src/HOL/Tools/Nitpick/nitpick_hol.ML
author blanchet
Tue Jan 03 18:33:17 2012 +0100 (2012-01-03)
changeset 46081 8f6465f7021b
parent 45980 af59825c40cf
child 46083 efeaa79f021b
permissions -rw-r--r--
ported mono calculus to handle "set" type constructors
     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      total_consts: bool option,
    31      needs: term list option,
    32      tac_timeout: Time.time option,
    33      evals: term list,
    34      case_names: (string * int) list,
    35      def_tables: const_table * const_table,
    36      nondef_table: const_table,
    37      nondefs: term list,
    38      simp_table: const_table Unsynchronized.ref,
    39      psimp_table: const_table,
    40      choice_spec_table: const_table,
    41      intro_table: const_table,
    42      ground_thm_table: term list Inttab.table,
    43      ersatz_table: (string * string) list,
    44      skolems: (string * string list) list Unsynchronized.ref,
    45      special_funs: special_fun list Unsynchronized.ref,
    46      unrolled_preds: unrolled list Unsynchronized.ref,
    47      wf_cache: wf_cache Unsynchronized.ref,
    48      constr_cache: (typ * styp list) list Unsynchronized.ref}
    49 
    50   datatype fixpoint_kind = Lfp | Gfp | NoFp
    51   datatype boxability =
    52     InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
    53 
    54   val name_sep : string
    55   val numeral_prefix : string
    56   val base_prefix : string
    57   val step_prefix : string
    58   val unrolled_prefix : string
    59   val ubfp_prefix : string
    60   val lbfp_prefix : string
    61   val quot_normal_prefix : string
    62   val skolem_prefix : string
    63   val special_prefix : string
    64   val uncurry_prefix : string
    65   val eval_prefix : string
    66   val iter_var_prefix : string
    67   val strip_first_name_sep : string -> string * string
    68   val original_name : string -> string
    69   val abs_var : indexname * typ -> term -> term
    70   val s_conj : term * term -> term
    71   val s_disj : term * term -> term
    72   val strip_any_connective : term -> term list * term
    73   val conjuncts_of : term -> term list
    74   val disjuncts_of : term -> term list
    75   val unarize_unbox_etc_type : typ -> typ
    76   val uniterize_unarize_unbox_etc_type : typ -> typ
    77   val string_for_type : Proof.context -> typ -> string
    78   val pretty_for_type : Proof.context -> typ -> Pretty.T
    79   val prefix_name : string -> string -> string
    80   val shortest_name : string -> string
    81   val short_name : string -> string
    82   val shorten_names_in_term : term -> term
    83   val strict_type_match : theory -> typ * typ -> bool
    84   val type_match : theory -> typ * typ -> bool
    85   val const_match : theory -> styp * styp -> bool
    86   val term_match : theory -> term * term -> bool
    87   val frac_from_term_pair : typ -> term -> term -> term
    88   val is_TFree : typ -> bool
    89   val is_fun_type : typ -> bool
    90   val is_set_like_type : typ -> bool
    91   val is_pair_type : typ -> bool
    92   val is_lfp_iterator_type : typ -> bool
    93   val is_gfp_iterator_type : typ -> bool
    94   val is_fp_iterator_type : typ -> bool
    95   val is_iterator_type : typ -> bool
    96   val is_boolean_type : typ -> bool
    97   val is_integer_type : typ -> bool
    98   val is_bit_type : typ -> bool
    99   val is_word_type : typ -> bool
   100   val is_integer_like_type : typ -> bool
   101   val is_record_type : typ -> bool
   102   val is_number_type : Proof.context -> typ -> bool
   103   val is_higher_order_type : typ -> bool
   104   val elem_type : typ -> typ
   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_ersatz :
   153     (string * string) list -> morphism -> Context.generic -> Context.generic
   154   val register_ersatz_global : (string * string) list -> theory -> theory
   155   val register_codatatype :
   156     typ -> string -> styp list -> morphism -> Context.generic -> Context.generic
   157   val register_codatatype_global :
   158     typ -> string -> styp list -> theory -> theory
   159   val unregister_codatatype :
   160     typ -> morphism -> Context.generic -> Context.generic
   161   val unregister_codatatype_global : typ -> theory -> theory
   162   val datatype_constrs : hol_context -> typ -> styp list
   163   val binarized_and_boxed_datatype_constrs :
   164     hol_context -> bool -> typ -> styp list
   165   val num_datatype_constrs : hol_context -> typ -> int
   166   val constr_name_for_sel_like : string -> string
   167   val binarized_and_boxed_constr_for_sel : hol_context -> bool -> styp -> styp
   168   val card_of_type : (typ * int) list -> typ -> int
   169   val bounded_card_of_type : int -> int -> (typ * int) list -> typ -> int
   170   val bounded_exact_card_of_type :
   171     hol_context -> typ list -> int -> int -> (typ * int) list -> typ -> int
   172   val typical_card_of_type : typ -> int
   173   val is_finite_type : hol_context -> typ -> bool
   174   val is_special_eligible_arg : bool -> typ list -> term -> bool
   175   val s_let :
   176     typ list -> string -> int -> typ -> typ -> (term -> term) -> term -> term
   177   val s_betapply : typ list -> term * term -> term
   178   val s_betapplys : typ list -> term * term list -> term
   179   val discriminate_value : hol_context -> styp -> term -> term
   180   val select_nth_constr_arg :
   181     Proof.context -> (typ option * bool) list -> styp -> term -> int -> typ
   182     -> term
   183   val construct_value :
   184     Proof.context -> (typ option * bool) list -> styp -> term list -> term
   185   val coerce_term : hol_context -> typ list -> typ -> typ -> term -> term
   186   val special_bounds : term list -> (indexname * typ) list
   187   val is_funky_typedef : Proof.context -> typ -> bool
   188   val all_defs_of : theory -> (term * term) list -> term list
   189   val all_nondefs_of : Proof.context -> (term * term) list -> term list
   190   val arity_of_built_in_const :
   191     theory -> (typ option * bool) list -> styp -> int option
   192   val is_built_in_const :
   193     theory -> (typ option * bool) list -> styp -> bool
   194   val term_under_def : term -> term
   195   val case_const_names :
   196     Proof.context -> (typ option * bool) list -> (string * int) list
   197   val unfold_defs_in_term : hol_context -> term -> term
   198   val const_def_tables :
   199     Proof.context -> (term * term) list -> term list
   200     -> const_table * const_table
   201   val const_nondef_table : term list -> const_table
   202   val const_simp_table : Proof.context -> (term * term) list -> const_table
   203   val const_psimp_table : Proof.context -> (term * term) list -> const_table
   204   val const_choice_spec_table :
   205     Proof.context -> (term * term) list -> const_table
   206   val inductive_intro_table :
   207     Proof.context -> (term * term) list -> const_table * const_table
   208     -> const_table
   209   val ground_theorem_table : theory -> term list Inttab.table
   210   val ersatz_table : Proof.context -> (string * string) list
   211   val add_simps : const_table Unsynchronized.ref -> string -> term list -> unit
   212   val inverse_axioms_for_rep_fun : Proof.context -> styp -> term list
   213   val optimized_typedef_axioms : Proof.context -> string * typ list -> term list
   214   val optimized_quot_type_axioms :
   215     Proof.context -> (typ option * bool) list -> string * typ list -> term list
   216   val def_of_const : theory -> const_table * const_table -> styp -> term option
   217   val fixpoint_kind_of_rhs : term -> fixpoint_kind
   218   val fixpoint_kind_of_const :
   219     theory -> const_table * const_table -> string * typ -> fixpoint_kind
   220   val is_real_inductive_pred : hol_context -> styp -> bool
   221   val is_constr_pattern : Proof.context -> term -> bool
   222   val is_constr_pattern_lhs : Proof.context -> term -> bool
   223   val is_constr_pattern_formula : Proof.context -> term -> bool
   224   val nondef_props_for_const :
   225     theory -> bool -> const_table -> styp -> term list
   226   val is_choice_spec_fun : hol_context -> styp -> bool
   227   val is_choice_spec_axiom : theory -> const_table -> term -> bool
   228   val is_real_equational_fun : hol_context -> styp -> bool
   229   val is_equational_fun_but_no_plain_def : hol_context -> styp -> bool
   230   val codatatype_bisim_axioms : hol_context -> typ -> term list
   231   val is_well_founded_inductive_pred : hol_context -> styp -> bool
   232   val unrolled_inductive_pred_const : hol_context -> bool -> styp -> term
   233   val equational_fun_axioms : hol_context -> styp -> term list
   234   val is_equational_fun_surely_complete : hol_context -> styp -> bool
   235   val merged_type_var_table_for_terms :
   236     theory -> term list -> (sort * string) list
   237   val merge_type_vars_in_term :
   238     theory -> bool -> (sort * string) list -> term -> term
   239   val ground_types_in_type : hol_context -> bool -> typ -> typ list
   240   val ground_types_in_terms : hol_context -> bool -> term list -> typ list
   241 end;
   242 
   243 structure Nitpick_HOL : NITPICK_HOL =
   244 struct
   245 
   246 open Nitpick_Util
   247 
   248 type const_table = term list Symtab.table
   249 type special_fun = (styp * int list * term list) * styp
   250 type unrolled = styp * styp
   251 type wf_cache = (styp * (bool * bool)) list
   252 
   253 type hol_context =
   254   {thy: theory,
   255    ctxt: Proof.context,
   256    max_bisim_depth: int,
   257    boxes: (typ option * bool option) list,
   258    stds: (typ option * bool) list,
   259    wfs: (styp option * bool option) list,
   260    user_axioms: bool option,
   261    debug: bool,
   262    whacks: term list,
   263    binary_ints: bool option,
   264    destroy_constrs: bool,
   265    specialize: bool,
   266    star_linear_preds: bool,
   267    total_consts: bool option,
   268    needs: term list option,
   269    tac_timeout: Time.time option,
   270    evals: term list,
   271    case_names: (string * int) list,
   272    def_tables: const_table * const_table,
   273    nondef_table: const_table,
   274    nondefs: term list,
   275    simp_table: const_table Unsynchronized.ref,
   276    psimp_table: const_table,
   277    choice_spec_table: const_table,
   278    intro_table: const_table,
   279    ground_thm_table: term list Inttab.table,
   280    ersatz_table: (string * string) list,
   281    skolems: (string * string list) list Unsynchronized.ref,
   282    special_funs: special_fun list Unsynchronized.ref,
   283    unrolled_preds: unrolled list Unsynchronized.ref,
   284    wf_cache: wf_cache Unsynchronized.ref,
   285    constr_cache: (typ * styp list) list Unsynchronized.ref}
   286 
   287 datatype fixpoint_kind = Lfp | Gfp | NoFp
   288 datatype boxability =
   289   InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
   290 
   291 structure Data = Generic_Data
   292 (
   293   type T = {frac_types: (string * (string * string) list) list,
   294             ersatz_table: (string * string) list,
   295             codatatypes: (string * (string * styp list)) list}
   296   val empty = {frac_types = [], ersatz_table = [], codatatypes = []}
   297   val extend = I
   298   fun merge ({frac_types = fs1, ersatz_table = et1, codatatypes = cs1},
   299              {frac_types = fs2, ersatz_table = et2, codatatypes = cs2}) : T =
   300     {frac_types = AList.merge (op =) (K true) (fs1, fs2),
   301      ersatz_table = AList.merge (op =) (K true) (et1, et2),
   302      codatatypes = AList.merge (op =) (K true) (cs1, cs2)}
   303 )
   304 
   305 val name_sep = "$"
   306 val numeral_prefix = nitpick_prefix ^ "num" ^ name_sep
   307 val sel_prefix = nitpick_prefix ^ "sel"
   308 val discr_prefix = nitpick_prefix ^ "is" ^ name_sep
   309 val set_prefix = nitpick_prefix ^ "set" ^ name_sep
   310 val lfp_iterator_prefix = nitpick_prefix ^ "lfpit" ^ name_sep
   311 val gfp_iterator_prefix = nitpick_prefix ^ "gfpit" ^ name_sep
   312 val unrolled_prefix = nitpick_prefix ^ "unroll" ^ name_sep
   313 val base_prefix = nitpick_prefix ^ "base" ^ name_sep
   314 val step_prefix = nitpick_prefix ^ "step" ^ name_sep
   315 val ubfp_prefix = nitpick_prefix ^ "ubfp" ^ name_sep
   316 val lbfp_prefix = nitpick_prefix ^ "lbfp" ^ name_sep
