src/HOL/Tools/Nitpick/nitpick_hol.ML
author huffman
Sun Mar 25 20:15:39 2012 +0200 (2012-03-25)
changeset 47108 2a1953f0d20d
parent 46819 9b38f8527510
child 47109 db5026631799
permissions -rw-r--r--
merged fork with new numeral representation (see NEWS)
     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_fun_or_set_type : typ -> bool
    92   val is_set_like_type : typ -> bool
    93   val is_pair_type : typ -> bool
    94   val is_lfp_iterator_type : typ -> bool
    95   val is_gfp_iterator_type : typ -> bool
    96   val is_fp_iterator_type : typ -> bool
    97   val is_iterator_type : typ -> bool
    98   val is_boolean_type : typ -> bool
    99   val is_integer_type : typ -> bool
   100   val is_bit_type : typ -> bool
   101   val is_word_type : typ -> bool
   102   val is_integer_like_type : typ -> bool
   103   val is_record_type : typ -> bool
   104   val is_number_type : Proof.context -> typ -> bool
   105   val is_higher_order_type : typ -> bool
   106   val elem_type : typ -> typ
   107   val pseudo_domain_type : typ -> typ
   108   val pseudo_range_type : typ -> typ
   109   val const_for_iterator_type : typ -> styp
   110   val strip_n_binders : int -> typ -> typ list * typ
   111   val nth_range_type : int -> typ -> typ
   112   val num_factors_in_type : typ -> int
   113   val num_binder_types : typ -> int
   114   val curried_binder_types : typ -> typ list
   115   val mk_flat_tuple : typ -> term list -> term
   116   val dest_n_tuple : int -> term -> term list
   117   val is_real_datatype : theory -> string -> bool
   118   val is_standard_datatype : theory -> (typ option * bool) list -> typ -> bool
   119   val is_codatatype : Proof.context -> typ -> bool
   120   val is_quot_type : Proof.context -> typ -> bool
   121   val is_pure_typedef : Proof.context -> typ -> bool
   122   val is_univ_typedef : Proof.context -> typ -> bool
   123   val is_datatype : Proof.context -> (typ option * bool) list -> typ -> bool
   124   val is_record_constr : styp -> bool
   125   val is_record_get : theory -> styp -> bool
   126   val is_record_update : theory -> styp -> bool
   127   val is_abs_fun : Proof.context -> styp -> bool
   128   val is_rep_fun : Proof.context -> styp -> bool
   129   val is_quot_abs_fun : Proof.context -> styp -> bool
   130   val is_quot_rep_fun : Proof.context -> styp -> bool
   131   val mate_of_rep_fun : Proof.context -> styp -> styp
   132   val is_constr_like : Proof.context -> styp -> bool
   133   val is_constr : Proof.context -> (typ option * bool) list -> styp -> bool
   134   val is_sel : string -> bool
   135   val is_sel_like_and_no_discr : string -> bool
   136   val box_type : hol_context -> boxability -> typ -> typ
   137   val binarize_nat_and_int_in_type : typ -> typ
   138   val binarize_nat_and_int_in_term : term -> term
   139   val discr_for_constr : styp -> styp
   140   val num_sels_for_constr_type : typ -> int
   141   val nth_sel_name_for_constr_name : string -> int -> string
   142   val nth_sel_for_constr : styp -> int -> styp
   143   val binarized_and_boxed_nth_sel_for_constr :
   144     hol_context -> bool -> styp -> int -> styp
   145   val sel_no_from_name : string -> int
   146   val close_form : term -> term
   147   val distinctness_formula : typ -> term list -> term
   148   val register_frac_type :
   149     string -> (string * string) list -> morphism -> Context.generic
   150     -> Context.generic
   151   val register_frac_type_global :
   152     string -> (string * string) list -> theory -> theory
   153   val unregister_frac_type :
   154     string -> morphism -> Context.generic -> Context.generic
   155   val unregister_frac_type_global : string -> theory -> theory
   156   val register_ersatz :
   157     (string * string) list -> morphism -> Context.generic -> Context.generic
   158   val register_ersatz_global : (string * string) list -> theory -> theory
   159   val register_codatatype :
   160     typ -> string -> styp list -> morphism -> Context.generic -> Context.generic
   161   val register_codatatype_global :
   162     typ -> string -> styp list -> theory -> theory
   163   val unregister_codatatype :
   164     typ -> morphism -> Context.generic -> Context.generic
