src/HOL/Tools/Nitpick/nitpick_hol.ML
author blanchet
Tue May 31 16:38:36 2011 +0200 (2011-05-31)
changeset 43085 0a2f5b86bdd7
parent 42958 034fc4d0c909
child 43827 62d64709af3b
permissions -rw-r--r--
first step in sharing more code between ATP and Metis translation
     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_codatatype :
   152     typ -> string -> styp list -> morphism -> Context.generic -> Context.generic
   153   val register_codatatype_global :
   154     typ -> string -> styp list -> theory -> theory
   155   val unregister_codatatype :
   156     typ -> morphism -> Context.generic -> Context.generic
   157   val unregister_codatatype_global : typ -> theory -> theory
   158   val datatype_constrs : hol_context -> typ -> styp list
   159   val binarized_and_boxed_datatype_constrs :
   160     hol_context -> bool -> typ -> styp list
   161   val num_datatype_constrs : hol_context -> typ -> int
   162   val constr_name_for_sel_like : string -> string
   163   val binarized_and_boxed_constr_for_sel : hol_context -> bool -> styp -> styp
   164   val card_of_type : (typ * int) list -> typ -> int
   165   val bounded_card_of_type : int -> int -> (typ * int) list -> typ -> int
   166   val bounded_exact_card_of_type :
   167     hol_context -> typ list -> int -> int -> (typ * int) list -> typ -> int
   168   val typical_card_of_type : typ -> int
   169   val is_finite_type : hol_context -> typ -> bool
   170   val is_special_eligible_arg : bool -> typ list -> term -> bool
   171   val s_let :
   172     typ list -> string -> int -> typ -> typ -> (term -> term) -> term -> term
   173   val s_betapply : typ list -> term * term -> term
   174   val s_betapplys : typ list -> term * term list -> term
   175   val discriminate_value : hol_context -> styp -> term -> term
   176   val select_nth_constr_arg :
   177     Proof.context -> (typ option * bool) list -> styp -> term -> int -> typ
   178     -> term
   179   val construct_value :
   180     Proof.context -> (typ option * bool) list -> styp -> term list -> term
   181   val coerce_term : hol_context -> typ list -> typ -> typ -> term -> term
   182   val special_bounds : term list -> (indexname * typ) list
   183   val is_funky_typedef : Proof.context -> typ -> bool
   184   val all_defs_of : theory -> (term * term) list -> term list
   185   val all_nondefs_of : Proof.context -> (term * term) list -> term list
   186   val arity_of_built_in_const :
   187     theory -> (typ option * bool) list -> styp -> int option
   188   val is_built_in_const :
   189     theory -> (typ option * bool) list -> styp -> bool
   190   val term_under_def : term -> term
   191   val case_const_names :
   192     Proof.context -> (typ option * bool) list -> (string * int) list
   193   val unfold_defs_in_term : hol_context -> term -> term
   194   val const_def_tables :
   195     Proof.context -> (term * term) list -> term list
   196     -> const_table * const_table
   197   val const_nondef_table : term list -> const_table
   198   val const_simp_table : Proof.context -> (term * term) list -> const_table
   199   val const_psimp_table : Proof.context -> (term * term) list -> const_table
   200   val const_choice_spec_table :
   201     Proof.context -> (term * term) list -> const_table
   202   val inductive_intro_table :
   203     Proof.context -> (term * term) list -> const_table * const_table
   204     -> const_table
   205   val ground_theorem_table : theory -> term list Inttab.table
   206   val ersatz_table : Proof.context -> (string * string) list
   207   val add_simps : const_table Unsynchronized.ref -> string -> term list -> unit
   208   val inverse_axioms_for_rep_fun : Proof.context -> styp -> term list
   209   val optimized_typedef_axioms : Proof.context -> string * typ list -> term list
   210   val optimized_quot_type_axioms :
   211     Proof.context -> (typ option * bool) list -> string * typ list -> term list
   212   val def_of_const : theory -> const_table * const_table -> styp -> term option
   213   val fixpoint_kind_of_rhs : term -> fixpoint_kind
   214   val fixpoint_kind_of_const :
   215     theory -> const_table * const_table -> string * typ -> fixpoint_kind
   216   val is_real_inductive_pred : hol_context -> styp -> bool
   217   val is_constr_pattern : Proof.context -> term -> bool
   218   val is_constr_pattern_lhs : Proof.context -> term -> bool
   219   val is_constr_pattern_formula : Proof.context -> term -> bool
   220   val nondef_props_for_const :
   221     theory -> bool -> const_table -> styp -> term list
   222   val is_choice_spec_fun : hol_context -> styp -> bool
   223   val is_choice_spec_axiom : theory -> const_table -> term -> bool
   224   val is_real_equational_fun : hol_context -> styp -> bool
   225   val is_equational_fun_but_no_plain_def : hol_context -> styp -> bool
   226   val codatatype_bisim_axioms : hol_context -> typ -> term list
   227   val is_well_founded_inductive_pred : hol_context -> styp -> bool
   228   val unrolled_inductive_pred_const : hol_context -> bool -> styp -> term
   229   val equational_fun_axioms : hol_context -> styp -> term list
   230   val is_equational_fun_surely_complete : hol_context -> styp -> bool
   231   val merged_type_var_table_for_terms :
   232     theory -> term list -> (sort * string) list
   233   val merge_type_vars_in_term :
   234     theory -> bool -> (sort * string) list -> term -> term
   235   val ground_types_in_type : hol_context -> bool -> typ -> typ list
   236   val ground_types_in_terms : hol_context -> bool -> term list -> typ list
   237 end;
   238 
   239 structure Nitpick_HOL : NITPICK_HOL =
   240 struct
   241 
   242 open Nitpick_Util
   243 
   244 type const_table = term list Symtab.table
   245 type special_fun = (styp * int list * term list) * styp
   246 type unrolled = styp * styp
   247 type wf_cache = (styp * (bool * bool)) list
   248 
   249 type hol_context =
   250   {thy: theory,
   251    ctxt: Proof.context,
   252    max_bisim_depth: int,
   253    boxes: (typ option * bool option) list,
   254    stds: (typ option * bool) list,
   255    wfs: (styp option * bool option) list,
   256    user_axioms: bool option,
   257    debug: bool,
   258    whacks: term list,
   259    binary_ints: bool option,
   260    destroy_constrs: bool,
   261    specialize: bool,
   262    star_linear_preds: bool,
   263    total_consts: bool option,
   264    needs: term list option,
   265    tac_timeout: Time.time option,
   266    evals: term list,
   267    case_names: (string * int) list,
   268    def_tables: const_table * const_table,
   269    nondef_table: const_table,
   270    nondefs: term list,
   271    simp_table: const_table Unsynchronized.ref,
   272    psimp_table: const_table,
   273    choice_spec_table: const_table,
   274    intro_table: const_table,
   275    ground_thm_table: term list Inttab.table,
   276    ersatz_table: (string * string) list,
   277    skolems: (string * string list) list Unsynchronized.ref,
   278    special_funs: special_fun list Unsynchronized.ref,
   279    unrolled_preds: unrolled list Unsynchronized.ref,
   280    wf_cache: wf_cache Unsynchronized.ref,
   281    constr_cache: (typ * styp list) list Unsynchronized.ref}
   282 
   283 datatype fixpoint_kind = Lfp | Gfp | NoFp
   284 datatype boxability =
   285   InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
   286 
   287 structure Data = Generic_Data
   288 (
   289   type T = {frac_types: (string * (string * string) list) list,
   290             codatatypes: (string * (string * styp list)) list}
   291   val empty = {frac_types = [], codatatypes = []}
   292   val extend = I
   293   fun merge ({frac_types = fs1, codatatypes = cs1},
   294              {frac_types = fs2, codatatypes = cs2}) : T =
   295     {frac_types = AList.merge (op =) (K true) (fs1, fs2),
   296      codatatypes = AList.merge (op =) (K true) (cs1, cs2)}
   297 )
   298 
   299 val name_sep = "$"
   300 val numeral_prefix = nitpick_prefix ^ "num" ^ name_sep
   301 val sel_prefix = nitpick_prefix ^ "sel"
   302 val discr_prefix = nitpick_prefix ^ "is" ^ name_sep
   303 val set_prefix = nitpick_prefix ^ "set" ^ name_sep
   304 val lfp_iterator_prefix = nitpick_prefix ^ "lfpit" ^ name_sep
   305 val gfp_iterator_prefix = nitpick_prefix ^ "gfpit" ^ name_sep
   306 val unrolled_prefix = nitpick_prefix ^ "unroll" ^ name_sep
   307 val base_prefix = nitpick_prefix ^ "base" ^ name_sep
   308 val step_prefix = nitpick_prefix ^ "step" ^ name_sep
   309 val ubfp_prefix = nitpick_prefix ^ "ubfp" ^ name_sep
   310 val lbfp_prefix = nitpick_prefix ^ "lbfp" ^ name_sep
   311 val quot_normal_prefix = nitpick_prefix ^ "qn" ^ name_sep
   312 val skolem_prefix = nitpick_prefix ^ "sk"
   313 val special_prefix = nitpick_prefix ^ "sp"
   314 val uncurry_prefix = nitpick_prefix ^ "unc"
   315 val eval_prefix = nitpick_prefix ^ "eval"
   316 val iter_var_prefix = "i"
   317 
   318 (** Constant/type information and term/type manipulation **)
   319 
   320 fun sel_prefix_for j = sel_prefix ^ string_of_int j ^ name_sep
   321 fun quot_normal_name_for_type ctxt T =
   322   quot_normal_prefix ^ unyxml (Syntax.string_of_typ ctxt T)
   323 
   324 val strip_first_name_sep =
   325   Substring.full #> Substring.position name_sep ##> Substring.triml 1
   326   #> pairself Substring.string
   327 fun original_name s =
   328   if String.isPrefix nitpick_prefix s then
   329     case strip_first_name_sep s of (s1, "") => s1 | (_, s2) => original_name s2
   330   else
   331     s
   332 
   333 fun s_conj (t1, @{const True}) = t1
   334   | s_conj (@{const True}, t2) = t2
   335   | s_conj (t1, t2) =
   336     if t1 = @{const False} orelse t2 = @{const False} then @{const False}
   337     else HOLogic.mk_conj (t1, t2)
   338 fun s_disj (t1, @{const False}) = t1
   339   | s_disj (@{const False}, t2) = t2
   340   | s_disj (t1, t2) =
   341     if t1 = @{const True} orelse t2 = @{const True} then @{const True}
   342     else HOLogic.mk_disj (t1, t2)
   343 
   344 fun strip_connective conn_t (t as (t0 $ t1 $ t2)) =
   345     if t0 = conn_t then strip_connective t0 t2 @ strip_connective t0 t1 else [t]
   346   | strip_connective _ t = [t]
   347 fun strip_any_connective (t as (t0 $ _ $ _)) =
   348     if t0 = @{const HOL.conj} orelse t0 = @{const HOL.disj} then
   349       (strip_connective t0 t, t0)
   350     else
   351       ([t], @{const Not})
   352   | strip_any_connective t = ([t], @{const Not})
   353 val conjuncts_of = strip_connective @{const HOL.conj}
   354 val disjuncts_of = strip_connective @{const HOL.disj}
   355 
   356 (* When you add constants to these lists, make sure to handle them in
   357    "Nitpick_Nut.nut_from_term", and perhaps in "Nitpick_Mono.consider_term" as
   358    well. *)
   359 val built_in_consts =
   360   [(@{const_name all}, 1),
   361    (@{const_name "=="}, 2),
   362    (@{const_name "==>"}, 2),
   363    (@{const_name Pure.conjunction}, 2),
   364    (@{const_name Trueprop}, 1),
   365    (@{const_name Not}, 1),
   366    (@{const_name False}, 0),
   367    (@{const_name True}, 0),
   368    (@{const_name All}, 1),
   369    (@{const_name Ex}, 1),
   370    (@{const_name HOL.eq}, 1),
   371    (@{const_name HOL.conj}, 2),
   372    (@{const_name HOL.disj}, 2),
   373    (@{const_name HOL.implies}, 2),
   374    (@{const_name If}, 3),
   375    (@{const_name Let}, 2),
   376    (@{const_name Pair}, 2),
   377    (@{const_name fst}, 1),
   378    (@{const_name snd}, 1),
   379    (@{const_name Id}, 0),
   380    (@{const_name converse}, 1),
   381    (@{const_name trancl}, 1),
   382    (@{const_name rel_comp}, 2),
   383    (@{const_name finite}, 1),
   384    (@{const_name unknown}, 0),
   385    (@{const_name is_unknown}, 1),
   386    (@{const_name safe_The}, 1),
   387    (@{const_name Frac}, 0),
   388    (@{const_name norm_frac}, 0)]
   389 val built_in_nat_consts =
   390   [(@{const_name Suc}, 0),
   391    (@{const_name nat}, 0),
   392    (@{const_name nat_gcd}, 0),
   393    (@{const_name nat_lcm}, 0)]
   394 val built_in_typed_consts =
   395   [((@{const_name zero_class.zero}, int_T), 0),
   396    ((@{const_name one_class.one}, int_T), 0),
   397    ((@{const_name plus_class.plus}, int_T --> int_T --> int_T), 0),
   398    ((@{const_name minus_class.minus}, int_T --> int_T --> int_T), 0),
   399    ((@{const_name times_class.times}, int_T --> int_T --> int_T), 0),
   400    ((@{const_name div_class.div}, int_T --> int_T --> int_T), 0),
   401    ((@{const_name uminus_class.uminus}, int_T --> int_T), 0),
   402    ((@{const_name ord_class.less}, int_T --> int_T --> bool_T), 2),
   403    ((@{const_name ord_class.less_eq}, int_T --> int_T --> bool_T), 2)]
   404 val built_in_typed_nat_consts =
   405   [((@{const_name zero_class.zero}, nat_T), 0),
   406    ((@{const_name one_class.one}, nat_T), 0),
   407    ((@{const_name plus_class.plus}, nat_T --> nat_T --> nat_T), 0),
   408    ((@{const_name minus_class.minus}, nat_T --> nat_T --> nat_T), 0),
   409    ((@{const_name times_class.times}, nat_T --> nat_T --> nat_T), 0),
   410    ((@{const_name div_class.div}, nat_T --> nat_T --> nat_T), 0),
   411    ((@{const_name ord_class.less}, nat_T --> nat_T --> bool_T), 2),