   317 val quot_normal_prefix = nitpick_prefix ^ "qn" ^ name_sep
   318 val skolem_prefix = nitpick_prefix ^ "sk"
   319 val special_prefix = nitpick_prefix ^ "sp"
   320 val uncurry_prefix = nitpick_prefix ^ "unc"
   321 val eval_prefix = nitpick_prefix ^ "eval"
   322 val iter_var_prefix = "i"
   323 
   324 (** Constant/type information and term/type manipulation **)
   325 
   326 fun sel_prefix_for j = sel_prefix ^ string_of_int j ^ name_sep
   327 fun quot_normal_name_for_type ctxt T =
   328   quot_normal_prefix ^ unyxml (Syntax.string_of_typ ctxt T)
   329 
   330 val strip_first_name_sep =
   331   Substring.full #> Substring.position name_sep ##> Substring.triml 1
   332   #> pairself Substring.string
   333 fun original_name s =
   334   if String.isPrefix nitpick_prefix s then
   335     case strip_first_name_sep s of (s1, "") => s1 | (_, s2) => original_name s2
   336   else
   337     s
   338 
   339 fun s_conj (t1, @{const True}) = t1
   340   | s_conj (@{const True}, t2) = t2
   341   | s_conj (t1, t2) =
   342     if t1 = @{const False} orelse t2 = @{const False} then @{const False}
   343     else HOLogic.mk_conj (t1, t2)
   344 fun s_disj (t1, @{const False}) = t1
   345   | s_disj (@{const False}, t2) = t2
   346   | s_disj (t1, t2) =
   347     if t1 = @{const True} orelse t2 = @{const True} then @{const True}
   348     else HOLogic.mk_disj (t1, t2)
   349 
   350 fun strip_connective conn_t (t as (t0 $ t1 $ t2)) =
   351     if t0 = conn_t then strip_connective t0 t2 @ strip_connective t0 t1 else [t]
   352   | strip_connective _ t = [t]
   353 fun strip_any_connective (t as (t0 $ _ $ _)) =
   354     if t0 = @{const HOL.conj} orelse t0 = @{const HOL.disj} then
   355       (strip_connective t0 t, t0)
   356     else
   357       ([t], @{const Not})
   358   | strip_any_connective t = ([t], @{const Not})
   359 val conjuncts_of = strip_connective @{const HOL.conj}
   360 val disjuncts_of = strip_connective @{const HOL.disj}
   361 
   362 (* When you add constants to these lists, make sure to handle them in
   363    "Nitpick_Nut.nut_from_term", and perhaps in "Nitpick_Mono.consider_term" as
   364    well. *)
   365 val built_in_consts =
   366   [(@{const_name all}, 1),
   367    (@{const_name "=="}, 2),
   368    (@{const_name "==>"}, 2),
   369    (@{const_name Pure.conjunction}, 2),
   370    (@{const_name Trueprop}, 1),
   371    (@{const_name Not}, 1),
   372    (@{const_name False}, 0),
   373    (@{const_name True}, 0),
   374    (@{const_name All}, 1),
   375    (@{const_name Ex}, 1),
   376    (@{const_name HOL.eq}, 1),
   377    (@{const_name HOL.conj}, 2),
   378    (@{const_name HOL.disj}, 2),
   379    (@{const_name HOL.implies}, 2),
   380    (@{const_name If}, 3),
   381    (@{const_name Let}, 2),
   382    (@{const_name Pair}, 2),
   383    (@{const_name fst}, 1),
   384    (@{const_name snd}, 1),
   385    (@{const_name Id}, 0),
   386    (@{const_name converse}, 1),
   387    (@{const_name trancl}, 1),
   388    (@{const_name rel_comp}, 2),
   389    (@{const_name finite}, 1),
   390    (@{const_name unknown}, 0),
   391    (@{const_name is_unknown}, 1),
   392    (@{const_name safe_The}, 1),
   393    (@{const_name Frac}, 0),
   394    (@{const_name norm_frac}, 0)]
   395 val built_in_nat_consts =
   396   [(@{const_name Suc}, 0),
   397    (@{const_name nat}, 0),
   398    (@{const_name nat_gcd}, 0),
   399    (@{const_name nat_lcm}, 0)]
   400 val built_in_typed_consts =
   401   [((@{const_name zero_class.zero}, int_T), 0),
   402    ((@{const_name one_class.one}, int_T), 0),
   403    ((@{const_name plus_class.plus}, int_T --> int_T --> int_T), 0),
   404    ((@{const_name minus_class.minus}, int_T --> int_T --> int_T), 0),
   405    ((@{const_name times_class.times}, int_T --> int_T --> int_T), 0),
   406    ((@{const_name div_class.div}, int_T --> int_T --> int_T), 0),
   407    ((@{const_name uminus_class.uminus}, int_T --> int_T), 0),
   408    ((@{const_name ord_class.less}, int_T --> int_T --> bool_T), 2),
   409    ((@{const_name ord_class.less_eq}, int_T --> int_T --> bool_T), 2)]
   410 val built_in_typed_nat_consts =
   411   [((@{const_name zero_class.zero}, nat_T), 0),
   412    ((@{const_name one_class.one}, nat_T), 0),
   413    ((@{const_name plus_class.plus}, nat_T --> nat_T --> nat_T), 0),
   414    ((@{const_name minus_class.minus}, nat_T --> nat_T --> nat_T), 0),
   415    ((@{const_name times_class.times}, nat_T --> nat_T --> nat_T), 0),
   416    ((@{const_name div_class.div}, nat_T --> nat_T --> nat_T), 0),
   417    ((@{const_name ord_class.less}, nat_T --> nat_T --> bool_T), 2),
   418    ((@{const_name ord_class.less_eq}, nat_T --> nat_T --> bool_T), 2),
   419    ((@{const_name of_nat}, nat_T --> int_T), 0)]
   420 val built_in_set_like_consts =
   421   [(@{const_name ord_class.less_eq}, 2)]
   422 
   423 fun unarize_type @{typ "unsigned_bit word"} = nat_T
   424   | unarize_type @{typ "signed_bit word"} = int_T
   425   | unarize_type (Type (s, Ts as _ :: _)) = Type (s, map unarize_type Ts)
   426   | unarize_type T = T
   427 fun unarize_unbox_etc_type (Type (@{type_name fun_box}, Ts)) =
   428     unarize_unbox_etc_type (Type (@{type_name fun}, Ts))
   429   | unarize_unbox_etc_type (Type (@{type_name pair_box}, Ts)) =
   430     Type (@{type_name prod}, map unarize_unbox_etc_type Ts)
   431   | unarize_unbox_etc_type @{typ "unsigned_bit word"} = nat_T
   432   | unarize_unbox_etc_type @{typ "signed_bit word"} = int_T
   433   | unarize_unbox_etc_type (Type (s, Ts as _ :: _)) =
   434     Type (s, map unarize_unbox_etc_type Ts)
   435   | unarize_unbox_etc_type T = T
   436 fun uniterize_type (Type (s, Ts as _ :: _)) = Type (s, map uniterize_type Ts)
   437   | uniterize_type @{typ bisim_iterator} = nat_T
   438   | uniterize_type T = T
   439 val uniterize_unarize_unbox_etc_type = uniterize_type o unarize_unbox_etc_type
   440 
   441 fun string_for_type ctxt = Syntax.string_of_typ ctxt o unarize_unbox_etc_type
   442 fun pretty_for_type ctxt = Syntax.pretty_typ ctxt o unarize_unbox_etc_type
   443 
   444 val prefix_name = Long_Name.qualify o Long_Name.base_name
   445 fun shortest_name s = List.last (space_explode "." s) handle List.Empty => ""
   446 val prefix_abs_vars = Term.map_abs_vars o prefix_name
   447 fun short_name s =
   448   case space_explode name_sep s of
   449     [_] => s |> String.isPrefix nitpick_prefix s ? unprefix nitpick_prefix
   450   | ss => map shortest_name ss |> space_implode "_"
   451 fun shorten_names_in_type (Type (s, Ts)) =
   452     Type (short_name s, map shorten_names_in_type Ts)
   453   | shorten_names_in_type T = T
   454 val shorten_names_in_term =
   455   map_aterms (fn Const (s, T) => Const (short_name s, T) | t => t)
   456   #> map_types shorten_names_in_type
   457 
   458 fun strict_type_match thy (T1, T2) =
   459   (Sign.typ_match thy (T2, T1) Vartab.empty; true)
   460   handle Type.TYPE_MATCH => false
   461 fun type_match thy = strict_type_match thy o pairself unarize_unbox_etc_type
   462 fun const_match thy ((s1, T1), (s2, T2)) =
   463   s1 = s2 andalso type_match thy (T1, T2)
   464 fun term_match thy (Const x1, Const x2) = const_match thy (x1, x2)
   465   | term_match thy (Free (s1, T1), Free (s2, T2)) =
   466     const_match thy ((shortest_name s1, T1), (shortest_name s2, T2))
   467   | term_match _ (t1, t2) = t1 aconv t2
   468 
   469 fun frac_from_term_pair T t1 t2 =
   470   case snd (HOLogic.dest_number t1) of
   471     0 => HOLogic.mk_number T 0
   472   | n1 => case snd (HOLogic.dest_number t2) of
   473             1 => HOLogic.mk_number T n1
   474           | n2 => Const (@{const_name divide}, T --> T --> T)
   475                   $ HOLogic.mk_number T n1 $ HOLogic.mk_number T n2
   476 
   477 fun is_TFree (TFree _) = true
   478   | is_TFree _ = false
   479 fun is_fun_type (Type (@{type_name fun}, _)) = true
   480   | is_fun_type _ = false
   481 fun is_set_like_type (Type (@{type_name fun}, [_, @{typ bool}])) = true
   482   | is_set_like_type (Type (@{type_name set}, _)) = true
   483   | is_set_like_type _ = false
   484 fun is_pair_type (Type (@{type_name prod}, _)) = true
   485   | is_pair_type _ = false
   486 fun is_lfp_iterator_type (Type (s, _)) = String.isPrefix lfp_iterator_prefix s
   487   | is_lfp_iterator_type _ = false
   488 fun is_gfp_iterator_type (Type (s, _)) = String.isPrefix gfp_iterator_prefix s
   489   | is_gfp_iterator_type _ = false
   490 val is_fp_iterator_type = is_lfp_iterator_type orf is_gfp_iterator_type
   491 fun is_iterator_type T =
   492   (T = @{typ bisim_iterator} orelse is_fp_iterator_type T)
   493 fun is_boolean_type T = (T = prop_T orelse T = bool_T)
   494 fun is_integer_type T = (T = nat_T orelse T = int_T)
   495 fun is_bit_type T = (T = @{typ unsigned_bit} orelse T = @{typ signed_bit})
   496 fun is_word_type (Type (@{type_name word}, _)) = true
   497   | is_word_type _ = false
   498 val is_integer_like_type = is_iterator_type orf is_integer_type orf is_word_type
   499 val is_record_type = not o null o Record.dest_recTs
   500 fun is_frac_type ctxt (Type (s, [])) =
   501     s |> AList.defined (op =) (#frac_types (Data.get (Context.Proof ctxt)))
   502   | is_frac_type _ _ = false
   503 fun is_number_type ctxt = is_integer_like_type orf is_frac_type ctxt
   504 fun is_higher_order_type (Type (@{type_name fun}, _)) = true
   505   | is_higher_order_type (Type (_, Ts)) = exists is_higher_order_type Ts
   506   | is_higher_order_type _ = false
   507 
   508 fun elem_type (Type (@{type_name set}, [T])) = T
   509   | elem_type T = raise TYPE ("Nitpick_HOL.elem_type", [T], [])
   510 
   511 fun iterator_type_for_const gfp (s, T) =
   512   Type ((if gfp then gfp_iterator_prefix else lfp_iterator_prefix) ^ s,
   513         binder_types T)
   514 fun const_for_iterator_type (Type (s, Ts)) =
   515     (strip_first_name_sep s |> snd, Ts ---> bool_T)
   516   | const_for_iterator_type T =
   517     raise TYPE ("Nitpick_HOL.const_for_iterator_type", [T], [])
   518 
   519 fun strip_n_binders 0 T = ([], T)
   520   | strip_n_binders n (Type (@{type_name fun}, [T1, T2])) =
   521     strip_n_binders (n - 1) T2 |>> cons T1
   522   | strip_n_binders n (Type (@{type_name fun_box}, Ts)) =
   523     strip_n_binders n (Type (@{type_name fun}, Ts))
   524   | strip_n_binders _ T = raise TYPE ("Nitpick_HOL.strip_n_binders", [T], [])
   525 val nth_range_type = snd oo strip_n_binders
   526 
   527 fun num_factors_in_type (Type (@{type_name prod}, [T1, T2])) =
   528     fold (Integer.add o num_factors_in_type) [T1, T2] 0
   529   | num_factors_in_type _ = 1
   530 fun num_binder_types (Type (@{type_name fun}, [_, T2])) =
   531     1 + num_binder_types T2
   532   | num_binder_types _ = 0
   533 val curried_binder_types = maps HOLogic.flatten_tupleT o binder_types
   534 fun maybe_curried_binder_types T =
   535   (if is_pair_type (body_type T) then binder_types else curried_binder_types) T
   536 
   537 fun mk_flat_tuple _ [t] = t
   538   | mk_flat_tuple (Type (@{type_name prod}, [T1, T2])) (t :: ts) =
   539     HOLogic.pair_const T1 T2 $ t $ (mk_flat_tuple T2 ts)
   540   | mk_flat_tuple T ts = raise TYPE ("Nitpick_HOL.mk_flat_tuple", [T], ts)
   541 fun dest_n_tuple 1 t = [t]
   542   | dest_n_tuple n t = HOLogic.dest_prod t ||> dest_n_tuple (n - 1) |> op ::
   543 
   544 type typedef_info =
   545   {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string,
   546    set_def: thm option, prop_of_Rep: thm, set_name: string,
   547    Abs_inverse: thm option, Rep_inverse: thm option}
   548 
   549 fun typedef_info ctxt s =
   550   if is_frac_type ctxt (Type (s, [])) then
   551     SOME {abs_type = Type (s, []), rep_type = @{typ "int * int"},
   552           Abs_name = @{const_name Abs_Frac}, Rep_name = @{const_name Rep_Frac},
   553           set_def = NONE, prop_of_Rep = @{prop "Rep_Frac x \<in> Collect Frac"}
   554                           |> Logic.varify_global,
   555           set_name = @{const_name Frac}, Abs_inverse = NONE, Rep_inverse = NONE}
   556   else case Typedef.get_info ctxt s of
   557     (* When several entries are returned, it shouldn't matter much which one
   558        we take (according to Florian Haftmann). *)
   559     (* The "Logic.varifyT_global" calls are a temporary hack because these
   560        types's type variables sometimes clash with locally fixed type variables.