   165   val unregister_codatatype_global : typ -> theory -> theory
   166   val datatype_constrs : hol_context -> typ -> styp list
   167   val binarized_and_boxed_datatype_constrs :
   168     hol_context -> bool -> typ -> styp list
   169   val num_datatype_constrs : hol_context -> typ -> int
   170   val constr_name_for_sel_like : string -> string
   171   val binarized_and_boxed_constr_for_sel : hol_context -> bool -> styp -> styp
   172   val card_of_type : (typ * int) list -> typ -> int
   173   val bounded_card_of_type : int -> int -> (typ * int) list -> typ -> int
   174   val bounded_exact_card_of_type :
   175     hol_context -> typ list -> int -> int -> (typ * int) list -> typ -> int
   176   val typical_card_of_type : typ -> int
   177   val is_finite_type : hol_context -> typ -> bool
   178   val is_special_eligible_arg : bool -> typ list -> term -> bool
   179   val s_let :
   180     typ list -> string -> int -> typ -> typ -> (term -> term) -> term -> term
   181   val s_betapply : typ list -> term * term -> term
   182   val s_betapplys : typ list -> term * term list -> term
   183   val discriminate_value : hol_context -> styp -> term -> term
   184   val select_nth_constr_arg :
   185     Proof.context -> (typ option * bool) list -> styp -> term -> int -> typ
   186     -> term
   187   val construct_value :
   188     Proof.context -> (typ option * bool) list -> styp -> term list -> term
   189   val coerce_term : hol_context -> typ list -> typ -> typ -> term -> term
   190   val special_bounds : term list -> (indexname * typ) list
   191   val is_funky_typedef : Proof.context -> typ -> bool
   192   val all_defs_of : theory -> (term * term) list -> term list
   193   val all_nondefs_of : Proof.context -> (term * term) list -> term list
   194   val arity_of_built_in_const :
   195     theory -> (typ option * bool) list -> styp -> int option
   196   val is_built_in_const :
   197     theory -> (typ option * bool) list -> styp -> bool
   198   val term_under_def : term -> term
   199   val case_const_names :
   200     Proof.context -> (typ option * bool) list -> (string * int) list
   201   val unfold_defs_in_term : hol_context -> term -> term
   202   val const_def_tables :
   203     Proof.context -> (term * term) list -> term list
   204     -> const_table * const_table
   205   val const_nondef_table : term list -> const_table
   206   val const_simp_table : Proof.context -> (term * term) list -> const_table
   207   val const_psimp_table : Proof.context -> (term * term) list -> const_table
   208   val const_choice_spec_table :
   209     Proof.context -> (term * term) list -> const_table
   210   val inductive_intro_table :
   211     Proof.context -> (term * term) list -> const_table * const_table
   212     -> const_table
   213   val ground_theorem_table : theory -> term list Inttab.table
   214   val ersatz_table : Proof.context -> (string * string) list
   215   val add_simps : const_table Unsynchronized.ref -> string -> term list -> unit
   216   val inverse_axioms_for_rep_fun : Proof.context -> styp -> term list
   217   val optimized_typedef_axioms : Proof.context -> string * typ list -> term list
   218   val optimized_quot_type_axioms :
   219     Proof.context -> (typ option * bool) list -> string * typ list -> term list
   220   val def_of_const : theory -> const_table * const_table -> styp -> term option
   221   val fixpoint_kind_of_rhs : term -> fixpoint_kind
   222   val fixpoint_kind_of_const :
   223     theory -> const_table * const_table -> string * typ -> fixpoint_kind
   224   val is_real_inductive_pred : hol_context -> styp -> bool
   225   val is_constr_pattern : Proof.context -> term -> bool
   226   val is_constr_pattern_lhs : Proof.context -> term -> bool
   227   val is_constr_pattern_formula : Proof.context -> term -> bool
   228   val nondef_props_for_const :
   229     theory -> bool -> const_table -> styp -> term list
   230   val is_choice_spec_fun : hol_context -> styp -> bool
   231   val is_choice_spec_axiom : theory -> const_table -> term -> bool
   232   val is_real_equational_fun : hol_context -> styp -> bool
   233   val is_equational_fun_but_no_plain_def : hol_context -> styp -> bool
   234   val codatatype_bisim_axioms : hol_context -> typ -> term list
