thy o rpair t1)
  1657                   (Inttab.lookup_list ground_thm_table (hash_term t1)) then
  1658           do_term depth Ts t2
  1659         else
  1660           do_const depth Ts t x [t1, t2, t3]
  1661       | Const x => do_const depth Ts t x []
  1662       | t1 $ t2 =>
  1663         (case strip_comb t of
  1664            (Const x, ts) => do_const depth Ts t x ts
  1665          | _ => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2))
  1666       | Bound _ => t
  1667       | Abs (s, T, body) => Abs (s, T, do_term depth (T :: Ts) body)
  1668       | _ => if member (term_match thy) whacks t then
  1669                Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1670              else
  1671                t
  1672     and select_nth_constr_arg_with_args _ _ (x as (_, T)) [] n res_T =
  1673         (Abs (Name.uu, body_type T,
  1674               select_nth_constr_arg ctxt stds x (Bound 0) n res_T), [])
  1675       | select_nth_constr_arg_with_args depth Ts x (t :: ts) n res_T =
  1676         (select_nth_constr_arg ctxt stds x (do_term depth Ts t) n res_T, ts)
  1677     and quot_rep_of depth Ts abs_T rep_T ts =
  1678       select_nth_constr_arg_with_args depth Ts
  1679           (@{const_name Quot}, rep_T --> abs_T) ts 0 rep_T
  1680     and do_const depth Ts t (x as (s, T)) ts =
  1681       if member (term_match thy) whacks (Const x) then
  1682         Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1683       else case AList.lookup (op =) ersatz_table s of
  1684         SOME s' =>
  1685         do_const (depth + 1) Ts (list_comb (Const (s', T), ts)) (s', T) ts
  1686       | NONE =>
  1687         let
  1688           fun def_inline_threshold () =
  1689             if is_boolean_type (nth_range_type (length ts) T) then
  1690               def_inline_threshold_for_booleans
  1691             else
  1692               def_inline_threshold_for_non_booleans
  1693           val (const, ts) =
  1694             if is_built_in_const thy stds x then
  1695               (Const x, ts)
  1696             else case AList.lookup (op =) case_names s of
  1697               SOME n =>
  1698               if length ts < n then
  1699                 (do_term depth Ts (eta_expand Ts t (n - length ts)), [])
  1700               else
  1701                 let
  1702                   val (dataT, res_T) = nth_range_type n T
  1703                                        |> pairf domain_type range_type
  1704                 in
  1705                   (optimized_case_def hol_ctxt dataT res_T
  1706                                       (map (do_term depth Ts) (take n ts)),
  1707                    drop n ts)
  1708                 end
  1709             | _ =>
  1710               if is_constr ctxt stds x then
  1711                 (Const x, ts)
  1712               else if is_stale_constr ctxt x then
  1713                 raise NOT_SUPPORTED ("(non-co)constructors of codatatypes \
  1714                                      \(\"" ^ s ^ "\")")
  1715               else if is_quot_abs_fun ctxt x then
  1716                 let
  1717                   val rep_T = domain_type T
  1718                   val abs_T = range_type T
  1719                 in
  1720                   (Abs (Name.uu, rep_T,
  1721                         Const (@{const_name Quot}, rep_T --> abs_T)
  1722                                $ (Const (quot_normal_name_for_type ctxt abs_T,
  1723                                          rep_T --> rep_T) $ Bound 0)), ts)
  1724                 end
  1725               else if is_quot_rep_fun ctxt x then
  1726                 quot_rep_of depth Ts (domain_type T) (range_type T) ts
  1727               else if is_record_get thy x then
  1728                 case length ts of
  1729                   0 => (do_term depth Ts (eta_expand Ts t 1), [])
  1730                 | _ => (optimized_record_get hol_ctxt s (domain_type T)