   412    ((@{const_name ord_class.less_eq}, nat_T --> nat_T --> bool_T), 2),
   413    ((@{const_name of_nat}, nat_T --> int_T), 0)]
   414 val built_in_set_consts =
   415   [(@{const_name ord_class.less_eq}, 2)]
   416 
   417 fun unarize_type @{typ "unsigned_bit word"} = nat_T
   418   | unarize_type @{typ "signed_bit word"} = int_T
   419   | unarize_type (Type (s, Ts as _ :: _)) = Type (s, map unarize_type Ts)
   420   | unarize_type T = T
   421 fun unarize_unbox_etc_type (Type (@{type_name fun_box}, Ts)) =
   422     unarize_unbox_etc_type (Type (@{type_name fun}, Ts))
   423   | unarize_unbox_etc_type (Type (@{type_name pair_box}, Ts)) =
   424     Type (@{type_name prod}, map unarize_unbox_etc_type Ts)
   425   | unarize_unbox_etc_type @{typ "unsigned_bit word"} = nat_T
   426   | unarize_unbox_etc_type @{typ "signed_bit word"} = int_T
   427   | unarize_unbox_etc_type (Type (s, Ts as _ :: _)) =
   428     Type (s, map unarize_unbox_etc_type Ts)
   429   | unarize_unbox_etc_type T = T
   430 fun uniterize_type (Type (s, Ts as _ :: _)) = Type (s, map uniterize_type Ts)
   431   | uniterize_type @{typ bisim_iterator} = nat_T
   432   | uniterize_type T = T
   433 val uniterize_unarize_unbox_etc_type = uniterize_type o unarize_unbox_etc_type
   434 
   435 fun string_for_type ctxt = Syntax.string_of_typ ctxt o unarize_unbox_etc_type
   436 fun pretty_for_type ctxt = Syntax.pretty_typ ctxt o unarize_unbox_etc_type
   437 
   438 val prefix_name = Long_Name.qualify o Long_Name.base_name
   439 fun shortest_name s = List.last (space_explode "." s) handle List.Empty => ""
   440 val prefix_abs_vars = Term.map_abs_vars o prefix_name
   441 fun short_name s =
   442   case space_explode name_sep s of
   443     [_] => s |> String.isPrefix nitpick_prefix s ? unprefix nitpick_prefix
   444   | ss => map shortest_name ss |> space_implode "_"
   445 fun shorten_names_in_type (Type (s, Ts)) =
   446     Type (short_name s, map shorten_names_in_type Ts)
   447   | shorten_names_in_type T = T
   448 val shorten_names_in_term =
   449   map_aterms (fn Const (s, T) => Const (short_name s, T) | t => t)
   450   #> map_types shorten_names_in_type
   451 
   452 fun strict_type_match thy (T1, T2) =
   453   (Sign.typ_match thy (T2, T1) Vartab.empty; true)
   454   handle Type.TYPE_MATCH => false
   455 fun type_match thy = strict_type_match thy o pairself unarize_unbox_etc_type
   456 fun const_match thy ((s1, T1), (s2, T2)) =
   457   s1 = s2 andalso type_match thy (T1, T2)
   458 fun term_match thy (Const x1, Const x2) = const_match thy (x1, x2)
   459   | term_match thy (Free (s1, T1), Free (s2, T2)) =
   460     const_match thy ((shortest_name s1, T1), (shortest_name s2, T2))
   461   | term_match _ (t1, t2) = t1 aconv t2
   462 
   463 fun frac_from_term_pair T t1 t2 =
   464   case snd (HOLogic.dest_number t1) of
   465     0 => HOLogic.mk_number T 0
   466   | n1 => case snd (HOLogic.dest_number t2) of
   467             1 => HOLogic.mk_number T n1
   468           | n2 => Const (@{const_name divide}, T --> T --> T)
   469                   $ HOLogic.mk_number T n1 $ HOLogic.mk_number T n2
   470 
   471 fun is_TFree (TFree _) = true
   472   | is_TFree _ = false
   473 fun is_fun_type (Type (@{type_name fun}, _)) = true
   474   | is_fun_type _ = false
   475 fun is_set_type (Type (@{type_name fun}, [_, @{typ bool}])) = true
   476   | is_set_type _ = false
   477 fun is_pair_type (Type (@{type_name prod}, _)) = true
   478   | is_pair_type _ = false
   479 fun is_lfp_iterator_type (Type (s, _)) = String.isPrefix lfp_iterator_prefix s
   480   | is_lfp_iterator_type _ = false
   481 fun is_gfp_iterator_type (Type (s, _)) = String.isPrefix gfp_iterator_prefix s
   482   | is_gfp_iterator_type _ = false
   483 val is_fp_iterator_type = is_lfp_iterator_type orf is_gfp_iterator_type
   484 fun is_iterator_type T =
   485   (T = @{typ bisim_iterator} orelse is_fp_iterator_type T)
   486 fun is_boolean_type T = (T = prop_T orelse T = bool_T)
   487 fun is_integer_type T = (T = nat_T orelse T = int_T)
   488 fun is_bit_type T = (T = @{typ unsigned_bit} orelse T = @{typ signed_bit})
   489 fun is_word_type (Type (@{type_name word}, _)) = true
   490   | is_word_type _ = false
   491 val is_integer_like_type = is_iterator_type orf is_integer_type orf is_word_type
   492 val is_record_type = not o null o Record.dest_recTs
   493 fun is_frac_type ctxt (Type (s, [])) =
   494     s |> AList.lookup (op =) (#frac_types (Data.get (Context.Proof ctxt)))
   495       |> these |> null |> not
   496   | is_frac_type _ _ = false
   497 fun is_number_type ctxt = is_integer_like_type orf is_frac_type ctxt
   498 fun is_higher_order_type (Type (@{type_name fun}, _)) = true
   499   | is_higher_order_type (Type (_, Ts)) = exists is_higher_order_type Ts
   500   | is_higher_order_type _ = false
   501 
   502 fun iterator_type_for_const gfp (s, T) =
   503   Type ((if gfp then gfp_iterator_prefix else lfp_iterator_prefix) ^ s,
   504         binder_types T)
   505 fun const_for_iterator_type (Type (s, Ts)) =
   506     (strip_first_name_sep s |> snd, Ts ---> bool_T)
   507   | const_for_iterator_type T =
   508     raise TYPE ("Nitpick_HOL.const_for_iterator_type", [T], [])
   509 
   510 fun strip_n_binders 0 T = ([], T)
   511   | strip_n_binders n (Type (@{type_name fun}, [T1, T2])) =
   512     strip_n_binders (n - 1) T2 |>> cons T1
   513   | strip_n_binders n (Type (@{type_name fun_box}, Ts)) =
   514     strip_n_binders n (Type (@{type_name fun}, Ts))
   515   | strip_n_binders _ T = raise TYPE ("Nitpick_HOL.strip_n_binders", [T], [])
   516 val nth_range_type = snd oo strip_n_binders
   517 
   518 fun num_factors_in_type (Type (@{type_name prod}, [T1, T2])) =
   519     fold (Integer.add o num_factors_in_type) [T1, T2] 0
   520   | num_factors_in_type _ = 1
   521 fun num_binder_types (Type (@{type_name fun}, [_, T2])) =
   522     1 + num_binder_types T2
   523   | num_binder_types _ = 0
   524 val curried_binder_types = maps HOLogic.flatten_tupleT o binder_types
   525 fun maybe_curried_binder_types T =
   526   (if is_pair_type (body_type T) then binder_types else curried_binder_types) T
   527 
   528 fun mk_flat_tuple _ [t] = t
   529   | mk_flat_tuple (Type (@{type_name prod}, [T1, T2])) (t :: ts) =
   530     HOLogic.pair_const T1 T2 $ t $ (mk_flat_tuple T2 ts)
   531   | mk_flat_tuple T ts = raise TYPE ("Nitpick_HOL.mk_flat_tuple", [T], ts)
   532 fun dest_n_tuple 1 t = [t]
   533   | dest_n_tuple n t = HOLogic.dest_prod t ||> dest_n_tuple (n - 1) |> op ::
   534 
   535 type typedef_info =
   536   {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string,
   537    set_def: thm option, prop_of_Rep: thm, set_name: string,
   538    Abs_inverse: thm option, Rep_inverse: thm option}
   539 
   540 fun typedef_info ctxt s =
   541   if is_frac_type ctxt (Type (s, [])) then
   542     SOME {abs_type = Type (s, []), rep_type = @{typ "int * int"},
   543           Abs_name = @{const_name Abs_Frac}, Rep_name = @{const_name Rep_Frac},
   544           set_def = NONE, prop_of_Rep = @{prop "Rep_Frac x \<in> Frac"}
   545                           |> Logic.varify_global,
   546           set_name = @{const_name Frac}, Abs_inverse = NONE, Rep_inverse = NONE}
   547   else case Typedef.get_info ctxt s of
   548     (* When several entries are returned, it shouldn't matter much which one
   549        we take (according to Florian Haftmann). *)
   550     (* The "Logic.varifyT_global" calls are a temporary hack because these
   551        types's type variables sometimes clash with locally fixed type variables.
   552        Remove these calls once "Typedef" is fully localized. *)
   553     ({abs_type, rep_type, Abs_name, Rep_name, ...},
   554      {set_def, Rep, Abs_inverse, Rep_inverse, ...}) :: _ =>
   555     SOME {abs_type = Logic.varifyT_global abs_type,
   556           rep_type = Logic.varifyT_global rep_type, Abs_name = Abs_name,
   557           Rep_name = Rep_name, set_def = set_def, prop_of_Rep = prop_of Rep,
   558           set_name = set_prefix ^ s, Abs_inverse = SOME Abs_inverse,
   559           Rep_inverse = SOME Rep_inverse}
   560   | _ => NONE
   561 
   562 val is_typedef = is_some oo typedef_info
   563 val is_real_datatype = is_some oo Datatype.get_info
   564 fun is_standard_datatype thy = the oo triple_lookup (type_match thy)
   565 
   566 (* FIXME: Use antiquotation for "code_numeral" below or detect "rep_datatype",
   567    e.g., by adding a field to "Datatype_Aux.info". *)
   568 fun is_basic_datatype thy stds s =
   569   member (op =) [@{type_name prod}, @{type_name bool}, @{type_name int},
   570                  "Code_Numeral.code_numeral"] s orelse
   571   (s = @{type_name nat} andalso is_standard_datatype thy stds nat_T)
   572 
   573 fun repair_constr_type ctxt body_T' T =
   574   varify_and_instantiate_type ctxt (body_type T) body_T' T
   575 
   576 fun register_frac_type_generic frac_s ersaetze generic =
   577   let
   578     val {frac_types, codatatypes} = Data.get generic
   579     val frac_types = AList.update (op =) (frac_s, ersaetze) frac_types
   580   in Data.put {frac_types = frac_types, codatatypes = codatatypes} generic end
   581 (* TODO: Consider morphism. *)
   582 fun register_frac_type frac_s ersaetze (_ : morphism) =
   583   register_frac_type_generic frac_s ersaetze
   584 val register_frac_type_global = Context.theory_map oo register_frac_type_generic
   585 
   586 fun unregister_frac_type_generic frac_s = register_frac_type_generic frac_s []
   587 (* TODO: Consider morphism. *)
   588 fun unregister_frac_type frac_s (_ : morphism) =
   589   unregister_frac_type_generic frac_s
   590 val unregister_frac_type_global =
   591   Context.theory_map o unregister_frac_type_generic
   592 
   593 fun register_codatatype_generic co_T case_name constr_xs generic =
   594   let
   595     val ctxt = Context.proof_of generic
   596     val thy = Context.theory_of generic
   597     val {frac_types, codatatypes} = Data.get generic
   598     val constr_xs = map (apsnd (repair_constr_type ctxt co_T)) constr_xs
   599     val (co_s, co_Ts) = dest_Type co_T
   600     val _ =
   601       if forall is_TFree co_Ts andalso not (has_duplicates (op =) co_Ts) andalso
   602          co_s <> @{type_name fun} andalso
   603          not (is_basic_datatype thy [(NONE, true)] co_s) then
   604         ()
   605       else
   606         raise TYPE ("Nitpick_HOL.register_codatatype_generic", [co_T], [])
   607     val codatatypes = AList.update (op =) (co_s, (case_name, constr_xs))
   608                                    codatatypes
   609   in Data.put {frac_types = frac_types, codatatypes = codatatypes} generic end
   610 (* TODO: Consider morphism. *)
   611 fun register_codatatype co_T case_name constr_xs (_ : morphism) =
   612   register_codatatype_generic co_T case_name constr_xs
   613 val register_codatatype_global =
   614   Context.theory_map ooo register_codatatype_generic
   615 
   616 fun unregister_codatatype_generic co_T = register_codatatype_generic co_T "" []
   617 (* TODO: Consider morphism. *)
   618 fun unregister_codatatype co_T (_ : morphism) =
   619   unregister_codatatype_generic co_T
   620 val unregister_codatatype_global =
   621   Context.theory_map o unregister_codatatype_generic
   622 
   623 fun is_codatatype ctxt (Type (s, _)) =
   624     s |> AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   625       |> Option.map snd |> these |> null |> not
   626   | is_codatatype _ _ = false
   627 fun is_real_quot_type thy (Type (s, _)) =
   628     is_some (Quotient_Info.quotdata_lookup_raw thy s)
   629   | is_real_quot_type _ _ = false
   630 fun is_quot_type ctxt T =
   631   let val thy = Proof_Context.theory_of ctxt in
   632     is_real_quot_type thy T andalso not (is_codatatype ctxt T)
   633   end
   634 fun is_pure_typedef ctxt (T as Type (s, _)) =
   635     let val thy = Proof_Context.theory_of ctxt in
   636       is_typedef ctxt s andalso
   637       not (is_real_datatype thy s orelse is_real_quot_type thy T orelse
   638            is_codatatype ctxt T orelse is_record_type T orelse
   639            is_integer_like_type T)
   640     end
   641   | is_pure_typedef _ _ = false
   642 fun is_univ_typedef ctxt (Type (s, _)) =
   643     (case typedef_info ctxt s of
   644        SOME {set_def, prop_of_Rep, ...} =>
   645        let
   646          val t_opt =
   647            case set_def of
   648              SOME thm => try (snd o Logic.dest_equals o prop_of) thm
   649            | NONE => try (snd o HOLogic.dest_mem o HOLogic.dest_Trueprop)
   650                          prop_of_Rep
   651        in
   652          case t_opt of
   653            SOME (Const (@{const_name top}, _)) => true
   654            (* "Multiset.multiset" *)
   655          | SOME (Const (@{const_name Collect}, _)
   656                  $ Abs (_, _, Const (@{const_name finite}, _) $ _)) => true
   657            (* "FinFun.finfun" *)
   658          | SOME (Const (@{const_name Collect}, _) $ Abs (_, _,
   659                      Const (@{const_name Ex}, _) $ Abs (_, _,
   660                          Const (@{const_name finite}, _) $ _))) => true
   661          | _ => false
   662        end