   235   val is_well_founded_inductive_pred : hol_context -> styp -> bool
   236   val unrolled_inductive_pred_const : hol_context -> bool -> styp -> term
   237   val equational_fun_axioms : hol_context -> styp -> term list
   238   val is_equational_fun_surely_complete : hol_context -> styp -> bool
   239   val merged_type_var_table_for_terms :
   240     theory -> term list -> (sort * string) list
   241   val merge_type_vars_in_term :
   242     theory -> bool -> (sort * string) list -> term -> term
   243   val ground_types_in_type : hol_context -> bool -> typ -> typ list
   244   val ground_types_in_terms : hol_context -> bool -> term list -> typ list
   245 end;
   246 
   247 structure Nitpick_HOL : NITPICK_HOL =
   248 struct
   249 
   250 open Nitpick_Util
   251 
   252 type const_table = term list Symtab.table
   253 type special_fun = (styp * int list * term list) * styp
   254 type unrolled = styp * styp
   255 type wf_cache = (styp * (bool * bool)) list
   256 
   257 type hol_context =
   258   {thy: theory,
   259    ctxt: Proof.context,
   260    max_bisim_depth: int,
   261    boxes: (typ option * bool option) list,
   262    stds: (typ option * bool) list,
   263    wfs: (styp option * bool option) list,
   264    user_axioms: bool option,
   265    debug: bool,
   266    whacks: term list,
   267    binary_ints: bool option,
   268    destroy_constrs: bool,
   269    specialize: bool,
   270    star_linear_preds: bool,
   271    total_consts: bool option,
   272    needs: term list option,
   273    tac_timeout: Time.time option,
   274    evals: term list,
   275    case_names: (string * int) list,
   276    def_tables: const_table * const_table,
   277    nondef_table: const_table,
   278    nondefs: term list,
   279    simp_table: const_table Unsynchronized.ref,
   280    psimp_table: const_table,
   281    choice_spec_table: const_table,
   282    intro_table: const_table,
   283    ground_thm_table: term list Inttab.table,
   284    ersatz_table: (string * string) list,
   285    skolems: (string * string list) list Unsynchronized.ref,
   286    special_funs: special_fun list Unsynchronized.ref,
   287    unrolled_preds: unrolled list Unsynchronized.ref,
   288    wf_cache: wf_cache Unsynchronized.ref,
   289    constr_cache: (typ * styp list) list Unsynchronized.ref}
   290 
   291 datatype fixpoint_kind = Lfp | Gfp | NoFp
   292 datatype boxability =
   293   InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
   294 
   295 structure Data = Generic_Data
   296 (
   297   type T = {frac_types: (string * (string * string) list) list,
   298             ersatz_table: (string * string) list,
   299             codatatypes: (string * (string * styp list)) list}
   300   val empty = {frac_types = [], ersatz_table = [], codatatypes = []}
   301   val extend = I
   302   fun merge ({frac_types = fs1, ersatz_table = et1, codatatypes = cs1},
   303              {frac_types = fs2, ersatz_table = et2, codatatypes = cs2}) : T =
   304     {frac_types = AList.merge (op =) (K true) (fs1, fs2),
   305      ersatz_table = AList.merge (op =) (K true) (et1, et2),
   306      codatatypes = AList.merge (op =) (K true) (cs1, cs2)}
   307 )
   308 
   309 val name_sep = "$"
   310 val numeral_prefix = nitpick_prefix ^ "num" ^ name_sep
   311 val sel_prefix = nitpick_prefix ^ "sel"
   312 val discr_prefix = nitpick_prefix ^ "is" ^ name_sep
   313 val set_prefix = nitpick_prefix ^ "set" ^ name_sep
   314 val lfp_iterator_prefix = nitpick_prefix ^ "lfpit" ^ name_sep
   315 val gfp_iterator_prefix = nitpick_prefix ^ "gfpit" ^ name_sep
   316 val unrolled_prefix = nitpick_prefix ^ "unroll" ^ name_sep
   317 val base_prefix = nitpick_prefix ^ "base" ^ name_sep
   318 val step_prefix = nitpick_prefix ^ "step" ^ name_sep
   319 val ubfp_prefix = nitpick_prefix ^ "ubfp" ^ name_sep
   320 val lbfp_prefix = nitpick_prefix ^ "lbfp" ^ name_sep
   321 val quot_normal_prefix = nitpick_prefix ^ "qn" ^ name_sep
   322 val skolem_prefix = nitpick_prefix ^ "sk"
   323 val special_prefix = nitpick_prefix ^ "sp"
   324 val uncurry_prefix = nitpick_prefix ^ "unc"
   325 val eval_prefix = nitpick_prefix ^ "eval"
   326 val iter_var_prefix = "i"
   327 