  1731                             (range_type T) (do_term depth Ts (hd ts)), tl ts)
  1732               else if is_record_update thy x then
  1733                 case length ts of
  1734                   2 => (optimized_record_update hol_ctxt
  1735                             (unsuffix Record.updateN s) (nth_range_type 2 T)
  1736                             (do_term depth Ts (hd ts))
  1737                             (do_term depth Ts (nth ts 1)), [])
  1738                 | n => (do_term depth Ts (eta_expand Ts t (2 - n)), [])
  1739               else if is_abs_fun ctxt x andalso
  1740                       is_quot_type ctxt (range_type T) then
  1741                 let
  1742                   val abs_T = range_type T
  1743                   val rep_T = domain_type (domain_type T)
  1744                   val eps_fun = Const (@{const_name Eps},
  1745                                        (rep_T --> bool_T) --> rep_T)
  1746                   val normal_fun =
  1747                     Const (quot_normal_name_for_type ctxt abs_T,
  1748                            rep_T --> rep_T)
  1749                   val abs_fun = Const (@{const_name Quot}, rep_T --> abs_T)
  1750                 in
  1751                   (Abs (Name.uu, rep_T --> bool_T,
  1752                         abs_fun $ (normal_fun $ (eps_fun $ Bound 0)))
  1753                    |> do_term (depth + 1) Ts, ts)
  1754                 end
  1755               else if is_rep_fun ctxt x then
  1756                 let val x' = mate_of_rep_fun ctxt x in
  1757                   if is_constr ctxt stds x' then
  1758                     select_nth_constr_arg_with_args depth Ts x' ts 0
  1759                                                     (range_type T)
  1760                   else if is_quot_type ctxt (domain_type T) then
  1761                     let
  1762                       val abs_T = domain_type T
  1763                       val rep_T = domain_type (range_type T)
  1764                       val (rep_fun, _) = quot_rep_of depth Ts abs_T rep_T []
  1765                       val (equiv_rel, _) =
  1766                         equiv_relation_for_quot_type thy abs_T
  1767                     in
  1768                       (Abs (Name.uu, abs_T, equiv_rel $ (rep_fun $ Bound 0)),
  1769                        ts)
  1770                     end
  1771                   else
  1772                     (Const x, ts)
  1773                 end
  1774               else if is_equational_fun_but_no_plain_def hol_ctxt x orelse
  1775                       is_choice_spec_fun hol_ctxt x then
  1776                 (Const x, ts)
  1777               else case def_of_const thy def_table x of
  1778                 SOME def =>
  1779                 if depth > unfold_max_depth then
  1780                   raise TOO_LARGE ("Nitpick_HOL.unfold_defs_in_term",
  1781                                    "too many nested definitions (" ^
  1782                                    string_of_int depth ^ ") while expanding " ^
  1783                                    quote s)
  1784                 else if s = @{const_name wfrec'} then
  1785                   (do_term (depth + 1) Ts (s_betapplys Ts (def, ts)), [])
  1786                 else if size_of_term def > def_inline_threshold () then
  1787                   (Const x, ts)
  1788                 else
  1789                   (do_term (depth + 1) Ts def, ts)
  1790               | NONE => (Const x, ts)
  1791         in
  1792           s_betapplys Ts (const, map (do_term depth Ts) ts)
  1793           |> s_beta_norm Ts
  1794         end
  1795   in do_term 0 [] end
  1796 
  1797 (** Axiom extraction/generation **)
  1798 
  1799 fun extensional_equal j (Type (@{type_name fun}, [dom_T, ran_T])) t1 t2 =
  1800     let val var_t = Var (("x", j), dom_T) in
  1801       extensional_equal (j + 1) ran_T (betapply (t1, var_t))