   561        Remove these calls once "Typedef" is fully localized. *)
   562     ({abs_type, rep_type, Abs_name, Rep_name, ...},
   563      {set_def, Rep, Abs_inverse, Rep_inverse, ...}) :: _ =>
   564     SOME {abs_type = Logic.varifyT_global abs_type,
   565           rep_type = Logic.varifyT_global rep_type, Abs_name = Abs_name,
   566           Rep_name = Rep_name, set_def = set_def, prop_of_Rep = prop_of Rep,
   567           set_name = set_prefix ^ s, Abs_inverse = SOME Abs_inverse,
   568           Rep_inverse = SOME Rep_inverse}
   569   | _ => NONE
   570 
   571 val is_typedef = is_some oo typedef_info
   572 val is_real_datatype = is_some oo Datatype.get_info
   573 fun is_standard_datatype thy = the oo triple_lookup (type_match thy)
   574 
   575 (* FIXME: Use antiquotation for "code_numeral" below or detect "rep_datatype",
   576    e.g., by adding a field to "Datatype_Aux.info". *)
   577 fun is_basic_datatype thy stds s =
   578   member (op =) [@{type_name prod}, @{type_name bool}, @{type_name int},
   579                  "Code_Numeral.code_numeral"] s orelse
   580   (s = @{type_name nat} andalso is_standard_datatype thy stds nat_T)
   581 
   582 fun repair_constr_type ctxt body_T' T =
   583   varify_and_instantiate_type ctxt (body_type T) body_T' T
   584 
   585 fun register_frac_type_generic frac_s ersaetze generic =
   586   let
   587     val {frac_types, ersatz_table, codatatypes} = Data.get generic
   588     val frac_types = AList.update (op =) (frac_s, ersaetze) frac_types
   589   in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
   590                codatatypes = codatatypes} generic end
   591 (* TODO: Consider morphism. *)
   592 fun register_frac_type frac_s ersaetze (_ : morphism) =
   593   register_frac_type_generic frac_s ersaetze
   594 val register_frac_type_global = Context.theory_map oo register_frac_type_generic
   595 
   596 fun unregister_frac_type_generic frac_s = register_frac_type_generic frac_s []
   597 (* TODO: Consider morphism. *)
   598 fun unregister_frac_type frac_s (_ : morphism) =
   599   unregister_frac_type_generic frac_s
   600 val unregister_frac_type_global =
   601   Context.theory_map o unregister_frac_type_generic
   602 
   603 fun register_ersatz_generic ersatz generic =
   604   let
   605     val {frac_types, ersatz_table, codatatypes} = Data.get generic
   606     val ersatz_table = AList.merge (op =) (K true) (ersatz_table, ersatz)
   607   in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
   608                codatatypes = codatatypes} generic end
   609 (* TODO: Consider morphism. *)
   610 fun register_ersatz ersatz (_ : morphism) =
   611   register_ersatz_generic ersatz
   612 val register_ersatz_global = Context.theory_map o register_ersatz_generic
   613 
   614 fun register_codatatype_generic co_T case_name constr_xs generic =
   615   let
   616     val ctxt = Context.proof_of generic
   617     val thy = Context.theory_of generic
   618     val {frac_types, ersatz_table, codatatypes} = Data.get generic
   619     val constr_xs = map (apsnd (repair_constr_type ctxt co_T)) constr_xs
   620     val (co_s, co_Ts) = dest_Type co_T
   621     val _ =
   622       if forall is_TFree co_Ts andalso not (has_duplicates (op =) co_Ts) andalso
   623          co_s <> @{type_name fun} andalso
   624          not (is_basic_datatype thy [(NONE, true)] co_s) then
   625         ()
   626       else
   627         raise TYPE ("Nitpick_HOL.register_codatatype_generic", [co_T], [])
   628     val codatatypes = AList.update (op =) (co_s, (case_name, constr_xs))
   629                                    codatatypes
   630   in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
   631                codatatypes = codatatypes} generic end
   632 (* TODO: Consider morphism. *)
   633 fun register_codatatype co_T case_name constr_xs (_ : morphism) =
   634   register_codatatype_generic co_T case_name constr_xs
   635 val register_codatatype_global =
   636   Context.theory_map ooo register_codatatype_generic
   637 
   638 fun unregister_codatatype_generic co_T = register_codatatype_generic co_T "" []
   639 (* TODO: Consider morphism. *)
   640 fun unregister_codatatype co_T (_ : morphism) =
   641   unregister_codatatype_generic co_T
   642 val unregister_codatatype_global =
   643   Context.theory_map o unregister_codatatype_generic
   644 
   645 fun is_codatatype ctxt (Type (s, _)) =
   646     s |> AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   647       |> Option.map snd |> these |> null |> not
   648   | is_codatatype _ _ = false
   649 fun is_real_quot_type ctxt (Type (s, _)) =
   650     is_some (Quotient_Info.lookup_quotients ctxt s)
   651   | is_real_quot_type _ _ = false
   652 fun is_quot_type ctxt T =
   653     is_real_quot_type ctxt T andalso not (is_codatatype ctxt T)
   654 fun is_pure_typedef ctxt (T as Type (s, _)) =
   655     let val thy = Proof_Context.theory_of ctxt in
   656       is_typedef ctxt s andalso
   657       not (is_real_datatype thy s orelse is_real_quot_type ctxt T orelse
   658            is_codatatype ctxt T orelse is_record_type T orelse
   659            is_integer_like_type T)
   660     end
   661   | is_pure_typedef _ _ = false
   662 fun is_univ_typedef ctxt (Type (s, _)) =
   663     (case typedef_info ctxt s of
   664        SOME {set_def, prop_of_Rep, ...} =>
   665        let
   666          val t_opt =
   667            case set_def of
   668              SOME thm => try (snd o Logic.dest_equals o prop_of) thm
   669            | NONE => try (snd o HOLogic.dest_mem o HOLogic.dest_Trueprop)
   670                          prop_of_Rep
   671        in
   672          case t_opt of
   673            SOME (Const (@{const_name top}, _)) => true
   674            (* "Multiset.multiset" *)
   675          | SOME (Const (@{const_name Collect}, _)
   676                  $ Abs (_, _, Const (@{const_name finite}, _) $ _)) => true
   677            (* "FinFun.finfun" *)
   678          | SOME (Const (@{const_name Collect}, _) $ Abs (_, _,
   679                      Const (@{const_name Ex}, _) $ Abs (_, _,
   680                          Const (@{const_name finite}, _) $ _))) => true
   681          | _ => false
   682        end
   683      | NONE => false)
   684   | is_univ_typedef _ _ = false
   685 fun is_datatype ctxt stds (T as Type (s, _)) =
   686     let val thy = Proof_Context.theory_of ctxt in
   687       (is_typedef ctxt s orelse is_codatatype ctxt T orelse
   688        T = @{typ ind} orelse is_real_quot_type ctxt T) andalso
   689       not (is_basic_datatype thy stds s)
   690     end
   691   | is_datatype _ _ _ = false
   692 
   693 fun all_record_fields thy T =
   694   let val (recs, more) = Record.get_extT_fields thy T in
   695     recs @ more :: all_record_fields thy (snd more)
   696   end
   697   handle TYPE _ => []
   698 fun is_record_constr (s, T) =
   699   String.isSuffix Record.extN s andalso
   700   let val dataT = body_type T in
   701     is_record_type dataT andalso
   702     s = unsuffix Record.ext_typeN (fst (dest_Type dataT)) ^ Record.extN
   703   end
   704 val num_record_fields = Integer.add 1 o length o fst oo Record.get_extT_fields
   705 fun no_of_record_field thy s T1 =
   706   find_index (curry (op =) s o fst)
   707              (Record.get_extT_fields thy T1 ||> single |> op @)
   708 fun is_record_get thy (s, Type (@{type_name fun}, [T1, _])) =
   709     exists (curry (op =) s o fst) (all_record_fields thy T1)
   710   | is_record_get _ _ = false
   711 fun is_record_update thy (s, T) =
   712   String.isSuffix Record.updateN s andalso
   713   exists (curry (op =) (unsuffix Record.updateN s) o fst)
   714          (all_record_fields thy (body_type T))
   715   handle TYPE _ => false
   716 fun is_abs_fun ctxt (s, Type (@{type_name fun}, [_, Type (s', _)])) =
   717     (case typedef_info ctxt s' of
   718        SOME {Abs_name, ...} => s = Abs_name
   719      | NONE => false)
   720   | is_abs_fun _ _ = false
   721 fun is_rep_fun ctxt (s, Type (@{type_name fun}, [Type (s', _), _])) =
   722     (case typedef_info ctxt s' of
   723        SOME {Rep_name, ...} => s = Rep_name
   724      | NONE => false)
   725   | is_rep_fun _ _ = false
   726 fun is_quot_abs_fun ctxt (x as (_, Type (@{type_name fun},
   727                                          [_, abs_T as Type (s', _)]))) =
   728     try (Quotient_Term.absrep_const_chk ctxt Quotient_Term.AbsF) s'
   729     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   730   | is_quot_abs_fun _ _ = false
   731 fun is_quot_rep_fun ctxt (x as (_, Type (@{type_name fun},
   732                                          [abs_T as Type (s', _), _]))) =
   733     try (Quotient_Term.absrep_const_chk ctxt Quotient_Term.RepF) s'
   734     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   735   | is_quot_rep_fun _ _ = false
   736 
   737 fun mate_of_rep_fun ctxt (x as (_, Type (@{type_name fun},
   738                                          [T1 as Type (s', _), T2]))) =
   739     (case typedef_info ctxt s' of
   740        SOME {Abs_name, ...} => (Abs_name, Type (@{type_name fun}, [T2, T1]))
   741      | NONE => raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x]))
   742   | mate_of_rep_fun _ x = raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x])
   743 fun rep_type_for_quot_type ctxt (T as Type (s, _)) =
   744     let
   745       val thy = Proof_Context.theory_of ctxt
   746       val {qtyp, rtyp, ...} = the (Quotient_Info.lookup_quotients ctxt s)
   747     in
   748       instantiate_type thy qtyp T rtyp
   749     end
   750   | rep_type_for_quot_type _ T =
   751     raise TYPE ("Nitpick_HOL.rep_type_for_quot_type", [T], [])
   752 fun equiv_relation_for_quot_type thy (Type (s, Ts)) =
   753     let
   754       val {qtyp, equiv_rel, equiv_thm, ...} = the (Quotient_Info.lookup_quotients thy s)
   755       val partial =
   756         case prop_of equiv_thm of
   757           @{const Trueprop} $ (Const (@{const_name equivp}, _) $ _) => false
   758         | @{const Trueprop} $ (Const (@{const_name part_equivp}, _) $ _) => true
   759         | _ => raise NOT_SUPPORTED "Ill-formed quotient type equivalence \
   760                                    \relation theorem"
   761       val Ts' = qtyp |> dest_Type |> snd
   762     in (subst_atomic_types (Ts' ~~ Ts) equiv_rel, partial) end
   763   | equiv_relation_for_quot_type _ T =
   764     raise TYPE ("Nitpick_HOL.equiv_relation_for_quot_type", [T], [])
   765 
   766 fun is_coconstr ctxt (s, T) =
   767   case body_type T of
   768     co_T as Type (co_s, _) =>
   769     let val {codatatypes, ...} = Data.get (Context.Proof ctxt) in
   770       exists (fn (s', T') => s = s' andalso repair_constr_type ctxt co_T T' = T)
   771              (AList.lookup (op =) codatatypes co_s |> Option.map snd |> these)
   772     end
   773   | _ => false
   774 fun is_constr_like ctxt (s, T) =
   775   member (op =) [@{const_name FunBox}, @{const_name PairBox},
   776                  @{const_name Quot}, @{const_name Zero_Rep},
   777                  @{const_name Suc_Rep}] s orelse
   778   let
   779     val thy = Proof_Context.theory_of ctxt
   780     val (x as (_, T)) = (s, unarize_unbox_etc_type T)
   781   in
   782     is_real_constr thy x orelse is_record_constr x orelse
   783     (is_abs_fun ctxt x andalso is_pure_typedef ctxt (range_type T)) orelse
   784     is_coconstr ctxt x
   785   end
   786 fun is_stale_constr ctxt (x as (_, T)) =
   787   is_codatatype ctxt (body_type T) andalso is_constr_like ctxt x andalso
   788   not (is_coconstr ctxt x)
   789 fun is_constr ctxt stds (x as (_, T)) =
   790   let val thy = Proof_Context.theory_of ctxt in
   791     is_constr_like ctxt x andalso
   792     not (is_basic_datatype thy stds
   793                          (fst (dest_Type (unarize_type (body_type T))))) andalso
   794     not (is_stale_constr ctxt x)
   795   end
   796 val is_sel = String.isPrefix discr_prefix orf String.isPrefix sel_prefix
   797 val is_sel_like_and_no_discr =
   798   String.isPrefix sel_prefix orf
   799   (member (op =) [@{const_name fst}, @{const_name snd}])
   800 
   801 fun in_fun_lhs_for InConstr = InSel
   802   | in_fun_lhs_for _ = InFunLHS
   803 fun in_fun_rhs_for InConstr = InConstr
   804   | in_fun_rhs_for InSel = InSel
   805   | in_fun_rhs_for InFunRHS1 = InFunRHS2
   806   | in_fun_rhs_for _ = InFunRHS1
   807 
   808 fun is_boxing_worth_it (hol_ctxt : hol_context) boxy T =
   809   case T of
   810     Type (@{type_name fun}, _) =>
   811     (boxy = InPair orelse boxy = InFunLHS) andalso
   812     not (is_boolean_type (body_type T))
   813   | Type (@{type_name prod}, Ts) =>
   814     boxy = InPair orelse boxy = InFunRHS1 orelse boxy = InFunRHS2 orelse
   815     ((boxy = InExpr orelse boxy = InFunLHS) andalso
   816      exists (is_boxing_worth_it hol_ctxt InPair)
   817             (map (box_type hol_ctxt InPair) Ts))
   818   | _ => false
   819 and should_box_type (hol_ctxt as {thy, boxes, ...}) boxy z =
   820   case triple_lookup (type_match thy) boxes (Type z) of
   821     SOME (SOME box_me) => box_me
   822   | _ => is_boxing_worth_it hol_ctxt boxy (Type z)
   823 and box_type hol_ctxt boxy T =
   824   case T of
   825     Type (z as (@{type_name fun}, [T1, T2])) =>
   826     if boxy <> InConstr andalso boxy <> InSel andalso
   827        should_box_type hol_ctxt boxy z then
   828       Type (@{type_name fun_box},
   829             [box_type hol_ctxt InFunLHS T1, box_type hol_ctxt InFunRHS1 T2])
   830     else
   831       box_type hol_ctxt (in_fun_lhs_for boxy) T1
   832       --> box_type hol_ctxt (in_fun_rhs_for boxy) T2
   833   | Type (z as (@{type_name prod}, Ts)) =>
   834     if boxy <> InConstr andalso boxy <> InSel
   835        andalso should_box_type hol_ctxt boxy z then
   836       Type (@{type_name pair_box}, map (box_type hol_ctxt InSel) Ts)
   837     else
   838       Type (@{type_name prod},
   839             map (box_type hol_ctxt
   840                           (if boxy = InConstr orelse boxy = InSel then boxy
   841                            else InPair)) Ts)
   842   | _ => T
   843 
   844 fun binarize_nat_and_int_in_type @{typ nat} = @{typ "unsigned_bit word"}