   328 (** Constant/type information and term/type manipulation **)
   329 
   330 fun sel_prefix_for j = sel_prefix ^ string_of_int j ^ name_sep
   331 fun quot_normal_name_for_type ctxt T =
   332   quot_normal_prefix ^ unyxml (Syntax.string_of_typ ctxt T)
   333 
   334 val strip_first_name_sep =
   335   Substring.full #> Substring.position name_sep ##> Substring.triml 1
   336   #> pairself Substring.string
   337 fun original_name s =
   338   if String.isPrefix nitpick_prefix s then
   339     case strip_first_name_sep s of (s1, "") => s1 | (_, s2) => original_name s2
   340   else
   341     s
   342 
   343 fun s_conj (t1, @{const True}) = t1
   344   | s_conj (@{const True}, t2) = t2
   345   | s_conj (t1, t2) =
   346     if t1 = @{const False} orelse t2 = @{const False} then @{const False}
   347     else HOLogic.mk_conj (t1, t2)
   348 fun s_disj (t1, @{const False}) = t1
   349   | s_disj (@{const False}, t2) = t2
   350   | s_disj (t1, t2) =
   351     if t1 = @{const True} orelse t2 = @{const True} then @{const True}
   352     else HOLogic.mk_disj (t1, t2)
   353 
   354 fun strip_connective conn_t (t as (t0 $ t1 $ t2)) =
   355     if t0 = conn_t then strip_connective t0 t2 @ strip_connective t0 t1 else [t]
   356   | strip_connective _ t = [t]
   357 fun strip_any_connective (t as (t0 $ _ $ _)) =
   358     if t0 = @{const HOL.conj} orelse t0 = @{const HOL.disj} then
   359       (strip_connective t0 t, t0)
   360     else
   361       ([t], @{const Not})
   362   | strip_any_connective t = ([t], @{const Not})
   363 val conjuncts_of = strip_connective @{const HOL.conj}
   364 val disjuncts_of = strip_connective @{const HOL.disj}
   365 
   366 (* When you add constants to these lists, make sure to handle them in
   367    "Nitpick_Nut.nut_from_term", and perhaps in "Nitpick_Mono.consider_term" as
   368    well. *)
   369 val built_in_consts =
   370   [(@{const_name all}, 1),
   371    (@{const_name "=="}, 2),
   372    (@{const_name "==>"}, 2),
   373    (@{const_name Pure.conjunction}, 2),
   374    (@{const_name Trueprop}, 1),
   375    (@{const_name Not}, 1),
   376    (@{const_name False}, 0),
   377    (@{const_name True}, 0),
   378    (@{const_name All}, 1),
   379    (@{const_name Ex}, 1),
   380    (@{const_name HOL.eq}, 1),
   381    (@{const_name HOL.conj}, 2),
   382    (@{const_name HOL.disj}, 2),
   383    (@{const_name HOL.implies}, 2),
   384    (@{const_name If}, 3),
   385    (@{const_name Let}, 2),
   386    (@{const_name Pair}, 2),
   387    (@{const_name fst}, 1),
   388    (@{const_name snd}, 1),
   389    (@{const_name Set.member}, 2),
   390    (@{const_name Collect}, 1),
   391    (@{const_name Id}, 0),
   392    (@{const_name converse}, 1),
   393    (@{const_name trancl}, 1),
   394    (@{const_name rel_comp}, 2),
   395    (@{const_name finite}, 1),
   396    (@{const_name unknown}, 0),
   397    (@{const_name is_unknown}, 1),
   398    (@{const_name safe_The}, 1),
   399    (@{const_name Frac}, 0),
   400    (@{const_name norm_frac}, 0)]
   401 val built_in_nat_consts =
   402   [(@{const_name Suc}, 0),
   403    (@{const_name nat}, 0),
   404    (@{const_name nat_gcd}, 0),
   405    (@{const_name nat_lcm}, 0)]
   406 val built_in_typed_consts =
   407   [((@{const_name zero_class.zero}, int_T), 0),
   408    ((@{const_name one_class.one}, int_T), 0),
   409    ((@{const_name plus_class.plus}, int_T --> int_T --> int_T), 0),
   410    ((@{const_name minus_class.minus}, int_T --> int_T --> int_T), 0),
   411    ((@{const_name times_class.times}, int_T --> int_T --> int_T), 0),
   412    ((@{const_name div_class.div}, int_T --> int_T --> int_T), 0),
   413    ((@{const_name uminus_class.uminus}, int_T --> int_T), 0),
   414    ((@{const_name ord_class.less}, int_T --> int_T --> bool_T), 2),
   415    ((@{const_name ord_class.less_eq}, int_T --> int_T --> bool_T), 2)]
   416 val built_in_typed_nat_consts =
   417   [((@{const_name zero_class.zero}, nat_T), 0),
   418    ((@{const_name one_class.one}, nat_T), 0),
   419    ((@{const_name plus_class.plus}, nat_T --> nat_T --> nat_T), 0),
   420    ((@{const_name minus_class.minus}, nat_T --> nat_T --> nat_T), 0),
   421    ((@{const_name times_class.times}, nat_T --> nat_T --> nat_T), 0),