  1802                         (betapply (t2, var_t))
  1803     end
  1804   | extensional_equal _ T t1 t2 =
  1805     Const (@{const_name HOL.eq}, T --> T --> bool_T) $ t1 $ t2
  1806 
  1807 fun equationalize_term ctxt tag t =
  1808   let
  1809     val j = maxidx_of_term t + 1
  1810     val (prems, concl) = Logic.strip_horn t
  1811   in
  1812     Logic.list_implies (prems,
  1813         case concl of
  1814           @{const Trueprop} $ (Const (@{const_name HOL.eq}, Type (_, [T, _]))
  1815                                $ t1 $ t2) =>
  1816           @{const Trueprop} $ extensional_equal j T t1 t2
  1817         | @{const Trueprop} $ t' =>
  1818           @{const Trueprop} $ HOLogic.mk_eq (t', @{const True})
  1819         | Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
  1820           @{const Trueprop} $ extensional_equal j T t1 t2
  1821         | _ => (warning ("Ignoring " ^ quote tag ^ " for non-equation" ^
  1822                          quote (Syntax.string_of_term ctxt t) ^ ".");
  1823                 raise SAME ()))
  1824     |> SOME
  1825   end
  1826   handle SAME () => NONE
  1827 
  1828 fun pair_for_prop t =
  1829   case term_under_def t of
  1830     Const (s, _) => (s, t)
  1831   | t' => raise TERM ("Nitpick_HOL.pair_for_prop", [t, t'])
  1832 
  1833 fun def_table_for get ctxt subst =
  1834   ctxt |> get |> map (pair_for_prop o subst_atomic subst)
  1835        |> AList.group (op =) |> Symtab.make
  1836 
  1837 fun const_def_table ctxt subst ts =
  1838   def_table_for (map prop_of o Nitpick_Defs.get) ctxt subst
  1839   |> fold (fn (s, t) => Symtab.map_default (s, []) (cons t))
  1840           (map pair_for_prop ts)
  1841 
  1842 fun paired_with_consts t = map (rpair t) (Term.add_const_names t [])
  1843 fun const_nondef_table ts =
  1844   fold (append o paired_with_consts) ts [] |> AList.group (op =) |> Symtab.make
  1845 
  1846 fun const_simp_table ctxt =
  1847   def_table_for (map_filter (equationalize_term ctxt "nitpick_simp" o prop_of)
  1848                  o Nitpick_Simps.get) ctxt
  1849 fun const_psimp_table ctxt =
  1850   def_table_for (map_filter (equationalize_term ctxt "nitpick_psimp" o prop_of)
  1851                  o Nitpick_Psimps.get) ctxt
  1852 
  1853 fun const_choice_spec_table ctxt subst =
  1854   map (subst_atomic subst o prop_of) (Nitpick_Choice_Specs.get ctxt)
  1855   |> const_nondef_table
  1856 
  1857 fun inductive_intro_table ctxt subst def_table =
  1858   let val thy = ProofContext.theory_of ctxt in
  1859     def_table_for
  1860         (maps (map (unfold_mutually_inductive_preds thy def_table o prop_of)
  1861                o snd o snd)
  1862          o filter (fn (cat, _) => cat = Spec_Rules.Inductive orelse
  1863                                   cat = Spec_Rules.Co_Inductive)
  1864          o Spec_Rules.get) ctxt subst
  1865   end
  1866 
  1867 fun ground_theorem_table thy =
  1868   fold ((fn @{const Trueprop} $ t1 =>
  1869             is_ground_term t1 ? Inttab.map_default (hash_term t1, []) (cons t1)
  1870           | _ => I) o prop_of o snd) (Global_Theory.all_thms_of thy) Inttab.empty
  1871 
  1872 (* TODO: Move to "Nitpick.thy" *)
  1873 val basic_ersatz_table =
  1874   [(@{const_name card}, @{const_name card'}),
  1875    (@{const_name setsum}, @{const_name setsum'}),
  1876    (@{const_name fold_graph}, @{const_name fold_graph'}),
  1877    (@{const_name wf}, @{const_name wf'}),
  1878    (@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
  1879    (@{const_name wfrec}, @{const_name wfrec'})]
  1880 
  1881 fun ersatz_table ctxt =
  1882  basic_ersatz_table
  1883  |> fold (append o snd) (#frac_types (Data.get (Context.Proof ctxt)))
  1884 
  1885 fun add_simps simp_table s eqs =
  1886   Unsynchronized.change simp_table
  1887       (Symtab.update (s, eqs @ these (Symtab.lookup (!simp_table) s)))
  1888 
  1889 fun inverse_axioms_for_rep_fun ctxt (x as (_, T)) =
