src/HOL/Tools/Nitpick/nitpick_hol.ML
author wenzelm
Wed Aug 17 18:05:31 2011 +0200 (2011-08-17)
changeset 44241 7943b69f0188
parent 44017 e828be67dfe7
child 45272 5995ab88a00f
permissions -rw-r--r--
modernized signature of Term.absfree/absdummy;
eliminated obsolete Term.list_abs_free;
     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_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 const_for_iterator_type : typ -> styp
   105   val strip_n_binders : int -> typ -> typ list * typ
   106   val nth_range_type : int -> typ -> typ
   107   val num_factors_in_type : typ -> int
   108   val num_binder_types : typ -> int
   109   val curried_binder_types : typ -> typ list
   110   val mk_flat_tuple : typ -> term list -> term
   111   val dest_n_tuple : int -> term -> term list
   112   val is_real_datatype : theory -> string -> bool
   113   val is_standard_datatype : theory -> (typ option * bool) list -> typ -> bool
   114   val is_codatatype : Proof.context -> typ -> bool
   115   val is_quot_type : Proof.context -> typ -> bool
   116   val is_pure_typedef : Proof.context -> typ -> bool
   117   val is_univ_typedef : Proof.context -> typ -> bool
   118   val is_datatype : Proof.context -> (typ option * bool) list -> typ -> bool
   119   val is_record_constr : styp -> bool
   120   val is_record_get : theory -> styp -> bool
   121   val is_record_update : theory -> styp -> bool
   122   val is_abs_fun : Proof.context -> styp -> bool
   123   val is_rep_fun : Proof.context -> styp -> bool
   124   val is_quot_abs_fun : Proof.context -> styp -> bool
   125   val is_quot_rep_fun : Proof.context -> styp -> bool
   126   val mate_of_rep_fun : Proof.context -> styp -> styp
   127   val is_constr_like : Proof.context -> styp -> bool
   128   val is_constr : Proof.context -> (typ option * bool) list -> styp -> bool
   129   val is_sel : string -> bool
   130   val is_sel_like_and_no_discr : string -> bool
   131   val box_type : hol_context -> boxability -> typ -> typ
   132   val binarize_nat_and_int_in_type : typ -> typ
   133   val binarize_nat_and_int_in_term : term -> term
   134   val discr_for_constr : styp -> styp
   135   val num_sels_for_constr_type : typ -> int
   136   val nth_sel_name_for_constr_name : string -> int -> string
   137   val nth_sel_for_constr : styp -> int -> styp
   138   val binarized_and_boxed_nth_sel_for_constr :
   139     hol_context -> bool -> styp -> int -> styp
   140   val sel_no_from_name : string -> int
   141   val close_form : term -> term
   142   val distinctness_formula : typ -> term list -> term
   143   val register_frac_type :
   144     string -> (string * string) list -> morphism -> Context.generic
   145     -> Context.generic
   146   val register_frac_type_global :
   147     string -> (string * string) list -> theory -> theory
   148   val unregister_frac_type :
   149     string -> morphism -> Context.generic -> Context.generic
   150   val unregister_frac_type_global : string -> theory -> theory
   151   val register_ersatz :
   152     (string * string) list -> morphism -> Context.generic -> Context.generic
   153   val register_ersatz_global : (string * string) list -> theory -> theory
   154   val register_codatatype :
   155     typ -> string -> styp list -> morphism -> Context.generic -> Context.generic
   156   val register_codatatype_global :
   157     typ -> string -> styp list -> theory -> theory
   158   val unregister_codatatype :
   159     typ -> morphism -> Context.generic -> Context.generic
   160   val unregister_codatatype_global : typ -> theory -> theory
   161   val datatype_constrs : hol_context -> typ -> styp list
   162   val binarized_and_boxed_datatype_constrs :
   163     hol_context -> bool -> typ -> styp list
   164   val num_datatype_constrs : hol_context -> typ -> int
   165   val constr_name_for_sel_like : string -> string
   166   val binarized_and_boxed_constr_for_sel : hol_context -> bool -> styp -> styp
   167   val card_of_type : (typ * int) list -> typ -> int
   168   val bounded_card_of_type : int -> int -> (typ * int) list -> typ -> int
   169   val bounded_exact_card_of_type :
   170     hol_context -> typ list -> int -> int -> (typ * int) list -> typ -> int
   171   val typical_card_of_type : typ -> int
   172   val is_finite_type : hol_context -> typ -> bool
   173   val is_special_eligible_arg : bool -> typ list -> term -> bool
   174   val s_let :
   175     typ list -> string -> int -> typ -> typ -> (term -> term) -> term -> term
   176   val s_betapply : typ list -> term * term -> term
   177   val s_betapplys : typ list -> term * term list -> term
   178   val discriminate_value : hol_context -> styp -> term -> term
   179   val select_nth_constr_arg :
   180     Proof.context -> (typ option * bool) list -> styp -> term -> int -> typ
   181     -> term
   182   val construct_value :
   183     Proof.context -> (typ option * bool) list -> styp -> term list -> term
   184   val coerce_term : hol_context -> typ list -> typ -> typ -> term -> term
   185   val special_bounds : term list -> (indexname * typ) list
   186   val is_funky_typedef : Proof.context -> typ -> bool
   187   val all_defs_of : theory -> (term * term) list -> term list
   188   val all_nondefs_of : Proof.context -> (term * term) list -> term list
   189   val arity_of_built_in_const :
   190     theory -> (typ option * bool) list -> styp -> int option
   191   val is_built_in_const :
   192     theory -> (typ option * bool) list -> styp -> bool
   193   val term_under_def : term -> term
   194   val case_const_names :
   195     Proof.context -> (typ option * bool) list -> (string * int) list
   196   val unfold_defs_in_term : hol_context -> term -> term
   197   val const_def_tables :
   198     Proof.context -> (term * term) list -> term list
   199     -> const_table * const_table
   200   val const_nondef_table : term list -> const_table
   201   val const_simp_table : Proof.context -> (term * term) list -> const_table
   202   val const_psimp_table : Proof.context -> (term * term) list -> const_table
   203   val const_choice_spec_table :
   204     Proof.context -> (term * term) list -> const_table
   205   val inductive_intro_table :
   206     Proof.context -> (term * term) list -> const_table * const_table
   207     -> const_table
   208   val ground_theorem_table : theory -> term list Inttab.table
   209   val ersatz_table : Proof.context -> (string * string) list
   210   val add_simps : const_table Unsynchronized.ref -> string -> term list -> unit
   211   val inverse_axioms_for_rep_fun : Proof.context -> styp -> term list
   212   val optimized_typedef_axioms : Proof.context -> string * typ list -> term list
   213   val optimized_quot_type_axioms :
   214     Proof.context -> (typ option * bool) list -> string * typ list -> term list
   215   val def_of_const : theory -> const_table * const_table -> styp -> term option
   216   val fixpoint_kind_of_rhs : term -> fixpoint_kind
   217   val fixpoint_kind_of_const :
   218     theory -> const_table * const_table -> string * typ -> fixpoint_kind
   219   val is_real_inductive_pred : hol_context -> styp -> bool
   220   val is_constr_pattern : Proof.context -> term -> bool
   221   val is_constr_pattern_lhs : Proof.context -> term -> bool
   222   val is_constr_pattern_formula : Proof.context -> term -> bool
   223   val nondef_props_for_const :
   224     theory -> bool -> const_table -> styp -> term list
   225   val is_choice_spec_fun : hol_context -> styp -> bool
   226   val is_choice_spec_axiom : theory -> const_table -> term -> bool
   227   val is_real_equational_fun : hol_context -> styp -> bool
   228   val is_equational_fun_but_no_plain_def : hol_context -> styp -> bool
   229   val codatatype_bisim_axioms : hol_context -> typ -> term list
   230   val is_well_founded_inductive_pred : hol_context -> styp -> bool
   231   val unrolled_inductive_pred_const : hol_context -> bool -> styp -> term
   232   val equational_fun_axioms : hol_context -> styp -> term list
   233   val is_equational_fun_surely_complete : hol_context -> styp -> bool
   234   val merged_type_var_table_for_terms :
   235     theory -> term list -> (sort * string) list
   236   val merge_type_vars_in_term :
   237     theory -> bool -> (sort * string) list -> term -> term
   238   val ground_types_in_type : hol_context -> bool -> typ -> typ list
   239   val ground_types_in_terms : hol_context -> bool -> term list -> typ list
   240 end;
   241 
   242 structure Nitpick_HOL : NITPICK_HOL =
   243 struct
   244 
   245 open Nitpick_Util
   246 
   247 type const_table = term list Symtab.table
   248 type special_fun = (styp * int list * term list) * styp
   249 type unrolled = styp * styp
   250 type wf_cache = (styp * (bool * bool)) list
   251 
   252 type hol_context =
   253   {thy: theory,
   254    ctxt: Proof.context,
   255    max_bisim_depth: int,
   256    boxes: (typ option * bool option) list,
   257    stds: (typ option * bool) list,
   258    wfs: (styp option * bool option) list,
   259    user_axioms: bool option,
   260    debug: bool,
   261    whacks: term list,
   262    binary_ints: bool option,
   263    destroy_constrs: bool,
   264    specialize: bool,
   265    star_linear_preds: bool,
   266    total_consts: bool option,
   267    needs: term list option,
   268    tac_timeout: Time.time option,
   269    evals: term list,
   270    case_names: (string * int) list,
   271    def_tables: const_table * const_table,
   272    nondef_table: const_table,
   273    nondefs: term list,
   274    simp_table: const_table Unsynchronized.ref,
   275    psimp_table: const_table,
   276    choice_spec_table: const_table,
   277    intro_table: const_table,
   278    ground_thm_table: term list Inttab.table,
   279    ersatz_table: (string * string) list,
   280    skolems: (string * string list) list Unsynchronized.ref,
   281    special_funs: special_fun list Unsynchronized.ref,
   282    unrolled_preds: unrolled list Unsynchronized.ref,
   283    wf_cache: wf_cache Unsynchronized.ref,
   284    constr_cache: (typ * styp list) list Unsynchronized.ref}
   285 
   286 datatype fixpoint_kind = Lfp | Gfp | NoFp
   287 datatype boxability =
   288   InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
   289 
   290 structure Data = Generic_Data
   291 (
   292   type T = {frac_types: (string * (string * string) list) list,
   293             ersatz_table: (string * string) list,
   294             codatatypes: (string * (string * styp list)) list}
   295   val empty = {frac_types = [], ersatz_table = [], codatatypes = []}
   296   val extend = I
   297   fun merge ({frac_types = fs1, ersatz_table = et1, codatatypes = cs1},
   298              {frac_types = fs2, ersatz_table = et2, codatatypes = cs2}) : T =
   299     {frac_types = AList.merge (op =) (K true) (fs1, fs2),
   300      ersatz_table = AList.merge (op =) (K true) (et1, et2),
   301      codatatypes = AList.merge (op =) (K true) (cs1, cs2)}
   302 )
   303 
   304 val name_sep = "$"
   305 val numeral_prefix = nitpick_prefix ^ "num" ^ name_sep
   306 val sel_prefix = nitpick_prefix ^ "sel"
   307 val discr_prefix = nitpick_prefix ^ "is" ^ name_sep
   308 val set_prefix = nitpick_prefix ^ "set" ^ name_sep
   309 val lfp_iterator_prefix = nitpick_prefix ^ "lfpit" ^ name_sep
   310 val gfp_iterator_prefix = nitpick_prefix ^ "gfpit" ^ name_sep
   311 val unrolled_prefix = nitpick_prefix ^ "unroll" ^ name_sep
   312 val base_prefix = nitpick_prefix ^ "base" ^ name_sep
   313 val step_prefix = nitpick_prefix ^ "step" ^ name_sep
   314 val ubfp_prefix = nitpick_prefix ^ "ubfp" ^ name_sep
   315 val lbfp_prefix = nitpick_prefix ^ "lbfp" ^ name_sep
   316 val quot_normal_prefix = nitpick_prefix ^ "qn" ^ name_sep
   317 val skolem_prefix = nitpick_prefix ^ "sk"
   318 val special_prefix = nitpick_prefix ^ "sp"
   319 val uncurry_prefix = nitpick_prefix ^ "unc"
   320 val eval_prefix = nitpick_prefix ^ "eval"