   663      | NONE => false)
   664   | is_univ_typedef _ _ = false
   665 fun is_datatype ctxt stds (T as Type (s, _)) =
   666     let val thy = Proof_Context.theory_of ctxt in
   667       (is_typedef ctxt s orelse is_codatatype ctxt T orelse
   668        T = @{typ ind} orelse is_real_quot_type thy T) andalso
   669       not (is_basic_datatype thy stds s)
   670     end
   671   | is_datatype _ _ _ = false
   672 
   673 fun all_record_fields thy T =
   674   let val (recs, more) = Record.get_extT_fields thy T in
   675     recs @ more :: all_record_fields thy (snd more)
   676   end
   677   handle TYPE _ => []
   678 fun is_record_constr (s, T) =
   679   String.isSuffix Record.extN s andalso
   680   let val dataT = body_type T in
   681     is_record_type dataT andalso
   682     s = unsuffix Record.ext_typeN (fst (dest_Type dataT)) ^ Record.extN
   683   end
   684 val num_record_fields = Integer.add 1 o length o fst oo Record.get_extT_fields
   685 fun no_of_record_field thy s T1 =
   686   find_index (curry (op =) s o fst)
   687              (Record.get_extT_fields thy T1 ||> single |> op @)
   688 fun is_record_get thy (s, Type (@{type_name fun}, [T1, _])) =
   689     exists (curry (op =) s o fst) (all_record_fields thy T1)
   690   | is_record_get _ _ = false
   691 fun is_record_update thy (s, T) =
   692   String.isSuffix Record.updateN s andalso
   693   exists (curry (op =) (unsuffix Record.updateN s) o fst)
   694          (all_record_fields thy (body_type T))
   695   handle TYPE _ => false
   696 fun is_abs_fun ctxt (s, Type (@{type_name fun}, [_, Type (s', _)])) =
   697     (case typedef_info ctxt s' of
   698        SOME {Abs_name, ...} => s = Abs_name
   699      | NONE => false)
   700   | is_abs_fun _ _ = false
   701 fun is_rep_fun ctxt (s, Type (@{type_name fun}, [Type (s', _), _])) =
   702     (case typedef_info ctxt s' of
   703        SOME {Rep_name, ...} => s = Rep_name
   704      | NONE => false)
   705   | is_rep_fun _ _ = false
   706 fun is_quot_abs_fun ctxt (x as (_, Type (@{type_name fun},
   707                                          [_, abs_T as Type (s', _)]))) =
   708     try (Quotient_Term.absrep_const_chk Quotient_Term.AbsF ctxt) s'
   709     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   710   | is_quot_abs_fun _ _ = false
   711 fun is_quot_rep_fun ctxt (x as (_, Type (@{type_name fun},
   712                                          [abs_T as Type (s', _), _]))) =
   713     try (Quotient_Term.absrep_const_chk Quotient_Term.RepF ctxt) s'
   714     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   715   | is_quot_rep_fun _ _ = false
   716 
   717 fun mate_of_rep_fun ctxt (x as (_, Type (@{type_name fun},
   718                                          [T1 as Type (s', _), T2]))) =
   719     (case typedef_info ctxt s' of
   720        SOME {Abs_name, ...} => (Abs_name, Type (@{type_name fun}, [T2, T1]))
   721      | NONE => raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x]))
   722   | mate_of_rep_fun _ x = raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x])
   723 fun rep_type_for_quot_type thy (T as Type (s, _)) =
   724     let val {qtyp, rtyp, ...} = Quotient_Info.quotdata_lookup thy s in
   725       instantiate_type thy qtyp T rtyp
   726     end
   727   | rep_type_for_quot_type _ T =
   728     raise TYPE ("Nitpick_HOL.rep_type_for_quot_type", [T], [])
   729 fun equiv_relation_for_quot_type thy (Type (s, Ts)) =
   730     let
   731       val {qtyp, equiv_rel, equiv_thm, ...} =
   732         Quotient_Info.quotdata_lookup thy s
   733       val partial =
   734         case prop_of equiv_thm of
   735           @{const Trueprop} $ (Const (@{const_name equivp}, _) $ _) => false
   736         | @{const Trueprop} $ (Const (@{const_name part_equivp}, _) $ _) => true
   737         | _ => raise NOT_SUPPORTED "Ill-formed quotient type equivalence \
   738                                    \relation theorem"
   739       val Ts' = qtyp |> dest_Type |> snd
   740     in (subst_atomic_types (Ts' ~~ Ts) equiv_rel, partial) end
   741   | equiv_relation_for_quot_type _ T =
   742     raise TYPE ("Nitpick_HOL.equiv_relation_for_quot_type", [T], [])
   743 
   744 fun is_coconstr ctxt (s, T) =
   745   case body_type T of
   746     co_T as Type (co_s, _) =>
   747     let val {codatatypes, ...} = Data.get (Context.Proof ctxt) in
   748       exists (fn (s', T') => s = s' andalso repair_constr_type ctxt co_T T' = T)
   749              (AList.lookup (op =) codatatypes co_s |> Option.map snd |> these)
   750     end
   751   | _ => false
   752 fun is_constr_like ctxt (s, T) =
   753   member (op =) [@{const_name FunBox}, @{const_name PairBox},
   754                  @{const_name Quot}, @{const_name Zero_Rep},
   755                  @{const_name Suc_Rep}] s orelse
   756   let
   757     val thy = Proof_Context.theory_of ctxt
   758     val (x as (_, T)) = (s, unarize_unbox_etc_type T)
   759   in
   760     is_real_constr thy x orelse is_record_constr x orelse
   761     (is_abs_fun ctxt x andalso is_pure_typedef ctxt (range_type T)) orelse
   762     is_coconstr ctxt x
   763   end
   764 fun is_stale_constr ctxt (x as (_, T)) =
   765   is_codatatype ctxt (body_type T) andalso is_constr_like ctxt x andalso
   766   not (is_coconstr ctxt x)
   767 fun is_constr ctxt stds (x as (_, T)) =
   768   let val thy = Proof_Context.theory_of ctxt in
   769     is_constr_like ctxt x andalso
   770     not (is_basic_datatype thy stds
   771                          (fst (dest_Type (unarize_type (body_type T))))) andalso
   772     not (is_stale_constr ctxt x)
   773   end
   774 val is_sel = String.isPrefix discr_prefix orf String.isPrefix sel_prefix
   775 val is_sel_like_and_no_discr =
   776   String.isPrefix sel_prefix orf
   777   (member (op =) [@{const_name fst}, @{const_name snd}])
   778 
   779 fun in_fun_lhs_for InConstr = InSel
   780   | in_fun_lhs_for _ = InFunLHS
   781 fun in_fun_rhs_for InConstr = InConstr
   782   | in_fun_rhs_for InSel = InSel
   783   | in_fun_rhs_for InFunRHS1 = InFunRHS2
   784   | in_fun_rhs_for _ = InFunRHS1
   785 
   786 fun is_boxing_worth_it (hol_ctxt : hol_context) boxy T =
   787   case T of
   788     Type (@{type_name fun}, _) =>
   789     (boxy = InPair orelse boxy = InFunLHS) andalso
   790     not (is_boolean_type (body_type T))
   791   | Type (@{type_name prod}, Ts) =>
   792     boxy = InPair orelse boxy = InFunRHS1 orelse boxy = InFunRHS2 orelse
   793     ((boxy = InExpr orelse boxy = InFunLHS) andalso
   794      exists (is_boxing_worth_it hol_ctxt InPair)
   795             (map (box_type hol_ctxt InPair) Ts))
   796   | _ => false
   797 and should_box_type (hol_ctxt as {thy, boxes, ...}) boxy z =
   798   case triple_lookup (type_match thy) boxes (Type z) of
   799     SOME (SOME box_me) => box_me
   800   | _ => is_boxing_worth_it hol_ctxt boxy (Type z)
   801 and box_type hol_ctxt boxy T =
   802   case T of
   803     Type (z as (@{type_name fun}, [T1, T2])) =>
   804     if boxy <> InConstr andalso boxy <> InSel andalso
   805        should_box_type hol_ctxt boxy z then
   806       Type (@{type_name fun_box},
   807             [box_type hol_ctxt InFunLHS T1, box_type hol_ctxt InFunRHS1 T2])
   808     else
   809       box_type hol_ctxt (in_fun_lhs_for boxy) T1
   810       --> box_type hol_ctxt (in_fun_rhs_for boxy) T2
   811   | Type (z as (@{type_name prod}, Ts)) =>
   812     if boxy <> InConstr andalso boxy <> InSel
   813        andalso should_box_type hol_ctxt boxy z then
   814       Type (@{type_name pair_box}, map (box_type hol_ctxt InSel) Ts)
   815     else
   816       Type (@{type_name prod},
   817             map (box_type hol_ctxt
   818                           (if boxy = InConstr orelse boxy = InSel then boxy
   819                            else InPair)) Ts)
   820   | _ => T
   821 
   822 fun binarize_nat_and_int_in_type @{typ nat} = @{typ "unsigned_bit word"}
   823   | binarize_nat_and_int_in_type @{typ int} = @{typ "signed_bit word"}
   824   | binarize_nat_and_int_in_type (Type (s, Ts)) =
   825     Type (s, map binarize_nat_and_int_in_type Ts)
   826   | binarize_nat_and_int_in_type T = T
   827 val binarize_nat_and_int_in_term = map_types binarize_nat_and_int_in_type
   828 
   829 fun discr_for_constr (s, T) = (discr_prefix ^ s, body_type T --> bool_T)
   830 
   831 fun num_sels_for_constr_type T = length (maybe_curried_binder_types T)
   832 fun nth_sel_name_for_constr_name s n =
   833   if s = @{const_name Pair} then
   834     if n = 0 then @{const_name fst} else @{const_name snd}
   835   else
   836     sel_prefix_for n ^ s
   837 fun nth_sel_for_constr x ~1 = discr_for_constr x
   838   | nth_sel_for_constr (s, T) n =
   839     (nth_sel_name_for_constr_name s n,
   840      body_type T --> nth (maybe_curried_binder_types T) n)
   841 fun binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize =
   842   apsnd ((binarize ? binarize_nat_and_int_in_type) o box_type hol_ctxt InSel)
   843   oo nth_sel_for_constr
   844 
   845 fun sel_no_from_name s =
   846   if String.isPrefix discr_prefix s then
   847     ~1
   848   else if String.isPrefix sel_prefix s then
   849     s |> unprefix sel_prefix |> Int.fromString |> the
   850   else if s = @{const_name snd} then
   851     1
   852   else
   853     0
   854 
   855 val close_form =
   856   let
   857     fun close_up zs zs' =
   858       fold (fn (z as ((s, _), T)) => fn t' =>
   859                Term.all T $ Abs (s, T, abstract_over (Var z, t')))
   860            (take (length zs' - length zs) zs')
   861     fun aux zs (@{const "==>"} $ t1 $ t2) =
   862         let val zs' = Term.add_vars t1 zs in
   863           close_up zs zs' (Logic.mk_implies (t1, aux zs' t2))
   864         end
   865       | aux zs t = close_up zs (Term.add_vars t zs) t
   866   in aux [] end
   867 
   868 fun distinctness_formula T =
   869   all_distinct_unordered_pairs_of
   870   #> map (fn (t1, t2) => @{const Not} $ (HOLogic.eq_const T $ t1 $ t2))
   871   #> List.foldr (s_conj o swap) @{const True}
   872 
   873 fun zero_const T = Const (@{const_name zero_class.zero}, T)
   874 fun suc_const T = Const (@{const_name Suc}, T --> T)
   875 
   876 fun uncached_datatype_constrs ({thy, ctxt, stds, ...} : hol_context)
   877                               (T as Type (s, Ts)) =
   878     (case AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   879                        s of
   880        SOME (_, xs' as (_ :: _)) => map (apsnd (repair_constr_type ctxt T)) xs'
   881      | _ =>
   882        if is_datatype ctxt stds T then
   883          case Datatype.get_info thy s of
   884            SOME {index, descr, ...} =>
   885            let
   886              val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the
   887            in
   888              map (apsnd (fn Us =>
   889                             map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))
   890                  constrs
   891            end
   892          | NONE =>
   893            if is_record_type T then
   894              let
   895                val s' = unsuffix Record.ext_typeN s ^ Record.extN
   896                val T' = (Record.get_extT_fields thy T
   897                         |> apsnd single |> uncurry append |> map snd) ---> T
   898              in [(s', T')] end
   899            else if is_real_quot_type thy T then
   900              [(@{const_name Quot}, rep_type_for_quot_type thy T --> T)]
   901            else case typedef_info ctxt s of
   902              SOME {abs_type, rep_type, Abs_name, ...} =>
   903              [(Abs_name,
   904                varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
   905            | NONE =>
   906              if T = @{typ ind} then
   907                [dest_Const @{const Zero_Rep}, dest_Const @{const Suc_Rep}]
   908              else
   909                []
   910        else
   911          [])
   912   | uncached_datatype_constrs _ _ = []
   913 fun datatype_constrs (hol_ctxt as {constr_cache, ...}) T =
   914   case AList.lookup (op =) (!constr_cache) T of
   915     SOME xs => xs
   916   | NONE =>
   917     let val xs = uncached_datatype_constrs hol_ctxt T in
   918       (Unsynchronized.change constr_cache (cons (T, xs)); xs)
   919     end
   920 fun binarized_and_boxed_datatype_constrs hol_ctxt binarize =
   921   map (apsnd ((binarize ? binarize_nat_and_int_in_type)
   922               o box_type hol_ctxt InConstr)) o datatype_constrs hol_ctxt
   923 val num_datatype_constrs = length oo datatype_constrs
   924 
   925 fun constr_name_for_sel_like @{const_name fst} = @{const_name Pair}
   926   | constr_name_for_sel_like @{const_name snd} = @{const_name Pair}
   927   | constr_name_for_sel_like s' = original_name s'
   928 fun binarized_and_boxed_constr_for_sel hol_ctxt binarize (s', T') =
   929   let val s = constr_name_for_sel_like s' in
   930     AList.lookup (op =)
   931         (binarized_and_boxed_datatype_constrs hol_ctxt binarize (domain_type T'))
   932         s
   933     |> the |> pair s
   934   end
   935 
   936 fun card_of_type assigns (Type (@{type_name fun}, [T1, T2])) =
   937     reasonable_power (card_of_type assigns T2) (card_of_type assigns T1)