  1890   let
  1891     val thy = ProofContext.theory_of ctxt
  1892     val abs_T = domain_type T
  1893   in
  1894     typedef_info ctxt (fst (dest_Type abs_T)) |> the
  1895     |> pairf #Abs_inverse #Rep_inverse
  1896     |> pairself (specialize_type thy x o prop_of o the)
  1897     ||> single |> op ::
  1898   end
  1899 fun optimized_typedef_axioms ctxt (abs_z as (abs_s, _)) =
  1900   let
  1901     val thy = ProofContext.theory_of ctxt
  1902     val abs_T = Type abs_z
  1903   in
  1904     if is_univ_typedef ctxt abs_T then
  1905       []
  1906     else case typedef_info ctxt abs_s of
  1907       SOME {abs_type, rep_type, Rep_name, prop_of_Rep, set_name, ...} =>
  1908       let
  1909         val rep_T = varify_and_instantiate_type ctxt abs_type abs_T rep_type
  1910         val rep_t = Const (Rep_name, abs_T --> rep_T)
  1911         val set_t = Const (set_name, rep_T --> bool_T)
  1912         val set_t' =
  1913           prop_of_Rep |> HOLogic.dest_Trueprop
  1914                       |> specialize_type thy (dest_Const rep_t)
  1915                       |> HOLogic.dest_mem |> snd
  1916       in
  1917         [HOLogic.all_const abs_T
  1918          $ Abs (Name.uu, abs_T, set_t $ (rep_t $ Bound 0))]
  1919         |> set_t <> set_t' ? cons (HOLogic.mk_eq (set_t, set_t'))
  1920         |> map HOLogic.mk_Trueprop
  1921       end
  1922     | NONE => []
  1923   end
  1924 fun optimized_quot_type_axioms ctxt stds abs_z =
  1925   let
  1926     val thy = ProofContext.theory_of ctxt
  1927     val abs_T = Type abs_z
  1928     val rep_T = rep_type_for_quot_type thy abs_T
  1929     val (equiv_rel, partial) = equiv_relation_for_quot_type thy abs_T
  1930     val a_var = Var (("a", 0), abs_T)
  1931     val x_var = Var (("x", 0), rep_T)
  1932     val y_var = Var (("y", 0), rep_T)
  1933     val x = (@{const_name Quot}, rep_T --> abs_T)
  1934     val sel_a_t = select_nth_constr_arg ctxt stds x a_var 0 rep_T
  1935     val normal_fun =
  1936       Const (quot_normal_name_for_type ctxt abs_T, rep_T --> rep_T)
  1937     val normal_x = normal_fun $ x_var
  1938     val normal_y = normal_fun $ y_var
  1939     val is_unknown_t = Const (@{const_name is_unknown}, rep_T --> bool_T)
  1940   in
  1941     [Logic.mk_equals (normal_fun $ sel_a_t, sel_a_t),
  1942      Logic.list_implies
  1943          ([@{const Not} $ (is_unknown_t $ normal_x),
  1944            @{const Not} $ (is_unknown_t $ normal_y),
  1945            equiv_rel $ x_var $ y_var] |> map HOLogic.mk_Trueprop,
  1946            Logic.mk_equals (normal_x, normal_y)),
  1947      Logic.list_implies
  1948          ([HOLogic.mk_Trueprop (@{const Not} $ (is_unknown_t $ normal_x)),
  1949            HOLogic.mk_Trueprop (@{const Not} $ HOLogic.mk_eq (normal_x, x_var))],
  1950           HOLogic.mk_Trueprop (equiv_rel $ x_var $ normal_x))]
  1951     |> partial ? cons (HOLogic.mk_Trueprop (equiv_rel $ sel_a_t $ sel_a_t))
  1952   end
  1953 
  1954 fun codatatype_bisim_axioms (hol_ctxt as {ctxt, stds, ...}) T =
  1955   let
  1956     val xs = datatype_constrs hol_ctxt T
  1957     val set_T = T --> bool_T
  1958     val iter_T = @{typ bisim_iterator}
  1959     val bisim_max = @{const bisim_iterator_max}
  1960     val n_var = Var (("n", 0), iter_T)
  1961     val n_var_minus_1 =
  1962       Const (@{const_name safe_The}, (iter_T --> bool_T) --> iter_T)
  1963       $ Abs ("m", iter_T, HOLogic.eq_const iter_T
  1964                           $ (suc_const iter_T $ Bound 0) $ n_var)
  1965     val x_var = Var (("x", 0), T)
  1966     val y_var = Var (("y", 0), T)
  1967     fun bisim_const T =
  1968       Const (@{const_name bisim}, iter_T --> T --> T --> bool_T)
  1969     fun nth_sub_bisim x n nth_T =
  1970       (if is_codatatype ctxt nth_T then bisim_const nth_T $ n_var_minus_1
  1971        else HOLogic.eq_const nth_T)
  1972       $ select_nth_constr_arg ctxt stds x x_var n nth_T