   321 val iter_var_prefix = "i"
   322 
   323 (** Constant/type information and term/type manipulation **)
   324 
   325 fun sel_prefix_for j = sel_prefix ^ string_of_int j ^ name_sep
   326 fun quot_normal_name_for_type ctxt T =
   327   quot_normal_prefix ^ unyxml (Syntax.string_of_typ ctxt T)
   328 
   329 val strip_first_name_sep =
   330   Substring.full #> Substring.position name_sep ##> Substring.triml 1
   331   #> pairself Substring.string
   332 fun original_name s =
   333   if String.isPrefix nitpick_prefix s then
   334     case strip_first_name_sep s of (s1, "") => s1 | (_, s2) => original_name s2
   335   else
   336     s
   337 
   338 fun s_conj (t1, @{const True}) = t1
   339   | s_conj (@{const True}, t2) = t2
   340   | s_conj (t1, t2) =
   341     if t1 = @{const False} orelse t2 = @{const False} then @{const False}
   342     else HOLogic.mk_conj (t1, t2)
   343 fun s_disj (t1, @{const False}) = t1
   344   | s_disj (@{const False}, t2) = t2
   345   | s_disj (t1, t2) =
   346     if t1 = @{const True} orelse t2 = @{const True} then @{const True}
   347     else HOLogic.mk_disj (t1, t2)
   348 
   349 fun strip_connective conn_t (t as (t0 $ t1 $ t2)) =
   350     if t0 = conn_t then strip_connective t0 t2 @ strip_connective t0 t1 else [t]
   351   | strip_connective _ t = [t]
   352 fun strip_any_connective (t as (t0 $ _ $ _)) =
   353     if t0 = @{const HOL.conj} orelse t0 = @{const HOL.disj} then
   354       (strip_connective t0 t, t0)
   355     else
   356       ([t], @{const Not})
   357   | strip_any_connective t = ([t], @{const Not})
   358 val conjuncts_of = strip_connective @{const HOL.conj}
   359 val disjuncts_of = strip_connective @{const HOL.disj}
   360 
   361 (* When you add constants to these lists, make sure to handle them in
   362    "Nitpick_Nut.nut_from_term", and perhaps in "Nitpick_Mono.consider_term" as
   363    well. *)
   364 val built_in_consts =
   365   [(@{const_name all}, 1),
   366    (@{const_name "=="}, 2),
   367    (@{const_name "==>"}, 2),
   368    (@{const_name Pure.conjunction}, 2),
   369    (@{const_name Trueprop}, 1),
   370    (@{const_name Not}, 1),
   371    (@{const_name False}, 0),
   372    (@{const_name True}, 0),
   373    (@{const_name All}, 1),
   374    (@{const_name Ex}, 1),
   375    (@{const_name HOL.eq}, 1),
   376    (@{const_name HOL.conj}, 2),
   377    (@{const_name HOL.disj}, 2),
   378    (@{const_name HOL.implies}, 2),
   379    (@{const_name If}, 3),
   380    (@{const_name Let}, 2),
   381    (@{const_name Pair}, 2),
   382    (@{const_name fst}, 1),
   383    (@{const_name snd}, 1),
   384    (@{const_name Id}, 0),
   385    (@{const_name converse}, 1),
   386    (@{const_name trancl}, 1),
   387    (@{const_name rel_comp}, 2),
   388    (@{const_name finite}, 1),
   389    (@{const_name unknown}, 0),
   390    (@{const_name is_unknown}, 1),
   391    (@{const_name safe_The}, 1),
   392    (@{const_name Frac}, 0),
   393    (@{const_name norm_frac}, 0)]
   394 val built_in_nat_consts =
   395   [(@{const_name Suc}, 0),
   396    (@{const_name nat}, 0),
   397    (@{const_name nat_gcd}, 0),
   398    (@{const_name nat_lcm}, 0)]
   399 val built_in_typed_consts =
   400   [((@{const_name zero_class.zero}, int_T), 0),
   401    ((@{const_name one_class.one}, int_T), 0),
   402    ((@{const_name plus_class.plus}, int_T --> int_T --> int_T), 0),
   403    ((@{const_name minus_class.minus}, int_T --> int_T --> int_T), 0),
   404    ((@{const_name times_class.times}, int_T --> int_T --> int_T), 0),
   405    ((@{const_name div_class.div}, int_T --> int_T --> int_T), 0),
   406    ((@{const_name uminus_class.uminus}, int_T --> int_T), 0),
   407    ((@{const_name ord_class.less}, int_T --> int_T --> bool_T), 2),
   408    ((@{const_name ord_class.less_eq}, int_T --> int_T --> bool_T), 2)]
   409 val built_in_typed_nat_consts =
   410   [((@{const_name zero_class.zero}, nat_T), 0),
   411    ((@{const_name one_class.one}, nat_T), 0),
   412    ((@{const_name plus_class.plus}, nat_T --> nat_T --> nat_T), 0),
   413    ((@{const_name minus_class.minus}, nat_T --> nat_T --> nat_T), 0),
   414    ((@{const_name times_class.times}, nat_T --> nat_T --> nat_T), 0),
   415    ((@{const_name div_class.div}, nat_T --> nat_T --> nat_T), 0),
   416    ((@{const_name ord_class.less}, nat_T --> nat_T --> bool_T), 2),
   417    ((@{const_name ord_class.less_eq}, nat_T --> nat_T --> bool_T), 2),
   418    ((@{const_name of_nat}, nat_T --> int_T), 0)]
   419 val built_in_set_consts =
   420   [(@{const_name ord_class.less_eq}, 2)]
   421 
   422 fun unarize_type @{typ "unsigned_bit word"} = nat_T
   423   | unarize_type @{typ "signed_bit word"} = int_T
   424   | unarize_type (Type (s, Ts as _ :: _)) = Type (s, map unarize_type Ts)
   425   | unarize_type T = T
   426 fun unarize_unbox_etc_type (Type (@{type_name fun_box}, Ts)) =
   427     unarize_unbox_etc_type (Type (@{type_name fun}, Ts))
   428   | unarize_unbox_etc_type (Type (@{type_name pair_box}, Ts)) =
   429     Type (@{type_name prod}, map unarize_unbox_etc_type Ts)
   430   | unarize_unbox_etc_type @{typ "unsigned_bit word"} = nat_T
   431   | unarize_unbox_etc_type @{typ "signed_bit word"} = int_T
   432   | unarize_unbox_etc_type (Type (s, Ts as _ :: _)) =
   433     Type (s, map unarize_unbox_etc_type Ts)
   434   | unarize_unbox_etc_type T = T
   435 fun uniterize_type (Type (s, Ts as _ :: _)) = Type (s, map uniterize_type Ts)
   436   | uniterize_type @{typ bisim_iterator} = nat_T
   437   | uniterize_type T = T
   438 val uniterize_unarize_unbox_etc_type = uniterize_type o unarize_unbox_etc_type
   439 
   440 fun string_for_type ctxt = Syntax.string_of_typ ctxt o unarize_unbox_etc_type
   441 fun pretty_for_type ctxt = Syntax.pretty_typ ctxt o unarize_unbox_etc_type
   442 
   443 val prefix_name = Long_Name.qualify o Long_Name.base_name
   444 fun shortest_name s = List.last (space_explode "." s) handle List.Empty => ""
   445 val prefix_abs_vars = Term.map_abs_vars o prefix_name
   446 fun short_name s =
   447   case space_explode name_sep s of
   448     [_] => s |> String.isPrefix nitpick_prefix s ? unprefix nitpick_prefix
   449   | ss => map shortest_name ss |> space_implode "_"
   450 fun shorten_names_in_type (Type (s, Ts)) =
   451     Type (short_name s, map shorten_names_in_type Ts)
   452   | shorten_names_in_type T = T
   453 val shorten_names_in_term =
   454   map_aterms (fn Const (s, T) => Const (short_name s, T) | t => t)
   455   #> map_types shorten_names_in_type
   456 
   457 fun strict_type_match thy (T1, T2) =
   458   (Sign.typ_match thy (T2, T1) Vartab.empty; true)
   459   handle Type.TYPE_MATCH => false
   460 fun type_match thy = strict_type_match thy o pairself unarize_unbox_etc_type
   461 fun const_match thy ((s1, T1), (s2, T2)) =
   462   s1 = s2 andalso type_match thy (T1, T2)
   463 fun term_match thy (Const x1, Const x2) = const_match thy (x1, x2)
   464   | term_match thy (Free (s1, T1), Free (s2, T2)) =
   465     const_match thy ((shortest_name s1, T1), (shortest_name s2, T2))
   466   | term_match _ (t1, t2) = t1 aconv t2
   467 
   468 fun frac_from_term_pair T t1 t2 =
   469   case snd (HOLogic.dest_number t1) of
   470     0 => HOLogic.mk_number T 0
   471   | n1 => case snd (HOLogic.dest_number t2) of
   472             1 => HOLogic.mk_number T n1
   473           | n2 => Const (@{const_name divide}, T --> T --> T)
   474                   $ HOLogic.mk_number T n1 $ HOLogic.mk_number T n2
   475 
   476 fun is_TFree (TFree _) = true
   477   | is_TFree _ = false
   478 fun is_fun_type (Type (@{type_name fun}, _)) = true
   479   | is_fun_type _ = false
   480 fun is_set_type (Type (@{type_name fun}, [_, @{typ bool}])) = true
   481   | is_set_type _ = false
   482 fun is_pair_type (Type (@{type_name prod}, _)) = true
   483   | is_pair_type _ = false
   484 fun is_lfp_iterator_type (Type (s, _)) = String.isPrefix lfp_iterator_prefix s
   485   | is_lfp_iterator_type _ = false
   486 fun is_gfp_iterator_type (Type (s, _)) = String.isPrefix gfp_iterator_prefix s
   487   | is_gfp_iterator_type _ = false
   488 val is_fp_iterator_type = is_lfp_iterator_type orf is_gfp_iterator_type
   489 fun is_iterator_type T =
   490   (T = @{typ bisim_iterator} orelse is_fp_iterator_type T)
   491 fun is_boolean_type T = (T = prop_T orelse T = bool_T)
   492 fun is_integer_type T = (T = nat_T orelse T = int_T)
   493 fun is_bit_type T = (T = @{typ unsigned_bit} orelse T = @{typ signed_bit})
   494 fun is_word_type (Type (@{type_name word}, _)) = true
   495   | is_word_type _ = false
   496 val is_integer_like_type = is_iterator_type orf is_integer_type orf is_word_type
   497 val is_record_type = not o null o Record.dest_recTs
   498 fun is_frac_type ctxt (Type (s, [])) =
   499     s |> AList.defined (op =) (#frac_types (Data.get (Context.Proof ctxt)))
   500   | is_frac_type _ _ = false
   501 fun is_number_type ctxt = is_integer_like_type orf is_frac_type ctxt
   502 fun is_higher_order_type (Type (@{type_name fun}, _)) = true
   503   | is_higher_order_type (Type (_, Ts)) = exists is_higher_order_type Ts
   504   | is_higher_order_type _ = false
   505 
   506 fun iterator_type_for_const gfp (s, T) =
   507   Type ((if gfp then gfp_iterator_prefix else lfp_iterator_prefix) ^ s,
   508         binder_types T)
   509 fun const_for_iterator_type (Type (s, Ts)) =
   510     (strip_first_name_sep s |> snd, Ts ---> bool_T)
   511   | const_for_iterator_type T =
   512     raise TYPE ("Nitpick_HOL.const_for_iterator_type", [T], [])
   513 
   514 fun strip_n_binders 0 T = ([], T)
   515   | strip_n_binders n (Type (@{type_name fun}, [T1, T2])) =
   516     strip_n_binders (n - 1) T2 |>> cons T1
   517   | strip_n_binders n (Type (@{type_name fun_box}, Ts)) =
   518     strip_n_binders n (Type (@{type_name fun}, Ts))
   519   | strip_n_binders _ T = raise TYPE ("Nitpick_HOL.strip_n_binders", [T], [])
   520 val nth_range_type = snd oo strip_n_binders
   521 
   522 fun num_factors_in_type (Type (@{type_name prod}, [T1, T2])) =
   523     fold (Integer.add o num_factors_in_type) [T1, T2] 0
   524   | num_factors_in_type _ = 1
   525 fun num_binder_types (Type (@{type_name fun}, [_, T2])) =
   526     1 + num_binder_types T2
   527   | num_binder_types _ = 0
   528 val curried_binder_types = maps HOLogic.flatten_tupleT o binder_types
   529 fun maybe_curried_binder_types T =
   530   (if is_pair_type (body_type T) then binder_types else curried_binder_types) T
   531 
   532 fun mk_flat_tuple _ [t] = t
   533   | mk_flat_tuple (Type (@{type_name prod}, [T1, T2])) (t :: ts) =
   534     HOLogic.pair_const T1 T2 $ t $ (mk_flat_tuple T2 ts)
   535   | mk_flat_tuple T ts = raise TYPE ("Nitpick_HOL.mk_flat_tuple", [T], ts)
   536 fun dest_n_tuple 1 t = [t]
   537   | dest_n_tuple n t = HOLogic.dest_prod t ||> dest_n_tuple (n - 1) |> op ::
   538 
   539 type typedef_info =
   540   {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string,
   541    set_def: thm option, prop_of_Rep: thm, set_name: string,
   542    Abs_inverse: thm option, Rep_inverse: thm option}
   543 
   544 fun typedef_info ctxt s =
   545   if is_frac_type ctxt (Type (s, [])) then
   546     SOME {abs_type = Type (s, []), rep_type = @{typ "int * int"},
   547           Abs_name = @{const_name Abs_Frac}, Rep_name = @{const_name Rep_Frac},
   548           set_def = NONE, prop_of_Rep = @{prop "Rep_Frac x \<in> Frac"}
   549                           |> Logic.varify_global,
   550           set_name = @{const_name Frac}, Abs_inverse = NONE, Rep_inverse = NONE}
   551   else case Typedef.get_info ctxt s of
   552     (* When several entries are returned, it shouldn't matter much which one
   553        we take (according to Florian Haftmann). *)
   554     (* The "Logic.varifyT_global" calls are a temporary hack because these
   555        types's type variables sometimes clash with locally fixed type variables.