   422    ((@{const_name div_class.div}, nat_T --> nat_T --> nat_T), 0),
   423    ((@{const_name ord_class.less}, nat_T --> nat_T --> bool_T), 2),
   424    ((@{const_name ord_class.less_eq}, nat_T --> nat_T --> bool_T), 2),
   425    ((@{const_name of_nat}, nat_T --> int_T), 0)]
   426 val built_in_set_like_consts =
   427   [(@{const_name ord_class.less_eq}, 2)]
   428 
   429 fun unarize_type @{typ "unsigned_bit word"} = nat_T
   430   | unarize_type @{typ "signed_bit word"} = int_T
   431   | unarize_type (Type (s, Ts as _ :: _)) = Type (s, map unarize_type Ts)
   432   | unarize_type T = T
   433 fun unarize_unbox_etc_type (Type (@{type_name fun_box}, Ts)) =
   434     unarize_unbox_etc_type (Type (@{type_name fun}, Ts))
   435   | unarize_unbox_etc_type (Type (@{type_name pair_box}, Ts)) =
   436     Type (@{type_name prod}, map unarize_unbox_etc_type Ts)
   437   | unarize_unbox_etc_type @{typ "unsigned_bit word"} = nat_T
   438   | unarize_unbox_etc_type @{typ "signed_bit word"} = int_T
   439   | unarize_unbox_etc_type (Type (s, Ts as _ :: _)) =
   440     Type (s, map unarize_unbox_etc_type Ts)
   441   | unarize_unbox_etc_type T = T
   442 fun uniterize_type (Type (s, Ts as _ :: _)) = Type (s, map uniterize_type Ts)
   443   | uniterize_type @{typ bisim_iterator} = nat_T
   444   | uniterize_type T = T
   445 val uniterize_unarize_unbox_etc_type = uniterize_type o unarize_unbox_etc_type
   446 
   447 fun string_for_type ctxt = Syntax.string_of_typ ctxt o unarize_unbox_etc_type
   448 fun pretty_for_type ctxt = Syntax.pretty_typ ctxt o unarize_unbox_etc_type
   449 
   450 val prefix_name = Long_Name.qualify o Long_Name.base_name
   451 fun shortest_name s = List.last (space_explode "." s) handle List.Empty => ""
   452 val prefix_abs_vars = Term.map_abs_vars o prefix_name
   453 fun short_name s =
   454   case space_explode name_sep s of
   455     [_] => s |> String.isPrefix nitpick_prefix s ? unprefix nitpick_prefix
   456   | ss => map shortest_name ss |> space_implode "_"
   457 fun shorten_names_in_type (Type (s, Ts)) =
   458     Type (short_name s, map shorten_names_in_type Ts)
   459   | shorten_names_in_type T = T
   460 val shorten_names_in_term =
   461   map_aterms (fn Const (s, T) => Const (short_name s, T) | t => t)
   462   #> map_types shorten_names_in_type
   463 
   464 fun strict_type_match thy (T1, T2) =
   465   (Sign.typ_match thy (T2, T1) Vartab.empty; true)
   466   handle Type.TYPE_MATCH => false
   467 fun type_match thy = strict_type_match thy o pairself unarize_unbox_etc_type
   468 fun const_match thy ((s1, T1), (s2, T2)) =
   469   s1 = s2 andalso type_match thy (T1, T2)
   470 fun term_match thy (Const x1, Const x2) = const_match thy (x1, x2)
   471   | term_match thy (Free (s1, T1), Free (s2, T2)) =
   472     const_match thy ((shortest_name s1, T1), (shortest_name s2, T2))
   473   | term_match _ (t1, t2) = t1 aconv t2
   474 
   475 fun frac_from_term_pair T t1 t2 =
   476   case snd (HOLogic.dest_number t1) of
   477     0 => HOLogic.mk_number T 0
   478   | n1 => case snd (HOLogic.dest_number t2) of
   479             1 => HOLogic.mk_number T n1
   480           | n2 => Const (@{const_name divide}, T --> T --> T)
   481                   $ HOLogic.mk_number T n1 $ HOLogic.mk_number T n2
   482 
   483 fun is_TFree (TFree _) = true
   484   | is_TFree _ = false
   485 fun is_fun_type (Type (@{type_name fun}, _)) = true
   486   | is_fun_type _ = false
   487 fun is_set_type (Type (@{type_name set}, _)) = true
   488   | is_set_type _ = false
   489 val is_fun_or_set_type = is_fun_type orf is_set_type
   490 fun is_set_like_type (Type (@{type_name fun}, [_, T'])) =
   491     (body_type T' = bool_T)
   492   | is_set_like_type (Type (@{type_name set}, _)) = true
   493   | is_set_like_type _ = false
   494 fun is_pair_type (Type (@{type_name prod}, _)) = true
   495   | is_pair_type _ = false
   496 fun is_lfp_iterator_type (Type (s, _)) = String.isPrefix lfp_iterator_prefix s
   497   | is_lfp_iterator_type _ = false
   498 fun is_gfp_iterator_type (Type (s, _)) = String.isPrefix gfp_iterator_prefix s
   499   | is_gfp_iterator_type _ = false
   500 val is_fp_iterator_type = is_lfp_iterator_type orf is_gfp_iterator_type
   501 fun is_iterator_type T =