   938   | card_of_type assigns (Type (@{type_name prod}, [T1, T2])) =
   939     card_of_type assigns T1 * card_of_type assigns T2
   940   | card_of_type _ (Type (@{type_name itself}, _)) = 1
   941   | card_of_type _ @{typ prop} = 2
   942   | card_of_type _ @{typ bool} = 2
   943   | card_of_type assigns T =
   944     case AList.lookup (op =) assigns T of
   945       SOME k => k
   946     | NONE => if T = @{typ bisim_iterator} then 0
   947               else raise TYPE ("Nitpick_HOL.card_of_type", [T], [])
   948 
   949 fun bounded_card_of_type max default_card assigns
   950                          (Type (@{type_name fun}, [T1, T2])) =
   951     let
   952       val k1 = bounded_card_of_type max default_card assigns T1
   953       val k2 = bounded_card_of_type max default_card assigns T2
   954     in
   955       if k1 = max orelse k2 = max then max
   956       else Int.min (max, reasonable_power k2 k1)
   957     end
   958   | bounded_card_of_type max default_card assigns
   959                          (Type (@{type_name prod}, [T1, T2])) =
   960     let
   961       val k1 = bounded_card_of_type max default_card assigns T1
   962       val k2 = bounded_card_of_type max default_card assigns T2
   963     in if k1 = max orelse k2 = max then max else Int.min (max, k1 * k2) end
   964   | bounded_card_of_type max default_card assigns T =
   965     Int.min (max, if default_card = ~1 then
   966                     card_of_type assigns T
   967                   else
   968                     card_of_type assigns T
   969                     handle TYPE ("Nitpick_HOL.card_of_type", _, _) =>
   970                            default_card)
   971 
   972 (* Similar to "ATP_Translate.tiny_card_of_type". *)
   973 fun bounded_exact_card_of_type hol_ctxt finitizable_dataTs max default_card
   974                                assigns T =
   975   let
   976     fun aux avoid T =
   977       (if member (op =) avoid T then
   978          0
   979        else if member (op =) finitizable_dataTs T then
   980          raise SAME ()
   981        else case T of
   982          Type (@{type_name fun}, [T1, T2]) =>
   983          (case (aux avoid T1, aux avoid T2) of
   984             (_, 1) => 1
   985           | (0, _) => 0
   986           | (_, 0) => 0
   987           | (k1, k2) =>
   988             if k1 >= max orelse k2 >= max then max
   989             else Int.min (max, reasonable_power k2 k1))
   990        | Type (@{type_name prod}, [T1, T2]) =>
   991          (case (aux avoid T1, aux avoid T2) of
   992             (0, _) => 0
   993           | (_, 0) => 0
   994           | (k1, k2) =>
   995             if k1 >= max orelse k2 >= max then max
   996             else Int.min (max, k1 * k2))
   997        | Type (@{type_name itself}, _) => 1
   998        | @{typ prop} => 2
   999        | @{typ bool} => 2
  1000        | Type _ =>
  1001          (case datatype_constrs hol_ctxt T of
  1002             [] => if is_integer_type T orelse is_bit_type T then 0
  1003                   else raise SAME ()
  1004           | constrs =>
  1005             let
  1006               val constr_cards =
  1007                 map (Integer.prod o map (aux (T :: avoid)) o binder_types o snd)
  1008                     constrs
  1009             in
  1010               if exists (curry (op =) 0) constr_cards then 0
  1011               else Int.min (max, Integer.sum constr_cards)
  1012             end)
  1013        | _ => raise SAME ())
  1014       handle SAME () =>
  1015              AList.lookup (op =) assigns T |> the_default default_card
  1016   in Int.min (max, aux [] T) end
  1017 
  1018 val typical_atomic_card = 4
  1019 val typical_card_of_type = bounded_card_of_type 16777217 typical_atomic_card []
  1020 
  1021 fun is_finite_type hol_ctxt T =
  1022   bounded_exact_card_of_type hol_ctxt [] 1 2 [] T > 0
  1023 
  1024 fun is_special_eligible_arg strict Ts t =
  1025   case map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t) of
  1026     [] => true
  1027   | bad_Ts =>
  1028     let
  1029       val bad_Ts_cost =
  1030         if strict then fold (curry (op *) o typical_card_of_type) bad_Ts 1
  1031         else fold (Integer.max o typical_card_of_type) bad_Ts 0
  1032       val T_cost = typical_card_of_type (fastype_of1 (Ts, t))
  1033     in (bad_Ts_cost, T_cost) |> (if strict then op < else op <=) end
  1034 
  1035 fun abs_var ((s, j), T) body = Abs (s, T, abstract_over (Var ((s, j), T), body))
  1036 
  1037 fun let_var s = (nitpick_prefix ^ s, 999)
  1038 val let_inline_threshold = 20
  1039 
  1040 fun s_let Ts s n abs_T body_T f t =
  1041   if (n - 1) * (size_of_term t - 1) <= let_inline_threshold orelse
  1042      is_special_eligible_arg false Ts t then
  1043     f t
  1044   else
  1045     let val z = (let_var s, abs_T) in
  1046       Const (@{const_name Let}, abs_T --> (abs_T --> body_T) --> body_T)
  1047       $ t $ abs_var z (incr_boundvars 1 (f (Var z)))
  1048     end
  1049 
  1050 fun loose_bvar1_count (Bound i, k) = if i = k then 1 else 0
  1051   | loose_bvar1_count (t1 $ t2, k) =
  1052     loose_bvar1_count (t1, k) + loose_bvar1_count (t2, k)
  1053   | loose_bvar1_count (Abs (_, _, t), k) = loose_bvar1_count (t, k + 1)
  1054   | loose_bvar1_count _ = 0
  1055 
  1056 fun s_betapply _ (t1 as Const (@{const_name "=="}, _) $ t1', t2) =
  1057     if t1' aconv t2 then @{prop True} else t1 $ t2
  1058   | s_betapply _ (t1 as Const (@{const_name HOL.eq}, _) $ t1', t2) =
  1059     if t1' aconv t2 then @{term True} else t1 $ t2
  1060   | s_betapply _ (Const (@{const_name If}, _) $ @{const True} $ t1', _) = t1'
  1061   | s_betapply _ (Const (@{const_name If}, _) $ @{const False} $ _, t2) = t2
  1062   | s_betapply Ts (Const (@{const_name Let},
  1063                           Type (_, [bound_T, Type (_, [_, body_T])]))
  1064                    $ t12 $ Abs (s, T, t13'), t2) =
  1065     let val body_T' = range_type body_T in
  1066       Const (@{const_name Let}, bound_T --> (bound_T --> body_T') --> body_T')
  1067       $ t12 $ Abs (s, T, s_betapply (T :: Ts) (t13', incr_boundvars 1 t2))
  1068     end
  1069   | s_betapply Ts (t1 as Abs (s1, T1, t1'), t2) =
  1070     (s_let Ts s1 (loose_bvar1_count (t1', 0)) T1 (fastype_of1 (T1 :: Ts, t1'))
  1071            (curry betapply t1) t2
  1072      (* FIXME: fix all "s_betapply []" calls *)
  1073      handle TERM _ => betapply (t1, t2)
  1074           | General.Subscript => betapply (t1, t2))
  1075   | s_betapply _ (t1, t2) = t1 $ t2
  1076 fun s_betapplys Ts = Library.foldl (s_betapply Ts)
  1077 
  1078 fun s_beta_norm Ts t =
  1079   let
  1080     fun aux _ (Var _) = raise Same.SAME
  1081       | aux Ts (Abs (s, T, t')) = Abs (s, T, aux (T :: Ts) t')
  1082       | aux Ts ((t1 as Abs _) $ t2) =
  1083         Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1084       | aux Ts (t1 $ t2) =
  1085         ((case aux Ts t1 of
  1086            t1 as Abs _ => Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1087          | t1 => t1 $ Same.commit (aux Ts) t2)
  1088         handle Same.SAME => t1 $ aux Ts t2)
  1089       | aux _ _ = raise Same.SAME
  1090   in aux Ts t handle Same.SAME => t end
  1091 
  1092 fun discr_term_for_constr hol_ctxt (x as (s, T)) =
  1093   let val dataT = body_type T in
  1094     if s = @{const_name Suc} then
  1095       Abs (Name.uu, dataT,
  1096            @{const Not} $ HOLogic.mk_eq (zero_const dataT, Bound 0))
  1097     else if num_datatype_constrs hol_ctxt dataT >= 2 then
  1098       Const (discr_for_constr x)
  1099     else
  1100       Abs (Name.uu, dataT, @{const True})
  1101   end
  1102 fun discriminate_value (hol_ctxt as {ctxt, ...}) x t =
  1103   case head_of t of
  1104     Const x' =>
  1105     if x = x' then @{const True}
  1106     else if is_constr_like ctxt x' then @{const False}
  1107     else s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1108   | _ => s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1109 
  1110 fun nth_arg_sel_term_for_constr thy stds (x as (s, T)) n =
  1111   let val (arg_Ts, dataT) = strip_type T in
  1112     if dataT = nat_T andalso is_standard_datatype thy stds nat_T then
  1113       @{term "%n::nat. n - 1"}
  1114     else if is_pair_type dataT then
  1115       Const (nth_sel_for_constr x n)
  1116     else
  1117       let
  1118         fun aux m (Type (@{type_name prod}, [T1, T2])) =
  1119             let
  1120               val (m, t1) = aux m T1
  1121               val (m, t2) = aux m T2
  1122             in (m, HOLogic.mk_prod (t1, t2)) end
  1123           | aux m T =
  1124             (m + 1, Const (nth_sel_name_for_constr_name s m, dataT --> T)
  1125                     $ Bound 0)
  1126         val m = fold (Integer.add o num_factors_in_type)
  1127                      (List.take (arg_Ts, n)) 0
  1128       in Abs ("x", dataT, aux m (nth arg_Ts n) |> snd) end
  1129   end
  1130 fun select_nth_constr_arg ctxt stds x t n res_T =
  1131   let val thy = Proof_Context.theory_of ctxt in
  1132     (case strip_comb t of
  1133        (Const x', args) =>
  1134        if x = x' then nth args n
  1135        else if is_constr_like ctxt x' then Const (@{const_name unknown}, res_T)
  1136        else raise SAME ()
  1137      | _ => raise SAME())
  1138     handle SAME () =>
  1139            s_betapply [] (nth_arg_sel_term_for_constr thy stds x n, t)
  1140   end
  1141 
  1142 fun construct_value _ _ x [] = Const x
  1143   | construct_value ctxt stds (x as (s, _)) args =
  1144     let val args = map Envir.eta_contract args in
  1145       case hd args of
  1146         Const (s', _) $ t =>
  1147         if is_sel_like_and_no_discr s' andalso
  1148            constr_name_for_sel_like s' = s andalso
  1149            forall (fn (n, t') =>
  1150                       select_nth_constr_arg ctxt stds x t n dummyT = t')
  1151                   (index_seq 0 (length args) ~~ args) then
  1152           t
  1153         else
  1154           list_comb (Const x, args)
  1155       | _ => list_comb (Const x, args)
  1156     end
  1157 
  1158 fun constr_expand (hol_ctxt as {ctxt, stds, ...}) T t =
  1159   (case head_of t of
  1160      Const x => if is_constr_like ctxt x then t else raise SAME ()
  1161    | _ => raise SAME ())
  1162   handle SAME () =>
  1163          let
  1164            val x' as (_, T') =
  1165              if is_pair_type T then
  1166                let val (T1, T2) = HOLogic.dest_prodT T in
  1167                  (@{const_name Pair}, T1 --> T2 --> T)
  1168                end
  1169              else
  1170                datatype_constrs hol_ctxt T |> hd
  1171            val arg_Ts = binder_types T'
  1172          in
  1173            list_comb (Const x', map2 (select_nth_constr_arg ctxt stds x' t)
  1174                                      (index_seq 0 (length arg_Ts)) arg_Ts)
  1175          end
  1176 
  1177 fun coerce_bound_no f j t =
  1178   case t of
  1179     t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2
  1180   | Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')
  1181   | Bound j' => if j' = j then f t else t
  1182   | _ => t
  1183 fun coerce_bound_0_in_term hol_ctxt new_T old_T =
  1184   old_T <> new_T ? coerce_bound_no (coerce_term hol_ctxt [new_T] old_T new_T) 0
  1185 and coerce_term (hol_ctxt as {ctxt, stds, ...}) Ts new_T old_T t =
  1186   if old_T = new_T then
  1187     t
  1188   else
  1189     case (new_T, old_T) of
  1190       (Type (new_s, new_Ts as [new_T1, new_T2]),
  1191        Type (@{type_name fun}, [old_T1, old_T2])) =>
  1192       (case eta_expand Ts t 1 of
  1193          Abs (s, _, t') =>
  1194          Abs (s, new_T1,
  1195               t' |> coerce_bound_0_in_term hol_ctxt new_T1 old_T1
  1196                  |> coerce_term hol_ctxt (new_T1 :: Ts) new_T2 old_T2)
  1197          |> Envir.eta_contract
  1198          |> new_s <> @{type_name fun}
  1199             ? construct_value ctxt stds
  1200                   (@{const_name FunBox},
  1201                    Type (@{type_name fun}, new_Ts) --> new_T)
  1202               o single
  1203        | t' => raise TERM ("Nitpick_HOL.coerce_term", [t']))
  1204     | (Type (new_s, new_Ts as [new_T1, new_T2]),
  1205        Type (old_s, old_Ts as [old_T1, old_T2])) =>
  1206       if old_s = @{type_name fun_box} orelse
  1207          old_s = @{type_name pair_box} orelse old_s = @{type_name prod} then
  1208         case constr_expand hol_ctxt old_T t of
  1209           Const (old_s, _) $ t1 =>
  1210           if new_s = @{type_name fun} then
  1211             coerce_term hol_ctxt Ts new_T (Type (@{type_name fun}, old_Ts)) t1
  1212           else
  1213             construct_value ctxt stds
  1214                 (old_s, Type (@{type_name fun}, new_Ts) --> new_T)
  1215                 [coerce_term hol_ctxt Ts (Type (@{type_name fun}, new_Ts))
  1216                              (Type (@{type_name fun}, old_Ts)) t1]
  1217         | Const _ $ t1 $ t2 =>
  1218           construct_value ctxt stds
  1219               (if new_s = @{type_name prod} then @{const_name Pair}
  1220                else @{const_name PairBox}, new_Ts ---> new_T)