   556        Remove these calls once "Typedef" is fully localized. *)
   557     ({abs_type, rep_type, Abs_name, Rep_name, ...},
   558      {set_def, Rep, Abs_inverse, Rep_inverse, ...}) :: _ =>
   559     SOME {abs_type = Logic.varifyT_global abs_type,
   560           rep_type = Logic.varifyT_global rep_type, Abs_name = Abs_name,
   561           Rep_name = Rep_name, set_def = set_def, prop_of_Rep = prop_of Rep,
   562           set_name = set_prefix ^ s, Abs_inverse = SOME Abs_inverse,
   563           Rep_inverse = SOME Rep_inverse}
   564   | _ => NONE
   565 
   566 val is_typedef = is_some oo typedef_info
   567 val is_real_datatype = is_some oo Datatype.get_info
   568 fun is_standard_datatype thy = the oo triple_lookup (type_match thy)
   569 
   570 (* FIXME: Use antiquotation for "code_numeral" below or detect "rep_datatype",
   571    e.g., by adding a field to "Datatype_Aux.info". *)
   572 fun is_basic_datatype thy stds s =
   573   member (op =) [@{type_name prod}, @{type_name bool}, @{type_name int},
   574                  "Code_Numeral.code_numeral"] s orelse
   575   (s = @{type_name nat} andalso is_standard_datatype thy stds nat_T)
   576 
   577 fun repair_constr_type ctxt body_T' T =
   578   varify_and_instantiate_type ctxt (body_type T) body_T' T
   579 
   580 fun register_frac_type_generic frac_s ersaetze generic =
   581   let
   582     val {frac_types, ersatz_table, codatatypes} = Data.get generic
   583     val frac_types = AList.update (op =) (frac_s, ersaetze) frac_types
   584   in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
   585                codatatypes = codatatypes} generic end
   586 (* TODO: Consider morphism. *)
   587 fun register_frac_type frac_s ersaetze (_ : morphism) =
   588   register_frac_type_generic frac_s ersaetze
   589 val register_frac_type_global = Context.theory_map oo register_frac_type_generic
   590 
   591 fun unregister_frac_type_generic frac_s = register_frac_type_generic frac_s []
   592 (* TODO: Consider morphism. *)
   593 fun unregister_frac_type frac_s (_ : morphism) =
   594   unregister_frac_type_generic frac_s
   595 val unregister_frac_type_global =
   596   Context.theory_map o unregister_frac_type_generic
   597 
   598 fun register_ersatz_generic ersatz generic =
   599   let
   600     val {frac_types, ersatz_table, codatatypes} = Data.get generic
   601     val ersatz_table = AList.merge (op =) (K true) (ersatz_table, ersatz)
   602   in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
   603                codatatypes = codatatypes} generic end
   604 (* TODO: Consider morphism. *)
   605 fun register_ersatz ersatz (_ : morphism) =
   606   register_ersatz_generic ersatz
   607 val register_ersatz_global = Context.theory_map o register_ersatz_generic
   608 
   609 fun register_codatatype_generic co_T case_name constr_xs generic =
   610   let
   611     val ctxt = Context.proof_of generic
   612     val thy = Context.theory_of generic
   613     val {frac_types, ersatz_table, codatatypes} = Data.get generic
   614     val constr_xs = map (apsnd (repair_constr_type ctxt co_T)) constr_xs
   615     val (co_s, co_Ts) = dest_Type co_T
   616     val _ =
   617       if forall is_TFree co_Ts andalso not (has_duplicates (op =) co_Ts) andalso
   618          co_s <> @{type_name fun} andalso
   619          not (is_basic_datatype thy [(NONE, true)] co_s) then
   620         ()
   621       else
   622         raise TYPE ("Nitpick_HOL.register_codatatype_generic", [co_T], [])
   623     val codatatypes = AList.update (op =) (co_s, (case_name, constr_xs))
   624                                    codatatypes
   625   in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
   626                codatatypes = codatatypes} generic end
   627 (* TODO: Consider morphism. *)
   628 fun register_codatatype co_T case_name constr_xs (_ : morphism) =
   629   register_codatatype_generic co_T case_name constr_xs
   630 val register_codatatype_global =
   631   Context.theory_map ooo register_codatatype_generic
   632 
   633 fun unregister_codatatype_generic co_T = register_codatatype_generic co_T "" []
   634 (* TODO: Consider morphism. *)
   635 fun unregister_codatatype co_T (_ : morphism) =
   636   unregister_codatatype_generic co_T
   637 val unregister_codatatype_global =
   638   Context.theory_map o unregister_codatatype_generic
   639 
   640 fun is_codatatype ctxt (Type (s, _)) =
   641     s |> AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   642       |> Option.map snd |> these |> null |> not
   643   | is_codatatype _ _ = false
   644 fun is_real_quot_type thy (Type (s, _)) =
   645     is_some (Quotient_Info.quotdata_lookup_raw thy s)
   646   | is_real_quot_type _ _ = false
   647 fun is_quot_type ctxt T =
   648   let val thy = Proof_Context.theory_of ctxt in
   649     is_real_quot_type thy T andalso not (is_codatatype ctxt T)
   650   end
   651 fun is_pure_typedef ctxt (T as Type (s, _)) =
   652     let val thy = Proof_Context.theory_of ctxt in
   653       is_typedef ctxt s andalso
   654       not (is_real_datatype thy s orelse is_real_quot_type thy T orelse
   655            is_codatatype ctxt T orelse is_record_type T orelse
   656            is_integer_like_type T)
   657     end
   658   | is_pure_typedef _ _ = false
   659 fun is_univ_typedef ctxt (Type (s, _)) =
   660     (case typedef_info ctxt s of
   661        SOME {set_def, prop_of_Rep, ...} =>
   662        let
   663          val t_opt =
   664            case set_def of
   665              SOME thm => try (snd o Logic.dest_equals o prop_of) thm
   666            | NONE => try (snd o HOLogic.dest_mem o HOLogic.dest_Trueprop)
   667                          prop_of_Rep
   668        in
   669          case t_opt of
   670            SOME (Const (@{const_name top}, _)) => true
   671            (* "Multiset.multiset" *)
   672          | SOME (Const (@{const_name Collect}, _)
   673                  $ Abs (_, _, Const (@{const_name finite}, _) $ _)) => true
   674            (* "FinFun.finfun" *)
   675          | SOME (Const (@{const_name Collect}, _) $ Abs (_, _,
   676                      Const (@{const_name Ex}, _) $ Abs (_, _,
   677                          Const (@{const_name finite}, _) $ _))) => true
   678          | _ => false
   679        end
   680      | NONE => false)
   681   | is_univ_typedef _ _ = false
   682 fun is_datatype ctxt stds (T as Type (s, _)) =
   683     let val thy = Proof_Context.theory_of ctxt in
   684       (is_typedef ctxt s orelse is_codatatype ctxt T orelse
   685        T = @{typ ind} orelse is_real_quot_type thy T) andalso
   686       not (is_basic_datatype thy stds s)
   687     end
   688   | is_datatype _ _ _ = false
   689 
   690 fun all_record_fields thy T =
   691   let val (recs, more) = Record.get_extT_fields thy T in
   692     recs @ more :: all_record_fields thy (snd more)
   693   end
   694   handle TYPE _ => []
   695 fun is_record_constr (s, T) =
   696   String.isSuffix Record.extN s andalso
   697   let val dataT = body_type T in
   698     is_record_type dataT andalso
   699     s = unsuffix Record.ext_typeN (fst (dest_Type dataT)) ^ Record.extN
   700   end
   701 val num_record_fields = Integer.add 1 o length o fst oo Record.get_extT_fields
   702 fun no_of_record_field thy s T1 =
   703   find_index (curry (op =) s o fst)
   704              (Record.get_extT_fields thy T1 ||> single |> op @)
   705 fun is_record_get thy (s, Type (@{type_name fun}, [T1, _])) =
   706     exists (curry (op =) s o fst) (all_record_fields thy T1)
   707   | is_record_get _ _ = false
   708 fun is_record_update thy (s, T) =
   709   String.isSuffix Record.updateN s andalso
   710   exists (curry (op =) (unsuffix Record.updateN s) o fst)
   711          (all_record_fields thy (body_type T))
   712   handle TYPE _ => false
   713 fun is_abs_fun ctxt (s, Type (@{type_name fun}, [_, Type (s', _)])) =
   714     (case typedef_info ctxt s' of
   715        SOME {Abs_name, ...} => s = Abs_name
   716      | NONE => false)
   717   | is_abs_fun _ _ = false
   718 fun is_rep_fun ctxt (s, Type (@{type_name fun}, [Type (s', _), _])) =
   719     (case typedef_info ctxt s' of
   720        SOME {Rep_name, ...} => s = Rep_name
   721      | NONE => false)
   722   | is_rep_fun _ _ = false
   723 fun is_quot_abs_fun ctxt (x as (_, Type (@{type_name fun},
   724                                          [_, abs_T as Type (s', _)]))) =
   725     try (Quotient_Term.absrep_const_chk Quotient_Term.AbsF ctxt) s'
   726     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   727   | is_quot_abs_fun _ _ = false
   728 fun is_quot_rep_fun ctxt (x as (_, Type (@{type_name fun},
   729                                          [abs_T as Type (s', _), _]))) =
   730     try (Quotient_Term.absrep_const_chk Quotient_Term.RepF ctxt) s'
   731     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   732   | is_quot_rep_fun _ _ = false
   733 
   734 fun mate_of_rep_fun ctxt (x as (_, Type (@{type_name fun},
   735                                          [T1 as Type (s', _), T2]))) =
   736     (case typedef_info ctxt s' of
   737        SOME {Abs_name, ...} => (Abs_name, Type (@{type_name fun}, [T2, T1]))
   738      | NONE => raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x]))
   739   | mate_of_rep_fun _ x = raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x])
   740 fun rep_type_for_quot_type thy (T as Type (s, _)) =
   741     let val {qtyp, rtyp, ...} = Quotient_Info.quotdata_lookup thy s in
   742       instantiate_type thy qtyp T rtyp
   743     end
   744   | rep_type_for_quot_type _ T =
   745     raise TYPE ("Nitpick_HOL.rep_type_for_quot_type", [T], [])
   746 fun equiv_relation_for_quot_type thy (Type (s, Ts)) =
   747     let
   748       val {qtyp, equiv_rel, equiv_thm, ...} =
   749         Quotient_Info.quotdata_lookup thy s
   750       val partial =
   751         case prop_of equiv_thm of
   752           @{const Trueprop} $ (Const (@{const_name equivp}, _) $ _) => false
   753         | @{const Trueprop} $ (Const (@{const_name part_equivp}, _) $ _) => true
   754         | _ => raise NOT_SUPPORTED "Ill-formed quotient type equivalence \
   755                                    \relation theorem"
   756       val Ts' = qtyp |> dest_Type |> snd
   757     in (subst_atomic_types (Ts' ~~ Ts) equiv_rel, partial) end
   758   | equiv_relation_for_quot_type _ T =
   759     raise TYPE ("Nitpick_HOL.equiv_relation_for_quot_type", [T], [])
   760 
   761 fun is_coconstr ctxt (s, T) =
   762   case body_type T of
   763     co_T as Type (co_s, _) =>
   764     let val {codatatypes, ...} = Data.get (Context.Proof ctxt) in
   765       exists (fn (s', T') => s = s' andalso repair_constr_type ctxt co_T T' = T)
   766              (AList.lookup (op =) codatatypes co_s |> Option.map snd |> these)
   767     end
   768   | _ => false
   769 fun is_constr_like ctxt (s, T) =
   770   member (op =) [@{const_name FunBox}, @{const_name PairBox},
   771                  @{const_name Quot}, @{const_name Zero_Rep},
   772                  @{const_name Suc_Rep}] s orelse
   773   let
   774     val thy = Proof_Context.theory_of ctxt
   775     val (x as (_, T)) = (s, unarize_unbox_etc_type T)
   776   in
   777     is_real_constr thy x orelse is_record_constr x orelse
   778     (is_abs_fun ctxt x andalso is_pure_typedef ctxt (range_type T)) orelse
   779     is_coconstr ctxt x
   780   end
   781 fun is_stale_constr ctxt (x as (_, T)) =
   782   is_codatatype ctxt (body_type T) andalso is_constr_like ctxt x andalso
   783   not (is_coconstr ctxt x)
   784 fun is_constr ctxt stds (x as (_, T)) =
   785   let val thy = Proof_Context.theory_of ctxt in
   786     is_constr_like ctxt x andalso
   787     not (is_basic_datatype thy stds
   788                          (fst (dest_Type (unarize_type (body_type T))))) andalso
   789     not (is_stale_constr ctxt x)
   790   end
   791 val is_sel = String.isPrefix discr_prefix orf String.isPrefix sel_prefix
   792 val is_sel_like_and_no_discr =
   793   String.isPrefix sel_prefix orf
   794   (member (op =) [@{const_name fst}, @{const_name snd}])
   795 
   796 fun in_fun_lhs_for InConstr = InSel
   797   | in_fun_lhs_for _ = InFunLHS
   798 fun in_fun_rhs_for InConstr = InConstr
   799   | in_fun_rhs_for InSel = InSel
   800   | in_fun_rhs_for InFunRHS1 = InFunRHS2
   801   | in_fun_rhs_for _ = InFunRHS1
   802 
   803 fun is_boxing_worth_it (hol_ctxt : hol_context) boxy T =
   804   case T of
   805     Type (@{type_name fun}, _) =>
   806     (boxy = InPair orelse boxy = InFunLHS) andalso
   807     not (is_boolean_type (body_type T))
   808   | Type (@{type_name prod}, Ts) =>
   809     boxy = InPair orelse boxy = InFunRHS1 orelse boxy = InFunRHS2 orelse
   810     ((boxy = InExpr orelse boxy = InFunLHS) andalso
   811      exists (is_boxing_worth_it hol_ctxt InPair)
   812             (map (box_type hol_ctxt InPair) Ts))
   813   | _ => false
   814 and should_box_type (hol_ctxt as {thy, boxes, ...}) boxy z =
   815   case triple_lookup (type_match thy) boxes (Type z) of
   816     SOME (SOME box_me) => box_me
   817   | _ => is_boxing_worth_it hol_ctxt boxy (Type z)
   818 and box_type hol_ctxt boxy T =
   819   case T of
   820     Type (z as (@{type_name fun}, [T1, T2])) =>
   821     if boxy <> InConstr andalso boxy <> InSel andalso
   822        should_box_type hol_ctxt boxy z then
   823       Type (@{type_name fun_box},
   824             [box_type hol_ctxt InFunLHS T1, box_type hol_ctxt InFunRHS1 T2])
   825     else
   826       box_type hol_ctxt (in_fun_lhs_for boxy) T1
   827       --> box_type hol_ctxt (in_fun_rhs_for boxy) T2
   828   | Type (z as (@{type_name prod}, Ts)) =>
   829     if boxy <> InConstr andalso boxy <> InSel
   830        andalso should_box_type hol_ctxt boxy z then
   831       Type (@{type_name pair_box}, map (box_type hol_ctxt InSel) Ts)
   832     else
   833       Type (@{type_name prod},
   834             map (box_type hol_ctxt
   835                           (if boxy = InConstr orelse boxy = InSel then boxy
   836                            else InPair)) Ts)
   837   | _ => T
   838 
   839 fun binarize_nat_and_int_in_type @{typ nat} = @{typ "unsigned_bit word"}
   840   | binarize_nat_and_int_in_type @{typ int} = @{typ "signed_bit word"}
   841   | binarize_nat_and_int_in_type (Type (s, Ts)) =