   845   | binarize_nat_and_int_in_type @{typ int} = @{typ "signed_bit word"}
   846   | binarize_nat_and_int_in_type (Type (s, Ts)) =
   847     Type (s, map binarize_nat_and_int_in_type Ts)
   848   | binarize_nat_and_int_in_type T = T
   849 val binarize_nat_and_int_in_term = map_types binarize_nat_and_int_in_type
   850 
   851 fun discr_for_constr (s, T) = (discr_prefix ^ s, body_type T --> bool_T)
   852 
   853 fun num_sels_for_constr_type T = length (maybe_curried_binder_types T)
   854 fun nth_sel_name_for_constr_name s n =
   855   if s = @{const_name Pair} then
   856     if n = 0 then @{const_name fst} else @{const_name snd}
   857   else
   858     sel_prefix_for n ^ s
   859 fun nth_sel_for_constr x ~1 = discr_for_constr x
   860   | nth_sel_for_constr (s, T) n =
   861     (nth_sel_name_for_constr_name s n,
   862      body_type T --> nth (maybe_curried_binder_types T) n)
   863 fun binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize =
   864   apsnd ((binarize ? binarize_nat_and_int_in_type) o box_type hol_ctxt InSel)
   865   oo nth_sel_for_constr
   866 
   867 fun sel_no_from_name s =
   868   if String.isPrefix discr_prefix s then
   869     ~1
   870   else if String.isPrefix sel_prefix s then
   871     s |> unprefix sel_prefix |> Int.fromString |> the
   872   else if s = @{const_name snd} then
   873     1
   874   else
   875     0
   876 
   877 val close_form =
   878   let
   879     fun close_up zs zs' =
   880       fold (fn (z as ((s, _), T)) => fn t' =>
   881                Term.all T $ Abs (s, T, abstract_over (Var z, t')))
   882            (take (length zs' - length zs) zs')
   883     fun aux zs (@{const "==>"} $ t1 $ t2) =
   884         let val zs' = Term.add_vars t1 zs in
   885           close_up zs zs' (Logic.mk_implies (t1, aux zs' t2))
   886         end
   887       | aux zs t = close_up zs (Term.add_vars t zs) t
   888   in aux [] end
   889 
   890 fun distinctness_formula T =
   891   all_distinct_unordered_pairs_of
   892   #> map (fn (t1, t2) => @{const Not} $ (HOLogic.eq_const T $ t1 $ t2))
   893   #> List.foldr (s_conj o swap) @{const True}
   894 
   895 fun zero_const T = Const (@{const_name zero_class.zero}, T)
   896 fun suc_const T = Const (@{const_name Suc}, T --> T)
   897 
   898 fun uncached_datatype_constrs ({thy, ctxt, stds, ...} : hol_context)
   899                               (T as Type (s, Ts)) =
   900     (case AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   901                        s of
   902        SOME (_, xs' as (_ :: _)) => map (apsnd (repair_constr_type ctxt T)) xs'
   903      | _ =>
   904        if is_datatype ctxt stds T then
   905          case Datatype.get_info thy s of
   906            SOME {index, descr, ...} =>
   907            let
   908              val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the
   909            in
   910              map (apsnd (fn Us =>
   911                             map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))
   912                  constrs
   913            end
   914          | NONE =>
   915            if is_record_type T then
   916              let
   917                val s' = unsuffix Record.ext_typeN s ^ Record.extN
   918                val T' = (Record.get_extT_fields thy T
   919                         |> apsnd single |> uncurry append |> map snd) ---> T
   920              in [(s', T')] end
   921            else if is_real_quot_type ctxt T then
   922              [(@{const_name Quot}, rep_type_for_quot_type ctxt T --> T)]
   923            else case typedef_info ctxt s of
   924              SOME {abs_type, rep_type, Abs_name, ...} =>
   925              [(Abs_name,
   926                varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
   927            | NONE =>
   928              if T = @{typ ind} then
   929                [dest_Const @{const Zero_Rep}, dest_Const @{const Suc_Rep}]
   930              else
   931                []
   932        else
   933          [])
   934   | uncached_datatype_constrs _ _ = []
   935 fun datatype_constrs (hol_ctxt as {constr_cache, ...}) T =
   936   case AList.lookup (op =) (!constr_cache) T of
   937     SOME xs => xs
   938   | NONE =>
   939     let val xs = uncached_datatype_constrs hol_ctxt T in
   940       (Unsynchronized.change constr_cache (cons (T, xs)); xs)
   941     end
   942 fun binarized_and_boxed_datatype_constrs hol_ctxt binarize =
   943   map (apsnd ((binarize ? binarize_nat_and_int_in_type)
   944               o box_type hol_ctxt InConstr)) o datatype_constrs hol_ctxt
   945 val num_datatype_constrs = length oo datatype_constrs
   946 
   947 fun constr_name_for_sel_like @{const_name fst} = @{const_name Pair}
   948   | constr_name_for_sel_like @{const_name snd} = @{const_name Pair}
   949   | constr_name_for_sel_like s' = original_name s'
   950 fun binarized_and_boxed_constr_for_sel hol_ctxt binarize (s', T') =
   951   let val s = constr_name_for_sel_like s' in
   952     AList.lookup (op =)
   953         (binarized_and_boxed_datatype_constrs hol_ctxt binarize (domain_type T'))
   954         s
   955     |> the |> pair s
   956   end
   957 
   958 fun card_of_type assigns (Type (@{type_name fun}, [T1, T2])) =
   959     reasonable_power (card_of_type assigns T2) (card_of_type assigns T1)
   960   | card_of_type assigns (Type (@{type_name prod}, [T1, T2])) =
   961     card_of_type assigns T1 * card_of_type assigns T2
   962   | card_of_type _ (Type (@{type_name itself}, _)) = 1
   963   | card_of_type _ @{typ prop} = 2
   964   | card_of_type _ @{typ bool} = 2
   965   | card_of_type assigns T =
   966     case AList.lookup (op =) assigns T of
   967       SOME k => k
   968     | NONE => if T = @{typ bisim_iterator} then 0
   969               else raise TYPE ("Nitpick_HOL.card_of_type", [T], [])
   970 
   971 fun bounded_card_of_type max default_card assigns
   972                          (Type (@{type_name fun}, [T1, T2])) =
   973     let
   974       val k1 = bounded_card_of_type max default_card assigns T1
   975       val k2 = bounded_card_of_type max default_card assigns T2
   976     in
   977       if k1 = max orelse k2 = max then max
   978       else Int.min (max, reasonable_power k2 k1)
   979     end
   980   | bounded_card_of_type max default_card assigns
   981                          (Type (@{type_name prod}, [T1, T2])) =
   982     let
   983       val k1 = bounded_card_of_type max default_card assigns T1
   984       val k2 = bounded_card_of_type max default_card assigns T2
   985     in if k1 = max orelse k2 = max then max else Int.min (max, k1 * k2) end
   986   | bounded_card_of_type max default_card assigns T =
   987     Int.min (max, if default_card = ~1 then
   988                     card_of_type assigns T
   989                   else
   990                     card_of_type assigns T
   991                     handle TYPE ("Nitpick_HOL.card_of_type", _, _) =>
   992                            default_card)
   993 
   994 (* Similar to "ATP_Translate.tiny_card_of_type". *)
   995 fun bounded_exact_card_of_type hol_ctxt finitizable_dataTs max default_card
   996                                assigns T =
   997   let
   998     fun aux avoid T =
   999       (if member (op =) avoid T then
  1000          0
  1001        else if member (op =) finitizable_dataTs T then
  1002          raise SAME ()
  1003        else case T of
  1004          Type (@{type_name fun}, [T1, T2]) =>
  1005          (case (aux avoid T1, aux avoid T2) of
  1006             (_, 1) => 1
  1007           | (0, _) => 0
  1008           | (_, 0) => 0
  1009           | (k1, k2) =>
  1010             if k1 >= max orelse k2 >= max then max
  1011             else Int.min (max, reasonable_power k2 k1))
  1012        | Type (@{type_name prod}, [T1, T2]) =>
  1013          (case (aux avoid T1, aux avoid T2) of
  1014             (0, _) => 0
  1015           | (_, 0) => 0
  1016           | (k1, k2) =>
  1017             if k1 >= max orelse k2 >= max then max
  1018             else Int.min (max, k1 * k2))
  1019        | Type (@{type_name itself}, _) => 1
  1020        | @{typ prop} => 2
  1021        | @{typ bool} => 2
  1022        | Type _ =>
  1023          (case datatype_constrs hol_ctxt T of
  1024             [] => if is_integer_type T orelse is_bit_type T then 0
  1025                   else raise SAME ()
  1026           | constrs =>
  1027             let
  1028               val constr_cards =
  1029                 map (Integer.prod o map (aux (T :: avoid)) o binder_types o snd)
  1030                     constrs
  1031             in
  1032               if exists (curry (op =) 0) constr_cards then 0
  1033               else Int.min (max, Integer.sum constr_cards)
  1034             end)
  1035        | _ => raise SAME ())
  1036       handle SAME () =>
  1037              AList.lookup (op =) assigns T |> the_default default_card
  1038   in Int.min (max, aux [] T) end
  1039 
  1040 val typical_atomic_card = 4
  1041 val typical_card_of_type = bounded_card_of_type 16777217 typical_atomic_card []
  1042 
  1043 fun is_finite_type hol_ctxt T =
  1044   bounded_exact_card_of_type hol_ctxt [] 1 2 [] T > 0
  1045 
  1046 fun is_special_eligible_arg strict Ts t =
  1047   case map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t) of
  1048     [] => true
  1049   | bad_Ts =>
  1050     let
  1051       val bad_Ts_cost =
  1052         if strict then fold (curry (op *) o typical_card_of_type) bad_Ts 1
  1053         else fold (Integer.max o typical_card_of_type) bad_Ts 0
  1054       val T_cost = typical_card_of_type (fastype_of1 (Ts, t))
  1055     in (bad_Ts_cost, T_cost) |> (if strict then op < else op <=) end
  1056 
  1057 fun abs_var ((s, j), T) body = Abs (s, T, abstract_over (Var ((s, j), T), body))
  1058 
  1059 fun let_var s = (nitpick_prefix ^ s, 999)
  1060 val let_inline_threshold = 20
  1061 
  1062 fun s_let Ts s n abs_T body_T f t =
  1063   if (n - 1) * (size_of_term t - 1) <= let_inline_threshold orelse
  1064      is_special_eligible_arg false Ts t then
  1065     f t
  1066   else
  1067     let val z = (let_var s, abs_T) in
  1068       Const (@{const_name Let}, abs_T --> (abs_T --> body_T) --> body_T)
  1069       $ t $ abs_var z (incr_boundvars 1 (f (Var z)))
  1070     end
  1071 
  1072 fun loose_bvar1_count (Bound i, k) = if i = k then 1 else 0
  1073   | loose_bvar1_count (t1 $ t2, k) =
  1074     loose_bvar1_count (t1, k) + loose_bvar1_count (t2, k)
  1075   | loose_bvar1_count (Abs (_, _, t), k) = loose_bvar1_count (t, k + 1)
  1076   | loose_bvar1_count _ = 0
  1077 
  1078 fun s_betapply _ (t1 as Const (@{const_name "=="}, _) $ t1', t2) =
  1079     if t1' aconv t2 then @{prop True} else t1 $ t2
  1080   | s_betapply _ (t1 as Const (@{const_name HOL.eq}, _) $ t1', t2) =
  1081     if t1' aconv t2 then @{term True} else t1 $ t2
  1082   | s_betapply _ (Const (@{const_name If}, _) $ @{const True} $ t1', _) = t1'
  1083   | s_betapply _ (Const (@{const_name If}, _) $ @{const False} $ _, t2) = t2
  1084   | s_betapply Ts (Const (@{const_name Let},
  1085                           Type (_, [bound_T, Type (_, [_, body_T])]))
  1086                    $ t12 $ Abs (s, T, t13'), t2) =
  1087     let val body_T' = range_type body_T in
  1088       Const (@{const_name Let}, bound_T --> (bound_T --> body_T') --> body_T')
  1089       $ t12 $ Abs (s, T, s_betapply (T :: Ts) (t13', incr_boundvars 1 t2))
  1090     end
  1091   | s_betapply Ts (t1 as Abs (s1, T1, t1'), t2) =
  1092     (s_let Ts s1 (loose_bvar1_count (t1', 0)) T1 (fastype_of1 (T1 :: Ts, t1'))
  1093            (curry betapply t1) t2
  1094      (* FIXME: fix all "s_betapply []" calls *)
  1095      handle TERM _ => betapply (t1, t2)
  1096           | General.Subscript => betapply (t1, t2))
  1097   | s_betapply _ (t1, t2) = t1 $ t2
  1098 fun s_betapplys Ts = Library.foldl (s_betapply Ts)
  1099 
  1100 fun s_beta_norm Ts t =
  1101   let
  1102     fun aux _ (Var _) = raise Same.SAME
  1103       | aux Ts (Abs (s, T, t')) = Abs (s, T, aux (T :: Ts) t')
  1104       | aux Ts ((t1 as Abs _) $ t2) =
  1105         Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1106       | aux Ts (t1 $ t2) =
  1107         ((case aux Ts t1 of
  1108            t1 as Abs _ => Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1109          | t1 => t1 $ Same.commit (aux Ts) t2)
  1110         handle Same.SAME => t1 $ aux Ts t2)
  1111       | aux _ _ = raise Same.SAME
  1112   in aux Ts t handle Same.SAME => t end
  1113 
  1114 fun discr_term_for_constr hol_ctxt (x as (s, T)) =
  1115   let val dataT = body_type T in
  1116     if s = @{const_name Suc} then
  1117       Abs (Name.uu, dataT,
  1118            @{const Not} $ HOLogic.mk_eq (zero_const dataT, Bound 0))
  1119     else if num_datatype_constrs hol_ctxt dataT >= 2 then
  1120       Const (discr_for_constr x)
  1121     else
  1122       Abs (Name.uu, dataT, @{const True})
  1123   end
  1124 fun discriminate_value (hol_ctxt as {ctxt, ...}) x t =
  1125   case head_of t of
  1126     Const x' =>
  1127     if x = x' then @{const True}
  1128     else if is_constr_like ctxt x' then @{const False}
  1129     else s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1130   | _ => s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1131 
  1132 fun nth_arg_sel_term_for_constr thy stds (x as (s, T)) n =
  1133   let val (arg_Ts, dataT) = strip_type T in
  1134     if dataT = nat_T andalso is_standard_datatype thy stds nat_T then
  1135       @{term "%n::nat. n - 1"}
  1136     else if is_pair_type dataT then
  1137       Const (nth_sel_for_constr x n)
  1138     else
  1139       let
  1140         fun aux m (Type (@{type_name prod}, [T1, T2])) =
  1141             let
  1142               val (m, t1) = aux m T1
  1143               val (m, t2) = aux m T2
  1144             in (m, HOLogic.mk_prod (t1, t2)) end
  1145           | aux m T =
  1146             (m + 1, Const (nth_sel_name_for_constr_name s m, dataT --> T)
  1147                     $ Bound 0)
  1148         val m = fold (Integer.add o num_factors_in_type)
  1149                      (List.take (arg_Ts, n)) 0
  1150       in Abs ("x", dataT, aux m (nth arg_Ts n) |> snd) end