  1221               (map3 (coerce_term hol_ctxt Ts) [new_T1, new_T2] [old_T1, old_T2]
  1222                     [t1, t2])
  1223         | t' => raise TERM ("Nitpick_HOL.coerce_term", [t'])
  1224       else
  1225         raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1226     | _ => raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1227 
  1228 fun is_ground_term (t1 $ t2) = is_ground_term t1 andalso is_ground_term t2
  1229   | is_ground_term (Const _) = true
  1230   | is_ground_term _ = false
  1231 
  1232 fun hashw_term (t1 $ t2) = hashw (hashw_term t1, hashw_term t2)
  1233   | hashw_term (Const (s, _)) = hashw_string (s, 0w0)
  1234   | hashw_term _ = 0w0
  1235 val hash_term = Word.toInt o hashw_term
  1236 
  1237 fun special_bounds ts =
  1238   fold Term.add_vars ts [] |> sort (Term_Ord.fast_indexname_ord o pairself fst)
  1239 
  1240 (* FIXME: detect "rep_datatype"? *)
  1241 fun is_funky_typedef_name ctxt s =
  1242   member (op =) [@{type_name unit}, @{type_name prod},
  1243                  @{type_name Sum_Type.sum}, @{type_name int}] s orelse
  1244   is_frac_type ctxt (Type (s, []))
  1245 fun is_funky_typedef ctxt (Type (s, _)) = is_funky_typedef_name ctxt s
  1246   | is_funky_typedef _ _ = false
  1247 fun is_typedef_axiom ctxt boring (@{const "==>"} $ _ $ t2) =
  1248     is_typedef_axiom ctxt boring t2
  1249   | is_typedef_axiom ctxt boring
  1250         (@{const Trueprop} $ (Const (@{const_name Typedef.type_definition}, _)
  1251          $ Const (_, Type (@{type_name fun}, [Type (s, _), _]))
  1252          $ Const _ $ _)) =
  1253     boring <> is_funky_typedef_name ctxt s andalso is_typedef ctxt s
  1254   | is_typedef_axiom _ _ _ = false
  1255 
  1256 fun all_defs_of thy subst =
  1257   let
  1258     val def_names =
  1259       thy |> Theory.defs_of
  1260           |> Defs.all_specifications_of
  1261           |> maps snd |> map_filter #def
  1262           |> Ord_List.make fast_string_ord
  1263   in
  1264     Theory.nodes_of thy
  1265     |> maps Thm.axioms_of
  1266     |> map (apsnd (subst_atomic subst o prop_of))
  1267     |> sort (fast_string_ord o pairself fst)
  1268     |> Ord_List.inter (fast_string_ord o apsnd fst) def_names
  1269     |> map snd
  1270   end
  1271 
  1272 (* Ideally we would check against "Complex_Main", not "Refute", but any theory
  1273    will do as long as it contains all the "axioms" and "axiomatization"
  1274    commands. *)
  1275 fun is_built_in_theory thy = Theory.subthy (thy, @{theory Refute})
  1276 
  1277 fun all_nondefs_of ctxt subst =
  1278   ctxt |> Spec_Rules.get
  1279        |> filter (curry (op =) Spec_Rules.Unknown o fst)
  1280        |> maps (snd o snd)
  1281        |> filter_out (is_built_in_theory o theory_of_thm)
  1282        |> map (subst_atomic subst o prop_of)
  1283 
  1284 fun arity_of_built_in_const thy stds (s, T) =
  1285   if s = @{const_name If} then
  1286     if nth_range_type 3 T = @{typ bool} then NONE else SOME 3
  1287   else
  1288     let val std_nats = is_standard_datatype thy stds nat_T in
  1289       case AList.lookup (op =)
  1290                     (built_in_consts
  1291                      |> std_nats ? append built_in_nat_consts) s of
  1292         SOME n => SOME n
  1293       | NONE =>
  1294         case AList.lookup (op =)
  1295                  (built_in_typed_consts
  1296                   |> std_nats ? append built_in_typed_nat_consts)
  1297                  (s, unarize_type T) of
  1298           SOME n => SOME n
  1299         | NONE =>
  1300           case s of
  1301             @{const_name zero_class.zero} =>
  1302             if is_iterator_type T then SOME 0 else NONE
  1303           | @{const_name Suc} =>
  1304             if is_iterator_type (domain_type T) then SOME 0 else NONE
  1305           | _ => if is_fun_type T andalso is_set_type (domain_type T) then
  1306                    AList.lookup (op =) built_in_set_consts s
  1307                  else
  1308                    NONE
  1309     end
  1310 val is_built_in_const = is_some ooo arity_of_built_in_const
  1311 
  1312 (* This function is designed to work for both real definition axioms and
  1313    simplification rules (equational specifications). *)
  1314 fun term_under_def t =
  1315   case t of
  1316     @{const "==>"} $ _ $ t2 => term_under_def t2
  1317   | Const (@{const_name "=="}, _) $ t1 $ _ => term_under_def t1
  1318   | @{const Trueprop} $ t1 => term_under_def t1
  1319   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => term_under_def t1
  1320   | Abs (_, _, t') => term_under_def t'
  1321   | t1 $ _ => term_under_def t1
  1322   | _ => t
  1323 
  1324 (* Here we crucially rely on "specialize_type" performing a preorder traversal
  1325    of the term, without which the wrong occurrence of a constant could be
  1326    matched in the face of overloading. *)
  1327 fun def_props_for_const thy stds table (x as (s, _)) =
  1328   if is_built_in_const thy stds x then
  1329     []
  1330   else
  1331     these (Symtab.lookup table s)
  1332     |> map_filter (try (specialize_type thy x))
  1333     |> filter (curry (op =) (Const x) o term_under_def)
  1334 
  1335 fun normalized_rhs_of t =
  1336   let
  1337     fun aux (v as Var _) (SOME t) = SOME (lambda v t)
  1338       | aux (c as Const (@{const_name TYPE}, _)) (SOME t) = SOME (lambda c t)
  1339       | aux _ _ = NONE
  1340     val (lhs, rhs) =
  1341       case t of
  1342         Const (@{const_name "=="}, _) $ t1 $ t2 => (t1, t2)
  1343       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ t2) =>
  1344         (t1, t2)
  1345       | _ => raise TERM ("Nitpick_HOL.normalized_rhs_of", [t])
  1346     val args = strip_comb lhs |> snd
  1347   in fold_rev aux args (SOME rhs) end
  1348 
  1349 fun get_def_of_const thy table (x as (s, _)) =
  1350   x |> def_props_for_const thy [(NONE, false)] table |> List.last
  1351     |> normalized_rhs_of |> Option.map (prefix_abs_vars s)
  1352   handle List.Empty => NONE
  1353 
  1354 fun def_of_const_ext thy (unfold_table, fallback_table) (x as (s, _)) =
  1355   if is_built_in_const thy [(NONE, false)] x orelse original_name s <> s then
  1356     NONE
  1357   else case get_def_of_const thy unfold_table x of
  1358     SOME def => SOME (true, def)
  1359   | NONE => get_def_of_const thy fallback_table x |> Option.map (pair false)
  1360 
  1361 val def_of_const = Option.map snd ooo def_of_const_ext
  1362 
  1363 fun fixpoint_kind_of_rhs (Abs (_, _, t)) = fixpoint_kind_of_rhs t
  1364   | fixpoint_kind_of_rhs (Const (@{const_name lfp}, _) $ Abs _) = Lfp
  1365   | fixpoint_kind_of_rhs (Const (@{const_name gfp}, _) $ Abs _) = Gfp
  1366   | fixpoint_kind_of_rhs _ = NoFp
  1367 
  1368 fun is_mutually_inductive_pred_def thy table t =
  1369   let
  1370     fun is_good_arg (Bound _) = true
  1371       | is_good_arg (Const (s, _)) =
  1372         s = @{const_name True} orelse s = @{const_name False} orelse
  1373         s = @{const_name undefined}
  1374       | is_good_arg _ = false
  1375   in
  1376     case t |> strip_abs_body |> strip_comb of
  1377       (Const x, ts as (_ :: _)) =>
  1378       (case def_of_const thy table x of
  1379          SOME t' => fixpoint_kind_of_rhs t' <> NoFp andalso
  1380                     forall is_good_arg ts
  1381        | NONE => false)
  1382     | _ => false
  1383   end
  1384 fun unfold_mutually_inductive_preds thy table =
  1385   map_aterms (fn t as Const x =>
  1386                  (case def_of_const thy table x of
  1387                     SOME t' =>
  1388                     let val t' = Envir.eta_contract t' in
  1389                       if is_mutually_inductive_pred_def thy table t' then t'
  1390                       else t
  1391                     end
  1392                  | NONE => t)
  1393                | t => t)
  1394 
  1395 fun case_const_names ctxt stds =
  1396   let val thy = Proof_Context.theory_of ctxt in
  1397     Symtab.fold (fn (dtype_s, {index, descr, case_name, ...}) =>
  1398                     if is_basic_datatype thy stds dtype_s then
  1399                       I
  1400                     else
  1401                       cons (case_name, AList.lookup (op =) descr index
  1402                                        |> the |> #3 |> length))
  1403                 (Datatype.get_all thy) [] @
  1404     map (apsnd length o snd) (#codatatypes (Data.get (Context.Proof ctxt)))
  1405   end
  1406 
  1407 fun fixpoint_kind_of_const thy table x =
  1408   if is_built_in_const thy [(NONE, false)] x then NoFp
  1409   else fixpoint_kind_of_rhs (the (def_of_const thy table x))
  1410   handle Option.Option => NoFp
  1411 
  1412 fun is_real_inductive_pred ({thy, stds, def_tables, intro_table, ...}
  1413                             : hol_context) x =
  1414   fixpoint_kind_of_const thy def_tables x <> NoFp andalso
  1415   not (null (def_props_for_const thy stds intro_table x))
  1416 fun is_inductive_pred hol_ctxt (x as (s, _)) =
  1417   is_real_inductive_pred hol_ctxt x orelse String.isPrefix ubfp_prefix s orelse
  1418   String.isPrefix lbfp_prefix s
  1419 
  1420 fun lhs_of_equation t =
  1421   case t of
  1422     Const (@{const_name all}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1423   | Const (@{const_name "=="}, _) $ t1 $ _ => SOME t1
  1424   | @{const "==>"} $ _ $ t2 => lhs_of_equation t2
  1425   | @{const Trueprop} $ t1 => lhs_of_equation t1
  1426   | Const (@{const_name All}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1427   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => SOME t1
  1428   | @{const HOL.implies} $ _ $ t2 => lhs_of_equation t2
  1429   | _ => NONE
  1430 fun is_constr_pattern _ (Bound _) = true
  1431   | is_constr_pattern _ (Var _) = true
  1432   | is_constr_pattern ctxt t =
  1433     case strip_comb t of
  1434       (Const x, args) =>
  1435       is_constr_like ctxt x andalso forall (is_constr_pattern ctxt) args
  1436     | _ => false
  1437 fun is_constr_pattern_lhs ctxt t =
  1438   forall (is_constr_pattern ctxt) (snd (strip_comb t))
  1439 fun is_constr_pattern_formula ctxt t =
  1440   case lhs_of_equation t of
  1441     SOME t' => is_constr_pattern_lhs ctxt t'
  1442   | NONE => false
  1443 
  1444 (* Similar to "specialize_type" but returns all matches rather than only the
  1445    first (preorder) match. *)
  1446 fun multi_specialize_type thy slack (s, T) t =
  1447   let
  1448     fun aux (Const (s', T')) ys =
  1449         if s = s' then
  1450           ys |> (if AList.defined (op =) ys T' then
  1451                    I
  1452                  else
  1453                    cons (T', monomorphic_term (Sign.typ_match thy (T', T)
  1454                                                               Vartab.empty) t)
  1455                    handle Type.TYPE_MATCH => I
  1456                         | TERM _ =>
  1457                           if slack then
  1458                             I
  1459                           else
  1460                             raise NOT_SUPPORTED
  1461                                       ("too much polymorphism in axiom \"" ^
  1462                                        Syntax.string_of_term_global thy t ^
  1463                                        "\" involving " ^ quote s))
  1464         else
  1465           ys
  1466       | aux _ ys = ys
  1467   in map snd (fold_aterms aux t []) end
  1468 fun nondef_props_for_const thy slack table (x as (s, _)) =
  1469   these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)
  1470 
  1471 fun unvarify_term (t1 $ t2) = unvarify_term t1 $ unvarify_term t2
  1472   | unvarify_term (Var ((s, 0), T)) = Free (s, T)
  1473   | unvarify_term (Abs (s, T, t')) = Abs (s, T, unvarify_term t')
  1474   | unvarify_term t = t
  1475 fun axiom_for_choice_spec thy =
  1476   unvarify_term
  1477   #> Object_Logic.atomize_term thy
  1478   #> Choice_Specification.close_form
  1479   #> HOLogic.mk_Trueprop
  1480 fun is_choice_spec_fun ({thy, def_tables, nondef_table, choice_spec_table, ...}
  1481                         : hol_context) x =
  1482   case nondef_props_for_const thy true choice_spec_table x of
  1483     [] => false
  1484   | ts => case def_of_const thy def_tables x of
  1485             SOME (Const (@{const_name Eps}, _) $ _) => true
  1486           | SOME _ => false
  1487           | NONE =>
  1488             let val ts' = nondef_props_for_const thy true nondef_table x in
  1489               length ts' = length ts andalso
  1490               forall (fn t =>
  1491                          exists (curry (op aconv) (axiom_for_choice_spec thy t))
  1492                                 ts') ts
  1493             end
  1494 
  1495 fun is_choice_spec_axiom thy choice_spec_table t =
  1496   Symtab.exists (fn (_, ts) =>
  1497                     exists (curry (op aconv) t o axiom_for_choice_spec thy) ts)
  1498                 choice_spec_table
  1499 
  1500 fun is_real_equational_fun ({thy, stds, simp_table, psimp_table, ...}
  1501                             : hol_context) x =
  1502   exists (fn table => not (null (def_props_for_const thy stds table x)))
  1503          [!simp_table, psimp_table]
  1504 fun is_equational_fun_but_no_plain_def hol_ctxt =