   502   (T = @{typ bisim_iterator} orelse is_fp_iterator_type T)
   503 fun is_boolean_type T = (T = prop_T orelse T = bool_T)
   504 fun is_integer_type T = (T = nat_T orelse T = int_T)
   505 fun is_bit_type T = (T = @{typ unsigned_bit} orelse T = @{typ signed_bit})
   506 fun is_word_type (Type (@{type_name word}, _)) = true
   507   | is_word_type _ = false
   508 val is_integer_like_type = is_iterator_type orf is_integer_type orf is_word_type
   509 val is_record_type = not o null o Record.dest_recTs
   510 fun is_frac_type ctxt (Type (s, [])) =
   511     s |> AList.defined (op =) (#frac_types (Data.get (Context.Proof ctxt)))
   512   | is_frac_type _ _ = false
   513 fun is_number_type ctxt = is_integer_like_type orf is_frac_type ctxt
   514 fun is_higher_order_type (Type (@{type_name fun}, _)) = true
   515   | is_higher_order_type (Type (@{type_name set}, _)) = true
   516   | is_higher_order_type (Type (_, Ts)) = exists is_higher_order_type Ts
   517   | is_higher_order_type _ = false
   518 
   519 fun elem_type (Type (@{type_name set}, [T'])) = T'
   520   | elem_type T = raise TYPE ("Nitpick_HOL.elem_type", [T], [])
   521 fun pseudo_domain_type (Type (@{type_name fun}, [T1, _])) = T1
   522   | pseudo_domain_type T = elem_type T
   523 fun pseudo_range_type (Type (@{type_name fun}, [_, T2])) = T2
   524   | pseudo_range_type (Type (@{type_name set}, _)) = bool_T
   525   | pseudo_range_type T = raise TYPE ("Nitpick_HOL.pseudo_range_type", [T], [])
   526 
   527 fun iterator_type_for_const gfp (s, T) =
   528   Type ((if gfp then gfp_iterator_prefix else lfp_iterator_prefix) ^ s,
   529         binder_types T)
   530 fun const_for_iterator_type (Type (s, Ts)) =
   531     (strip_first_name_sep s |> snd, Ts ---> bool_T)
   532   | const_for_iterator_type T =
   533     raise TYPE ("Nitpick_HOL.const_for_iterator_type", [T], [])
   534 
   535 fun strip_n_binders 0 T = ([], T)
   536   | strip_n_binders n (Type (@{type_name fun}, [T1, T2])) =
   537     strip_n_binders (n - 1) T2 |>> cons T1
   538   | strip_n_binders n (Type (@{type_name fun_box}, Ts)) =
   539     strip_n_binders n (Type (@{type_name fun}, Ts))
   540   | strip_n_binders _ T = raise TYPE ("Nitpick_HOL.strip_n_binders", [T], [])
   541 val nth_range_type = snd oo strip_n_binders
   542 
   543 fun num_factors_in_type (Type (@{type_name prod}, [T1, T2])) =
   544     fold (Integer.add o num_factors_in_type) [T1, T2] 0
   545   | num_factors_in_type _ = 1
   546 fun num_binder_types (Type (@{type_name fun}, [_, T2])) =
   547     1 + num_binder_types T2
   548   | num_binder_types _ = 0
   549 val curried_binder_types = maps HOLogic.flatten_tupleT o binder_types
   550 fun maybe_curried_binder_types T =
   551   (if is_pair_type (body_type T) then binder_types else curried_binder_types) T
   552 
   553 fun mk_flat_tuple _ [t] = t
   554   | mk_flat_tuple (Type (@{type_name prod}, [T1, T2])) (t :: ts) =
   555     HOLogic.pair_const T1 T2 $ t $ (mk_flat_tuple T2 ts)
   556   | mk_flat_tuple T ts = raise TYPE ("Nitpick_HOL.mk_flat_tuple", [T], ts)
   557 fun dest_n_tuple 1 t = [t]
   558   | dest_n_tuple n t = HOLogic.dest_prod t ||> dest_n_tuple (n - 1) |> op ::
   559 
   560 type typedef_info =
   561   {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string,
   562    set_def: thm option, prop_of_Rep: thm, set_name: string,
   563    Abs_inverse: thm option, Rep_inverse: thm option}
   564 
   565 fun typedef_info ctxt s =
   566   if is_frac_type ctxt (Type (s, [])) then
   567     SOME {abs_type = Type (s, []), rep_type = @{typ "int * int"},
   568           Abs_name = @{const_name Abs_Frac}, Rep_name = @{const_name Rep_Frac},
   569           set_def = NONE, prop_of_Rep = @{prop "Rep_Frac x \<in> Collect Frac"}
   570                           |> Logic.varify_global,
   571           set_name = @{const_name Frac}, Abs_inverse = NONE, Rep_inverse = NONE}
   572   else case Typedef.get_info ctxt s of
   573     (* When several entries are returned, it shouldn't matter much which one
   574        we take (according to Florian Haftmann). *)
   575     (* The "Logic.varifyT_global" calls are a temporary hack because these
   576        types's type variables sometimes clash with locally fixed type variables.