  1151   end
  1152 fun select_nth_constr_arg ctxt stds x t n res_T =
  1153   let val thy = Proof_Context.theory_of ctxt in
  1154     (case strip_comb t of
  1155        (Const x', args) =>
  1156        if x = x' then nth args n
  1157        else if is_constr_like ctxt x' then Const (@{const_name unknown}, res_T)
  1158        else raise SAME ()
  1159      | _ => raise SAME())
  1160     handle SAME () =>
  1161            s_betapply [] (nth_arg_sel_term_for_constr thy stds x n, t)
  1162   end
  1163 
  1164 fun construct_value _ _ x [] = Const x
  1165   | construct_value ctxt stds (x as (s, _)) args =
  1166     let val args = map Envir.eta_contract args in
  1167       case hd args of
  1168         Const (s', _) $ t =>
  1169         if is_sel_like_and_no_discr s' andalso
  1170            constr_name_for_sel_like s' = s andalso
  1171            forall (fn (n, t') =>
  1172                       select_nth_constr_arg ctxt stds x t n dummyT = t')
  1173                   (index_seq 0 (length args) ~~ args) then
  1174           t
  1175         else
  1176           list_comb (Const x, args)
  1177       | _ => list_comb (Const x, args)
  1178     end
  1179 
  1180 fun constr_expand (hol_ctxt as {ctxt, stds, ...}) T t =
  1181   (case head_of t of
  1182      Const x => if is_constr_like ctxt x then t else raise SAME ()
  1183    | _ => raise SAME ())
  1184   handle SAME () =>
  1185          let
  1186            val x' as (_, T') =
  1187              if is_pair_type T then
  1188                let val (T1, T2) = HOLogic.dest_prodT T in
  1189                  (@{const_name Pair}, T1 --> T2 --> T)
  1190                end
  1191              else
  1192                datatype_constrs hol_ctxt T |> hd
  1193            val arg_Ts = binder_types T'
  1194          in
  1195            list_comb (Const x', map2 (select_nth_constr_arg ctxt stds x' t)
  1196                                      (index_seq 0 (length arg_Ts)) arg_Ts)
  1197          end
  1198 
  1199 fun coerce_bound_no f j t =
  1200   case t of
  1201     t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2
  1202   | Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')
  1203   | Bound j' => if j' = j then f t else t
  1204   | _ => t
  1205 fun coerce_bound_0_in_term hol_ctxt new_T old_T =
  1206   old_T <> new_T ? coerce_bound_no (coerce_term hol_ctxt [new_T] old_T new_T) 0
  1207 and coerce_term (hol_ctxt as {ctxt, stds, ...}) Ts new_T old_T t =
  1208   if old_T = new_T then
  1209     t
  1210   else
  1211     case (new_T, old_T) of
  1212       (Type (new_s, new_Ts as [new_T1, new_T2]),
  1213        Type (@{type_name fun}, [old_T1, old_T2])) =>
  1214       (case eta_expand Ts t 1 of
  1215          Abs (s, _, t') =>
  1216          Abs (s, new_T1,
  1217               t' |> coerce_bound_0_in_term hol_ctxt new_T1 old_T1
  1218                  |> coerce_term hol_ctxt (new_T1 :: Ts) new_T2 old_T2)
  1219          |> Envir.eta_contract
  1220          |> new_s <> @{type_name fun}
  1221             ? construct_value ctxt stds
  1222                   (@{const_name FunBox},
  1223                    Type (@{type_name fun}, new_Ts) --> new_T)
  1224               o single
  1225        | t' => raise TERM ("Nitpick_HOL.coerce_term", [t']))
  1226     | (Type (new_s, new_Ts as [new_T1, new_T2]),
  1227        Type (old_s, old_Ts as [old_T1, old_T2])) =>
  1228       if old_s = @{type_name fun_box} orelse
  1229          old_s = @{type_name pair_box} orelse old_s = @{type_name prod} then
  1230         case constr_expand hol_ctxt old_T t of
  1231           Const (old_s, _) $ t1 =>
  1232           if new_s = @{type_name fun} then
  1233             coerce_term hol_ctxt Ts new_T (Type (@{type_name fun}, old_Ts)) t1
  1234           else
  1235             construct_value ctxt stds
  1236                 (old_s, Type (@{type_name fun}, new_Ts) --> new_T)
  1237                 [coerce_term hol_ctxt Ts (Type (@{type_name fun}, new_Ts))
  1238                              (Type (@{type_name fun}, old_Ts)) t1]
  1239         | Const _ $ t1 $ t2 =>
  1240           construct_value ctxt stds
  1241               (if new_s = @{type_name prod} then @{const_name Pair}
  1242                else @{const_name PairBox}, new_Ts ---> new_T)
  1243               (map3 (coerce_term hol_ctxt Ts) [new_T1, new_T2] [old_T1, old_T2]
  1244                     [t1, t2])
  1245         | t' => raise TERM ("Nitpick_HOL.coerce_term", [t'])
  1246       else
  1247         raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1248     | _ => raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1249 
  1250 fun is_ground_term (t1 $ t2) = is_ground_term t1 andalso is_ground_term t2
  1251   | is_ground_term (Const _) = true
  1252   | is_ground_term _ = false
  1253 
  1254 fun special_bounds ts =
  1255   fold Term.add_vars ts [] |> sort (Term_Ord.fast_indexname_ord o pairself fst)
  1256 
  1257 (* FIXME: detect "rep_datatype"? *)
  1258 fun is_funky_typedef_name ctxt s =
  1259   member (op =) [@{type_name unit}, @{type_name prod},
  1260                  @{type_name Sum_Type.sum}, @{type_name int}] s orelse
  1261   is_frac_type ctxt (Type (s, []))
  1262 fun is_funky_typedef ctxt (Type (s, _)) = is_funky_typedef_name ctxt s
  1263   | is_funky_typedef _ _ = false
  1264 fun is_typedef_axiom ctxt boring (@{const "==>"} $ _ $ t2) =
  1265     is_typedef_axiom ctxt boring t2
  1266   | is_typedef_axiom ctxt boring
  1267         (@{const Trueprop} $ (Const (@{const_name Typedef.type_definition}, _)
  1268          $ Const (_, Type (@{type_name fun}, [Type (s, _), _]))
  1269          $ Const _ $ _)) =
  1270     boring <> is_funky_typedef_name ctxt s andalso is_typedef ctxt s
  1271   | is_typedef_axiom _ _ _ = false
  1272 
  1273 fun all_defs_of thy subst =
  1274   let
  1275     val def_names =
  1276       thy |> Theory.defs_of
  1277           |> Defs.all_specifications_of
  1278           |> maps snd |> map_filter #def
  1279           |> Ord_List.make fast_string_ord
  1280   in
  1281     Theory.nodes_of thy
  1282     |> maps Thm.axioms_of
  1283     |> map (apsnd (subst_atomic subst o prop_of))
  1284     |> sort (fast_string_ord o pairself fst)
  1285     |> Ord_List.inter (fast_string_ord o apsnd fst) def_names
  1286     |> map snd
  1287   end
  1288 
  1289 (* Ideally we would check against "Complex_Main", not "Refute", but any theory
  1290    will do as long as it contains all the "axioms" and "axiomatization"
  1291    commands. *)
  1292 fun is_built_in_theory thy = Theory.subthy (thy, @{theory Refute})
  1293 
  1294 fun all_nondefs_of ctxt subst =
  1295   ctxt |> Spec_Rules.get
  1296        |> filter (curry (op =) Spec_Rules.Unknown o fst)
  1297        |> maps (snd o snd)
  1298        |> filter_out (is_built_in_theory o theory_of_thm)
  1299        |> map (subst_atomic subst o prop_of)
  1300 
  1301 fun arity_of_built_in_const thy stds (s, T) =
  1302   if s = @{const_name If} then
  1303     if nth_range_type 3 T = @{typ bool} then NONE else SOME 3
  1304   else
  1305     let val std_nats = is_standard_datatype thy stds nat_T in
  1306       case AList.lookup (op =)
  1307                     (built_in_consts
  1308                      |> std_nats ? append built_in_nat_consts) s of
  1309         SOME n => SOME n
  1310       | NONE =>
  1311         case AList.lookup (op =)
  1312                  (built_in_typed_consts
  1313                   |> std_nats ? append built_in_typed_nat_consts)
  1314                  (s, unarize_type T) of
  1315           SOME n => SOME n
  1316         | NONE =>
  1317           case s of
  1318             @{const_name zero_class.zero} =>
  1319             if is_iterator_type T then SOME 0 else NONE
  1320           | @{const_name Suc} =>
  1321             if is_iterator_type (domain_type T) then SOME 0 else NONE
  1322           | _ => if is_fun_type T andalso is_set_like_type (domain_type T) then
  1323                    AList.lookup (op =) built_in_set_like_consts s
  1324                  else
  1325                    NONE
  1326     end
  1327 val is_built_in_const = is_some ooo arity_of_built_in_const
  1328 
  1329 (* This function is designed to work for both real definition axioms and
  1330    simplification rules (equational specifications). *)
  1331 fun term_under_def t =
  1332   case t of
  1333     @{const "==>"} $ _ $ t2 => term_under_def t2
  1334   | Const (@{const_name "=="}, _) $ t1 $ _ => term_under_def t1
  1335   | @{const Trueprop} $ t1 => term_under_def t1
  1336   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => term_under_def t1
  1337   | Abs (_, _, t') => term_under_def t'
  1338   | t1 $ _ => term_under_def t1
  1339   | _ => t
  1340 
  1341 (* Here we crucially rely on "specialize_type" performing a preorder traversal
  1342    of the term, without which the wrong occurrence of a constant could be
  1343    matched in the face of overloading. *)
  1344 fun def_props_for_const thy stds table (x as (s, _)) =
  1345   if is_built_in_const thy stds x then
  1346     []
  1347   else
  1348     these (Symtab.lookup table s)
  1349     |> map_filter (try (specialize_type thy x))
  1350     |> filter (curry (op =) (Const x) o term_under_def)
  1351 
  1352 fun normalized_rhs_of t =
  1353   let
  1354     fun aux (v as Var _) (SOME t) = SOME (lambda v t)
  1355       | aux (c as Const (@{const_name TYPE}, _)) (SOME t) = SOME (lambda c t)
  1356       | aux _ _ = NONE
  1357     val (lhs, rhs) =
  1358       case t of
  1359         Const (@{const_name "=="}, _) $ t1 $ t2 => (t1, t2)
  1360       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ t2) =>
  1361         (t1, t2)
  1362       | _ => raise TERM ("Nitpick_HOL.normalized_rhs_of", [t])
  1363     val args = strip_comb lhs |> snd
  1364   in fold_rev aux args (SOME rhs) end
  1365 
  1366 fun get_def_of_const thy table (x as (s, _)) =
  1367   x |> def_props_for_const thy [(NONE, false)] table |> List.last
  1368     |> normalized_rhs_of |> Option.map (prefix_abs_vars s)
  1369   handle List.Empty => NONE
  1370 
  1371 fun def_of_const_ext thy (unfold_table, fallback_table) (x as (s, _)) =
  1372   if is_built_in_const thy [(NONE, false)] x orelse original_name s <> s then
  1373     NONE
  1374   else case get_def_of_const thy unfold_table x of
  1375     SOME def => SOME (true, def)
  1376   | NONE => get_def_of_const thy fallback_table x |> Option.map (pair false)
  1377 
  1378 val def_of_const = Option.map snd ooo def_of_const_ext
  1379 
  1380 fun fixpoint_kind_of_rhs (Abs (_, _, t)) = fixpoint_kind_of_rhs t
  1381   | fixpoint_kind_of_rhs (Const (@{const_name lfp}, _) $ Abs _) = Lfp
  1382   | fixpoint_kind_of_rhs (Const (@{const_name gfp}, _) $ Abs _) = Gfp
  1383   | fixpoint_kind_of_rhs _ = NoFp
  1384 
  1385 fun is_mutually_inductive_pred_def thy table t =
  1386   let
  1387     fun is_good_arg (Bound _) = true
  1388       | is_good_arg (Const (s, _)) =
  1389         s = @{const_name True} orelse s = @{const_name False} orelse
  1390         s = @{const_name undefined}
  1391       | is_good_arg _ = false
  1392   in
  1393     case t |> strip_abs_body |> strip_comb of
  1394       (Const x, ts as (_ :: _)) =>
  1395       (case def_of_const thy table x of
  1396          SOME t' => fixpoint_kind_of_rhs t' <> NoFp andalso
  1397                     forall is_good_arg ts
  1398        | NONE => false)
  1399     | _ => false
  1400   end
  1401 fun unfold_mutually_inductive_preds thy table =
  1402   map_aterms (fn t as Const x =>
  1403                  (case def_of_const thy table x of
  1404                     SOME t' =>
  1405                     let val t' = Envir.eta_contract t' in
  1406                       if is_mutually_inductive_pred_def thy table t' then t'
  1407                       else t
  1408                     end
  1409                  | NONE => t)
  1410                | t => t)
  1411 
  1412 fun case_const_names ctxt stds =
  1413   let val thy = Proof_Context.theory_of ctxt in
  1414     Symtab.fold (fn (dtype_s, {index, descr, case_name, ...}) =>
  1415                     if is_basic_datatype thy stds dtype_s then
  1416                       I
  1417                     else
  1418                       cons (case_name, AList.lookup (op =) descr index
  1419                                        |> the |> #3 |> length))
  1420                 (Datatype.get_all thy) [] @
  1421     map (apsnd length o snd) (#codatatypes (Data.get (Context.Proof ctxt)))
  1422   end
  1423 
  1424 fun fixpoint_kind_of_const thy table x =
  1425   if is_built_in_const thy [(NONE, false)] x then NoFp
  1426   else fixpoint_kind_of_rhs (the (def_of_const thy table x))
  1427   handle Option.Option => NoFp
  1428 
  1429 fun is_real_inductive_pred ({thy, stds, def_tables, intro_table, ...}
  1430                             : hol_context) x =
  1431   fixpoint_kind_of_const thy def_tables x <> NoFp andalso
  1432   not (null (def_props_for_const thy stds intro_table x))
  1433 fun is_inductive_pred hol_ctxt (x as (s, _)) =
  1434   is_real_inductive_pred hol_ctxt x orelse String.isPrefix ubfp_prefix s orelse
  1435   String.isPrefix lbfp_prefix s
  1436 
  1437 fun lhs_of_equation t =
  1438   case t of
  1439     Const (@{const_name all}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1440   | Const (@{const_name "=="}, _) $ t1 $ _ => SOME t1
  1441   | @{const "==>"} $ _ $ t2 => lhs_of_equation t2
  1442   | @{const Trueprop} $ t1 => lhs_of_equation t1
  1443   | Const (@{const_name All}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1444   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => SOME t1
  1445   | @{const HOL.implies} $ _ $ t2 => lhs_of_equation t2
  1446   | _ => NONE
  1447 fun is_constr_pattern _ (Bound _) = true
  1448   | is_constr_pattern _ (Var _) = true
  1449   | is_constr_pattern ctxt t =
  1450     case strip_comb t of
  1451       (Const x, args) =>
  1452       is_constr_like ctxt x andalso forall (is_constr_pattern ctxt) args
  1453     | _ => false
  1454 fun is_constr_pattern_lhs ctxt t =
  1455   forall (is_constr_pattern ctxt) (snd (strip_comb t))
  1456 fun is_constr_pattern_formula ctxt t =
  1457   case lhs_of_equation t of