  1505   is_real_equational_fun hol_ctxt orf is_inductive_pred hol_ctxt
  1506 
  1507 (** Constant unfolding **)
  1508 
  1509 fun constr_case_body ctxt stds Ts (func_t, (x as (_, T))) =
  1510   let val arg_Ts = binder_types T in
  1511     s_betapplys Ts (func_t, map2 (select_nth_constr_arg ctxt stds x (Bound 0))
  1512                                  (index_seq 0 (length arg_Ts)) arg_Ts)
  1513   end
  1514 fun add_constr_case res_T (body_t, guard_t) res_t =
  1515   if res_T = bool_T then
  1516     s_conj (HOLogic.mk_imp (guard_t, body_t), res_t)
  1517   else
  1518     Const (@{const_name If}, bool_T --> res_T --> res_T --> res_T)
  1519     $ guard_t $ body_t $ res_t
  1520 fun optimized_case_def (hol_ctxt as {ctxt, stds, ...}) Ts dataT res_T func_ts =
  1521   let
  1522     val xs = datatype_constrs hol_ctxt dataT
  1523     val cases =
  1524       func_ts ~~ xs
  1525       |> map (fn (func_t, x) =>
  1526                  (constr_case_body ctxt stds (dataT :: Ts)
  1527                                    (incr_boundvars 1 func_t, x),
  1528                   discriminate_value hol_ctxt x (Bound 0)))
  1529       |> AList.group (op aconv)
  1530       |> map (apsnd (List.foldl s_disj @{const False}))
  1531       |> sort (int_ord o pairself (size_of_term o snd))
  1532       |> rev
  1533   in
  1534     if res_T = bool_T then
  1535       if forall (member (op =) [@{const False}, @{const True}] o fst) cases then
  1536         case cases of
  1537           [(body_t, _)] => body_t
  1538         | [_, (@{const True}, head_t2)] => head_t2
  1539         | [_, (@{const False}, head_t2)] => @{const Not} $ head_t2
  1540         | _ => raise BAD ("Nitpick_HOL.optimized_case_def", "impossible cases")
  1541       else
  1542         @{const True} |> fold_rev (add_constr_case res_T) cases
  1543     else
  1544       fst (hd cases) |> fold_rev (add_constr_case res_T) (tl cases)
  1545   end
  1546   |> curry absdummy dataT
  1547 
  1548 fun optimized_record_get (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T res_T t =
  1549   let val constr_x = hd (datatype_constrs hol_ctxt rec_T) in
  1550     case no_of_record_field thy s rec_T of
  1551       ~1 => (case rec_T of
  1552                Type (_, Ts as _ :: _) =>
  1553                let
  1554                  val rec_T' = List.last Ts
  1555                  val j = num_record_fields thy rec_T - 1
  1556                in
  1557                  select_nth_constr_arg ctxt stds constr_x t j res_T
  1558                  |> optimized_record_get hol_ctxt s rec_T' res_T
  1559                end
  1560              | _ => raise TYPE ("Nitpick_HOL.optimized_record_get", [rec_T],
  1561                                 []))
  1562     | j => select_nth_constr_arg ctxt stds constr_x t j res_T
  1563   end
  1564 fun optimized_record_update (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T fun_t
  1565                             rec_t =
  1566   let
  1567     val constr_x as (_, constr_T) = hd (datatype_constrs hol_ctxt rec_T)
  1568     val Ts = binder_types constr_T
  1569     val n = length Ts
  1570     val special_j = no_of_record_field thy s rec_T
  1571     val ts =
  1572       map2 (fn j => fn T =>
  1573                let val t = select_nth_constr_arg ctxt stds constr_x rec_t j T in
  1574                  if j = special_j then
  1575                    s_betapply [] (fun_t, t)
  1576                  else if j = n - 1 andalso special_j = ~1 then
  1577                    optimized_record_update hol_ctxt s
  1578                        (rec_T |> dest_Type |> snd |> List.last) fun_t t
  1579                  else
  1580                    t
  1581                end) (index_seq 0 n) Ts
  1582   in list_comb (Const constr_x, ts) end
  1583 
  1584 (* Prevents divergence in case of cyclic or infinite definition dependencies. *)
  1585 val unfold_max_depth = 255
  1586 
  1587 (* Inline definitions or define as an equational constant? Booleans tend to
  1588    benefit more from inlining, due to the polarity analysis. (However, if
  1589    "total_consts" is set, the polarity analysis is likely not to be so
  1590    crucial.) *)
  1591 val def_inline_threshold_for_booleans = 60
  1592 val def_inline_threshold_for_non_booleans = 20
  1593 
  1594 fun unfold_defs_in_term
  1595         (hol_ctxt as {thy, ctxt, stds, whacks, total_consts, case_names,
  1596                       def_tables, ground_thm_table, ersatz_table, ...}) =
  1597   let
  1598     fun do_term depth Ts t =
  1599       case t of
  1600         (t0 as Const (@{const_name Int.number_class.number_of},
  1601                       Type (@{type_name fun}, [_, ran_T]))) $ t1 =>
  1602         ((if is_number_type ctxt ran_T then
  1603             let
  1604               val j = t1 |> HOLogic.dest_numeral
  1605                          |> ran_T = nat_T ? Integer.max 0
  1606               val s = numeral_prefix ^ signed_string_of_int j
  1607             in
  1608               if is_integer_like_type ran_T then
  1609                 if is_standard_datatype thy stds ran_T then
  1610                   Const (s, ran_T)
  1611                 else
  1612                   funpow j (curry (op $) (suc_const ran_T)) (zero_const ran_T)
  1613               else
  1614                 do_term depth Ts (Const (@{const_name of_int}, int_T --> ran_T)
  1615                                   $ Const (s, int_T))
  1616             end
  1617             handle TERM _ => raise SAME ()
  1618           else
  1619             raise SAME ())
  1620          handle SAME () =>
  1621                 s_betapply [] (do_term depth Ts t0, do_term depth Ts t1))
  1622       | Const (@{const_name refl_on}, T) $ Const (@{const_name top}, _) $ t2 =>
  1623         do_const depth Ts t (@{const_name refl'}, range_type T) [t2]
  1624       | (t0 as Const (@{const_name Sigma}, Type (_, [T1, Type (_, [T2, T3])])))
  1625         $ t1 $ (t2 as Abs (_, _, t2')) =>
  1626         if loose_bvar1 (t2', 0) then
  1627           s_betapplys Ts (do_term depth Ts t0, map (do_term depth Ts) [t1, t2])
  1628         else
  1629           do_term depth Ts
  1630                   (Const (@{const_name prod}, T1 --> range_type T2 --> T3)
  1631                    $ t1 $ incr_boundvars ~1 t2')
  1632       | Const (x as (@{const_name distinct},
  1633                Type (@{type_name fun}, [Type (@{type_name list}, [T']), _])))
  1634         $ (t1 as _ $ _) =>
  1635         (t1 |> HOLogic.dest_list |> distinctness_formula T'
  1636          handle TERM _ => do_const depth Ts t x [t1])
  1637       | Const (x as (@{const_name If}, _)) $ t1 $ t2 $ t3 =>
  1638         if is_ground_term t1 andalso
  1639            exists (Pattern.matches thy o rpair t1)
  1640                   (Inttab.lookup_list ground_thm_table (hash_term t1)) then
  1641           do_term depth Ts t2
  1642         else
  1643           do_const depth Ts t x [t1, t2, t3]
  1644       | Const (@{const_name Let}, _) $ t1 $ t2 =>
  1645         s_betapply Ts (pairself (do_term depth Ts) (t2, t1))
  1646       | Const x => do_const depth Ts t x []
  1647       | t1 $ t2 =>
  1648         (case strip_comb t of
  1649            (Const x, ts) => do_const depth Ts t x ts
  1650          | _ => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2))
  1651       | Bound _ => t
  1652       | Abs (s, T, body) => Abs (s, T, do_term depth (T :: Ts) body)
  1653       | _ => if member (term_match thy) whacks t then
  1654                Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1655              else
  1656                t
  1657     and select_nth_constr_arg_with_args _ _ (x as (_, T)) [] n res_T =
  1658         (Abs (Name.uu, body_type T,
  1659               select_nth_constr_arg ctxt stds x (Bound 0) n res_T), [])
  1660       | select_nth_constr_arg_with_args depth Ts x (t :: ts) n res_T =
  1661         (select_nth_constr_arg ctxt stds x (do_term depth Ts t) n res_T, ts)
  1662     and quot_rep_of depth Ts abs_T rep_T ts =
  1663       select_nth_constr_arg_with_args depth Ts
  1664           (@{const_name Quot}, rep_T --> abs_T) ts 0 rep_T
  1665     and do_const depth Ts t (x as (s, T)) ts =
  1666       if member (term_match thy) whacks (Const x) then
  1667         Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1668       else case AList.lookup (op =) ersatz_table s of
  1669         SOME s' =>
  1670         do_const (depth + 1) Ts (list_comb (Const (s', T), ts)) (s', T) ts
  1671       | NONE =>
  1672         let
  1673           fun def_inline_threshold () =
  1674             if is_boolean_type (nth_range_type (length ts) T) andalso
  1675                total_consts <> SOME true then
  1676               def_inline_threshold_for_booleans
  1677             else
  1678               def_inline_threshold_for_non_booleans
  1679           val (const, ts) =
  1680             if is_built_in_const thy stds x then
  1681               (Const x, ts)
  1682             else case AList.lookup (op =) case_names s of
  1683               SOME n =>
  1684               if length ts < n then
  1685                 (do_term depth Ts (eta_expand Ts t (n - length ts)), [])
  1686               else
  1687                 let
  1688                   val (dataT, res_T) = nth_range_type n T
  1689                                        |> pairf domain_type range_type
  1690                 in
  1691                   (optimized_case_def hol_ctxt Ts dataT res_T
  1692                                       (map (do_term depth Ts) (take n ts)),
  1693                    drop n ts)
  1694                 end
  1695             | _ =>
  1696               if is_constr ctxt stds x then
  1697                 (Const x, ts)
  1698               else if is_stale_constr ctxt x then
  1699                 raise NOT_SUPPORTED ("(non-co)constructors of codatatypes \
  1700                                      \(\"" ^ s ^ "\")")
  1701               else if is_quot_abs_fun ctxt x then
  1702                 let
  1703                   val rep_T = domain_type T
  1704                   val abs_T = range_type T
  1705                 in
  1706                   (Abs (Name.uu, rep_T,
  1707                         Const (@{const_name Quot}, rep_T --> abs_T)
  1708                                $ (Const (quot_normal_name_for_type ctxt abs_T,
  1709                                          rep_T --> rep_T) $ Bound 0)), ts)
  1710                 end
  1711               else if is_quot_rep_fun ctxt x then
  1712                 quot_rep_of depth Ts (domain_type T) (range_type T) ts
  1713               else if is_record_get thy x then
  1714                 case length ts of
  1715                   0 => (do_term depth Ts (eta_expand Ts t 1), [])
  1716                 | _ => (optimized_record_get hol_ctxt s (domain_type T)
  1717                             (range_type T) (do_term depth Ts (hd ts)), tl ts)
  1718               else if is_record_update thy x then
  1719                 case length ts of
  1720                   2 => (optimized_record_update hol_ctxt
  1721                             (unsuffix Record.updateN s) (nth_range_type 2 T)
  1722                             (do_term depth Ts (hd ts))
  1723                             (do_term depth Ts (nth ts 1)), [])
  1724                 | n => (do_term depth Ts (eta_expand Ts t (2 - n)), [])
  1725               else if is_abs_fun ctxt x andalso
  1726                       is_quot_type ctxt (range_type T) then
  1727                 let
  1728                   val abs_T = range_type T
  1729                   val rep_T = domain_type (domain_type T)
  1730                   val eps_fun = Const (@{const_name Eps},
  1731                                        (rep_T --> bool_T) --> rep_T)
  1732                   val normal_fun =
  1733                     Const (quot_normal_name_for_type ctxt abs_T,
  1734                            rep_T --> rep_T)
  1735                   val abs_fun = Const (@{const_name Quot}, rep_T --> abs_T)
  1736                 in
  1737                   (Abs (Name.uu, rep_T --> bool_T,
  1738                         abs_fun $ (normal_fun $ (eps_fun $ Bound 0)))
  1739                    |> do_term (depth + 1) Ts, ts)
  1740                 end
  1741               else if is_rep_fun ctxt x then
  1742                 let val x' = mate_of_rep_fun ctxt x in
  1743                   if is_constr ctxt stds x' then
  1744                     select_nth_constr_arg_with_args depth Ts x' ts 0
  1745                                                     (range_type T)
  1746                   else if is_quot_type ctxt (domain_type T) then
  1747                     let
  1748                       val abs_T = domain_type T
  1749                       val rep_T = domain_type (range_type T)
  1750                       val (rep_fun, _) = quot_rep_of depth Ts abs_T rep_T []
  1751                       val (equiv_rel, _) =
  1752                         equiv_relation_for_quot_type thy abs_T
  1753                     in
  1754                       (Abs (Name.uu, abs_T, equiv_rel $ (rep_fun $ Bound 0)),
  1755                        ts)
  1756                     end
  1757                   else
  1758                     (Const x, ts)