  1973       $ select_nth_constr_arg ctxt stds x y_var n nth_T
  1974     fun case_func (x as (_, T)) =
  1975       let
  1976         val arg_Ts = binder_types T
  1977         val core_t =
  1978           discriminate_value hol_ctxt x y_var ::
  1979           map2 (nth_sub_bisim x) (index_seq 0 (length arg_Ts)) arg_Ts
  1980           |> foldr1 s_conj
  1981       in List.foldr absdummy core_t arg_Ts end
  1982   in
  1983     [HOLogic.mk_imp
  1984        (HOLogic.mk_disj (HOLogic.eq_const iter_T $ n_var $ zero_const iter_T,
  1985             s_betapply [] (optimized_case_def hol_ctxt T bool_T
  1986                                               (map case_func xs), x_var)),
  1987         bisim_const T $ n_var $ x_var $ y_var),
  1988      HOLogic.eq_const set_T $ (bisim_const T $ bisim_max $ x_var)
  1989      $ (Const (@{const_name insert}, T --> set_T --> set_T)
  1990         $ x_var $ Const (@{const_name bot_class.bot}, set_T))]
  1991     |> map HOLogic.mk_Trueprop
  1992   end
  1993 
  1994 exception NO_TRIPLE of unit
  1995 
  1996 fun triple_for_intro_rule thy x t =
  1997   let
  1998     val prems = Logic.strip_imp_prems t |> map (Object_Logic.atomize_term thy)
  1999     val concl = Logic.strip_imp_concl t |> Object_Logic.atomize_term thy
  2000     val (main, side) = List.partition (exists_Const (curry (op =) x)) prems
  2001     val is_good_head = curry (op =) (Const x) o head_of
  2002   in
  2003     if forall is_good_head main then (side, main, concl) else raise NO_TRIPLE ()
  2004   end
  2005 
  2006 val tuple_for_args = HOLogic.mk_tuple o snd o strip_comb
  2007 fun wf_constraint_for rel side concl main =
  2008   let
  2009     val core = HOLogic.mk_mem (HOLogic.mk_prod
  2010                                (pairself tuple_for_args (main, concl)), Var rel)
  2011     val t = List.foldl HOLogic.mk_imp core side
  2012     val vars = filter_out (curry (op =) rel) (Term.add_vars t [])
  2013   in
  2014     Library.foldl (fn (t', ((x, j), T)) =>
  2015                       HOLogic.all_const T
  2016                       $ Abs (x, T, abstract_over (Var ((x, j), T), t')))
  2017                   (t, vars)
  2018   end
  2019 fun wf_constraint_for_triple rel (side, main, concl) =
  2020   map (wf_constraint_for rel side concl) main |> foldr1 s_conj
  2021 
  2022 fun terminates_by ctxt timeout goal tac =
  2023   can (SINGLE (Classical.safe_tac (claset_of ctxt)) #> the
  2024        #> SINGLE (DETERM_TIMEOUT timeout
  2025                                  (tac ctxt (auto_tac (clasimpset_of ctxt))))
  2026        #> the #> Goal.finish ctxt) goal
  2027 
  2028 val max_cached_wfs = 50
  2029 val cached_timeout =
  2030   Synchronized.var "Nitpick_HOL.cached_timeout" (SOME Time.zeroTime)
  2031 val cached_wf_props =
  2032   Synchronized.var "Nitpick_HOL.cached_wf_props" ([] : (term * bool) list)
  2033 
  2034 val termination_tacs = [Lexicographic_Order.lex_order_tac true,
  2035                         ScnpReconstruct.sizechange_tac]
  2036 
  2037 fun uncached_is_well_founded_inductive_pred
  2038         ({thy, ctxt, stds, debug, tac_timeout, intro_table, ...} : hol_context)
  2039         (x as (_, T)) =
  2040   case def_props_for_const thy stds intro_table x of
  2041     [] => raise TERM ("Nitpick_HOL.uncached_is_well_founded_inductive",
  2042                       [Const x])
  2043   | intro_ts =>
  2044     (case map (triple_for_intro_rule thy x) intro_ts
  2045           |> filter_out (null o #2) of
  2046        [] => true
  2047      | triples =>
  2048        let
  2049          val binders_T = HOLogic.mk_tupleT (binder_types T)
  2050          val rel_T = HOLogic.mk_prodT (binders_T, binders_T) --> bool_T
  2051          val j = fold Integer.max (map maxidx_of_term intro_ts) 0 + 1
  2052          val rel = (("R", j), rel_T)
  2053          val prop = Const (@{const_name wf}, rel_T --> bool_T) $ Var rel ::