  1458     SOME t' => is_constr_pattern_lhs ctxt t'
  1459   | NONE => false
  1460 
  1461 (* Similar to "specialize_type" but returns all matches rather than only the
  1462    first (preorder) match. *)
  1463 fun multi_specialize_type thy slack (s, T) t =
  1464   let
  1465     fun aux (Const (s', T')) ys =
  1466         if s = s' then
  1467           ys |> (if AList.defined (op =) ys T' then
  1468                    I
  1469                  else
  1470                    cons (T', monomorphic_term (Sign.typ_match thy (T', T)
  1471                                                               Vartab.empty) t)
  1472                    handle Type.TYPE_MATCH => I
  1473                         | TERM _ =>
  1474                           if slack then
  1475                             I
  1476                           else
  1477                             raise NOT_SUPPORTED
  1478                                       ("too much polymorphism in axiom \"" ^
  1479                                        Syntax.string_of_term_global thy t ^
  1480                                        "\" involving " ^ quote s))
  1481         else
  1482           ys
  1483       | aux _ ys = ys
  1484   in map snd (fold_aterms aux t []) end
  1485 fun nondef_props_for_const thy slack table (x as (s, _)) =
  1486   these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)
  1487 
  1488 fun unvarify_term (t1 $ t2) = unvarify_term t1 $ unvarify_term t2
  1489   | unvarify_term (Var ((s, 0), T)) = Free (s, T)
  1490   | unvarify_term (Abs (s, T, t')) = Abs (s, T, unvarify_term t')
  1491   | unvarify_term t = t
  1492 fun axiom_for_choice_spec thy =
  1493   unvarify_term
  1494   #> Object_Logic.atomize_term thy
  1495   #> Choice_Specification.close_form
  1496   #> HOLogic.mk_Trueprop
  1497 fun is_choice_spec_fun ({thy, def_tables, nondef_table, choice_spec_table, ...}
  1498                         : hol_context) x =
  1499   case nondef_props_for_const thy true choice_spec_table x of
  1500     [] => false
  1501   | ts => case def_of_const thy def_tables x of
  1502             SOME (Const (@{const_name Eps}, _) $ _) => true
  1503           | SOME _ => false
  1504           | NONE =>
  1505             let val ts' = nondef_props_for_const thy true nondef_table x in
  1506               length ts' = length ts andalso
  1507               forall (fn t =>
  1508                          exists (curry (op aconv) (axiom_for_choice_spec thy t))
  1509                                 ts') ts
  1510             end
  1511 
  1512 fun is_choice_spec_axiom thy choice_spec_table t =
  1513   Symtab.exists (fn (_, ts) =>
  1514                     exists (curry (op aconv) t o axiom_for_choice_spec thy) ts)
  1515                 choice_spec_table
  1516 
  1517 fun is_real_equational_fun ({thy, stds, simp_table, psimp_table, ...}
  1518                             : hol_context) x =
  1519   exists (fn table => not (null (def_props_for_const thy stds table x)))
  1520          [!simp_table, psimp_table]
  1521 fun is_equational_fun_but_no_plain_def hol_ctxt =
  1522   is_real_equational_fun hol_ctxt orf is_inductive_pred hol_ctxt
  1523 
  1524 (** Constant unfolding **)
  1525 
  1526 fun constr_case_body ctxt stds Ts (func_t, (x as (_, T))) =
  1527   let val arg_Ts = binder_types T in
  1528     s_betapplys Ts (func_t, map2 (select_nth_constr_arg ctxt stds x (Bound 0))
  1529                                  (index_seq 0 (length arg_Ts)) arg_Ts)
  1530   end
  1531 fun add_constr_case res_T (body_t, guard_t) res_t =
  1532   if res_T = bool_T then
  1533     s_conj (HOLogic.mk_imp (guard_t, body_t), res_t)
  1534   else
  1535     Const (@{const_name If}, bool_T --> res_T --> res_T --> res_T)
  1536     $ guard_t $ body_t $ res_t
  1537 fun optimized_case_def (hol_ctxt as {ctxt, stds, ...}) Ts dataT res_T func_ts =
  1538   let
  1539     val xs = datatype_constrs hol_ctxt dataT
  1540     val cases =
  1541       func_ts ~~ xs
  1542       |> map (fn (func_t, x) =>
  1543                  (constr_case_body ctxt stds (dataT :: Ts)
  1544                                    (incr_boundvars 1 func_t, x),
  1545                   discriminate_value hol_ctxt x (Bound 0)))
  1546       |> AList.group (op aconv)
  1547       |> map (apsnd (List.foldl s_disj @{const False}))
  1548       |> sort (int_ord o pairself (size_of_term o snd))
  1549       |> rev
  1550   in
  1551     if res_T = bool_T then
  1552       if forall (member (op =) [@{const False}, @{const True}] o fst) cases then
  1553         case cases of
  1554           [(body_t, _)] => body_t
  1555         | [_, (@{const True}, head_t2)] => head_t2
  1556         | [_, (@{const False}, head_t2)] => @{const Not} $ head_t2
  1557         | _ => raise BAD ("Nitpick_HOL.optimized_case_def", "impossible cases")
  1558       else
  1559         @{const True} |> fold_rev (add_constr_case res_T) cases
  1560     else
  1561       fst (hd cases) |> fold_rev (add_constr_case res_T) (tl cases)
  1562   end
  1563   |> absdummy dataT
  1564 
  1565 fun optimized_record_get (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T res_T t =
  1566   let val constr_x = hd (datatype_constrs hol_ctxt rec_T) in
  1567     case no_of_record_field thy s rec_T of
  1568       ~1 => (case rec_T of
  1569                Type (_, Ts as _ :: _) =>
  1570                let
  1571                  val rec_T' = List.last Ts
  1572                  val j = num_record_fields thy rec_T - 1
  1573                in
  1574                  select_nth_constr_arg ctxt stds constr_x t j res_T
  1575                  |> optimized_record_get hol_ctxt s rec_T' res_T
  1576                end
  1577              | _ => raise TYPE ("Nitpick_HOL.optimized_record_get", [rec_T],
  1578                                 []))
  1579     | j => select_nth_constr_arg ctxt stds constr_x t j res_T
  1580   end
  1581 fun optimized_record_update (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T fun_t
  1582                             rec_t =
  1583   let
  1584     val constr_x as (_, constr_T) = hd (datatype_constrs hol_ctxt rec_T)
  1585     val Ts = binder_types constr_T
  1586     val n = length Ts
  1587     val special_j = no_of_record_field thy s rec_T
  1588     val ts =
  1589       map2 (fn j => fn T =>
  1590                let val t = select_nth_constr_arg ctxt stds constr_x rec_t j T in
  1591                  if j = special_j then
  1592                    s_betapply [] (fun_t, t)
  1593                  else if j = n - 1 andalso special_j = ~1 then
  1594                    optimized_record_update hol_ctxt s
  1595                        (rec_T |> dest_Type |> snd |> List.last) fun_t t
  1596                  else
  1597                    t
  1598                end) (index_seq 0 n) Ts
  1599   in list_comb (Const constr_x, ts) end
  1600 
  1601 (* Prevents divergence in case of cyclic or infinite definition dependencies. *)
  1602 val unfold_max_depth = 255
  1603 
  1604 (* Inline definitions or define as an equational constant? Booleans tend to
  1605    benefit more from inlining, due to the polarity analysis. (However, if
  1606    "total_consts" is set, the polarity analysis is likely not to be so
  1607    crucial.) *)
  1608 val def_inline_threshold_for_booleans = 60
  1609 val def_inline_threshold_for_non_booleans = 20
  1610 
  1611 fun unfold_defs_in_term
  1612         (hol_ctxt as {thy, ctxt, stds, whacks, total_consts, case_names,
  1613                       def_tables, ground_thm_table, ersatz_table, ...}) =
  1614   let
  1615     fun do_term depth Ts t =
  1616       case t of
  1617         (t0 as Const (@{const_name Int.number_class.number_of},
  1618                       Type (@{type_name fun}, [_, ran_T]))) $ t1 =>
  1619         ((if is_number_type ctxt ran_T then
  1620             let
  1621               val j = t1 |> HOLogic.dest_numeral
  1622                          |> ran_T = nat_T ? Integer.max 0
  1623               val s = numeral_prefix ^ signed_string_of_int j
  1624             in
  1625               if is_integer_like_type ran_T then
  1626                 if is_standard_datatype thy stds ran_T then
  1627                   Const (s, ran_T)
  1628                 else
  1629                   funpow j (curry (op $) (suc_const ran_T)) (zero_const ran_T)
  1630               else
  1631                 do_term depth Ts (Const (@{const_name of_int}, int_T --> ran_T)
  1632                                   $ Const (s, int_T))
  1633             end
  1634             handle TERM _ => raise SAME ()
  1635           else
  1636             raise SAME ())
  1637          handle SAME () =>
  1638                 s_betapply [] (do_term depth Ts t0, do_term depth Ts t1))
  1639       | Const (@{const_name refl_on}, T) $ Const (@{const_name top}, _) $ t2 =>
  1640         do_const depth Ts t (@{const_name refl'}, range_type T) [t2]
  1641       | (t0 as Const (@{const_name Sigma}, Type (_, [T1, Type (_, [T2, T3])])))
  1642         $ t1 $ (t2 as Abs (_, _, t2')) =>
  1643         if loose_bvar1 (t2', 0) then
  1644           s_betapplys Ts (do_term depth Ts t0, map (do_term depth Ts) [t1, t2])
  1645         else
  1646           do_term depth Ts
  1647                   (Const (@{const_name prod}, T1 --> range_type T2 --> T3)
  1648                    $ t1 $ incr_boundvars ~1 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 (@{const_name Let}, _) $ t1 $ t2 =>
  1662         s_betapply Ts (pairself (do_term depth Ts) (t2, t1))
  1663       | Const x => do_const depth Ts t x []
  1664       | t1 $ t2 =>
  1665         (case strip_comb t of
  1666            (Const x, ts) => do_const depth Ts t x ts
  1667          | _ => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2))
  1668       | Bound _ => t
  1669       | Abs (s, T, body) => Abs (s, T, do_term depth (T :: Ts) body)
  1670       | _ => if member (term_match thy) whacks t then
  1671                Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1672              else
  1673                t
  1674     and select_nth_constr_arg_with_args _ _ (x as (_, T)) [] n res_T =
  1675         (Abs (Name.uu, body_type T,
  1676               select_nth_constr_arg ctxt stds x (Bound 0) n res_T), [])
  1677       | select_nth_constr_arg_with_args depth Ts x (t :: ts) n res_T =
  1678         (select_nth_constr_arg ctxt stds x (do_term depth Ts t) n res_T, ts)
  1679     and quot_rep_of depth Ts abs_T rep_T ts =
  1680       select_nth_constr_arg_with_args depth Ts
  1681           (@{const_name Quot}, rep_T --> abs_T) ts 0 rep_T
  1682     and do_const depth Ts t (x as (s, T)) ts =
  1683       if member (term_match thy) whacks (Const x) then
  1684         Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1685       else case AList.lookup (op =) ersatz_table s of
  1686         SOME s' =>
  1687         do_const (depth + 1) Ts (list_comb (Const (s', T), ts)) (s', T) ts
  1688       | NONE =>
  1689         let
  1690           fun def_inline_threshold () =
  1691             if is_boolean_type (nth_range_type (length ts) T) andalso
  1692                total_consts <> SOME true then
  1693               def_inline_threshold_for_booleans
  1694             else
  1695               def_inline_threshold_for_non_booleans
  1696           val (const, ts) =
  1697             if is_built_in_const thy stds x then
  1698               (Const x, ts)
  1699             else case AList.lookup (op =) case_names s of
  1700               SOME n =>
  1701               if length ts < n then
  1702                 (do_term depth Ts (eta_expand Ts t (n - length ts)), [])
  1703               else
  1704                 let
  1705                   val (dataT, res_T) = nth_range_type n T
  1706                                        |> pairf domain_type range_type
  1707                 in
  1708                   (optimized_case_def hol_ctxt Ts dataT res_T
  1709                                       (map (do_term depth Ts) (take n ts)),
  1710                    drop n ts)
  1711                 end
  1712             | _ =>
  1713               if is_constr ctxt stds x then
  1714                 (Const x, ts)
  1715               else if is_stale_constr ctxt x then
  1716                 raise NOT_SUPPORTED ("(non-co)constructors of codatatypes \
  1717                                      \(\"" ^ s ^ "\")")
  1718               else if is_quot_abs_fun ctxt x then
  1719                 let
  1720                   val rep_T = domain_type T
  1721                   val abs_T = range_type T
  1722                 in
  1723                   (Abs (Name.uu, rep_T,
  1724                         Const (@{const_name Quot}, rep_T --> abs_T)
  1725                                $ (Const (quot_normal_name_for_type ctxt abs_T,
  1726                                          rep_T --> rep_T) $ Bound 0)), ts)
  1727                 end
  1728               else if is_quot_rep_fun ctxt x then
  1729                 quot_rep_of depth Ts (domain_type T) (range_type T) ts
  1730               else if is_record_get thy x then
  1731                 case length ts of
  1732                   0 => (do_term depth Ts (eta_expand Ts t 1), [])
  1733                 | _ => (optimized_record_get hol_ctxt s (domain_type T)
  1734                             (range_type T) (do_term depth Ts (hd ts)), tl ts)
  1735               else if is_record_update thy x then
  1736                 case length ts of
  1737                   2 => (optimized_record_update hol_ctxt
  1738                             (unsuffix Record.updateN s) (nth_range_type 2 T)
  1739                             (do_term depth Ts (hd ts))
  1740                             (do_term depth Ts (nth ts 1)), [])
  1741                 | n => (do_term depth Ts (eta_expand Ts t (2 - n)), [])
  1742               else if is_abs_fun ctxt x andalso
  1743                       is_quot_type ctxt (range_type T) then
  1744                 let
  1745                   val abs_T = range_type T
  1746                   val rep_T = domain_type (domain_type T)
  1747                   val eps_fun = Const (@{const_name Eps},
  1748                                        (rep_T --> bool_T) --> rep_T)
  1749                   val normal_fun =