   842     Type (s, map binarize_nat_and_int_in_type Ts)
   843   | binarize_nat_and_int_in_type T = T
   844 val binarize_nat_and_int_in_term = map_types binarize_nat_and_int_in_type
   845 
   846 fun discr_for_constr (s, T) = (discr_prefix ^ s, body_type T --> bool_T)
   847 
   848 fun num_sels_for_constr_type T = length (maybe_curried_binder_types T)
   849 fun nth_sel_name_for_constr_name s n =
   850   if s = @{const_name Pair} then
   851     if n = 0 then @{const_name fst} else @{const_name snd}
   852   else
   853     sel_prefix_for n ^ s
   854 fun nth_sel_for_constr x ~1 = discr_for_constr x
   855   | nth_sel_for_constr (s, T) n =
   856     (nth_sel_name_for_constr_name s n,
   857      body_type T --> nth (maybe_curried_binder_types T) n)
   858 fun binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize =
   859   apsnd ((binarize ? binarize_nat_and_int_in_type) o box_type hol_ctxt InSel)
   860   oo nth_sel_for_constr
   861 
   862 fun sel_no_from_name s =
   863   if String.isPrefix discr_prefix s then
   864     ~1
   865   else if String.isPrefix sel_prefix s then
   866     s |> unprefix sel_prefix |> Int.fromString |> the
   867   else if s = @{const_name snd} then
   868     1
   869   else
   870     0
   871 
   872 val close_form =
   873   let
   874     fun close_up zs zs' =
   875       fold (fn (z as ((s, _), T)) => fn t' =>
   876                Term.all T $ Abs (s, T, abstract_over (Var z, t')))
   877            (take (length zs' - length zs) zs')
   878     fun aux zs (@{const "==>"} $ t1 $ t2) =
   879         let val zs' = Term.add_vars t1 zs in
   880           close_up zs zs' (Logic.mk_implies (t1, aux zs' t2))
   881         end
   882       | aux zs t = close_up zs (Term.add_vars t zs) t
   883   in aux [] end
   884 
   885 fun distinctness_formula T =
   886   all_distinct_unordered_pairs_of
   887   #> map (fn (t1, t2) => @{const Not} $ (HOLogic.eq_const T $ t1 $ t2))
   888   #> List.foldr (s_conj o swap) @{const True}
   889 
   890 fun zero_const T = Const (@{const_name zero_class.zero}, T)
   891 fun suc_const T = Const (@{const_name Suc}, T --> T)
   892 
   893 fun uncached_datatype_constrs ({thy, ctxt, stds, ...} : hol_context)
   894                               (T as Type (s, Ts)) =
   895     (case AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   896                        s of
   897        SOME (_, xs' as (_ :: _)) => map (apsnd (repair_constr_type ctxt T)) xs'
   898      | _ =>
   899        if is_datatype ctxt stds T then
   900          case Datatype.get_info thy s of
   901            SOME {index, descr, ...} =>
   902            let
   903              val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the
   904            in
   905              map (apsnd (fn Us =>
   906                             map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))
   907                  constrs
   908            end
   909          | NONE =>
   910            if is_record_type T then
   911              let
   912                val s' = unsuffix Record.ext_typeN s ^ Record.extN
   913                val T' = (Record.get_extT_fields thy T
   914                         |> apsnd single |> uncurry append |> map snd) ---> T
   915              in [(s', T')] end
   916            else if is_real_quot_type thy T then
   917              [(@{const_name Quot}, rep_type_for_quot_type thy T --> T)]
   918            else case typedef_info ctxt s of
   919              SOME {abs_type, rep_type, Abs_name, ...} =>
   920              [(Abs_name,
   921                varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
   922            | NONE =>
   923              if T = @{typ ind} then
   924                [dest_Const @{const Zero_Rep}, dest_Const @{const Suc_Rep}]
   925              else
   926                []
   927        else
   928          [])
   929   | uncached_datatype_constrs _ _ = []
   930 fun datatype_constrs (hol_ctxt as {constr_cache, ...}) T =
   931   case AList.lookup (op =) (!constr_cache) T of
   932     SOME xs => xs
   933   | NONE =>
   934     let val xs = uncached_datatype_constrs hol_ctxt T in
   935       (Unsynchronized.change constr_cache (cons (T, xs)); xs)
   936     end
   937 fun binarized_and_boxed_datatype_constrs hol_ctxt binarize =
   938   map (apsnd ((binarize ? binarize_nat_and_int_in_type)
   939               o box_type hol_ctxt InConstr)) o datatype_constrs hol_ctxt
   940 val num_datatype_constrs = length oo datatype_constrs
   941 
   942 fun constr_name_for_sel_like @{const_name fst} = @{const_name Pair}
   943   | constr_name_for_sel_like @{const_name snd} = @{const_name Pair}
   944   | constr_name_for_sel_like s' = original_name s'
   945 fun binarized_and_boxed_constr_for_sel hol_ctxt binarize (s', T') =
   946   let val s = constr_name_for_sel_like s' in
   947     AList.lookup (op =)
   948         (binarized_and_boxed_datatype_constrs hol_ctxt binarize (domain_type T'))
   949         s
   950     |> the |> pair s
   951   end
   952 
   953 fun card_of_type assigns (Type (@{type_name fun}, [T1, T2])) =
   954     reasonable_power (card_of_type assigns T2) (card_of_type assigns T1)
   955   | card_of_type assigns (Type (@{type_name prod}, [T1, T2])) =
   956     card_of_type assigns T1 * card_of_type assigns T2
   957   | card_of_type _ (Type (@{type_name itself}, _)) = 1
   958   | card_of_type _ @{typ prop} = 2
   959   | card_of_type _ @{typ bool} = 2
   960   | card_of_type assigns T =
   961     case AList.lookup (op =) assigns T of
   962       SOME k => k
   963     | NONE => if T = @{typ bisim_iterator} then 0
   964               else raise TYPE ("Nitpick_HOL.card_of_type", [T], [])
   965 
   966 fun bounded_card_of_type max default_card assigns
   967                          (Type (@{type_name fun}, [T1, T2])) =
   968     let
   969       val k1 = bounded_card_of_type max default_card assigns T1
   970       val k2 = bounded_card_of_type max default_card assigns T2
   971     in
   972       if k1 = max orelse k2 = max then max
   973       else Int.min (max, reasonable_power k2 k1)
   974     end
   975   | bounded_card_of_type max default_card assigns
   976                          (Type (@{type_name prod}, [T1, T2])) =
   977     let
   978       val k1 = bounded_card_of_type max default_card assigns T1
   979       val k2 = bounded_card_of_type max default_card assigns T2
   980     in if k1 = max orelse k2 = max then max else Int.min (max, k1 * k2) end
   981   | bounded_card_of_type max default_card assigns T =
   982     Int.min (max, if default_card = ~1 then
   983                     card_of_type assigns T
   984                   else
   985                     card_of_type assigns T
   986                     handle TYPE ("Nitpick_HOL.card_of_type", _, _) =>
   987                            default_card)
   988 
   989 (* Similar to "ATP_Translate.tiny_card_of_type". *)
   990 fun bounded_exact_card_of_type hol_ctxt finitizable_dataTs max default_card
   991                                assigns T =
   992   let
   993     fun aux avoid T =
   994       (if member (op =) avoid T then
   995          0
   996        else if member (op =) finitizable_dataTs T then
   997          raise SAME ()
   998        else case T of
   999          Type (@{type_name fun}, [T1, T2]) =>
  1000          (case (aux avoid T1, aux avoid T2) of
  1001             (_, 1) => 1
  1002           | (0, _) => 0
  1003           | (_, 0) => 0
  1004           | (k1, k2) =>
  1005             if k1 >= max orelse k2 >= max then max
  1006             else Int.min (max, reasonable_power k2 k1))
  1007        | Type (@{type_name prod}, [T1, T2]) =>
  1008          (case (aux avoid T1, aux avoid T2) of
  1009             (0, _) => 0
  1010           | (_, 0) => 0
  1011           | (k1, k2) =>
  1012             if k1 >= max orelse k2 >= max then max
  1013             else Int.min (max, k1 * k2))
  1014        | Type (@{type_name itself}, _) => 1
  1015        | @{typ prop} => 2
  1016        | @{typ bool} => 2
  1017        | Type _ =>
  1018          (case datatype_constrs hol_ctxt T of
  1019             [] => if is_integer_type T orelse is_bit_type T then 0
  1020                   else raise SAME ()
  1021           | constrs =>
  1022             let
  1023               val constr_cards =
  1024                 map (Integer.prod o map (aux (T :: avoid)) o binder_types o snd)
  1025                     constrs
  1026             in
  1027               if exists (curry (op =) 0) constr_cards then 0
  1028               else Int.min (max, Integer.sum constr_cards)
  1029             end)
  1030        | _ => raise SAME ())
  1031       handle SAME () =>
  1032              AList.lookup (op =) assigns T |> the_default default_card
  1033   in Int.min (max, aux [] T) end
  1034 
  1035 val typical_atomic_card = 4
  1036 val typical_card_of_type = bounded_card_of_type 16777217 typical_atomic_card []
  1037 
  1038 fun is_finite_type hol_ctxt T =
  1039   bounded_exact_card_of_type hol_ctxt [] 1 2 [] T > 0
  1040 
  1041 fun is_special_eligible_arg strict Ts t =
  1042   case map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t) of
  1043     [] => true
  1044   | bad_Ts =>
  1045     let
  1046       val bad_Ts_cost =
  1047         if strict then fold (curry (op *) o typical_card_of_type) bad_Ts 1
  1048         else fold (Integer.max o typical_card_of_type) bad_Ts 0
  1049       val T_cost = typical_card_of_type (fastype_of1 (Ts, t))
  1050     in (bad_Ts_cost, T_cost) |> (if strict then op < else op <=) end
  1051 
  1052 fun abs_var ((s, j), T) body = Abs (s, T, abstract_over (Var ((s, j), T), body))
  1053 
  1054 fun let_var s = (nitpick_prefix ^ s, 999)
  1055 val let_inline_threshold = 20
  1056 
  1057 fun s_let Ts s n abs_T body_T f t =
  1058   if (n - 1) * (size_of_term t - 1) <= let_inline_threshold orelse
  1059      is_special_eligible_arg false Ts t then
  1060     f t
  1061   else
  1062     let val z = (let_var s, abs_T) in
  1063       Const (@{const_name Let}, abs_T --> (abs_T --> body_T) --> body_T)
  1064       $ t $ abs_var z (incr_boundvars 1 (f (Var z)))
  1065     end
  1066 
  1067 fun loose_bvar1_count (Bound i, k) = if i = k then 1 else 0
  1068   | loose_bvar1_count (t1 $ t2, k) =
  1069     loose_bvar1_count (t1, k) + loose_bvar1_count (t2, k)
  1070   | loose_bvar1_count (Abs (_, _, t), k) = loose_bvar1_count (t, k + 1)
  1071   | loose_bvar1_count _ = 0
  1072 
  1073 fun s_betapply _ (t1 as Const (@{const_name "=="}, _) $ t1', t2) =
  1074     if t1' aconv t2 then @{prop True} else t1 $ t2
  1075   | s_betapply _ (t1 as Const (@{const_name HOL.eq}, _) $ t1', t2) =
  1076     if t1' aconv t2 then @{term True} else t1 $ t2
  1077   | s_betapply _ (Const (@{const_name If}, _) $ @{const True} $ t1', _) = t1'
  1078   | s_betapply _ (Const (@{const_name If}, _) $ @{const False} $ _, t2) = t2
  1079   | s_betapply Ts (Const (@{const_name Let},
  1080                           Type (_, [bound_T, Type (_, [_, body_T])]))
  1081                    $ t12 $ Abs (s, T, t13'), t2) =
  1082     let val body_T' = range_type body_T in
  1083       Const (@{const_name Let}, bound_T --> (bound_T --> body_T') --> body_T')
  1084       $ t12 $ Abs (s, T, s_betapply (T :: Ts) (t13', incr_boundvars 1 t2))
  1085     end
  1086   | s_betapply Ts (t1 as Abs (s1, T1, t1'), t2) =
  1087     (s_let Ts s1 (loose_bvar1_count (t1', 0)) T1 (fastype_of1 (T1 :: Ts, t1'))
  1088            (curry betapply t1) t2
  1089      (* FIXME: fix all "s_betapply []" calls *)
  1090      handle TERM _ => betapply (t1, t2)
  1091           | General.Subscript => betapply (t1, t2))
  1092   | s_betapply _ (t1, t2) = t1 $ t2
  1093 fun s_betapplys Ts = Library.foldl (s_betapply Ts)
  1094 
  1095 fun s_beta_norm Ts t =
  1096   let
  1097     fun aux _ (Var _) = raise Same.SAME
  1098       | aux Ts (Abs (s, T, t')) = Abs (s, T, aux (T :: Ts) t')
  1099       | aux Ts ((t1 as Abs _) $ t2) =
  1100         Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1101       | aux Ts (t1 $ t2) =
  1102         ((case aux Ts t1 of
  1103            t1 as Abs _ => Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1104          | t1 => t1 $ Same.commit (aux Ts) t2)
  1105         handle Same.SAME => t1 $ aux Ts t2)
  1106       | aux _ _ = raise Same.SAME
  1107   in aux Ts t handle Same.SAME => t end
  1108 
  1109 fun discr_term_for_constr hol_ctxt (x as (s, T)) =
  1110   let val dataT = body_type T in
  1111     if s = @{const_name Suc} then
  1112       Abs (Name.uu, dataT,
  1113            @{const Not} $ HOLogic.mk_eq (zero_const dataT, Bound 0))
  1114     else if num_datatype_constrs hol_ctxt dataT >= 2 then
  1115       Const (discr_for_constr x)
  1116     else
  1117       Abs (Name.uu, dataT, @{const True})
  1118   end
  1119 fun discriminate_value (hol_ctxt as {ctxt, ...}) x t =
  1120   case head_of t of
  1121     Const x' =>
  1122     if x = x' then @{const True}
  1123     else if is_constr_like ctxt x' then @{const False}
  1124     else s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1125   | _ => s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1126 
  1127 fun nth_arg_sel_term_for_constr thy stds (x as (s, T)) n =
  1128   let val (arg_Ts, dataT) = strip_type T in
  1129     if dataT = nat_T andalso is_standard_datatype thy stds nat_T then
  1130       @{term "%n::nat. n - 1"}
  1131     else if is_pair_type dataT then
  1132       Const (nth_sel_for_constr x n)
  1133     else
  1134       let
  1135         fun aux m (Type (@{type_name prod}, [T1, T2])) =
  1136             let
  1137               val (m, t1) = aux m T1
  1138               val (m, t2) = aux m T2
  1139             in (m, HOLogic.mk_prod (t1, t2)) end
  1140           | aux m T =
  1141             (m + 1, Const (nth_sel_name_for_constr_name s m, dataT --> T)
  1142                     $ Bound 0)
  1143         val m = fold (Integer.add o num_factors_in_type)
  1144                      (List.take (arg_Ts, n)) 0
  1145       in Abs ("x", dataT, aux m (nth arg_Ts n) |> snd) end
  1146   end
  1147 fun select_nth_constr_arg ctxt stds x t n res_T =
  1148   let val thy = Proof_Context.theory_of ctxt in
  1149     (case strip_comb t of
  1150        (Const x', args) =>
  1151        if x = x' then nth args n
  1152        else if is_constr_like ctxt x' then Const (@{const_name unknown}, res_T)