  1759                 end
  1760               else if is_equational_fun_but_no_plain_def hol_ctxt x orelse
  1761                       is_choice_spec_fun hol_ctxt x then
  1762                 (Const x, ts)
  1763               else case def_of_const_ext thy def_tables x of
  1764                 SOME (unfold, def) =>
  1765                 if depth > unfold_max_depth then
  1766                   raise TOO_LARGE ("Nitpick_HOL.unfold_defs_in_term",
  1767                                    "too many nested definitions (" ^
  1768                                    string_of_int depth ^ ") while expanding " ^
  1769                                    quote s)
  1770                 else if s = @{const_name wfrec'} then
  1771                   (do_term (depth + 1) Ts (s_betapplys Ts (def, ts)), [])
  1772                 else if not unfold andalso
  1773                      size_of_term def > def_inline_threshold () then
  1774                   (Const x, ts)
  1775                 else
  1776                   (do_term (depth + 1) Ts def, ts)
  1777               | NONE => (Const x, ts)
  1778         in
  1779           s_betapplys Ts (const, map (do_term depth Ts) ts)
  1780           |> s_beta_norm Ts
  1781         end
  1782   in do_term 0 [] end
  1783 
  1784 (** Axiom extraction/generation **)
  1785 
  1786 fun extensional_equal j (Type (@{type_name fun}, [dom_T, ran_T])) t1 t2 =
  1787     let val var_t = Var (("x", j), dom_T) in
  1788       extensional_equal (j + 1) ran_T (betapply (t1, var_t))
  1789                         (betapply (t2, var_t))
  1790     end
  1791   | extensional_equal _ T t1 t2 =
  1792     Const (@{const_name HOL.eq}, T --> T --> bool_T) $ t1 $ t2
  1793 
  1794 fun equationalize_term ctxt tag t =
  1795   let
  1796     val j = maxidx_of_term t + 1
  1797     val (prems, concl) = Logic.strip_horn t
  1798   in
  1799     Logic.list_implies (prems,
  1800         case concl of
  1801           @{const Trueprop} $ (Const (@{const_name HOL.eq}, Type (_, [T, _]))
  1802                                $ t1 $ t2) =>
  1803           @{const Trueprop} $ extensional_equal j T t1 t2
  1804         | @{const Trueprop} $ t' =>
  1805           @{const Trueprop} $ HOLogic.mk_eq (t', @{const True})
  1806         | Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
  1807           @{const Trueprop} $ extensional_equal j T t1 t2
  1808         | _ => (warning ("Ignoring " ^ quote tag ^ " for non-equation" ^
  1809                          quote (Syntax.string_of_term ctxt t) ^ ".");
  1810                 raise SAME ()))
  1811     |> SOME
  1812   end
  1813   handle SAME () => NONE
  1814 
  1815 fun pair_for_prop t =
  1816   case term_under_def t of
  1817     Const (s, _) => (s, t)
  1818   | t' => raise TERM ("Nitpick_HOL.pair_for_prop", [t, t'])
  1819 
  1820 fun def_table_for get ctxt subst =
  1821   ctxt |> get |> map (pair_for_prop o subst_atomic subst)
  1822        |> AList.group (op =) |> Symtab.make
  1823 
  1824 fun const_def_tables ctxt subst ts =
  1825   (def_table_for (map prop_of o Nitpick_Unfolds.get) ctxt subst,
  1826    fold (fn (s, t) => Symtab.map_default (s, []) (cons t))
  1827         (map pair_for_prop ts) Symtab.empty)
  1828 
  1829 fun paired_with_consts t = map (rpair t) (Term.add_const_names t [])
  1830 fun const_nondef_table ts =
  1831   fold (append o paired_with_consts) ts [] |> AList.group (op =) |> Symtab.make
  1832 
  1833 fun const_simp_table ctxt =
  1834   def_table_for (map_filter (equationalize_term ctxt "nitpick_simp" o prop_of)
  1835                  o Nitpick_Simps.get) ctxt
  1836 fun const_psimp_table ctxt =
  1837   def_table_for (map_filter (equationalize_term ctxt "nitpick_psimp" o prop_of)
  1838                  o Nitpick_Psimps.get) ctxt
  1839 
  1840 fun const_choice_spec_table ctxt subst =
  1841   map (subst_atomic subst o prop_of) (Nitpick_Choice_Specs.get ctxt)
  1842   |> const_nondef_table
  1843 
  1844 fun inductive_intro_table ctxt subst def_tables =
  1845   let val thy = Proof_Context.theory_of ctxt in
  1846     def_table_for
  1847         (maps (map (unfold_mutually_inductive_preds thy def_tables o prop_of)
  1848                o snd o snd)
  1849          o filter (fn (cat, _) => cat = Spec_Rules.Inductive orelse
  1850                                   cat = Spec_Rules.Co_Inductive)
  1851          o Spec_Rules.get) ctxt subst
  1852   end
  1853 
  1854 fun ground_theorem_table thy =
  1855   fold ((fn @{const Trueprop} $ t1 =>
  1856             is_ground_term t1 ? Inttab.map_default (hash_term t1, []) (cons t1)
  1857           | _ => I) o prop_of o snd) (Global_Theory.all_thms_of thy) Inttab.empty
  1858 
  1859 (* TODO: Move to "Nitpick.thy" *)
  1860 val basic_ersatz_table =
  1861   [(@{const_name card}, @{const_name card'}),
  1862    (@{const_name setsum}, @{const_name setsum'}),
  1863    (@{const_name fold_graph}, @{const_name fold_graph'}),
  1864    (@{const_name wf}, @{const_name wf'}),
  1865    (@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
  1866    (@{const_name wfrec}, @{const_name wfrec'})]
  1867 
  1868 fun ersatz_table ctxt =
  1869  basic_ersatz_table
  1870  |> fold (append o snd) (#frac_types (Data.get (Context.Proof ctxt)))
  1871 
  1872 fun add_simps simp_table s eqs =
  1873   Unsynchronized.change simp_table
  1874       (Symtab.update (s, eqs @ these (Symtab.lookup (!simp_table) s)))
  1875 
  1876 fun inverse_axioms_for_rep_fun ctxt (x as (_, T)) =
  1877   let
  1878     val thy = Proof_Context.theory_of ctxt
  1879     val abs_T = domain_type T
  1880   in
  1881     typedef_info ctxt (fst (dest_Type abs_T)) |> the
  1882     |> pairf #Abs_inverse #Rep_inverse
  1883     |> pairself (specialize_type thy x o prop_of o the)
  1884     ||> single |> op ::
  1885   end
  1886 fun optimized_typedef_axioms ctxt (abs_z as (abs_s, _)) =
  1887   let
  1888     val thy = Proof_Context.theory_of ctxt
  1889     val abs_T = Type abs_z
  1890   in
  1891     if is_univ_typedef ctxt abs_T then
  1892       []
  1893     else case typedef_info ctxt abs_s of
  1894       SOME {abs_type, rep_type, Rep_name, prop_of_Rep, set_name, ...} =>
  1895       let
  1896         val rep_T = varify_and_instantiate_type ctxt abs_type abs_T rep_type
  1897         val rep_t = Const (Rep_name, abs_T --> rep_T)
  1898         val set_t = Const (set_name, rep_T --> bool_T)
  1899         val set_t' =
  1900           prop_of_Rep |> HOLogic.dest_Trueprop
  1901                       |> specialize_type thy (dest_Const rep_t)
  1902                       |> HOLogic.dest_mem |> snd
  1903       in
  1904         [HOLogic.all_const abs_T
  1905          $ Abs (Name.uu, abs_T, set_t $ (rep_t $ Bound 0))]
  1906         |> set_t <> set_t' ? cons (HOLogic.mk_eq (set_t, set_t'))
  1907         |> map HOLogic.mk_Trueprop
  1908       end
  1909     | NONE => []
  1910   end
  1911 fun optimized_quot_type_axioms ctxt stds abs_z =
  1912   let
  1913     val thy = Proof_Context.theory_of ctxt
  1914     val abs_T = Type abs_z
  1915     val rep_T = rep_type_for_quot_type thy abs_T
  1916     val (equiv_rel, partial) = equiv_relation_for_quot_type thy abs_T
  1917     val a_var = Var (("a", 0), abs_T)
  1918     val x_var = Var (("x", 0), rep_T)
  1919     val y_var = Var (("y", 0), rep_T)
  1920     val x = (@{const_name Quot}, rep_T --> abs_T)
  1921     val sel_a_t = select_nth_constr_arg ctxt stds x a_var 0 rep_T
  1922     val normal_fun =
  1923       Const (quot_normal_name_for_type ctxt abs_T, rep_T --> rep_T)
  1924     val normal_x = normal_fun $ x_var
  1925     val normal_y = normal_fun $ y_var
  1926     val is_unknown_t = Const (@{const_name is_unknown}, rep_T --> bool_T)
  1927   in
  1928     [Logic.mk_equals (normal_fun $ sel_a_t, sel_a_t),
  1929      Logic.list_implies
  1930          ([@{const Not} $ (is_unknown_t $ normal_x),
  1931            @{const Not} $ (is_unknown_t $ normal_y),
  1932            equiv_rel $ x_var $ y_var] |> map HOLogic.mk_Trueprop,
  1933            Logic.mk_equals (normal_x, normal_y)),
  1934      Logic.list_implies
  1935          ([HOLogic.mk_Trueprop (@{const Not} $ (is_unknown_t $ normal_x)),
  1936            HOLogic.mk_Trueprop (@{const Not} $ HOLogic.mk_eq (normal_x, x_var))],
  1937           HOLogic.mk_Trueprop (equiv_rel $ x_var $ normal_x))]
  1938     |> partial ? cons (HOLogic.mk_Trueprop (equiv_rel $ sel_a_t $ sel_a_t))
  1939   end
  1940 
  1941 fun codatatype_bisim_axioms (hol_ctxt as {ctxt, stds, ...}) T =
  1942   let
  1943     val xs = datatype_constrs hol_ctxt T
  1944     val set_T = T --> bool_T
  1945     val iter_T = @{typ bisim_iterator}
  1946     val bisim_max = @{const bisim_iterator_max}
  1947     val n_var = Var (("n", 0), iter_T)
  1948     val n_var_minus_1 =
  1949       Const (@{const_name safe_The}, (iter_T --> bool_T) --> iter_T)
  1950       $ Abs ("m", iter_T, HOLogic.eq_const iter_T
  1951                           $ (suc_const iter_T $ Bound 0) $ n_var)
  1952     val x_var = Var (("x", 0), T)
  1953     val y_var = Var (("y", 0), T)
  1954     fun bisim_const T =
  1955       Const (@{const_name bisim}, iter_T --> T --> T --> bool_T)
  1956     fun nth_sub_bisim x n nth_T =
  1957       (if is_codatatype ctxt nth_T then bisim_const nth_T $ n_var_minus_1
  1958        else HOLogic.eq_const nth_T)
  1959       $ select_nth_constr_arg ctxt stds x x_var n nth_T
  1960       $ select_nth_constr_arg ctxt stds x y_var n nth_T
  1961     fun case_func (x as (_, T)) =
  1962       let
  1963         val arg_Ts = binder_types T
  1964         val core_t =
  1965           discriminate_value hol_ctxt x y_var ::
  1966           map2 (nth_sub_bisim x) (index_seq 0 (length arg_Ts)) arg_Ts
  1967           |> foldr1 s_conj
  1968       in List.foldr absdummy core_t arg_Ts end
  1969   in
  1970     [HOLogic.mk_imp
  1971        (HOLogic.mk_disj (HOLogic.eq_const iter_T $ n_var $ zero_const iter_T,
  1972             s_betapply [] (optimized_case_def hol_ctxt [] T bool_T
  1973                                               (map case_func xs), x_var)),
  1974         bisim_const T $ n_var $ x_var $ y_var),
  1975      HOLogic.eq_const set_T $ (bisim_const T $ bisim_max $ x_var)
  1976      $ (Const (@{const_name insert}, T --> set_T --> set_T)
  1977         $ x_var $ Const (@{const_name bot_class.bot}, set_T))]
  1978     |> map HOLogic.mk_Trueprop
  1979   end
  1980 
  1981 exception NO_TRIPLE of unit
  1982 
  1983 fun triple_for_intro_rule thy x t =
  1984   let
  1985     val prems = Logic.strip_imp_prems t |> map (Object_Logic.atomize_term thy)
  1986     val concl = Logic.strip_imp_concl t |> Object_Logic.atomize_term thy
  1987     val (main, side) = List.partition (exists_Const (curry (op =) x)) prems
  1988     val is_good_head = curry (op =) (Const x) o head_of
  1989   in
  1990     if forall is_good_head main then (side, main, concl) else raise NO_TRIPLE ()
  1991   end
  1992 
  1993 val tuple_for_args = HOLogic.mk_tuple o snd o strip_comb
  1994 fun wf_constraint_for rel side concl main =
  1995   let
  1996     val core = HOLogic.mk_mem (HOLogic.mk_prod
  1997                                (pairself tuple_for_args (main, concl)), Var rel)
  1998     val t = List.foldl HOLogic.mk_imp core side
  1999     val vars = filter_out (curry (op =) rel) (Term.add_vars t [])
  2000   in
  2001     Library.foldl (fn (t', ((x, j), T)) =>
  2002                       HOLogic.all_const T
  2003                       $ Abs (x, T, abstract_over (Var ((x, j), T), t')))
  2004                   (t, vars)
  2005   end
  2006 fun wf_constraint_for_triple rel (side, main, concl) =
  2007   map (wf_constraint_for rel side concl) main |> foldr1 s_conj
  2008 
  2009 fun terminates_by ctxt timeout goal tac =
  2010   can (SINGLE (Classical.safe_tac ctxt) #> the
  2011        #> SINGLE (DETERM_TIMEOUT timeout (tac ctxt (auto_tac ctxt)))
  2012        #> the #> Goal.finish ctxt) goal
  2013 
  2014 val max_cached_wfs = 50
  2015 val cached_timeout =
  2016   Synchronized.var "Nitpick_HOL.cached_timeout" (SOME Time.zeroTime)
  2017 val cached_wf_props =
  2018   Synchronized.var "Nitpick_HOL.cached_wf_props" ([] : (term * bool) list)
  2019 
  2020 val termination_tacs = [Lexicographic_Order.lex_order_tac true,
  2021                         ScnpReconstruct.sizechange_tac]
  2022 
  2023 fun uncached_is_well_founded_inductive_pred
  2024         ({thy, ctxt, stds, debug, tac_timeout, intro_table, ...} : hol_context)
  2025         (x as (_, T)) =
  2026   case def_props_for_const thy stds intro_table x of
  2027     [] => raise TERM ("Nitpick_HOL.uncached_is_well_founded_inductive",
  2028                       [Const x])
  2029   | intro_ts =>
  2030     (case map (triple_for_intro_rule thy x) intro_ts
  2031           |> filter_out (null o #2) of
  2032        [] => true
  2033      | triples =>
  2034        let
  2035          val binders_T = HOLogic.mk_tupleT (binder_types T)
  2036          val rel_T = HOLogic.mk_prodT (binders_T, binders_T) --> bool_T
  2037          val j = fold Integer.max (map maxidx_of_term intro_ts) 0 + 1
  2038          val rel = (("R", j), rel_T)
  2039          val prop = Const (@{const_name wf}, rel_T --> bool_T) $ Var rel ::