  2054                     map (wf_constraint_for_triple rel) triples
  2055                     |> foldr1 s_conj |> HOLogic.mk_Trueprop
  2056          val _ = if debug then
  2057                    priority ("Wellfoundedness goal: " ^
  2058                              Syntax.string_of_term ctxt prop ^ ".")
  2059                  else
  2060                    ()
  2061        in
  2062          if tac_timeout = Synchronized.value cached_timeout andalso
  2063             length (Synchronized.value cached_wf_props) < max_cached_wfs then
  2064            ()
  2065          else
  2066            (Synchronized.change cached_wf_props (K []);
  2067             Synchronized.change cached_timeout (K tac_timeout));
  2068          case AList.lookup (op =) (Synchronized.value cached_wf_props) prop of
  2069            SOME wf => wf
  2070          | NONE =>
  2071            let
  2072              val goal = prop |> cterm_of thy |> Goal.init
  2073              val wf = exists (terminates_by ctxt tac_timeout goal)
  2074                              termination_tacs
  2075            in Synchronized.change cached_wf_props (cons (prop, wf)); wf end
  2076        end)
  2077     handle List.Empty => false | NO_TRIPLE () => false
  2078 
  2079 (* The type constraint below is a workaround for a Poly/ML crash. *)
  2080 
  2081 fun is_well_founded_inductive_pred
  2082         (hol_ctxt as {thy, wfs, def_table, wf_cache, ...} : hol_context)
  2083         (x as (s, _)) =
  2084   case triple_lookup (const_match thy) wfs x of
  2085     SOME (SOME b) => b
  2086   | _ => s = @{const_name Nats} orelse s = @{const_name fold_graph'} orelse
  2087          case AList.lookup (op =) (!wf_cache) x of
  2088            SOME (_, wf) => wf
  2089          | NONE =>
  2090            let
  2091              val gfp = (fixpoint_kind_of_const thy def_table x = Gfp)
  2092              val wf = uncached_is_well_founded_inductive_pred hol_ctxt x
  2093            in
  2094              Unsynchronized.change wf_cache (cons (x, (gfp, wf))); wf
  2095            end
  2096 
  2097 fun ap_curry [_] _ t = t
  2098   | ap_curry arg_Ts tuple_T t =
  2099     let val n = length arg_Ts in
  2100       list_abs (map (pair "c") arg_Ts,
  2101                 incr_boundvars n t
  2102                 $ mk_flat_tuple tuple_T (map Bound (n - 1 downto 0)))
  2103     end
  2104 
  2105 fun num_occs_of_bound_in_term j (t1 $ t2) =
  2106     op + (pairself (num_occs_of_bound_in_term j) (t1, t2))
  2107   | num_occs_of_bound_in_term j (Abs (_, _, t')) =
  2108     num_occs_of_bound_in_term (j + 1) t'
  2109   | num_occs_of_bound_in_term j (Bound j') = if j' = j then 1 else 0
  2110   | num_occs_of_bound_in_term _ _ = 0
  2111 
  2112 val is_linear_inductive_pred_def =
  2113   let
  2114     fun do_disjunct j (Const (@{const_name Ex}, _) $ Abs (_, _, t2)) =
  2115         do_disjunct (j + 1) t2
  2116       | do_disjunct j t =
  2117         case num_occs_of_bound_in_term j t of
  2118           0 => true
  2119         | 1 => exists (curry (op =) (Bound j) o head_of) (conjuncts_of t)
  2120         | _ => false
  2121     fun do_lfp_def (Const (@{const_name lfp}, _) $ t2) =
  2122         let val (xs, body) = strip_abs t2 in
  2123           case length xs of
  2124             1 => false
  2125           | n => forall (do_disjunct (n - 1)) (disjuncts_of body)
  2126         end
  2127       | do_lfp_def _ = false
  2128   in do_lfp_def o strip_abs_body end
  2129 
  2130 fun n_ptuple_paths 0 = []
  2131   | n_ptuple_paths 1 = []
  2132   | n_ptuple_paths n = [] :: map (cons 2) (n_ptuple_paths (n - 1))
  2133 val ap_n_split = HOLogic.mk_psplits o n_ptuple_paths
  2134 
  2135 val linear_pred_base_and_step_rhss =
  2136   let
  2137     fun aux (Const (@{const_name lfp}, _) $ t2) =
  2138         let
  2139           val (xs, body) = strip_abs t2
  2140           val arg_Ts = map snd (tl xs)
  2141           val tuple_T = HOLogic.mk_tupleT arg_Ts
  2142           val j = length arg_Ts
  2143           fun repair_rec j (Const (@{const_name Ex}, T1) $ Abs (s2, T2, t2')) =
  2144               Const (@{const_name Ex}, T1)
  2145               $ Abs (s2, T2, repair_rec (j + 1) t2')
  2146             | repair_rec j (@{const HOL.conj} $ t1 $ t2) =
  2147               @{const HOL.conj} $ repair_rec j t1 $ repair_rec j t2
  2148             | repair_rec j t =
  2149               let val (head, args) = strip_comb t in
  2150                 if head = Bound j then
  2151                   HOLogic.eq_const tuple_T $ Bound j
  2152                   $ mk_flat_tuple tuple_T args