  1153        else raise SAME ()
  1154      | _ => raise SAME())
  1155     handle SAME () =>
  1156            s_betapply [] (nth_arg_sel_term_for_constr thy stds x n, t)
  1157   end
  1158 
  1159 fun construct_value _ _ x [] = Const x
  1160   | construct_value ctxt stds (x as (s, _)) args =
  1161     let val args = map Envir.eta_contract args in
  1162       case hd args of
  1163         Const (s', _) $ t =>
  1164         if is_sel_like_and_no_discr s' andalso
  1165            constr_name_for_sel_like s' = s andalso
  1166            forall (fn (n, t') =>
  1167                       select_nth_constr_arg ctxt stds x t n dummyT = t')
  1168                   (index_seq 0 (length args) ~~ args) then
  1169           t
  1170         else
  1171           list_comb (Const x, args)
  1172       | _ => list_comb (Const x, args)
  1173     end
  1174 
  1175 fun constr_expand (hol_ctxt as {ctxt, stds, ...}) T t =
  1176   (case head_of t of
  1177      Const x => if is_constr_like ctxt x then t else raise SAME ()
  1178    | _ => raise SAME ())
  1179   handle SAME () =>
  1180          let
  1181            val x' as (_, T') =
  1182              if is_pair_type T then
  1183                let val (T1, T2) = HOLogic.dest_prodT T in
  1184                  (@{const_name Pair}, T1 --> T2 --> T)
  1185                end
  1186              else
  1187                datatype_constrs hol_ctxt T |> hd
  1188            val arg_Ts = binder_types T'
  1189          in
  1190            list_comb (Const x', map2 (select_nth_constr_arg ctxt stds x' t)
  1191                                      (index_seq 0 (length arg_Ts)) arg_Ts)
  1192          end
  1193 
  1194 fun coerce_bound_no f j t =
  1195   case t of
  1196     t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2
  1197   | Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')
  1198   | Bound j' => if j' = j then f t else t
  1199   | _ => t
  1200 fun coerce_bound_0_in_term hol_ctxt new_T old_T =
  1201   old_T <> new_T ? coerce_bound_no (coerce_term hol_ctxt [new_T] old_T new_T) 0
  1202 and coerce_term (hol_ctxt as {ctxt, stds, ...}) Ts new_T old_T t =
  1203   if old_T = new_T then
  1204     t
  1205   else
  1206     case (new_T, old_T) of
  1207       (Type (new_s, new_Ts as [new_T1, new_T2]),
  1208        Type (@{type_name fun}, [old_T1, old_T2])) =>
  1209       (case eta_expand Ts t 1 of
  1210          Abs (s, _, t') =>
  1211          Abs (s, new_T1,
  1212               t' |> coerce_bound_0_in_term hol_ctxt new_T1 old_T1
  1213                  |> coerce_term hol_ctxt (new_T1 :: Ts) new_T2 old_T2)
  1214          |> Envir.eta_contract
  1215          |> new_s <> @{type_name fun}
  1216             ? construct_value ctxt stds
  1217                   (@{const_name FunBox},
  1218                    Type (@{type_name fun}, new_Ts) --> new_T)
  1219               o single
  1220        | t' => raise TERM ("Nitpick_HOL.coerce_term", [t']))
  1221     | (Type (new_s, new_Ts as [new_T1, new_T2]),
  1222        Type (old_s, old_Ts as [old_T1, old_T2])) =>
  1223       if old_s = @{type_name fun_box} orelse
  1224          old_s = @{type_name pair_box} orelse old_s = @{type_name prod} then
  1225         case constr_expand hol_ctxt old_T t of
  1226           Const (old_s, _) $ t1 =>
  1227           if new_s = @{type_name fun} then
  1228             coerce_term hol_ctxt Ts new_T (Type (@{type_name fun}, old_Ts)) t1
  1229           else
  1230             construct_value ctxt stds
  1231                 (old_s, Type (@{type_name fun}, new_Ts) --> new_T)
  1232                 [coerce_term hol_ctxt Ts (Type (@{type_name fun}, new_Ts))
  1233                              (Type (@{type_name fun}, old_Ts)) t1]
  1234         | Const _ $ t1 $ t2 =>
  1235           construct_value ctxt stds
  1236               (if new_s = @{type_name prod} then @{const_name Pair}
  1237                else @{const_name PairBox}, new_Ts ---> new_T)
  1238               (map3 (coerce_term hol_ctxt Ts) [new_T1, new_T2] [old_T1, old_T2]
  1239                     [t1, t2])
  1240         | t' => raise TERM ("Nitpick_HOL.coerce_term", [t'])
  1241       else
  1242         raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1243     | _ => raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1244 
  1245 fun is_ground_term (t1 $ t2) = is_ground_term t1 andalso is_ground_term t2
  1246   | is_ground_term (Const _) = true
  1247   | is_ground_term _ = false
  1248 
  1249 fun special_bounds ts =
  1250   fold Term.add_vars ts [] |> sort (Term_Ord.fast_indexname_ord o pairself fst)
  1251 
  1252 (* FIXME: detect "rep_datatype"? *)
  1253 fun is_funky_typedef_name ctxt s =
  1254   member (op =) [@{type_name unit}, @{type_name prod},
  1255                  @{type_name Sum_Type.sum}, @{type_name int}] s orelse
  1256   is_frac_type ctxt (Type (s, []))
  1257 fun is_funky_typedef ctxt (Type (s, _)) = is_funky_typedef_name ctxt s
  1258   | is_funky_typedef _ _ = false
  1259 fun is_typedef_axiom ctxt boring (@{const "==>"} $ _ $ t2) =
  1260     is_typedef_axiom ctxt boring t2
  1261   | is_typedef_axiom ctxt boring
  1262         (@{const Trueprop} $ (Const (@{const_name Typedef.type_definition}, _)
  1263          $ Const (_, Type (@{type_name fun}, [Type (s, _), _]))
  1264          $ Const _ $ _)) =
  1265     boring <> is_funky_typedef_name ctxt s andalso is_typedef ctxt s
  1266   | is_typedef_axiom _ _ _ = false
  1267 
  1268 fun all_defs_of thy subst =
  1269   let
  1270     val def_names =
  1271       thy |> Theory.defs_of
  1272           |> Defs.all_specifications_of
  1273           |> maps snd |> map_filter #def
  1274           |> Ord_List.make fast_string_ord
  1275   in
  1276     Theory.nodes_of thy
  1277     |> maps Thm.axioms_of
  1278     |> map (apsnd (subst_atomic subst o prop_of))
  1279     |> sort (fast_string_ord o pairself fst)
  1280     |> Ord_List.inter (fast_string_ord o apsnd fst) def_names
  1281     |> map snd
  1282   end
  1283 
  1284 (* Ideally we would check against "Complex_Main", not "Refute", but any theory
  1285    will do as long as it contains all the "axioms" and "axiomatization"
  1286    commands. *)
  1287 fun is_built_in_theory thy = Theory.subthy (thy, @{theory Refute})
  1288 
  1289 fun all_nondefs_of ctxt subst =
  1290   ctxt |> Spec_Rules.get
  1291        |> filter (curry (op =) Spec_Rules.Unknown o fst)
  1292        |> maps (snd o snd)
  1293        |> filter_out (is_built_in_theory o theory_of_thm)
  1294        |> map (subst_atomic subst o prop_of)
  1295 
  1296 fun arity_of_built_in_const thy stds (s, T) =
  1297   if s = @{const_name If} then
  1298     if nth_range_type 3 T = @{typ bool} then NONE else SOME 3
  1299   else
  1300     let val std_nats = is_standard_datatype thy stds nat_T in
  1301       case AList.lookup (op =)
  1302                     (built_in_consts
  1303                      |> std_nats ? append built_in_nat_consts) s of
  1304         SOME n => SOME n
  1305       | NONE =>
  1306         case AList.lookup (op =)
  1307                  (built_in_typed_consts
  1308                   |> std_nats ? append built_in_typed_nat_consts)
  1309                  (s, unarize_type T) of
  1310           SOME n => SOME n
  1311         | NONE =>
  1312           case s of
  1313             @{const_name zero_class.zero} =>
  1314             if is_iterator_type T then SOME 0 else NONE
  1315           | @{const_name Suc} =>
  1316             if is_iterator_type (domain_type T) then SOME 0 else NONE
  1317           | _ => if is_fun_type T andalso is_set_type (domain_type T) then
  1318                    AList.lookup (op =) built_in_set_consts s
  1319                  else
  1320                    NONE
  1321     end
  1322 val is_built_in_const = is_some ooo arity_of_built_in_const
  1323 
  1324 (* This function is designed to work for both real definition axioms and
  1325    simplification rules (equational specifications). *)
  1326 fun term_under_def t =
  1327   case t of
  1328     @{const "==>"} $ _ $ t2 => term_under_def t2
  1329   | Const (@{const_name "=="}, _) $ t1 $ _ => term_under_def t1
  1330   | @{const Trueprop} $ t1 => term_under_def t1
  1331   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => term_under_def t1
  1332   | Abs (_, _, t') => term_under_def t'
  1333   | t1 $ _ => term_under_def t1
  1334   | _ => t
  1335 
  1336 (* Here we crucially rely on "specialize_type" performing a preorder traversal
  1337    of the term, without which the wrong occurrence of a constant could be
  1338    matched in the face of overloading. *)
  1339 fun def_props_for_const thy stds table (x as (s, _)) =
  1340   if is_built_in_const thy stds x then
  1341     []
  1342   else
  1343     these (Symtab.lookup table s)
  1344     |> map_filter (try (specialize_type thy x))
  1345     |> filter (curry (op =) (Const x) o term_under_def)
  1346 
  1347 fun normalized_rhs_of t =
  1348   let
  1349     fun aux (v as Var _) (SOME t) = SOME (lambda v t)
  1350       | aux (c as Const (@{const_name TYPE}, _)) (SOME t) = SOME (lambda c t)
  1351       | aux _ _ = NONE
  1352     val (lhs, rhs) =
  1353       case t of
  1354         Const (@{const_name "=="}, _) $ t1 $ t2 => (t1, t2)
  1355       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ t2) =>
  1356         (t1, t2)
  1357       | _ => raise TERM ("Nitpick_HOL.normalized_rhs_of", [t])
  1358     val args = strip_comb lhs |> snd
  1359   in fold_rev aux args (SOME rhs) end
  1360 
  1361 fun get_def_of_const thy table (x as (s, _)) =
  1362   x |> def_props_for_const thy [(NONE, false)] table |> List.last
  1363     |> normalized_rhs_of |> Option.map (prefix_abs_vars s)
  1364   handle List.Empty => NONE
  1365 
  1366 fun def_of_const_ext thy (unfold_table, fallback_table) (x as (s, _)) =
  1367   if is_built_in_const thy [(NONE, false)] x orelse original_name s <> s then
  1368     NONE
  1369   else case get_def_of_const thy unfold_table x of
  1370     SOME def => SOME (true, def)
  1371   | NONE => get_def_of_const thy fallback_table x |> Option.map (pair false)
  1372 
  1373 val def_of_const = Option.map snd ooo def_of_const_ext
  1374 
  1375 fun fixpoint_kind_of_rhs (Abs (_, _, t)) = fixpoint_kind_of_rhs t
  1376   | fixpoint_kind_of_rhs (Const (@{const_name lfp}, _) $ Abs _) = Lfp
  1377   | fixpoint_kind_of_rhs (Const (@{const_name gfp}, _) $ Abs _) = Gfp
  1378   | fixpoint_kind_of_rhs _ = NoFp
  1379 
  1380 fun is_mutually_inductive_pred_def thy table t =
  1381   let
  1382     fun is_good_arg (Bound _) = true
  1383       | is_good_arg (Const (s, _)) =
  1384         s = @{const_name True} orelse s = @{const_name False} orelse
  1385         s = @{const_name undefined}
  1386       | is_good_arg _ = false
  1387   in
  1388     case t |> strip_abs_body |> strip_comb of
  1389       (Const x, ts as (_ :: _)) =>
  1390       (case def_of_const thy table x of
  1391          SOME t' => fixpoint_kind_of_rhs t' <> NoFp andalso
  1392                     forall is_good_arg ts
  1393        | NONE => false)
  1394     | _ => false
  1395   end
  1396 fun unfold_mutually_inductive_preds thy table =
  1397   map_aterms (fn t as Const x =>
  1398                  (case def_of_const thy table x of
  1399                     SOME t' =>
  1400                     let val t' = Envir.eta_contract t' in
  1401                       if is_mutually_inductive_pred_def thy table t' then t'
  1402                       else t
  1403                     end
  1404                  | NONE => t)
  1405                | t => t)
  1406 
  1407 fun case_const_names ctxt stds =
  1408   let val thy = Proof_Context.theory_of ctxt in
  1409     Symtab.fold (fn (dtype_s, {index, descr, case_name, ...}) =>
  1410                     if is_basic_datatype thy stds dtype_s then
  1411                       I
  1412                     else
  1413                       cons (case_name, AList.lookup (op =) descr index
  1414                                        |> the |> #3 |> length))
  1415                 (Datatype.get_all thy) [] @
  1416     map (apsnd length o snd) (#codatatypes (Data.get (Context.Proof ctxt)))
  1417   end
  1418 
  1419 fun fixpoint_kind_of_const thy table x =
  1420   if is_built_in_const thy [(NONE, false)] x then NoFp
  1421   else fixpoint_kind_of_rhs (the (def_of_const thy table x))
  1422   handle Option.Option => NoFp
  1423 
  1424 fun is_real_inductive_pred ({thy, stds, def_tables, intro_table, ...}
  1425                             : hol_context) x =
  1426   fixpoint_kind_of_const thy def_tables x <> NoFp andalso
  1427   not (null (def_props_for_const thy stds intro_table x))
  1428 fun is_inductive_pred hol_ctxt (x as (s, _)) =
  1429   is_real_inductive_pred hol_ctxt x orelse String.isPrefix ubfp_prefix s orelse
  1430   String.isPrefix lbfp_prefix s
  1431 
  1432 fun lhs_of_equation t =
  1433   case t of
  1434     Const (@{const_name all}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1435   | Const (@{const_name "=="}, _) $ t1 $ _ => SOME t1
  1436   | @{const "==>"} $ _ $ t2 => lhs_of_equation t2
  1437   | @{const Trueprop} $ t1 => lhs_of_equation t1
  1438   | Const (@{const_name All}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1439   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => SOME t1
  1440   | @{const HOL.implies} $ _ $ t2 => lhs_of_equation t2
  1441   | _ => NONE
  1442 fun is_constr_pattern _ (Bound _) = true
  1443   | is_constr_pattern _ (Var _) = true
  1444   | is_constr_pattern ctxt t =
  1445     case strip_comb t of
  1446       (Const x, args) =>
  1447       is_constr_like ctxt x andalso forall (is_constr_pattern ctxt) args
  1448     | _ => false
  1449 fun is_constr_pattern_lhs ctxt t =
  1450   forall (is_constr_pattern ctxt) (snd (strip_comb t))
  1451 fun is_constr_pattern_formula ctxt t =
  1452   case lhs_of_equation t of
  1453     SOME t' => is_constr_pattern_lhs ctxt t'
  1454   | NONE => false
  1455 
  1456 (* Similar to "specialize_type" but returns all matches rather than only the
  1457    first (preorder) match. *)
  1458 fun multi_specialize_type thy slack (s, T) t =
  1459   let
  1460     fun aux (Const (s', T')) ys =
  1461         if s = s' then
  1462           ys |> (if AList.defined (op =) ys T' then
  1463                    I