  2040                     map (wf_constraint_for_triple rel) triples
  2041                     |> foldr1 s_conj |> HOLogic.mk_Trueprop
  2042          val _ = if debug then
  2043                    Output.urgent_message ("Wellfoundedness goal: " ^
  2044                              Syntax.string_of_term ctxt prop ^ ".")
  2045                  else
  2046                    ()
  2047        in
  2048          if tac_timeout = Synchronized.value cached_timeout andalso
  2049             length (Synchronized.value cached_wf_props) < max_cached_wfs then
  2050            ()
  2051          else
  2052            (Synchronized.change cached_wf_props (K []);
  2053             Synchronized.change cached_timeout (K tac_timeout));
  2054          case AList.lookup (op =) (Synchronized.value cached_wf_props) prop of
  2055            SOME wf => wf
  2056          | NONE =>
  2057            let
  2058              val goal = prop |> cterm_of thy |> Goal.init
  2059              val wf = exists (terminates_by ctxt tac_timeout goal)
  2060                              termination_tacs
  2061            in Synchronized.change cached_wf_props (cons (prop, wf)); wf end
  2062        end)
  2063     handle List.Empty => false | NO_TRIPLE () => false
  2064 
  2065 (* The type constraint below is a workaround for a Poly/ML crash. *)
  2066 
  2067 fun is_well_founded_inductive_pred
  2068         (hol_ctxt as {thy, wfs, def_tables, wf_cache, ...} : hol_context)
  2069         (x as (s, _)) =
  2070   case triple_lookup (const_match thy) wfs x of
  2071     SOME (SOME b) => b
  2072   | _ => s = @{const_name Nats} orelse s = @{const_name fold_graph'} orelse
  2073          case AList.lookup (op =) (!wf_cache) x of
  2074            SOME (_, wf) => wf
  2075          | NONE =>
  2076            let
  2077              val gfp = (fixpoint_kind_of_const thy def_tables x = Gfp)
  2078              val wf = uncached_is_well_founded_inductive_pred hol_ctxt x
  2079            in
  2080              Unsynchronized.change wf_cache (cons (x, (gfp, wf))); wf
  2081            end
  2082 
  2083 fun ap_curry [_] _ t = t
  2084   | ap_curry arg_Ts tuple_T t =
  2085     let val n = length arg_Ts in
  2086       list_abs (map (pair "c") arg_Ts,
  2087                 incr_boundvars n t
  2088                 $ mk_flat_tuple tuple_T (map Bound (n - 1 downto 0)))
  2089     end
  2090 
  2091 fun num_occs_of_bound_in_term j (t1 $ t2) =
  2092     op + (pairself (num_occs_of_bound_in_term j) (t1, t2))
  2093   | num_occs_of_bound_in_term j (Abs (_, _, t')) =
  2094     num_occs_of_bound_in_term (j + 1) t'
  2095   | num_occs_of_bound_in_term j (Bound j') = if j' = j then 1 else 0
  2096   | num_occs_of_bound_in_term _ _ = 0
  2097 
  2098 val is_linear_inductive_pred_def =
  2099   let
  2100     fun do_disjunct j (Const (@{const_name Ex}, _) $ Abs (_, _, t2)) =
  2101         do_disjunct (j + 1) t2
  2102       | do_disjunct j t =
  2103         case num_occs_of_bound_in_term j t of
  2104           0 => true
  2105         | 1 => exists (curry (op =) (Bound j) o head_of) (conjuncts_of t)
  2106         | _ => false
  2107     fun do_lfp_def (Const (@{const_name lfp}, _) $ t2) =
  2108         let val (xs, body) = strip_abs t2 in
  2109           case length xs of
  2110             1 => false
  2111           | n => forall (do_disjunct (n - 1)) (disjuncts_of body)
  2112         end
  2113       | do_lfp_def _ = false
  2114   in do_lfp_def o strip_abs_body end
  2115 
  2116 fun n_ptuple_paths 0 = []
  2117   | n_ptuple_paths 1 = []
  2118   | n_ptuple_paths n = [] :: map (cons 2) (n_ptuple_paths (n - 1))
  2119 val ap_n_split = HOLogic.mk_psplits o n_ptuple_paths
  2120 
  2121 val linear_pred_base_and_step_rhss =
  2122   let
  2123     fun aux (Const (@{const_name lfp}, _) $ t2) =
  2124         let
  2125           val (xs, body) = strip_abs t2
  2126           val arg_Ts = map snd (tl xs)
  2127           val tuple_T = HOLogic.mk_tupleT arg_Ts
  2128           val j = length arg_Ts
  2129           fun repair_rec j (Const (@{const_name Ex}, T1) $ Abs (s2, T2, t2')) =
  2130               Const (@{const_name Ex}, T1)
  2131               $ Abs (s2, T2, repair_rec (j + 1) t2')
  2132             | repair_rec j (@{const HOL.conj} $ t1 $ t2) =
  2133               @{const HOL.conj} $ repair_rec j t1 $ repair_rec j t2
  2134             | repair_rec j t =
  2135               let val (head, args) = strip_comb t in
  2136                 if head = Bound j then
  2137                   HOLogic.eq_const tuple_T $ Bound j
  2138                   $ mk_flat_tuple tuple_T args
  2139                 else
  2140                   t
  2141               end
  2142           val (nonrecs, recs) =
  2143             List.partition (curry (op =) 0 o num_occs_of_bound_in_term j)
  2144                            (disjuncts_of body)
  2145           val base_body = nonrecs |> List.foldl s_disj @{const False}
  2146           val step_body = recs |> map (repair_rec j)
  2147                                |> List.foldl s_disj @{const False}
  2148         in
  2149           (list_abs (tl xs, incr_bv (~1, j, base_body))
  2150            |> ap_n_split (length arg_Ts) tuple_T bool_T,
  2151            Abs ("y", tuple_T, list_abs (tl xs, step_body)
  2152                               |> ap_n_split (length arg_Ts) tuple_T bool_T))
  2153         end
  2154       | aux t =
  2155         raise TERM ("Nitpick_HOL.linear_pred_base_and_step_rhss.aux", [t])
  2156   in aux end
  2157 
  2158 fun starred_linear_pred_const (hol_ctxt as {simp_table, ...}) (s, T) def =
  2159   let
  2160     val j = maxidx_of_term def + 1
  2161     val (outer, fp_app) = strip_abs def
  2162     val outer_bounds = map Bound (length outer - 1 downto 0)
  2163     val outer_vars = map (fn (s, T) => Var ((s, j), T)) outer
  2164     val fp_app = subst_bounds (rev outer_vars, fp_app)
  2165     val (outer_Ts, rest_T) = strip_n_binders (length outer) T
  2166     val tuple_arg_Ts = strip_type rest_T |> fst
  2167     val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
  2168     val set_T = tuple_T --> bool_T
  2169     val curried_T = tuple_T --> set_T
  2170     val uncurried_T = Type (@{type_name prod}, [tuple_T, tuple_T]) --> bool_T
  2171     val (base_rhs, step_rhs) = linear_pred_base_and_step_rhss fp_app
  2172     val base_x as (base_s, _) = (base_prefix ^ s, outer_Ts ---> set_T)
  2173     val base_eq = HOLogic.mk_eq (list_comb (Const base_x, outer_vars), base_rhs)
  2174                   |> HOLogic.mk_Trueprop
  2175     val _ = add_simps simp_table base_s [base_eq]
  2176     val step_x as (step_s, _) = (step_prefix ^ s, outer_Ts ---> curried_T)
  2177     val step_eq = HOLogic.mk_eq (list_comb (Const step_x, outer_vars), step_rhs)
  2178                   |> HOLogic.mk_Trueprop
  2179     val _ = add_simps simp_table step_s [step_eq]
  2180   in
  2181     list_abs (outer,
  2182               Const (@{const_name Image}, uncurried_T --> set_T --> set_T)
  2183               $ (Const (@{const_name rtrancl}, uncurried_T --> uncurried_T)
  2184                  $ (Const (@{const_name prod_case}, curried_T --> uncurried_T)
  2185                     $ list_comb (Const step_x, outer_bounds)))
  2186               $ list_comb (Const base_x, outer_bounds)
  2187               |> ap_curry tuple_arg_Ts tuple_T)
  2188     |> unfold_defs_in_term hol_ctxt
  2189   end
  2190 
  2191 fun is_good_starred_linear_pred_type (Type (@{type_name fun}, Ts)) =
  2192     forall (not o (is_fun_type orf is_pair_type)) Ts
  2193   | is_good_starred_linear_pred_type _ = false
  2194 
  2195 fun unrolled_inductive_pred_const (hol_ctxt as {thy, star_linear_preds,
  2196                                                 def_tables, simp_table, ...})
  2197                                   gfp (x as (s, T)) =
  2198   let
  2199     val iter_T = iterator_type_for_const gfp x
  2200     val x' as (s', _) = (unrolled_prefix ^ s, iter_T --> T)
  2201     val unrolled_const = Const x' $ zero_const iter_T
  2202     val def = the (def_of_const thy def_tables x)
  2203   in
  2204     if is_equational_fun_but_no_plain_def hol_ctxt x' then
  2205       unrolled_const (* already done *)
  2206     else if not gfp andalso star_linear_preds andalso
  2207          is_linear_inductive_pred_def def andalso
  2208          is_good_starred_linear_pred_type T then
  2209       starred_linear_pred_const hol_ctxt x def
  2210     else
  2211       let
  2212         val j = maxidx_of_term def + 1
  2213         val (outer, fp_app) = strip_abs def
  2214         val outer_bounds = map Bound (length outer - 1 downto 0)
  2215         val cur = Var ((iter_var_prefix, j + 1), iter_T)
  2216         val next = suc_const iter_T $ cur
  2217         val rhs =
  2218           case fp_app of
  2219             Const _ $ t =>
  2220             s_betapply [] (t, list_comb (Const x', next :: outer_bounds))
  2221           | _ => raise TERM ("Nitpick_HOL.unrolled_inductive_pred_const",
  2222                              [fp_app])
  2223         val (inner, naked_rhs) = strip_abs rhs
  2224         val all = outer @ inner
  2225         val bounds = map Bound (length all - 1 downto 0)
  2226         val vars = map (fn (s, T) => Var ((s, j), T)) all
  2227         val eq = HOLogic.mk_eq (list_comb (Const x', cur :: bounds), naked_rhs)
  2228                  |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2229         val _ = add_simps simp_table s' [eq]
  2230       in unrolled_const end
  2231   end
  2232 
  2233 fun raw_inductive_pred_axiom ({thy, def_tables, ...} : hol_context) x =
  2234   let
  2235     val def = the (def_of_const thy def_tables x)
  2236     val (outer, fp_app) = strip_abs def
  2237     val outer_bounds = map Bound (length outer - 1 downto 0)
  2238     val rhs =
  2239       case fp_app of
  2240         Const _ $ t => s_betapply [] (t, list_comb (Const x, outer_bounds))
  2241       | _ => raise TERM ("Nitpick_HOL.raw_inductive_pred_axiom", [fp_app])
  2242     val (inner, naked_rhs) = strip_abs rhs
  2243     val all = outer @ inner
  2244     val bounds = map Bound (length all - 1 downto 0)
  2245     val j = maxidx_of_term def + 1
  2246     val vars = map (fn (s, T) => Var ((s, j), T)) all
  2247   in
  2248     HOLogic.mk_eq (list_comb (Const x, bounds), naked_rhs)
  2249     |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2250   end
  2251 fun inductive_pred_axiom hol_ctxt (x as (s, T)) =
  2252   if String.isPrefix ubfp_prefix s orelse String.isPrefix lbfp_prefix s then
  2253     let val x' = (strip_first_name_sep s |> snd, T) in
  2254       raw_inductive_pred_axiom hol_ctxt x' |> subst_atomic [(Const x', Const x)]
  2255     end
  2256   else
  2257     raw_inductive_pred_axiom hol_ctxt x
  2258 
  2259 fun equational_fun_axioms (hol_ctxt as {thy, ctxt, stds, def_tables, simp_table,
  2260                                         psimp_table, ...}) x =
  2261   case def_props_for_const thy stds (!simp_table) x of
  2262     [] => (case def_props_for_const thy stds psimp_table x of
  2263              [] => (if is_inductive_pred hol_ctxt x then
  2264                       [inductive_pred_axiom hol_ctxt x]
  2265                     else case def_of_const thy def_tables x of
  2266                       SOME def =>
  2267                       @{const Trueprop} $ HOLogic.mk_eq (Const x, def)
  2268                       |> equationalize_term ctxt "" |> the |> single
  2269                     | NONE => [])
  2270            | psimps => psimps)
  2271   | simps => simps
  2272 fun is_equational_fun_surely_complete hol_ctxt x =
  2273   case equational_fun_axioms hol_ctxt x of
  2274     [@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)] =>
  2275     strip_comb t1 |> snd |> forall is_Var
  2276   | _ => false
  2277 
  2278 (** Type preprocessing **)
  2279 
  2280 fun merged_type_var_table_for_terms thy ts =
  2281   let
  2282     fun add (s, S) table =
  2283       table
  2284       |> (case AList.lookup (Sign.subsort thy o swap) table S of
  2285             SOME _ => I
  2286           | NONE =>
  2287             filter_out (fn (S', _) => Sign.subsort thy (S, S'))
  2288             #> cons (S, s))
  2289     val tfrees = [] |> fold Term.add_tfrees ts
  2290                     |> sort (string_ord o pairself fst)
  2291   in [] |> fold add tfrees |> rev end
  2292 
  2293 fun merge_type_vars_in_term thy merge_type_vars table =
  2294   merge_type_vars
  2295   ? map_types (map_atyps
  2296         (fn TFree (_, S) =>
  2297             TFree (table |> find_first (fn (S', _) => Sign.subsort thy (S', S))
  2298                          |> the |> swap)
  2299           | T => T))
  2300 
  2301 fun add_ground_types hol_ctxt binarize =
  2302   let
  2303     fun aux T accum =
  2304       case T of
  2305         Type (@{type_name fun}, Ts) => fold aux Ts accum
  2306       | Type (@{type_name prod}, Ts) => fold aux Ts accum
  2307       | Type (@{type_name itself}, [T1]) => aux T1 accum
  2308       | Type (_, Ts) =>
  2309         if member (op =) (@{typ prop} :: @{typ bool} :: accum) T then
  2310           accum
  2311         else
  2312           T :: accum
  2313           |> fold aux (case binarized_and_boxed_datatype_constrs hol_ctxt
  2314                                                                  binarize T of
  2315                          [] => Ts
  2316                        | xs => map snd xs)
  2317       | _ => insert (op =) T accum
  2318   in aux end
  2319 
  2320 fun ground_types_in_type hol_ctxt binarize T =
  2321   add_ground_types hol_ctxt binarize T []
  2322 fun ground_types_in_terms hol_ctxt binarize ts =
  2323   fold (fold_types (add_ground_types hol_ctxt binarize)) ts []
  2324 
  2325 end;