  2153                 else
  2154                   t
  2155               end
  2156           val (nonrecs, recs) =
  2157             List.partition (curry (op =) 0 o num_occs_of_bound_in_term j)
  2158                            (disjuncts_of body)
  2159           val base_body = nonrecs |> List.foldl s_disj @{const False}
  2160           val step_body = recs |> map (repair_rec j)
  2161                                |> List.foldl s_disj @{const False}
  2162         in
  2163           (list_abs (tl xs, incr_bv (~1, j, base_body))
  2164            |> ap_n_split (length arg_Ts) tuple_T bool_T,
  2165            Abs ("y", tuple_T, list_abs (tl xs, step_body)
  2166                               |> ap_n_split (length arg_Ts) tuple_T bool_T))
  2167         end
  2168       | aux t =
  2169         raise TERM ("Nitpick_HOL.linear_pred_base_and_step_rhss.aux", [t])
  2170   in aux end
  2171 
  2172 fun starred_linear_pred_const (hol_ctxt as {simp_table, ...}) (s, T) def =
  2173   let
  2174     val j = maxidx_of_term def + 1
  2175     val (outer, fp_app) = strip_abs def
  2176     val outer_bounds = map Bound (length outer - 1 downto 0)
  2177     val outer_vars = map (fn (s, T) => Var ((s, j), T)) outer
  2178     val fp_app = subst_bounds (rev outer_vars, fp_app)
  2179     val (outer_Ts, rest_T) = strip_n_binders (length outer) T
  2180     val tuple_arg_Ts = strip_type rest_T |> fst
  2181     val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
  2182     val set_T = tuple_T --> bool_T
  2183     val curried_T = tuple_T --> set_T
  2184     val uncurried_T = Type (@{type_name prod}, [tuple_T, tuple_T]) --> bool_T
  2185     val (base_rhs, step_rhs) = linear_pred_base_and_step_rhss fp_app
  2186     val base_x as (base_s, _) = (base_prefix ^ s, outer_Ts ---> set_T)
  2187     val base_eq = HOLogic.mk_eq (list_comb (Const base_x, outer_vars), base_rhs)
  2188                   |> HOLogic.mk_Trueprop
  2189     val _ = add_simps simp_table base_s [base_eq]
  2190     val step_x as (step_s, _) = (step_prefix ^ s, outer_Ts ---> curried_T)
  2191     val step_eq = HOLogic.mk_eq (list_comb (Const step_x, outer_vars), step_rhs)
  2192                   |> HOLogic.mk_Trueprop
  2193     val _ = add_simps simp_table step_s [step_eq]
  2194   in
  2195     list_abs (outer,
  2196               Const (@{const_name Image}, uncurried_T --> set_T --> set_T)
  2197               $ (Const (@{const_name rtrancl}, uncurried_T --> uncurried_T)
  2198                  $ (Const (@{const_name prod_case}, curried_T --> uncurried_T)
  2199                     $ list_comb (Const step_x, outer_bounds)))
  2200               $ list_comb (Const base_x, outer_bounds)
  2201               |> ap_curry tuple_arg_Ts tuple_T)
  2202     |> unfold_defs_in_term hol_ctxt
  2203   end
  2204 
  2205 fun is_good_starred_linear_pred_type (Type (@{type_name fun}, Ts)) =
  2206     forall (not o (is_fun_type orf is_pair_type)) Ts
  2207   | is_good_starred_linear_pred_type _ = false
  2208 
  2209 fun unrolled_inductive_pred_const (hol_ctxt as {thy, star_linear_preds,
  2210                                                 def_table, simp_table, ...})
  2211                                   gfp (x as (s, T)) =
  2212   let
  2213     val iter_T = iterator_type_for_const gfp x
  2214     val x' as (s', _) = (unrolled_prefix ^ s, iter_T --> T)
  2215     val unrolled_const = Const x' $ zero_const iter_T
  2216     val def = the (def_of_const thy def_table x)
  2217   in
  2218     if is_equational_fun_but_no_plain_def hol_ctxt x' then
  2219       unrolled_const (* already done *)
  2220     else if not gfp andalso star_linear_preds andalso
  2221          is_linear_inductive_pred_def def andalso
  2222          is_good_starred_linear_pred_type T then
  2223       starred_linear_pred_const hol_ctxt x def
  2224     else
  2225       let
  2226         val j = maxidx_of_term def + 1
  2227         val (outer, fp_app) = strip_abs def
  2228         val outer_bounds = map Bound (length outer - 1 downto 0)
  2229         val cur = Var ((iter_var_prefix, j + 1), iter_T)
  2230         val next = suc_const iter_T $ cur
  2231         val rhs =
  2232           case fp_app of
  2233             Const _ $ t =>
  2234             s_betapply [] (t, list_comb (Const x', next :: outer_bounds))
  2235           | _ => raise TERM ("Nitpick_HOL.unrolled_inductive_pred_const",
  2236                              [fp_app])
  2237         val (inner, naked_rhs) = strip_abs rhs
  2238         val all = outer @ inner
  2239         val bounds = map Bound (length all - 1 downto 0)