  1464                  else
  1465                    cons (T', monomorphic_term (Sign.typ_match thy (T', T)
  1466                                                               Vartab.empty) t)
  1467                    handle Type.TYPE_MATCH => I
  1468                         | TERM _ =>
  1469                           if slack then
  1470                             I
  1471                           else
  1472                             raise NOT_SUPPORTED
  1473                                       ("too much polymorphism in axiom \"" ^
  1474                                        Syntax.string_of_term_global thy t ^
  1475                                        "\" involving " ^ quote s))
  1476         else
  1477           ys
  1478       | aux _ ys = ys
  1479   in map snd (fold_aterms aux t []) end
  1480 fun nondef_props_for_const thy slack table (x as (s, _)) =
  1481   these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)
  1482 
  1483 fun unvarify_term (t1 $ t2) = unvarify_term t1 $ unvarify_term t2
  1484   | unvarify_term (Var ((s, 0), T)) = Free (s, T)
  1485   | unvarify_term (Abs (s, T, t')) = Abs (s, T, unvarify_term t')
  1486   | unvarify_term t = t
  1487 fun axiom_for_choice_spec thy =
  1488   unvarify_term
  1489   #> Object_Logic.atomize_term thy
  1490   #> Choice_Specification.close_form
  1491   #> HOLogic.mk_Trueprop
  1492 fun is_choice_spec_fun ({thy, def_tables, nondef_table, choice_spec_table, ...}
  1493                         : hol_context) x =
  1494   case nondef_props_for_const thy true choice_spec_table x of
  1495     [] => false
  1496   | ts => case def_of_const thy def_tables x of
  1497             SOME (Const (@{const_name Eps}, _) $ _) => true
  1498           | SOME _ => false
  1499           | NONE =>
  1500             let val ts' = nondef_props_for_const thy true nondef_table x in
  1501               length ts' = length ts andalso
  1502               forall (fn t =>
  1503                          exists (curry (op aconv) (axiom_for_choice_spec thy t))
  1504                                 ts') ts
  1505             end
  1506 
  1507 fun is_choice_spec_axiom thy choice_spec_table t =
  1508   Symtab.exists (fn (_, ts) =>
  1509                     exists (curry (op aconv) t o axiom_for_choice_spec thy) ts)
  1510                 choice_spec_table
  1511 
  1512 fun is_real_equational_fun ({thy, stds, simp_table, psimp_table, ...}
  1513                             : hol_context) x =
  1514   exists (fn table => not (null (def_props_for_const thy stds table x)))
  1515          [!simp_table, psimp_table]
  1516 fun is_equational_fun_but_no_plain_def hol_ctxt =
  1517   is_real_equational_fun hol_ctxt orf is_inductive_pred hol_ctxt
  1518 
  1519 (** Constant unfolding **)
  1520 
  1521 fun constr_case_body ctxt stds Ts (func_t, (x as (_, T))) =
  1522   let val arg_Ts = binder_types T in
  1523     s_betapplys Ts (func_t, map2 (select_nth_constr_arg ctxt stds x (Bound 0))
  1524                                  (index_seq 0 (length arg_Ts)) arg_Ts)
  1525   end
  1526 fun add_constr_case res_T (body_t, guard_t) res_t =
  1527   if res_T = bool_T then
  1528     s_conj (HOLogic.mk_imp (guard_t, body_t), res_t)
  1529   else
  1530     Const (@{const_name If}, bool_T --> res_T --> res_T --> res_T)
  1531     $ guard_t $ body_t $ res_t
  1532 fun optimized_case_def (hol_ctxt as {ctxt, stds, ...}) Ts dataT res_T func_ts =
  1533   let
  1534     val xs = datatype_constrs hol_ctxt dataT
  1535     val cases =
  1536       func_ts ~~ xs
  1537       |> map (fn (func_t, x) =>
  1538                  (constr_case_body ctxt stds (dataT :: Ts)
  1539                                    (incr_boundvars 1 func_t, x),
  1540                   discriminate_value hol_ctxt x (Bound 0)))
  1541       |> AList.group (op aconv)
  1542       |> map (apsnd (List.foldl s_disj @{const False}))
  1543       |> sort (int_ord o pairself (size_of_term o snd))
  1544       |> rev
  1545   in
  1546     if res_T = bool_T then
  1547       if forall (member (op =) [@{const False}, @{const True}] o fst) cases then
  1548         case cases of
  1549           [(body_t, _)] => body_t
  1550         | [_, (@{const True}, head_t2)] => head_t2
  1551         | [_, (@{const False}, head_t2)] => @{const Not} $ head_t2
  1552         | _ => raise BAD ("Nitpick_HOL.optimized_case_def", "impossible cases")
  1553       else
  1554         @{const True} |> fold_rev (add_constr_case res_T) cases
  1555     else
  1556       fst (hd cases) |> fold_rev (add_constr_case res_T) (tl cases)
  1557   end
  1558   |> absdummy dataT
  1559 
  1560 fun optimized_record_get (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T res_T t =
  1561   let val constr_x = hd (datatype_constrs hol_ctxt rec_T) in
  1562     case no_of_record_field thy s rec_T of
  1563       ~1 => (case rec_T of
  1564                Type (_, Ts as _ :: _) =>
  1565                let
  1566                  val rec_T' = List.last Ts
  1567                  val j = num_record_fields thy rec_T - 1
  1568                in
  1569                  select_nth_constr_arg ctxt stds constr_x t j res_T
  1570                  |> optimized_record_get hol_ctxt s rec_T' res_T
  1571                end
  1572              | _ => raise TYPE ("Nitpick_HOL.optimized_record_get", [rec_T],
  1573                                 []))
  1574     | j => select_nth_constr_arg ctxt stds constr_x t j res_T
  1575   end
  1576 fun optimized_record_update (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T fun_t
  1577                             rec_t =
  1578   let
  1579     val constr_x as (_, constr_T) = hd (datatype_constrs hol_ctxt rec_T)
  1580     val Ts = binder_types constr_T
  1581     val n = length Ts
  1582     val special_j = no_of_record_field thy s rec_T
  1583     val ts =
  1584       map2 (fn j => fn T =>
  1585                let val t = select_nth_constr_arg ctxt stds constr_x rec_t j T in
  1586                  if j = special_j then
  1587                    s_betapply [] (fun_t, t)
  1588                  else if j = n - 1 andalso special_j = ~1 then
  1589                    optimized_record_update hol_ctxt s
  1590                        (rec_T |> dest_Type |> snd |> List.last) fun_t t
  1591                  else
  1592                    t
  1593                end) (index_seq 0 n) Ts
  1594   in list_comb (Const constr_x, ts) end
  1595 
  1596 (* Prevents divergence in case of cyclic or infinite definition dependencies. *)
  1597 val unfold_max_depth = 255
  1598 
  1599 (* Inline definitions or define as an equational constant? Booleans tend to
  1600    benefit more from inlining, due to the polarity analysis. (However, if
  1601    "total_consts" is set, the polarity analysis is likely not to be so
  1602    crucial.) *)
  1603 val def_inline_threshold_for_booleans = 60
  1604 val def_inline_threshold_for_non_booleans = 20
  1605 
  1606 fun unfold_defs_in_term
  1607         (hol_ctxt as {thy, ctxt, stds, whacks, total_consts, case_names,
  1608                       def_tables, ground_thm_table, ersatz_table, ...}) =
  1609   let
  1610     fun do_term depth Ts t =
  1611       case t of
  1612         (t0 as Const (@{const_name Int.number_class.number_of},
  1613                       Type (@{type_name fun}, [_, ran_T]))) $ t1 =>
  1614         ((if is_number_type ctxt ran_T then
  1615             let
  1616               val j = t1 |> HOLogic.dest_numeral
  1617                          |> ran_T = nat_T ? Integer.max 0
  1618               val s = numeral_prefix ^ signed_string_of_int j
  1619             in
  1620               if is_integer_like_type ran_T then
  1621                 if is_standard_datatype thy stds ran_T then
  1622                   Const (s, ran_T)
  1623                 else
  1624                   funpow j (curry (op $) (suc_const ran_T)) (zero_const ran_T)
  1625               else
  1626                 do_term depth Ts (Const (@{const_name of_int}, int_T --> ran_T)
  1627                                   $ Const (s, int_T))
  1628             end
  1629             handle TERM _ => raise SAME ()
  1630           else
  1631             raise SAME ())
  1632          handle SAME () =>
  1633                 s_betapply [] (do_term depth Ts t0, do_term depth Ts t1))
  1634       | Const (@{const_name refl_on}, T) $ Const (@{const_name top}, _) $ t2 =>
  1635         do_const depth Ts t (@{const_name refl'}, range_type T) [t2]
  1636       | (t0 as Const (@{const_name Sigma}, Type (_, [T1, Type (_, [T2, T3])])))
  1637         $ t1 $ (t2 as Abs (_, _, t2')) =>
  1638         if loose_bvar1 (t2', 0) then
  1639           s_betapplys Ts (do_term depth Ts t0, map (do_term depth Ts) [t1, t2])
  1640         else
  1641           do_term depth Ts
  1642                   (Const (@{const_name prod}, T1 --> range_type T2 --> T3)
  1643                    $ t1 $ incr_boundvars ~1 t2')
  1644       | Const (x as (@{const_name distinct},
  1645                Type (@{type_name fun}, [Type (@{type_name list}, [T']), _])))
  1646         $ (t1 as _ $ _) =>
  1647         (t1 |> HOLogic.dest_list |> distinctness_formula T'
  1648          handle TERM _ => do_const depth Ts t x [t1])
  1649       | Const (x as (@{const_name If}, _)) $ t1 $ t2 $ t3 =>
  1650         if is_ground_term t1 andalso
  1651            exists (Pattern.matches thy o rpair t1)
  1652                   (Inttab.lookup_list ground_thm_table (hash_term t1)) then
  1653           do_term depth Ts t2
  1654         else
  1655           do_const depth Ts t x [t1, t2, t3]
  1656       | Const (@{const_name Let}, _) $ t1 $ t2 =>
  1657         s_betapply Ts (pairself (do_term depth Ts) (t2, t1))
  1658       | Const x => do_const depth Ts t x []
  1659       | t1 $ t2 =>
  1660         (case strip_comb t of
  1661            (Const x, ts) => do_const depth Ts t x ts
  1662          | _ => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2))
  1663       | Bound _ => t
  1664       | Abs (s, T, body) => Abs (s, T, do_term depth (T :: Ts) body)
  1665       | _ => if member (term_match thy) whacks t then
  1666                Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1667              else
  1668                t
  1669     and select_nth_constr_arg_with_args _ _ (x as (_, T)) [] n res_T =
  1670         (Abs (Name.uu, body_type T,
  1671               select_nth_constr_arg ctxt stds x (Bound 0) n res_T), [])
  1672       | select_nth_constr_arg_with_args depth Ts x (t :: ts) n res_T =
  1673         (select_nth_constr_arg ctxt stds x (do_term depth Ts t) n res_T, ts)
  1674     and quot_rep_of depth Ts abs_T rep_T ts =
  1675       select_nth_constr_arg_with_args depth Ts
  1676           (@{const_name Quot}, rep_T --> abs_T) ts 0 rep_T
  1677     and do_const depth Ts t (x as (s, T)) ts =
  1678       if member (term_match thy) whacks (Const x) then
  1679         Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1680       else case AList.lookup (op =) ersatz_table s of
  1681         SOME s' =>
  1682         do_const (depth + 1) Ts (list_comb (Const (s', T), ts)) (s', T) ts
  1683       | NONE =>
  1684         let
  1685           fun def_inline_threshold () =
  1686             if is_boolean_type (nth_range_type (length ts) T) andalso
  1687                total_consts <> SOME true then
  1688               def_inline_threshold_for_booleans
  1689             else
  1690               def_inline_threshold_for_non_booleans
  1691           val (const, ts) =
  1692             if is_built_in_const thy stds x then
  1693               (Const x, ts)
  1694             else case AList.lookup (op =) case_names s of
  1695               SOME n =>
  1696               if length ts < n then
  1697                 (do_term depth Ts (eta_expand Ts t (n - length ts)), [])
  1698               else
  1699                 let
  1700                   val (dataT, res_T) = nth_range_type n T
  1701                                        |> pairf domain_type range_type
  1702                 in
  1703                   (optimized_case_def hol_ctxt Ts dataT res_T
  1704                                       (map (do_term depth Ts) (take n ts)),
  1705                    drop n ts)
  1706                 end
  1707             | _ =>
  1708               if is_constr ctxt stds x then
  1709                 (Const x, ts)
  1710               else if is_stale_constr ctxt x then
  1711                 raise NOT_SUPPORTED ("(non-co)constructors of codatatypes \
  1712                                      \(\"" ^ s ^ "\")")
  1713               else if is_quot_abs_fun ctxt x then
  1714                 let
  1715                   val rep_T = domain_type T
  1716                   val abs_T = range_type T
  1717                 in
  1718                   (Abs (Name.uu, rep_T,
  1719                         Const (@{const_name Quot}, rep_T --> abs_T)
  1720                                $ (Const (quot_normal_name_for_type ctxt abs_T,
  1721                                          rep_T --> rep_T) $ Bound 0)), ts)
  1722                 end
  1723               else if is_quot_rep_fun ctxt x then
  1724                 quot_rep_of depth Ts (domain_type T) (range_type T) ts
  1725               else if is_record_get thy x then
  1726                 case length ts of
  1727                   0 => (do_term depth Ts (eta_expand Ts t 1), [])
  1728                 | _ => (optimized_record_get hol_ctxt s (domain_type T)
  1729                             (range_type T) (do_term depth Ts (hd ts)), tl ts)
  1730               else if is_record_update thy x then
  1731                 case length ts of
  1732                   2 => (optimized_record_update hol_ctxt
  1733                             (unsuffix Record.updateN s) (nth_range_type 2 T)
  1734                             (do_term depth Ts (hd ts))
  1735                             (do_term depth Ts (nth ts 1)), [])
  1736                 | n => (do_term depth Ts (eta_expand Ts t (2 - n)), [])
  1737               else if is_abs_fun ctxt x andalso
  1738                       is_quot_type ctxt (range_type T) then
  1739                 let
  1740                   val abs_T = range_type T
  1741                   val rep_T = domain_type (domain_type T)
  1742                   val eps_fun = Const (@{const_name Eps},