  1750                     Const (quot_normal_name_for_type ctxt abs_T,
  1751                            rep_T --> rep_T)
  1752                   val abs_fun = Const (@{const_name Quot}, rep_T --> abs_T)
  1753                 in
  1754                   (Abs (Name.uu, rep_T --> bool_T,
  1755                         abs_fun $ (normal_fun $ (eps_fun $ Bound 0)))
  1756                    |> do_term (depth + 1) Ts, ts)
  1757                 end
  1758               else if is_rep_fun ctxt x then
  1759                 let val x' = mate_of_rep_fun ctxt x in
  1760                   if is_constr ctxt stds x' then
  1761                     select_nth_constr_arg_with_args depth Ts x' ts 0
  1762                                                     (range_type T)
  1763                   else if is_quot_type ctxt (domain_type T) then
  1764                     let
  1765                       val abs_T = domain_type T
  1766                       val rep_T = domain_type (range_type T)
  1767                       val (rep_fun, _) = quot_rep_of depth Ts abs_T rep_T []
  1768                       val (equiv_rel, _) =
  1769                         equiv_relation_for_quot_type ctxt abs_T
  1770                     in
  1771                       (Abs (Name.uu, abs_T, equiv_rel $ (rep_fun $ Bound 0)),
  1772                        ts)
  1773                     end
  1774                   else
  1775                     (Const x, ts)
  1776                 end
  1777               else if is_equational_fun_but_no_plain_def hol_ctxt x orelse
  1778                       is_choice_spec_fun hol_ctxt x then
  1779                 (Const x, ts)
  1780               else case def_of_const_ext thy def_tables x of
  1781                 SOME (unfold, def) =>
  1782                 if depth > unfold_max_depth then
  1783                   raise TOO_LARGE ("Nitpick_HOL.unfold_defs_in_term",
  1784                                    "too many nested definitions (" ^
  1785                                    string_of_int depth ^ ") while expanding " ^
  1786                                    quote s)
  1787                 else if s = "Wfrec.wfrec'" (* FIXME unchecked! *) then
  1788                   (do_term (depth + 1) Ts (s_betapplys Ts (def, ts)), [])
  1789                 else if not unfold andalso
  1790                      size_of_term def > def_inline_threshold () then
  1791                   (Const x, ts)
  1792                 else
  1793                   (do_term (depth + 1) Ts def, ts)
  1794               | NONE => (Const x, ts)
  1795         in
  1796           s_betapplys Ts (const, map (do_term depth Ts) ts)
  1797           |> s_beta_norm Ts
  1798         end
  1799   in do_term 0 [] end
  1800 
  1801 (** Axiom extraction/generation **)
  1802 
  1803 fun extensional_equal j (Type (@{type_name fun}, [dom_T, ran_T])) t1 t2 =
  1804     let val var_t = Var (("x", j), dom_T) in
  1805       extensional_equal (j + 1) ran_T (betapply (t1, var_t))
  1806                         (betapply (t2, var_t))
  1807     end
  1808   | extensional_equal _ T t1 t2 =
  1809     Const (@{const_name HOL.eq}, T --> T --> bool_T) $ t1 $ t2
  1810 
  1811 fun equationalize_term ctxt tag t =
  1812   let
  1813     val j = maxidx_of_term t + 1
  1814     val (prems, concl) = Logic.strip_horn t
  1815   in
  1816     Logic.list_implies (prems,
  1817         case concl of
  1818           @{const Trueprop} $ (Const (@{const_name HOL.eq}, Type (_, [T, _]))
  1819                                $ t1 $ t2) =>
  1820           @{const Trueprop} $ extensional_equal j T t1 t2
  1821         | @{const Trueprop} $ t' =>
  1822           @{const Trueprop} $ HOLogic.mk_eq (t', @{const True})
  1823         | Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
  1824           @{const Trueprop} $ extensional_equal j T t1 t2
  1825         | _ => (warning ("Ignoring " ^ quote tag ^ " for non-equation" ^
  1826                          quote (Syntax.string_of_term ctxt t) ^ ".");
  1827                 raise SAME ()))
  1828     |> SOME
  1829   end
  1830   handle SAME () => NONE
  1831 
  1832 fun pair_for_prop t =
  1833   case term_under_def t of
  1834     Const (s, _) => (s, t)
  1835   | t' => raise TERM ("Nitpick_HOL.pair_for_prop", [t, t'])
  1836 
  1837 fun def_table_for get ctxt subst =
  1838   ctxt |> get |> map (pair_for_prop o subst_atomic subst)
  1839        |> AList.group (op =) |> Symtab.make
  1840 
  1841 fun const_def_tables ctxt subst ts =
  1842   (def_table_for (map prop_of o Nitpick_Unfolds.get) ctxt subst,
  1843    fold (fn (s, t) => Symtab.map_default (s, []) (cons t))
  1844         (map pair_for_prop ts) Symtab.empty)
  1845 
  1846 fun paired_with_consts t = map (rpair t) (Term.add_const_names t [])
  1847 fun const_nondef_table ts =
  1848   fold (append o paired_with_consts) ts [] |> AList.group (op =) |> Symtab.make
  1849 
  1850 fun const_simp_table ctxt =
  1851   def_table_for (map_filter (equationalize_term ctxt "nitpick_simp" o prop_of)
  1852                  o Nitpick_Simps.get) ctxt
  1853 fun const_psimp_table ctxt =
  1854   def_table_for (map_filter (equationalize_term ctxt "nitpick_psimp" o prop_of)
  1855                  o Nitpick_Psimps.get) ctxt
  1856 
  1857 fun const_choice_spec_table ctxt subst =
  1858   map (subst_atomic subst o prop_of) (Nitpick_Choice_Specs.get ctxt)
  1859   |> const_nondef_table
  1860 
  1861 fun inductive_intro_table ctxt subst def_tables =
  1862   let val thy = Proof_Context.theory_of ctxt in
  1863     def_table_for
  1864         (maps (map (unfold_mutually_inductive_preds thy def_tables o prop_of)
  1865                o snd o snd)
  1866          o filter (fn (cat, _) => cat = Spec_Rules.Inductive orelse
  1867                                   cat = Spec_Rules.Co_Inductive)
  1868          o Spec_Rules.get) ctxt subst
  1869   end
  1870 
  1871 fun ground_theorem_table thy =
  1872   fold ((fn @{const Trueprop} $ t1 =>
  1873             is_ground_term t1 ? Inttab.map_default (hash_term t1, []) (cons t1)
  1874           | _ => I) o prop_of o snd) (Global_Theory.all_thms_of thy) Inttab.empty
  1875 
  1876 fun ersatz_table ctxt =
  1877  #ersatz_table (Data.get (Context.Proof ctxt))
  1878  |> fold (append o snd) (#frac_types (Data.get (Context.Proof ctxt)))
  1879 
  1880 fun add_simps simp_table s eqs =
  1881   Unsynchronized.change simp_table
  1882       (Symtab.update (s, eqs @ these (Symtab.lookup (!simp_table) s)))
  1883 
  1884 fun inverse_axioms_for_rep_fun ctxt (x as (_, T)) =
  1885   let
  1886     val thy = Proof_Context.theory_of ctxt
  1887     val abs_T = domain_type T
  1888   in
  1889     typedef_info ctxt (fst (dest_Type abs_T)) |> the
  1890     |> pairf #Abs_inverse #Rep_inverse
  1891     |> pairself (specialize_type thy x o prop_of o the)
  1892     ||> single |> op ::
  1893   end
  1894 fun optimized_typedef_axioms ctxt (abs_z as (abs_s, _)) =
  1895   let
  1896     val thy = Proof_Context.theory_of ctxt
  1897     val abs_T = Type abs_z
  1898   in
  1899     if is_univ_typedef ctxt abs_T then
  1900       []
  1901     else case typedef_info ctxt abs_s of
  1902       SOME {abs_type, rep_type, Rep_name, prop_of_Rep, set_name, ...} =>
  1903       let
  1904         val rep_T = varify_and_instantiate_type ctxt abs_type abs_T rep_type
  1905         val rep_t = Const (Rep_name, abs_T --> rep_T)
  1906         val set_t = Const (set_name, rep_T --> bool_T)
  1907         val set_t' =
  1908           prop_of_Rep |> HOLogic.dest_Trueprop
  1909                       |> specialize_type thy (dest_Const rep_t)
  1910                       |> HOLogic.dest_mem |> snd
  1911       in
  1912         [HOLogic.all_const abs_T
  1913          $ Abs (Name.uu, abs_T, set_t $ (rep_t $ Bound 0))]
  1914         |> set_t <> set_t' ? cons (HOLogic.mk_eq (set_t, set_t'))
  1915         |> map HOLogic.mk_Trueprop
  1916       end
  1917     | NONE => []
  1918   end
  1919 fun optimized_quot_type_axioms ctxt stds abs_z =
  1920   let
  1921     val abs_T = Type abs_z
  1922     val rep_T = rep_type_for_quot_type ctxt abs_T
  1923     val (equiv_rel, partial) = equiv_relation_for_quot_type ctxt abs_T
  1924     val a_var = Var (("a", 0), abs_T)
  1925     val x_var = Var (("x", 0), rep_T)
  1926     val y_var = Var (("y", 0), rep_T)
  1927     val x = (@{const_name Quot}, rep_T --> abs_T)
  1928     val sel_a_t = select_nth_constr_arg ctxt stds x a_var 0 rep_T
  1929     val normal_fun =
  1930       Const (quot_normal_name_for_type ctxt abs_T, rep_T --> rep_T)
  1931     val normal_x = normal_fun $ x_var
  1932     val normal_y = normal_fun $ y_var
  1933     val is_unknown_t = Const (@{const_name is_unknown}, rep_T --> bool_T)
  1934   in
  1935     [Logic.mk_equals (normal_fun $ sel_a_t, sel_a_t),
  1936      Logic.list_implies
  1937          ([@{const Not} $ (is_unknown_t $ normal_x),
  1938            @{const Not} $ (is_unknown_t $ normal_y),
  1939            equiv_rel $ x_var $ y_var] |> map HOLogic.mk_Trueprop,
  1940            Logic.mk_equals (normal_x, normal_y)),
  1941      Logic.list_implies
  1942          ([HOLogic.mk_Trueprop (@{const Not} $ (is_unknown_t $ normal_x)),
  1943            HOLogic.mk_Trueprop (@{const Not} $ HOLogic.mk_eq (normal_x, x_var))],
  1944           HOLogic.mk_Trueprop (equiv_rel $ x_var $ normal_x))]
  1945     |> partial ? cons (HOLogic.mk_Trueprop (equiv_rel $ sel_a_t $ sel_a_t))
  1946   end
  1947 
  1948 fun codatatype_bisim_axioms (hol_ctxt as {ctxt, stds, ...}) T =
  1949   let
  1950     val xs = datatype_constrs hol_ctxt T
  1951     val set_T = T --> bool_T
  1952     val iter_T = @{typ bisim_iterator}
  1953     val bisim_max = @{const bisim_iterator_max}
  1954     val n_var = Var (("n", 0), iter_T)
  1955     val n_var_minus_1 =
  1956       Const (@{const_name safe_The}, (iter_T --> bool_T) --> iter_T)
  1957       $ Abs ("m", iter_T, HOLogic.eq_const iter_T
  1958                           $ (suc_const iter_T $ Bound 0) $ n_var)
  1959     val x_var = Var (("x", 0), T)
  1960     val y_var = Var (("y", 0), T)
  1961     fun bisim_const T =
  1962       Const (@{const_name bisim}, iter_T --> T --> T --> bool_T)
  1963     fun nth_sub_bisim x n nth_T =
  1964       (if is_codatatype ctxt nth_T then bisim_const nth_T $ n_var_minus_1
  1965        else HOLogic.eq_const nth_T)
  1966       $ select_nth_constr_arg ctxt stds x x_var n nth_T
  1967       $ select_nth_constr_arg ctxt stds x y_var n nth_T
  1968     fun case_func (x as (_, T)) =
  1969       let
  1970         val arg_Ts = binder_types T
  1971         val core_t =
  1972           discriminate_value hol_ctxt x y_var ::
  1973           map2 (nth_sub_bisim x) (index_seq 0 (length arg_Ts)) arg_Ts
  1974           |> foldr1 s_conj
  1975       in fold_rev absdummy arg_Ts core_t end
  1976   in
  1977     [HOLogic.mk_imp
  1978        (HOLogic.mk_disj (HOLogic.eq_const iter_T $ n_var $ zero_const iter_T,
  1979             s_betapply [] (optimized_case_def hol_ctxt [] T bool_T
  1980                                               (map case_func xs), x_var)),
  1981         bisim_const T $ n_var $ x_var $ y_var),
  1982      HOLogic.eq_const set_T $ (bisim_const T $ bisim_max $ x_var)
  1983      $ (Const (@{const_name insert}, T --> set_T --> set_T)
  1984         $ x_var $ Const (@{const_name bot_class.bot}, set_T))]
  1985     |> map HOLogic.mk_Trueprop
  1986   end
  1987 
  1988 exception NO_TRIPLE of unit
  1989 
  1990 fun triple_for_intro_rule thy x t =
  1991   let
  1992     val prems = Logic.strip_imp_prems t |> map (Object_Logic.atomize_term thy)
  1993     val concl = Logic.strip_imp_concl t |> Object_Logic.atomize_term thy
  1994     val (main, side) = List.partition (exists_Const (curry (op =) x)) prems
  1995     val is_good_head = curry (op =) (Const x) o head_of
  1996   in
  1997     if forall is_good_head main then (side, main, concl) else raise NO_TRIPLE ()
  1998   end
  1999 
  2000 val tuple_for_args = HOLogic.mk_tuple o snd o strip_comb
  2001 fun wf_constraint_for rel side concl main =
  2002   let
  2003     val core = HOLogic.mk_mem (HOLogic.mk_prod
  2004                                (pairself tuple_for_args (main, concl)), Var rel)
  2005     val t = List.foldl HOLogic.mk_imp core side
  2006     val vars = filter_out (curry (op =) rel) (Term.add_vars t [])
  2007   in
  2008     Library.foldl (fn (t', ((x, j), T)) =>
  2009                       HOLogic.all_const T
  2010                       $ Abs (x, T, abstract_over (Var ((x, j), T), t')))
  2011                   (t, vars)
  2012   end
  2013 fun wf_constraint_for_triple rel (side, main, concl) =
  2014   map (wf_constraint_for rel side concl) main |> foldr1 s_conj
  2015 
  2016 fun terminates_by ctxt timeout goal tac =
  2017   can (SINGLE (Classical.safe_tac ctxt) #> the
  2018        #> SINGLE (DETERM_TIMEOUT timeout (tac ctxt (auto_tac ctxt)))
  2019        #> the #> Goal.finish ctxt) goal
  2020 
  2021 val max_cached_wfs = 50
  2022 val cached_timeout =
  2023   Synchronized.var "Nitpick_HOL.cached_timeout" (SOME Time.zeroTime)
  2024 val cached_wf_props =
  2025   Synchronized.var "Nitpick_HOL.cached_wf_props" ([] : (term * bool) list)
  2026 
  2027 val termination_tacs = [Lexicographic_Order.lex_order_tac true,
  2028                         ScnpReconstruct.sizechange_tac]
  2029 
  2030 fun uncached_is_well_founded_inductive_pred
  2031         ({thy, ctxt, stds, debug, tac_timeout, intro_table, ...} : hol_context)
  2032         (x as (_, T)) =
  2033   case def_props_for_const thy stds intro_table x of
  2034     [] => raise TERM ("Nitpick_HOL.uncached_is_well_founded_inductive",
  2035                       [Const x])
  2036   | intro_ts =>
  2037     (case map (triple_for_intro_rule thy x) intro_ts
  2038           |> filter_out (null o #2) of
  2039        [] => true
  2040      | triples =>
  2041        let
  2042          val binders_T = HOLogic.mk_tupleT (binder_types T)
  2043          val rel_T = HOLogic.mk_prodT (binders_T, binders_T) --> bool_T
  2044          val j = fold Integer.max (map maxidx_of_term intro_ts) 0 + 1
  2045          val rel = (("R", j), rel_T)
  2046          val prop = Const (@{const_name wf}, rel_T --> bool_T) $ Var rel ::