  2240         val vars = map (fn (s, T) => Var ((s, j), T)) all
  2241         val eq = HOLogic.mk_eq (list_comb (Const x', cur :: bounds), naked_rhs)
  2242                  |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2243         val _ = add_simps simp_table s' [eq]
  2244       in unrolled_const end
  2245   end
  2246 
  2247 fun raw_inductive_pred_axiom ({thy, def_table, ...} : hol_context) x =
  2248   let
  2249     val def = the (def_of_const thy def_table x)
  2250     val (outer, fp_app) = strip_abs def
  2251     val outer_bounds = map Bound (length outer - 1 downto 0)
  2252     val rhs =
  2253       case fp_app of
  2254         Const _ $ t => s_betapply [] (t, list_comb (Const x, outer_bounds))
  2255       | _ => raise TERM ("Nitpick_HOL.raw_inductive_pred_axiom", [fp_app])
  2256     val (inner, naked_rhs) = strip_abs rhs
  2257     val all = outer @ inner
  2258     val bounds = map Bound (length all - 1 downto 0)
  2259     val j = maxidx_of_term def + 1
  2260     val vars = map (fn (s, T) => Var ((s, j), T)) all
  2261   in
  2262     HOLogic.mk_eq (list_comb (Const x, bounds), naked_rhs)
  2263     |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2264   end
  2265 fun inductive_pred_axiom hol_ctxt (x as (s, T)) =
  2266   if String.isPrefix ubfp_prefix s orelse String.isPrefix lbfp_prefix s then
  2267     let val x' = (strip_first_name_sep s |> snd, T) in
  2268       raw_inductive_pred_axiom hol_ctxt x' |> subst_atomic [(Const x', Const x)]
  2269     end
  2270   else
  2271     raw_inductive_pred_axiom hol_ctxt x
  2272 
  2273 fun equational_fun_axioms (hol_ctxt as {thy, ctxt, stds, def_table, simp_table,
  2274                                         psimp_table, ...}) x =
  2275   case def_props_for_const thy stds (!simp_table) x of
  2276     [] => (case def_props_for_const thy stds psimp_table x of
  2277              [] => (if is_inductive_pred hol_ctxt x then
  2278                       [inductive_pred_axiom hol_ctxt x]
  2279                     else case def_of_const thy def_table x of
  2280                       SOME def =>
  2281                       @{const Trueprop} $ HOLogic.mk_eq (Const x, def)
  2282                       |> equationalize_term ctxt "" |> the |> single
  2283                     | NONE => [])
  2284            | psimps => psimps)
  2285   | simps => simps
  2286 fun is_equational_fun_surely_complete hol_ctxt x =
  2287   case equational_fun_axioms hol_ctxt x of
  2288     [@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)] =>
  2289     strip_comb t1 |> snd |> forall is_Var
  2290   | _ => false
  2291 
  2292 (** Type preprocessing **)
  2293 
  2294 fun merged_type_var_table_for_terms thy ts =
  2295   let
  2296     fun add (s, S) table =
  2297       table
  2298       |> (case AList.lookup (Sign.subsort thy o swap) table S of
  2299             SOME _ => I
  2300           | NONE =>
  2301             filter_out (fn (S', _) => Sign.subsort thy (S, S'))
  2302             #> cons (S, s))
  2303     val tfrees = [] |> fold Term.add_tfrees ts
  2304                     |> sort (string_ord o pairself fst)
  2305   in [] |> fold add tfrees |> rev end
  2306 
  2307 fun merge_type_vars_in_term thy merge_type_vars table =
  2308   merge_type_vars
  2309   ? map_types (map_atyps
  2310         (fn TFree (_, S) =>
  2311             TFree (table |> find_first (fn (S', _) => Sign.subsort thy (S', S))
  2312                          |> the |> swap)
  2313           | T => T))
  2314 
  2315 fun add_ground_types hol_ctxt binarize =
  2316   let
  2317     fun aux T accum =
  2318       case T of
  2319         Type (@{type_name fun}, Ts) => fold aux Ts accum
  2320       | Type (@{type_name prod}, Ts) => fold aux Ts accum
  2321       | Type (@{type_name itself}, [T1]) => aux T1 accum
  2322       | Type (_, Ts) =>
  2323         if member (op =) (@{typ prop} :: @{typ bool} :: accum) T then
  2324           accum
  2325         else
  2326           T :: accum
  2327           |> fold aux (case binarized_and_boxed_datatype_constrs hol_ctxt
  2328                                                                  binarize T of
  2329                          [] => Ts
  2330                        | xs => map snd xs)
  2331       | _ => insert (op =) T accum
  2332   in aux end
  2333 
  2334 fun ground_types_in_type hol_ctxt binarize T =
  2335   add_ground_types hol_ctxt binarize T []
  2336 fun ground_types_in_terms hol_ctxt binarize ts =
  2337   fold (fold_types (add_ground_types hol_ctxt binarize)) ts []
  2338 
  2339 end;