   577        Remove these calls once "Typedef" is fully localized. *)
   578     ({abs_type, rep_type, Abs_name, Rep_name, ...},
   579      {set_def, Rep, Abs_inverse, Rep_inverse, ...}) :: _ =>
   580     SOME {abs_type = Logic.varifyT_global abs_type,
   581           rep_type = Logic.varifyT_global rep_type, Abs_name = Abs_name,
   582           Rep_name = Rep_name, set_def = set_def, prop_of_Rep = prop_of Rep,
   583           set_name = set_prefix ^ s, Abs_inverse = SOME Abs_inverse,
   584           Rep_inverse = SOME Rep_inverse}
   585   | _ => NONE
   586 
   587 val is_typedef = is_some oo typedef_info
   588 val is_real_datatype = is_some oo Datatype.get_info
   589 fun is_standard_datatype thy = the oo triple_lookup (type_match thy)
   590 
   591 (* FIXME: Use antiquotation for "code_numeral" below or detect "rep_datatype",
   592    e.g., by adding a field to "Datatype_Aux.info". *)
   593 fun is_basic_datatype thy stds s =
   594   member (op =) [@{type_name prod}, @{type_name set}, @{type_name bool},
   595                  @{type_name int}, "Code_Numeral.code_numeral"] s orelse
   596   (s = @{type_name nat} andalso is_standard_datatype thy stds nat_T)
   597 
   598 fun repair_constr_type ctxt body_T' T =
   599   varify_and_instantiate_type ctxt (body_type T) body_T' T
   600 
   601 fun register_frac_type_generic frac_s ersaetze generic =
   602   let
   603     val {frac_types, ersatz_table, codatatypes} = Data.get generic
   604     val frac_types = AList.update (op =) (frac_s, ersaetze) frac_types
   605   in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
   606                codatatypes = codatatypes} generic end
   607 (* TODO: Consider morphism. *)
   608 fun register_frac_type frac_s ersaetze (_ : morphism) =
   609   register_frac_type_generic frac_s ersaetze
   610 val register_frac_type_global = Context.theory_map oo register_frac_type_generic
   611 
   612 fun unregister_frac_type_generic frac_s = register_frac_type_generic frac_s []
   613 (* TODO: Consider morphism. *)
   614 fun unregister_frac_type frac_s (_ : morphism) =
   615   unregister_frac_type_generic frac_s
   616 val unregister_frac_type_global =
   617   Context.theory_map o unregister_frac_type_generic
   618 
   619 fun register_ersatz_generic ersatz generic =
   620   let
   621     val {frac_types, ersatz_table, codatatypes} = Data.get generic
   622     val ersatz_table = AList.merge (op =) (K true) (ersatz_table, ersatz)
   623   in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
   624                codatatypes = codatatypes} generic end
   625 (* TODO: Consider morphism. *)
   626 fun register_ersatz ersatz (_ : morphism) =
   627   register_ersatz_generic ersatz
   628 val register_ersatz_global = Context.theory_map o register_ersatz_generic
   629 
   630 fun register_codatatype_generic co_T case_name constr_xs generic =
   631   let
   632     val ctxt = Context.proof_of generic
   633     val thy = Context.theory_of generic
   634     val {frac_types, ersatz_table, codatatypes} = Data.get generic
   635     val constr_xs = map (apsnd (repair_constr_type ctxt co_T)) constr_xs
   636     val (co_s, co_Ts) = dest_Type co_T
   637     val _ =
   638       if forall is_TFree co_Ts andalso not (has_duplicates (op =) co_Ts) andalso
   639          co_s <> @{type_name fun} andalso
   640          not (is_basic_datatype thy [(NONE, true)] co_s) then
   641         ()
   642       else
   643         raise TYPE ("Nitpick_HOL.register_codatatype_generic", [co_T], [])
   644     val codatatypes = AList.update (op =) (co_s, (case_name, constr_xs))
   645                                    codatatypes
   646   in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
   647                codatatypes = codatatypes} generic end
   648 (* TODO: Consider morphism. *)
   649 fun register_codatatype co_T case_name constr_xs (_ : morphism) =
   650   register_codatatype_generic co_T case_name constr_xs
   651 val register_codatatype_global =
   652   Context.theory_map ooo register_codatatype_generic
   653 
   654 fun unregister_codatatype_generic co_T = register_codatatype_generic co_T "" []
   655 (* TODO: Consider morphism. *)
   656 fun unregister_codatatype co_T (_ : morphism) =
   657   unregister_codatatype_generic co_T
   658 val unregister_codatatype_global =
   659   Context.theory_map o unregister_codatatype_generic
   660 
   661 fun is_codatatype ctxt (Type (s, _)) =
   662     s |> AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   663       |> Option.map snd |> these |> null |> not
   664   | is_codatatype _ _ = false
   665 fun is_real_quot_type ctxt (Type (s, _)) =
   666     is_some (Quotient_Info.lookup_quotients ctxt s)
   667   | is_real_quot_type _ _ = false
   668 fun is_quot_type ctxt T =