  1743                                        (rep_T --> bool_T) --> rep_T)
  1744                   val normal_fun =
  1745                     Const (quot_normal_name_for_type ctxt abs_T,
  1746                            rep_T --> rep_T)
  1747                   val abs_fun = Const (@{const_name Quot}, rep_T --> abs_T)
  1748                 in
  1749                   (Abs (Name.uu, rep_T --> bool_T,
  1750                         abs_fun $ (normal_fun $ (eps_fun $ Bound 0)))
  1751                    |> do_term (depth + 1) Ts, ts)
  1752                 end
  1753               else if is_rep_fun ctxt x then
  1754                 let val x' = mate_of_rep_fun ctxt x in
  1755                   if is_constr ctxt stds x' then
  1756                     select_nth_constr_arg_with_args depth Ts x' ts 0
  1757                                                     (range_type T)
  1758                   else if is_quot_type ctxt (domain_type T) then
  1759                     let
  1760                       val abs_T = domain_type T
  1761                       val rep_T = domain_type (range_type T)
  1762                       val (rep_fun, _) = quot_rep_of depth Ts abs_T rep_T []
  1763                       val (equiv_rel, _) =
  1764                         equiv_relation_for_quot_type thy abs_T
  1765                     in
  1766                       (Abs (Name.uu, abs_T, equiv_rel $ (rep_fun $ Bound 0)),
  1767                        ts)
  1768                     end
  1769                   else
  1770                     (Const x, ts)
  1771                 end
  1772               else if is_equational_fun_but_no_plain_def hol_ctxt x orelse
  1773                       is_choice_spec_fun hol_ctxt x then
  1774                 (Const x, ts)
  1775               else case def_of_const_ext thy def_tables x of
  1776                 SOME (unfold, def) =>
  1777                 if depth > unfold_max_depth then
  1778                   raise TOO_LARGE ("Nitpick_HOL.unfold_defs_in_term",
  1779                                    "too many nested definitions (" ^
  1780                                    string_of_int depth ^ ") while expanding " ^
  1781                                    quote s)
  1782                 else if s = "Wfrec.wfrec'" (* FIXME unchecked! *) then
  1783                   (do_term (depth + 1) Ts (s_betapplys Ts (def, ts)), [])
  1784                 else if not unfold andalso
  1785                      size_of_term def > def_inline_threshold () then
  1786                   (Const x, ts)
  1787                 else
  1788                   (do_term (depth + 1) Ts def, ts)
  1789               | NONE => (Const x, ts)
  1790         in
  1791           s_betapplys Ts (const, map (do_term depth Ts) ts)
  1792           |> s_beta_norm Ts
  1793         end
  1794   in do_term 0 [] end
  1795 
  1796 (** Axiom extraction/generation **)
  1797 
  1798 fun extensional_equal j (Type (@{type_name fun}, [dom_T, ran_T])) t1 t2 =
  1799     let val var_t = Var (("x", j), dom_T) in
  1800       extensional_equal (j + 1) ran_T (betapply (t1, var_t))
  1801                         (betapply (t2, var_t))
  1802     end
  1803   | extensional_equal _ T t1 t2 =
  1804     Const (@{const_name HOL.eq}, T --> T --> bool_T) $ t1 $ t2
  1805 
  1806 fun equationalize_term ctxt tag t =
  1807   let
  1808     val j = maxidx_of_term t + 1
  1809     val (prems, concl) = Logic.strip_horn t
  1810   in
  1811     Logic.list_implies (prems,
  1812         case concl of
  1813           @{const Trueprop} $ (Const (@{const_name HOL.eq}, Type (_, [T, _]))
  1814                                $ t1 $ t2) =>
  1815           @{const Trueprop} $ extensional_equal j T t1 t2
  1816         | @{const Trueprop} $ t' =>
  1817           @{const Trueprop} $ HOLogic.mk_eq (t', @{const True})
  1818         | Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
  1819           @{const Trueprop} $ extensional_equal j T t1 t2
  1820         | _ => (warning ("Ignoring " ^ quote tag ^ " for non-equation" ^
  1821                          quote (Syntax.string_of_term ctxt t) ^ ".");
  1822                 raise SAME ()))
  1823     |> SOME
  1824   end
  1825   handle SAME () => NONE
  1826 
  1827 fun pair_for_prop t =
  1828   case term_under_def t of
  1829     Const (s, _) => (s, t)
  1830   | t' => raise TERM ("Nitpick_HOL.pair_for_prop", [t, t'])
  1831 
  1832 fun def_table_for get ctxt subst =
  1833   ctxt |> get |> map (pair_for_prop o subst_atomic subst)
  1834        |> AList.group (op =) |> Symtab.make
  1835 
  1836 fun const_def_tables ctxt subst ts =
  1837   (def_table_for (map prop_of o Nitpick_Unfolds.get) ctxt subst,
  1838    fold (fn (s, t) => Symtab.map_default (s, []) (cons t))
  1839         (map pair_for_prop ts) Symtab.empty)
  1840 
  1841 fun paired_with_consts t = map (rpair t) (Term.add_const_names t [])
  1842 fun const_nondef_table ts =
  1843   fold (append o paired_with_consts) ts [] |> AList.group (op =) |> Symtab.make
  1844 
  1845 fun const_simp_table ctxt =
  1846   def_table_for (map_filter (equationalize_term ctxt "nitpick_simp" o prop_of)
  1847                  o Nitpick_Simps.get) ctxt
  1848 fun const_psimp_table ctxt =
  1849   def_table_for (map_filter (equationalize_term ctxt "nitpick_psimp" o prop_of)
  1850                  o Nitpick_Psimps.get) ctxt
  1851 
  1852 fun const_choice_spec_table ctxt subst =
  1853   map (subst_atomic subst o prop_of) (Nitpick_Choice_Specs.get ctxt)
  1854   |> const_nondef_table
  1855 
  1856 fun inductive_intro_table ctxt subst def_tables =
  1857   let val thy = Proof_Context.theory_of ctxt in
  1858     def_table_for
  1859         (maps (map (unfold_mutually_inductive_preds thy def_tables o prop_of)
  1860                o snd o snd)
  1861          o filter (fn (cat, _) => cat = Spec_Rules.Inductive orelse
  1862                                   cat = Spec_Rules.Co_Inductive)
  1863          o Spec_Rules.get) ctxt subst
  1864   end
  1865 
  1866 fun ground_theorem_table thy =
  1867   fold ((fn @{const Trueprop} $ t1 =>
  1868             is_ground_term t1 ? Inttab.map_default (hash_term t1, []) (cons t1)
  1869           | _ => I) o prop_of o snd) (Global_Theory.all_thms_of thy) Inttab.empty
  1870 
  1871 fun ersatz_table ctxt =
  1872  #ersatz_table (Data.get (Context.Proof ctxt))
  1873  |> fold (append o snd) (#frac_types (Data.get (Context.Proof ctxt)))
  1874 
  1875 fun add_simps simp_table s eqs =
  1876   Unsynchronized.change simp_table
  1877       (Symtab.update (s, eqs @ these (Symtab.lookup (!simp_table) s)))
  1878 
  1879 fun inverse_axioms_for_rep_fun ctxt (x as (_, T)) =
  1880   let
  1881     val thy = Proof_Context.theory_of ctxt
  1882     val abs_T = domain_type T
  1883   in
  1884     typedef_info ctxt (fst (dest_Type abs_T)) |> the
  1885     |> pairf #Abs_inverse #Rep_inverse
  1886     |> pairself (specialize_type thy x o prop_of o the)
  1887     ||> single |> op ::
  1888   end
  1889 fun optimized_typedef_axioms ctxt (abs_z as (abs_s, _)) =
  1890   let
  1891     val thy = Proof_Context.theory_of ctxt
  1892     val abs_T = Type abs_z
  1893   in
  1894     if is_univ_typedef ctxt abs_T then
  1895       []
  1896     else case typedef_info ctxt abs_s of
  1897       SOME {abs_type, rep_type, Rep_name, prop_of_Rep, set_name, ...} =>
  1898       let
  1899         val rep_T = varify_and_instantiate_type ctxt abs_type abs_T rep_type
  1900         val rep_t = Const (Rep_name, abs_T --> rep_T)
  1901         val set_t = Const (set_name, rep_T --> bool_T)
  1902         val set_t' =
  1903           prop_of_Rep |> HOLogic.dest_Trueprop
  1904                       |> specialize_type thy (dest_Const rep_t)
  1905                       |> HOLogic.dest_mem |> snd
  1906       in
  1907         [HOLogic.all_const abs_T
  1908          $ Abs (Name.uu, abs_T, set_t $ (rep_t $ Bound 0))]
  1909         |> set_t <> set_t' ? cons (HOLogic.mk_eq (set_t, set_t'))
  1910         |> map HOLogic.mk_Trueprop
  1911       end
  1912     | NONE => []
  1913   end
  1914 fun optimized_quot_type_axioms ctxt stds abs_z =
  1915   let
  1916     val thy = Proof_Context.theory_of ctxt
  1917     val abs_T = Type abs_z
  1918     val rep_T = rep_type_for_quot_type thy abs_T
  1919     val (equiv_rel, partial) = equiv_relation_for_quot_type thy abs_T
  1920     val a_var = Var (("a", 0), abs_T)
  1921     val x_var = Var (("x", 0), rep_T)
  1922     val y_var = Var (("y", 0), rep_T)
  1923     val x = (@{const_name Quot}, rep_T --> abs_T)
  1924     val sel_a_t = select_nth_constr_arg ctxt stds x a_var 0 rep_T
  1925     val normal_fun =
  1926       Const (quot_normal_name_for_type ctxt abs_T, rep_T --> rep_T)
  1927     val normal_x = normal_fun $ x_var
  1928     val normal_y = normal_fun $ y_var
  1929     val is_unknown_t = Const (@{const_name is_unknown}, rep_T --> bool_T)
  1930   in
  1931     [Logic.mk_equals (normal_fun $ sel_a_t, sel_a_t),
  1932      Logic.list_implies
  1933          ([@{const Not} $ (is_unknown_t $ normal_x),
  1934            @{const Not} $ (is_unknown_t $ normal_y),
  1935            equiv_rel $ x_var $ y_var] |> map HOLogic.mk_Trueprop,
  1936            Logic.mk_equals (normal_x, normal_y)),
  1937      Logic.list_implies
  1938          ([HOLogic.mk_Trueprop (@{const Not} $ (is_unknown_t $ normal_x)),
  1939            HOLogic.mk_Trueprop (@{const Not} $ HOLogic.mk_eq (normal_x, x_var))],
  1940           HOLogic.mk_Trueprop (equiv_rel $ x_var $ normal_x))]
  1941     |> partial ? cons (HOLogic.mk_Trueprop (equiv_rel $ sel_a_t $ sel_a_t))
  1942   end
  1943 
  1944 fun codatatype_bisim_axioms (hol_ctxt as {ctxt, stds, ...}) T =
  1945   let
  1946     val xs = datatype_constrs hol_ctxt T
  1947     val set_T = T --> bool_T
  1948     val iter_T = @{typ bisim_iterator}
  1949     val bisim_max = @{const bisim_iterator_max}
  1950     val n_var = Var (("n", 0), iter_T)
  1951     val n_var_minus_1 =
  1952       Const (@{const_name safe_The}, (iter_T --> bool_T) --> iter_T)
  1953       $ Abs ("m", iter_T, HOLogic.eq_const iter_T
  1954                           $ (suc_const iter_T $ Bound 0) $ n_var)
  1955     val x_var = Var (("x", 0), T)
  1956     val y_var = Var (("y", 0), T)
  1957     fun bisim_const T =
  1958       Const (@{const_name bisim}, iter_T --> T --> T --> bool_T)
  1959     fun nth_sub_bisim x n nth_T =
  1960       (if is_codatatype ctxt nth_T then bisim_const nth_T $ n_var_minus_1
  1961        else HOLogic.eq_const nth_T)
  1962       $ select_nth_constr_arg ctxt stds x x_var n nth_T
  1963       $ select_nth_constr_arg ctxt stds x y_var n nth_T
  1964     fun case_func (x as (_, T)) =
  1965       let
  1966         val arg_Ts = binder_types T
  1967         val core_t =
  1968           discriminate_value hol_ctxt x y_var ::
  1969           map2 (nth_sub_bisim x) (index_seq 0 (length arg_Ts)) arg_Ts
  1970           |> foldr1 s_conj
  1971       in fold_rev absdummy arg_Ts core_t end
  1972   in
  1973     [HOLogic.mk_imp
  1974        (HOLogic.mk_disj (HOLogic.eq_const iter_T $ n_var $ zero_const iter_T,
  1975             s_betapply [] (optimized_case_def hol_ctxt [] T bool_T
  1976                                               (map case_func xs), x_var)),
  1977         bisim_const T $ n_var $ x_var $ y_var),
  1978      HOLogic.eq_const set_T $ (bisim_const T $ bisim_max $ x_var)
  1979      $ (Const (@{const_name insert}, T --> set_T --> set_T)
  1980         $ x_var $ Const (@{const_name bot_class.bot}, set_T))]
  1981     |> map HOLogic.mk_Trueprop
  1982   end
  1983 
  1984 exception NO_TRIPLE of unit
  1985 
  1986 fun triple_for_intro_rule thy x t =
  1987   let
  1988     val prems = Logic.strip_imp_prems t |> map (Object_Logic.atomize_term thy)
  1989     val concl = Logic.strip_imp_concl t |> Object_Logic.atomize_term thy
  1990     val (main, side) = List.partition (exists_Const (curry (op =) x)) prems
  1991     val is_good_head = curry (op =) (Const x) o head_of
  1992   in
  1993     if forall is_good_head main then (side, main, concl) else raise NO_TRIPLE ()
  1994   end
  1995 
  1996 val tuple_for_args = HOLogic.mk_tuple o snd o strip_comb
  1997 fun wf_constraint_for rel side concl main =
  1998   let
  1999     val core = HOLogic.mk_mem (HOLogic.mk_prod
  2000                                (pairself tuple_for_args (main, concl)), Var rel)
  2001     val t = List.foldl HOLogic.mk_imp core side
  2002     val vars = filter_out (curry (op =) rel) (Term.add_vars t [])
  2003   in
  2004     Library.foldl (fn (t', ((x, j), T)) =>
  2005                       HOLogic.all_const T
  2006                       $ Abs (x, T, abstract_over (Var ((x, j), T), t')))
  2007                   (t, vars)
  2008   end
  2009 fun wf_constraint_for_triple rel (side, main, concl) =
  2010   map (wf_constraint_for rel side concl) main |> foldr1 s_conj
  2011 
  2012 fun terminates_by ctxt timeout goal tac =
  2013   can (SINGLE (Classical.safe_tac ctxt) #> the
  2014        #> SINGLE (DETERM_TIMEOUT timeout (tac ctxt (auto_tac ctxt)))
  2015        #> the #> Goal.finish ctxt) goal
  2016 
  2017 val max_cached_wfs = 50
  2018 val cached_timeout =
  2019   Synchronized.var "Nitpick_HOL.cached_timeout" (SOME Time.zeroTime)
  2020 val cached_wf_props =
  2021   Synchronized.var "Nitpick_HOL.cached_wf_props" ([] : (term * bool) list)
  2022 
  2023 val termination_tacs = [Lexicographic_Order.lex_order_tac true,
  2024                         ScnpReconstruct.sizechange_tac]
  2025 
  2026 fun uncached_is_well_founded_inductive_pred
  2027         ({thy, ctxt, stds, debug, tac_timeout, intro_table, ...} : hol_context)
  2028         (x as (_, T)) =
  2029   case def_props_for_const thy stds intro_table x of
  2030     [] => raise TERM ("Nitpick_HOL.uncached_is_well_founded_inductive",
  2031                       [Const x])
  2032   | intro_ts =>
  2033     (case map (triple_for_intro_rule thy x) intro_ts
  2034           |> filter_out (null o #2) of
  2035        [] => true
  2036      | triples =>
  2037        let
  2038          val binders_T = HOLogic.mk_tupleT (binder_types T)
  2039          val rel_T = HOLogic.mk_prodT (binders_T, binders_T) --> bool_T
  2040          val j = fold Integer.max (map maxidx_of_term intro_ts) 0 + 1
  2041          val rel = (("R", j), rel_T)
  2042          val prop = Const (@{const_name wf}, rel_T --> bool_T) $ Var rel ::