  2047                     map (wf_constraint_for_triple rel) triples
  2048                     |> foldr1 s_conj |> HOLogic.mk_Trueprop
  2049          val _ = if debug then
  2050                    Output.urgent_message ("Wellfoundedness goal: " ^
  2051                              Syntax.string_of_term ctxt prop ^ ".")
  2052                  else
  2053                    ()
  2054        in
  2055          if tac_timeout = Synchronized.value cached_timeout andalso
  2056             length (Synchronized.value cached_wf_props) < max_cached_wfs then
  2057            ()
  2058          else
  2059            (Synchronized.change cached_wf_props (K []);
  2060             Synchronized.change cached_timeout (K tac_timeout));
  2061          case AList.lookup (op =) (Synchronized.value cached_wf_props) prop of
  2062            SOME wf => wf
  2063          | NONE =>
  2064            let
  2065              val goal = prop |> cterm_of thy |> Goal.init
  2066              val wf = exists (terminates_by ctxt tac_timeout goal)
  2067                              termination_tacs
  2068            in Synchronized.change cached_wf_props (cons (prop, wf)); wf end
  2069        end)
  2070     handle List.Empty => false | NO_TRIPLE () => false
  2071 
  2072 (* The type constraint below is a workaround for a Poly/ML crash. *)
  2073 
  2074 fun is_well_founded_inductive_pred
  2075         (hol_ctxt as {thy, wfs, def_tables, wf_cache, ...} : hol_context)
  2076         (x as (s, _)) =
  2077   case triple_lookup (const_match thy) wfs x of
  2078     SOME (SOME b) => b
  2079   | _ => s = @{const_name Nats} orelse s = @{const_name fold_graph'} orelse
  2080          case AList.lookup (op =) (!wf_cache) x of
  2081            SOME (_, wf) => wf
  2082          | NONE =>
  2083            let
  2084              val gfp = (fixpoint_kind_of_const thy def_tables x = Gfp)
  2085              val wf = uncached_is_well_founded_inductive_pred hol_ctxt x
  2086            in
  2087              Unsynchronized.change wf_cache (cons (x, (gfp, wf))); wf
  2088            end
  2089 
  2090 fun ap_curry [_] _ t = t
  2091   | ap_curry arg_Ts tuple_T t =
  2092     let val n = length arg_Ts in
  2093       list_abs (map (pair "c") arg_Ts,
  2094                 incr_boundvars n t
  2095                 $ mk_flat_tuple tuple_T (map Bound (n - 1 downto 0)))
  2096     end
  2097 
  2098 fun num_occs_of_bound_in_term j (t1 $ t2) =
  2099     op + (pairself (num_occs_of_bound_in_term j) (t1, t2))
  2100   | num_occs_of_bound_in_term j (Abs (_, _, t')) =
  2101     num_occs_of_bound_in_term (j + 1) t'
  2102   | num_occs_of_bound_in_term j (Bound j') = if j' = j then 1 else 0
  2103   | num_occs_of_bound_in_term _ _ = 0
  2104 
  2105 val is_linear_inductive_pred_def =
  2106   let
  2107     fun do_disjunct j (Const (@{const_name Ex}, _) $ Abs (_, _, t2)) =
  2108         do_disjunct (j + 1) t2
  2109       | do_disjunct j t =
  2110         case num_occs_of_bound_in_term j t of
  2111           0 => true
  2112         | 1 => exists (curry (op =) (Bound j) o head_of) (conjuncts_of t)
  2113         | _ => false
  2114     fun do_lfp_def (Const (@{const_name lfp}, _) $ t2) =
  2115         let val (xs, body) = strip_abs t2 in
  2116           case length xs of
  2117             1 => false
  2118           | n => forall (do_disjunct (n - 1)) (disjuncts_of body)
  2119         end
  2120       | do_lfp_def _ = false
  2121   in do_lfp_def o strip_abs_body end
  2122 
  2123 fun n_ptuple_paths 0 = []
  2124   | n_ptuple_paths 1 = []
  2125   | n_ptuple_paths n = [] :: map (cons 2) (n_ptuple_paths (n - 1))
  2126 val ap_n_split = HOLogic.mk_psplits o n_ptuple_paths
  2127 
  2128 val linear_pred_base_and_step_rhss =
  2129   let
  2130     fun aux (Const (@{const_name lfp}, _) $ t2) =
  2131         let
  2132           val (xs, body) = strip_abs t2
  2133           val arg_Ts = map snd (tl xs)
  2134           val tuple_T = HOLogic.mk_tupleT arg_Ts
  2135           val j = length arg_Ts
  2136           fun repair_rec j (Const (@{const_name Ex}, T1) $ Abs (s2, T2, t2')) =
  2137               Const (@{const_name Ex}, T1)
  2138               $ Abs (s2, T2, repair_rec (j + 1) t2')
  2139             | repair_rec j (@{const HOL.conj} $ t1 $ t2) =
  2140               @{const HOL.conj} $ repair_rec j t1 $ repair_rec j t2
  2141             | repair_rec j t =
  2142               let val (head, args) = strip_comb t in
  2143                 if head = Bound j then
  2144                   HOLogic.eq_const tuple_T $ Bound j
  2145                   $ mk_flat_tuple tuple_T args
  2146                 else
  2147                   t
  2148               end
  2149           val (nonrecs, recs) =
  2150             List.partition (curry (op =) 0 o num_occs_of_bound_in_term j)
  2151                            (disjuncts_of body)
  2152           val base_body = nonrecs |> List.foldl s_disj @{const False}
  2153           val step_body = recs |> map (repair_rec j)
  2154                                |> List.foldl s_disj @{const False}
  2155         in
  2156           (list_abs (tl xs, incr_bv (~1, j, base_body))
  2157            |> ap_n_split (length arg_Ts) tuple_T bool_T,
  2158            Abs ("y", tuple_T, list_abs (tl xs, step_body)
  2159                               |> ap_n_split (length arg_Ts) tuple_T bool_T))
  2160         end
  2161       | aux t =
  2162         raise TERM ("Nitpick_HOL.linear_pred_base_and_step_rhss.aux", [t])
  2163   in aux end
  2164 
  2165 fun starred_linear_pred_const (hol_ctxt as {simp_table, ...}) (s, T) def =
  2166   let
  2167     val j = maxidx_of_term def + 1
  2168     val (outer, fp_app) = strip_abs def
  2169     val outer_bounds = map Bound (length outer - 1 downto 0)
  2170     val outer_vars = map (fn (s, T) => Var ((s, j), T)) outer
  2171     val fp_app = subst_bounds (rev outer_vars, fp_app)
  2172     val (outer_Ts, rest_T) = strip_n_binders (length outer) T
  2173     val tuple_arg_Ts = strip_type rest_T |> fst
  2174     val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
  2175     val set_T = tuple_T --> bool_T
  2176     val curried_T = tuple_T --> set_T
  2177     val uncurried_T = Type (@{type_name prod}, [tuple_T, tuple_T]) --> bool_T
  2178     val (base_rhs, step_rhs) = linear_pred_base_and_step_rhss fp_app
  2179     val base_x as (base_s, _) = (base_prefix ^ s, outer_Ts ---> set_T)
  2180     val base_eq = HOLogic.mk_eq (list_comb (Const base_x, outer_vars), base_rhs)
  2181                   |> HOLogic.mk_Trueprop
  2182     val _ = add_simps simp_table base_s [base_eq]
  2183     val step_x as (step_s, _) = (step_prefix ^ s, outer_Ts ---> curried_T)
  2184     val step_eq = HOLogic.mk_eq (list_comb (Const step_x, outer_vars), step_rhs)
  2185                   |> HOLogic.mk_Trueprop
  2186     val _ = add_simps simp_table step_s [step_eq]
  2187   in
  2188     list_abs (outer,
  2189               Const (@{const_name Image}, uncurried_T --> set_T --> set_T)
  2190               $ (Const (@{const_name rtrancl}, uncurried_T --> uncurried_T)
  2191                  $ (Const (@{const_name prod_case}, curried_T --> uncurried_T)
  2192                     $ list_comb (Const step_x, outer_bounds)))
  2193               $ list_comb (Const base_x, outer_bounds)
  2194               |> ap_curry tuple_arg_Ts tuple_T)
  2195     |> unfold_defs_in_term hol_ctxt
  2196   end
  2197 
  2198 fun is_good_starred_linear_pred_type (Type (@{type_name fun}, Ts)) =
  2199     forall (not o (is_fun_type orf is_pair_type)) Ts
  2200   | is_good_starred_linear_pred_type _ = false
  2201 
  2202 fun unrolled_inductive_pred_const (hol_ctxt as {thy, star_linear_preds,
  2203                                                 def_tables, simp_table, ...})
  2204                                   gfp (x as (s, T)) =
  2205   let
  2206     val iter_T = iterator_type_for_const gfp x
  2207     val x' as (s', _) = (unrolled_prefix ^ s, iter_T --> T)
  2208     val unrolled_const = Const x' $ zero_const iter_T
  2209     val def = the (def_of_const thy def_tables x)
  2210   in
  2211     if is_equational_fun_but_no_plain_def hol_ctxt x' then
  2212       unrolled_const (* already done *)
  2213     else if not gfp andalso star_linear_preds andalso
  2214          is_linear_inductive_pred_def def andalso
  2215          is_good_starred_linear_pred_type T then
  2216       starred_linear_pred_const hol_ctxt x def
  2217     else
  2218       let
  2219         val j = maxidx_of_term def + 1
  2220         val (outer, fp_app) = strip_abs def
  2221         val outer_bounds = map Bound (length outer - 1 downto 0)
  2222         val cur = Var ((iter_var_prefix, j + 1), iter_T)
  2223         val next = suc_const iter_T $ cur
  2224         val rhs =
  2225           case fp_app of
  2226             Const _ $ t =>
  2227             s_betapply [] (t, list_comb (Const x', next :: outer_bounds))
  2228           | _ => raise TERM ("Nitpick_HOL.unrolled_inductive_pred_const",
  2229                              [fp_app])
  2230         val (inner, naked_rhs) = strip_abs rhs
  2231         val all = outer @ inner
  2232         val bounds = map Bound (length all - 1 downto 0)
  2233         val vars = map (fn (s, T) => Var ((s, j), T)) all
  2234         val eq = HOLogic.mk_eq (list_comb (Const x', cur :: bounds), naked_rhs)
  2235                  |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2236         val _ = add_simps simp_table s' [eq]
  2237       in unrolled_const end
  2238   end
  2239 
  2240 fun raw_inductive_pred_axiom ({thy, def_tables, ...} : hol_context) x =
  2241   let
  2242     val def = the (def_of_const thy def_tables x)
  2243     val (outer, fp_app) = strip_abs def
  2244     val outer_bounds = map Bound (length outer - 1 downto 0)
  2245     val rhs =
  2246       case fp_app of
  2247         Const _ $ t => s_betapply [] (t, list_comb (Const x, outer_bounds))
  2248       | _ => raise TERM ("Nitpick_HOL.raw_inductive_pred_axiom", [fp_app])
  2249     val (inner, naked_rhs) = strip_abs rhs
  2250     val all = outer @ inner
  2251     val bounds = map Bound (length all - 1 downto 0)
  2252     val j = maxidx_of_term def + 1
  2253     val vars = map (fn (s, T) => Var ((s, j), T)) all
  2254   in
  2255     HOLogic.mk_eq (list_comb (Const x, bounds), naked_rhs)
  2256     |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2257   end
  2258 fun inductive_pred_axiom hol_ctxt (x as (s, T)) =
  2259   if String.isPrefix ubfp_prefix s orelse String.isPrefix lbfp_prefix s then
  2260     let val x' = (strip_first_name_sep s |> snd, T) in
  2261       raw_inductive_pred_axiom hol_ctxt x' |> subst_atomic [(Const x', Const x)]
  2262     end
  2263   else
  2264     raw_inductive_pred_axiom hol_ctxt x
  2265 
  2266 fun equational_fun_axioms (hol_ctxt as {thy, ctxt, stds, def_tables, simp_table,
  2267                                         psimp_table, ...}) x =
  2268   case def_props_for_const thy stds (!simp_table) x of
  2269     [] => (case def_props_for_const thy stds psimp_table x of
  2270              [] => (if is_inductive_pred hol_ctxt x then
  2271                       [inductive_pred_axiom hol_ctxt x]
  2272                     else case def_of_const thy def_tables x of
  2273                       SOME def =>
  2274                       @{const Trueprop} $ HOLogic.mk_eq (Const x, def)
  2275                       |> equationalize_term ctxt "" |> the |> single
  2276                     | NONE => [])
  2277            | psimps => psimps)
  2278   | simps => simps
  2279 fun is_equational_fun_surely_complete hol_ctxt x =
  2280   case equational_fun_axioms hol_ctxt x of
  2281     [@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)] =>
  2282     strip_comb t1 |> snd |> forall is_Var
  2283   | _ => false
  2284 
  2285 (** Type preprocessing **)
  2286 
  2287 fun merged_type_var_table_for_terms thy ts =
  2288   let
  2289     fun add (s, S) table =
  2290       table
  2291       |> (case AList.lookup (Sign.subsort thy o swap) table S of
  2292             SOME _ => I
  2293           | NONE =>
  2294             filter_out (fn (S', _) => Sign.subsort thy (S, S'))
  2295             #> cons (S, s))
  2296     val tfrees = [] |> fold Term.add_tfrees ts
  2297                     |> sort (string_ord o pairself fst)
  2298   in [] |> fold add tfrees |> rev end
  2299 
  2300 fun merge_type_vars_in_term thy merge_type_vars table =
  2301   merge_type_vars
  2302   ? map_types (map_atyps
  2303         (fn TFree (_, S) =>
  2304             TFree (table |> find_first (fn (S', _) => Sign.subsort thy (S', S))
  2305                          |> the |> swap)
  2306           | T => T))
  2307 
  2308 fun add_ground_types hol_ctxt binarize =
  2309   let
  2310     fun aux T accum =
  2311       case T of
  2312         Type (@{type_name fun}, Ts) => fold aux Ts accum
  2313       | Type (@{type_name prod}, Ts) => fold aux Ts accum
  2314       | Type (@{type_name itself}, [T1]) => aux T1 accum
  2315       | Type (_, Ts) =>
  2316         if member (op =) (@{typ prop} :: @{typ bool} :: accum) T then
  2317           accum
  2318         else
  2319           T :: accum
  2320           |> fold aux (case binarized_and_boxed_datatype_constrs hol_ctxt
  2321                                                                  binarize T of
  2322                          [] => Ts
  2323                        | xs => map snd xs)
  2324       | _ => insert (op =) T accum
  2325   in aux end
  2326 
  2327 fun ground_types_in_type hol_ctxt binarize T =
  2328   add_ground_types hol_ctxt binarize T []
  2329 fun ground_types_in_terms hol_ctxt binarize ts =
  2330   fold (fold_types (add_ground_types hol_ctxt binarize)) ts []
  2331 
  2332 end;