  2043                     map (wf_constraint_for_triple rel) triples
  2044                     |> foldr1 s_conj |> HOLogic.mk_Trueprop
  2045          val _ = if debug then
  2046                    Output.urgent_message ("Wellfoundedness goal: " ^
  2047                              Syntax.string_of_term ctxt prop ^ ".")
  2048                  else
  2049                    ()
  2050        in
  2051          if tac_timeout = Synchronized.value cached_timeout andalso
  2052             length (Synchronized.value cached_wf_props) < max_cached_wfs then
  2053            ()
  2054          else
  2055            (Synchronized.change cached_wf_props (K []);
  2056             Synchronized.change cached_timeout (K tac_timeout));
  2057          case AList.lookup (op =) (Synchronized.value cached_wf_props) prop of
  2058            SOME wf => wf
  2059          | NONE =>
  2060            let
  2061              val goal = prop |> cterm_of thy |> Goal.init
  2062              val wf = exists (terminates_by ctxt tac_timeout goal)
  2063                              termination_tacs
  2064            in Synchronized.change cached_wf_props (cons (prop, wf)); wf end
  2065        end)
  2066     handle List.Empty => false | NO_TRIPLE () => false
  2067 
  2068 (* The type constraint below is a workaround for a Poly/ML crash. *)
  2069 
  2070 fun is_well_founded_inductive_pred
  2071         (hol_ctxt as {thy, wfs, def_tables, wf_cache, ...} : hol_context)
  2072         (x as (s, _)) =
  2073   case triple_lookup (const_match thy) wfs x of
  2074     SOME (SOME b) => b
  2075   | _ => s = @{const_name Nats} orelse s = @{const_name fold_graph'} orelse
  2076          case AList.lookup (op =) (!wf_cache) x of
  2077            SOME (_, wf) => wf
  2078          | NONE =>
  2079            let
  2080              val gfp = (fixpoint_kind_of_const thy def_tables x = Gfp)
  2081              val wf = uncached_is_well_founded_inductive_pred hol_ctxt x
  2082            in
  2083              Unsynchronized.change wf_cache (cons (x, (gfp, wf))); wf
  2084            end
  2085 
  2086 fun ap_curry [_] _ t = t
  2087   | ap_curry arg_Ts tuple_T t =
  2088     let val n = length arg_Ts in
  2089       list_abs (map (pair "c") arg_Ts,
  2090                 incr_boundvars n t
  2091                 $ mk_flat_tuple tuple_T (map Bound (n - 1 downto 0)))
  2092     end
  2093 
  2094 fun num_occs_of_bound_in_term j (t1 $ t2) =
  2095     op + (pairself (num_occs_of_bound_in_term j) (t1, t2))
  2096   | num_occs_of_bound_in_term j (Abs (_, _, t')) =
  2097     num_occs_of_bound_in_term (j + 1) t'
  2098   | num_occs_of_bound_in_term j (Bound j') = if j' = j then 1 else 0
  2099   | num_occs_of_bound_in_term _ _ = 0
  2100 
  2101 val is_linear_inductive_pred_def =
  2102   let
  2103     fun do_disjunct j (Const (@{const_name Ex}, _) $ Abs (_, _, t2)) =
  2104         do_disjunct (j + 1) t2
  2105       | do_disjunct j t =
  2106         case num_occs_of_bound_in_term j t of
  2107           0 => true
  2108         | 1 => exists (curry (op =) (Bound j) o head_of) (conjuncts_of t)
  2109         | _ => false
  2110     fun do_lfp_def (Const (@{const_name lfp}, _) $ t2) =
  2111         let val (xs, body) = strip_abs t2 in
  2112           case length xs of
  2113             1 => false
  2114           | n => forall (do_disjunct (n - 1)) (disjuncts_of body)
  2115         end
  2116       | do_lfp_def _ = false
  2117   in do_lfp_def o strip_abs_body end
  2118 
  2119 fun n_ptuple_paths 0 = []
  2120   | n_ptuple_paths 1 = []
  2121   | n_ptuple_paths n = [] :: map (cons 2) (n_ptuple_paths (n - 1))
  2122 val ap_n_split = HOLogic.mk_psplits o n_ptuple_paths
  2123 
  2124 val linear_pred_base_and_step_rhss =
  2125   let
  2126     fun aux (Const (@{const_name lfp}, _) $ t2) =
  2127         let
  2128           val (xs, body) = strip_abs t2
  2129           val arg_Ts = map snd (tl xs)
  2130           val tuple_T = HOLogic.mk_tupleT arg_Ts
  2131           val j = length arg_Ts
  2132           fun repair_rec j (Const (@{const_name Ex}, T1) $ Abs (s2, T2, t2')) =
  2133               Const (@{const_name Ex}, T1)
  2134               $ Abs (s2, T2, repair_rec (j + 1) t2')
  2135             | repair_rec j (@{const HOL.conj} $ t1 $ t2) =
  2136               @{const HOL.conj} $ repair_rec j t1 $ repair_rec j t2
  2137             | repair_rec j t =
  2138               let val (head, args) = strip_comb t in
  2139                 if head = Bound j then
  2140                   HOLogic.eq_const tuple_T $ Bound j
  2141                   $ mk_flat_tuple tuple_T args
  2142                 else
  2143                   t
  2144               end
  2145           val (nonrecs, recs) =
  2146             List.partition (curry (op =) 0 o num_occs_of_bound_in_term j)
  2147                            (disjuncts_of body)
  2148           val base_body = nonrecs |> List.foldl s_disj @{const False}
  2149           val step_body = recs |> map (repair_rec j)
  2150                                |> List.foldl s_disj @{const False}
  2151         in
  2152           (list_abs (tl xs, incr_bv (~1, j, base_body))
  2153            |> ap_n_split (length arg_Ts) tuple_T bool_T,
  2154            Abs ("y", tuple_T, list_abs (tl xs, step_body)
  2155                               |> ap_n_split (length arg_Ts) tuple_T bool_T))
  2156         end
  2157       | aux t =
  2158         raise TERM ("Nitpick_HOL.linear_pred_base_and_step_rhss.aux", [t])
  2159   in aux end
  2160 
  2161 fun starred_linear_pred_const (hol_ctxt as {simp_table, ...}) (s, T) def =
  2162   let
  2163     val j = maxidx_of_term def + 1
  2164     val (outer, fp_app) = strip_abs def
  2165     val outer_bounds = map Bound (length outer - 1 downto 0)
  2166     val outer_vars = map (fn (s, T) => Var ((s, j), T)) outer
  2167     val fp_app = subst_bounds (rev outer_vars, fp_app)
  2168     val (outer_Ts, rest_T) = strip_n_binders (length outer) T
  2169     val tuple_arg_Ts = strip_type rest_T |> fst
  2170     val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
  2171     val set_T = tuple_T --> bool_T
  2172     val curried_T = tuple_T --> set_T
  2173     val uncurried_T = Type (@{type_name prod}, [tuple_T, tuple_T]) --> bool_T
  2174     val (base_rhs, step_rhs) = linear_pred_base_and_step_rhss fp_app
  2175     val base_x as (base_s, _) = (base_prefix ^ s, outer_Ts ---> set_T)
  2176     val base_eq = HOLogic.mk_eq (list_comb (Const base_x, outer_vars), base_rhs)
  2177                   |> HOLogic.mk_Trueprop
  2178     val _ = add_simps simp_table base_s [base_eq]
  2179     val step_x as (step_s, _) = (step_prefix ^ s, outer_Ts ---> curried_T)
  2180     val step_eq = HOLogic.mk_eq (list_comb (Const step_x, outer_vars), step_rhs)
  2181                   |> HOLogic.mk_Trueprop
  2182     val _ = add_simps simp_table step_s [step_eq]
  2183   in
  2184     list_abs (outer,
  2185               Const (@{const_name Image}, uncurried_T --> set_T --> set_T)
  2186               $ (Const (@{const_name rtrancl}, uncurried_T --> uncurried_T)
  2187                  $ (Const (@{const_name prod_case}, curried_T --> uncurried_T)
  2188                     $ list_comb (Const step_x, outer_bounds)))
  2189               $ list_comb (Const base_x, outer_bounds)
  2190               |> ap_curry tuple_arg_Ts tuple_T)
  2191     |> unfold_defs_in_term hol_ctxt
  2192   end
  2193 
  2194 fun is_good_starred_linear_pred_type (Type (@{type_name fun}, Ts)) =
  2195     forall (not o (is_fun_type orf is_pair_type)) Ts
  2196   | is_good_starred_linear_pred_type _ = false
  2197 
  2198 fun unrolled_inductive_pred_const (hol_ctxt as {thy, star_linear_preds,
  2199                                                 def_tables, simp_table, ...})
  2200                                   gfp (x as (s, T)) =
  2201   let
  2202     val iter_T = iterator_type_for_const gfp x
  2203     val x' as (s', _) = (unrolled_prefix ^ s, iter_T --> T)
  2204     val unrolled_const = Const x' $ zero_const iter_T
  2205     val def = the (def_of_const thy def_tables x)
  2206   in
  2207     if is_equational_fun_but_no_plain_def hol_ctxt x' then
  2208       unrolled_const (* already done *)
  2209     else if not gfp andalso star_linear_preds andalso
  2210          is_linear_inductive_pred_def def andalso
  2211          is_good_starred_linear_pred_type T then
  2212       starred_linear_pred_const hol_ctxt x def
  2213     else
  2214       let
  2215         val j = maxidx_of_term def + 1
  2216         val (outer, fp_app) = strip_abs def
  2217         val outer_bounds = map Bound (length outer - 1 downto 0)
  2218         val cur = Var ((iter_var_prefix, j + 1), iter_T)
  2219         val next = suc_const iter_T $ cur
  2220         val rhs =
  2221           case fp_app of
  2222             Const _ $ t =>
  2223             s_betapply [] (t, list_comb (Const x', next :: outer_bounds))
  2224           | _ => raise TERM ("Nitpick_HOL.unrolled_inductive_pred_const",
  2225                              [fp_app])
  2226         val (inner, naked_rhs) = strip_abs rhs
  2227         val all = outer @ inner
  2228         val bounds = map Bound (length all - 1 downto 0)
  2229         val vars = map (fn (s, T) => Var ((s, j), T)) all
  2230         val eq = HOLogic.mk_eq (list_comb (Const x', cur :: bounds), naked_rhs)
  2231                  |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2232         val _ = add_simps simp_table s' [eq]
  2233       in unrolled_const end
  2234   end
  2235 
  2236 fun raw_inductive_pred_axiom ({thy, def_tables, ...} : hol_context) x =
  2237   let
  2238     val def = the (def_of_const thy def_tables x)
  2239     val (outer, fp_app) = strip_abs def
  2240     val outer_bounds = map Bound (length outer - 1 downto 0)
  2241     val rhs =
  2242       case fp_app of
  2243         Const _ $ t => s_betapply [] (t, list_comb (Const x, outer_bounds))
  2244       | _ => raise TERM ("Nitpick_HOL.raw_inductive_pred_axiom", [fp_app])
  2245     val (inner, naked_rhs) = strip_abs rhs
  2246     val all = outer @ inner
  2247     val bounds = map Bound (length all - 1 downto 0)
  2248     val j = maxidx_of_term def + 1
  2249     val vars = map (fn (s, T) => Var ((s, j), T)) all
  2250   in
  2251     HOLogic.mk_eq (list_comb (Const x, bounds), naked_rhs)
  2252     |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2253   end
  2254 fun inductive_pred_axiom hol_ctxt (x as (s, T)) =
  2255   if String.isPrefix ubfp_prefix s orelse String.isPrefix lbfp_prefix s then
  2256     let val x' = (strip_first_name_sep s |> snd, T) in
  2257       raw_inductive_pred_axiom hol_ctxt x' |> subst_atomic [(Const x', Const x)]
  2258     end
  2259   else
  2260     raw_inductive_pred_axiom hol_ctxt x
  2261 
  2262 fun equational_fun_axioms (hol_ctxt as {thy, ctxt, stds, def_tables, simp_table,
  2263                                         psimp_table, ...}) x =
  2264   case def_props_for_const thy stds (!simp_table) x of
  2265     [] => (case def_props_for_const thy stds psimp_table x of
  2266              [] => (if is_inductive_pred hol_ctxt x then
  2267                       [inductive_pred_axiom hol_ctxt x]
  2268                     else case def_of_const thy def_tables x of
  2269                       SOME def =>
  2270                       @{const Trueprop} $ HOLogic.mk_eq (Const x, def)
  2271                       |> equationalize_term ctxt "" |> the |> single
  2272                     | NONE => [])
  2273            | psimps => psimps)
  2274   | simps => simps
  2275 fun is_equational_fun_surely_complete hol_ctxt x =
  2276   case equational_fun_axioms hol_ctxt x of
  2277     [@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)] =>
  2278     strip_comb t1 |> snd |> forall is_Var
  2279   | _ => false
  2280 
  2281 (** Type preprocessing **)
  2282 
  2283 fun merged_type_var_table_for_terms thy ts =
  2284   let
  2285     fun add (s, S) table =
  2286       table
  2287       |> (case AList.lookup (Sign.subsort thy o swap) table S of
  2288             SOME _ => I
  2289           | NONE =>
  2290             filter_out (fn (S', _) => Sign.subsort thy (S, S'))
  2291             #> cons (S, s))
  2292     val tfrees = [] |> fold Term.add_tfrees ts
  2293                     |> sort (string_ord o pairself fst)
  2294   in [] |> fold add tfrees |> rev end
  2295 
  2296 fun merge_type_vars_in_term thy merge_type_vars table =
  2297   merge_type_vars
  2298   ? map_types (map_atyps
  2299         (fn TFree (_, S) =>
  2300             TFree (table |> find_first (fn (S', _) => Sign.subsort thy (S', S))
  2301                          |> the |> swap)
  2302           | T => T))
  2303 
  2304 fun add_ground_types hol_ctxt binarize =
  2305   let
  2306     fun aux T accum =
  2307       case T of
  2308         Type (@{type_name fun}, Ts) => fold aux Ts accum
  2309       | Type (@{type_name prod}, Ts) => fold aux Ts accum
  2310       | Type (@{type_name itself}, [T1]) => aux T1 accum
  2311       | Type (_, Ts) =>
  2312         if member (op =) (@{typ prop} :: @{typ bool} :: accum) T then
  2313           accum
  2314         else
  2315           T :: accum
  2316           |> fold aux (case binarized_and_boxed_datatype_constrs hol_ctxt
  2317                                                                  binarize T of
  2318                          [] => Ts
  2319                        | xs => map snd xs)
  2320       | _ => insert (op =) T accum
  2321   in aux end
  2322 
  2323 fun ground_types_in_type hol_ctxt binarize T =
  2324   add_ground_types hol_ctxt binarize T []
  2325 fun ground_types_in_terms hol_ctxt binarize ts =
  2326   fold (fold_types (add_ground_types hol_ctxt binarize)) ts []
  2327 
  2328 end;