   669     is_real_quot_type ctxt T andalso not (is_codatatype ctxt T)
   670 fun is_pure_typedef ctxt (T as Type (s, _)) =
   671     let val thy = Proof_Context.theory_of ctxt in
   672       is_typedef ctxt s andalso
   673       not (is_real_datatype thy s orelse is_real_quot_type ctxt T orelse
   674            is_codatatype ctxt T orelse is_record_type T orelse
   675            is_integer_like_type T)
   676     end
   677   | is_pure_typedef _ _ = false
   678 fun is_univ_typedef ctxt (Type (s, _)) =
   679     (case typedef_info ctxt s of
   680        SOME {set_def, prop_of_Rep, ...} =>
   681        let
   682          val t_opt =
   683            case set_def of
   684              SOME thm => try (snd o Logic.dest_equals o prop_of) thm
   685            | NONE => try (snd o HOLogic.dest_mem o HOLogic.dest_Trueprop)
   686                          prop_of_Rep
   687        in
   688          case t_opt of
   689            SOME (Const (@{const_name top}, _)) => true
   690            (* "Multiset.multiset" *)
   691          | SOME (Const (@{const_name Collect}, _)
   692                  $ Abs (_, _, Const (@{const_name finite}, _) $ _)) => true
   693            (* "FinFun.finfun" *)
   694          | SOME (Const (@{const_name Collect}, _) $ Abs (_, _,
   695                      Const (@{const_name Ex}, _) $ Abs (_, _,
   696                          Const (@{const_name finite}, _) $ _))) => true
   697          | _ => false
   698        end
   699      | NONE => false)
   700   | is_univ_typedef _ _ = false
   701 fun is_datatype ctxt stds (T as Type (s, _)) =
   702     let val thy = Proof_Context.theory_of ctxt in
   703       (is_typedef ctxt s orelse is_codatatype ctxt T orelse
   704        T = @{typ ind} orelse is_real_quot_type ctxt T) andalso
   705       not (is_basic_datatype thy stds s)
   706     end
   707   | is_datatype _ _ _ = false
   708 
   709 fun all_record_fields thy T =
   710   let val (recs, more) = Record.get_extT_fields thy T in
   711     recs @ more :: all_record_fields thy (snd more)
   712   end
   713   handle TYPE _ => []
   714 fun is_record_constr (s, T) =
   715   String.isSuffix Record.extN s andalso
   716   let val dataT = body_type T in
   717     is_record_type dataT andalso
   718     s = unsuffix Record.ext_typeN (fst (dest_Type dataT)) ^ Record.extN
   719   end
   720 val num_record_fields = Integer.add 1 o length o fst oo Record.get_extT_fields
   721 fun no_of_record_field thy s T1 =
   722   find_index (curry (op =) s o fst)
   723              (Record.get_extT_fields thy T1 ||> single |> op @)
   724 fun is_record_get thy (s, Type (@{type_name fun}, [T1, _])) =
   725     exists (curry (op =) s o fst) (all_record_fields thy T1)
   726   | is_record_get _ _ = false
   727 fun is_record_update thy (s, T) =
   728   String.isSuffix Record.updateN s andalso
   729   exists (curry (op =) (unsuffix Record.updateN s) o fst)
   730          (all_record_fields thy (body_type T))
   731   handle TYPE _ => false
   732 fun is_abs_fun ctxt (s, Type (@{type_name fun}, [_, Type (s', _)])) =
   733     (case typedef_info ctxt s' of
   734        SOME {Abs_name, ...} => s = Abs_name
   735      | NONE => false)
   736   | is_abs_fun _ _ = false
   737 fun is_rep_fun ctxt (s, Type (@{type_name fun}, [Type (s', _), _])) =
   738     (case typedef_info ctxt s' of
   739        SOME {Rep_name, ...} => s = Rep_name
   740      | NONE => false)
   741   | is_rep_fun _ _ = false
   742 fun is_quot_abs_fun ctxt (x as (_, Type (@{type_name fun},
   743                                          [_, abs_T as Type (s', _)]))) =
   744     try (Quotient_Term.absrep_const_chk ctxt Quotient_Term.AbsF) s'
   745     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   746   | is_quot_abs_fun _ _ = false
   747 fun is_quot_rep_fun ctxt (s, Type (@{type_name fun},
   748                                    [abs_T as Type (abs_s, _), _])) =
   749     (case try (Quotient_Term.absrep_const_chk ctxt Quotient_Term.RepF) abs_s of
   750        SOME (Const (s', _)) => s = s' andalso not (is_codatatype ctxt abs_T)
   751      | NONE => false)
   752   | is_quot_rep_fun _ _ = false
   753 
   754 fun mate_of_rep_fun ctxt (x as (_, Type (@{type_name fun},
   755                                          [T1 as Type (s', _), T2]))) =
   756     (case typedef_info ctxt s' of
   757        SOME {Abs_name, ...} => (Abs_name, Type (@{type_name fun}, [T2, T1]))
   758      | NONE => raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x]))
   759   | mate_of_rep_fun _ x = raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x])
   760 fun rep_type_for_quot_type ctxt (T as Type (s, _)) =
   761     let
   762       val thy = Proof_Context.theory_of ctxt
   763       val {qtyp, rtyp, ...} = the (Quotient_Info.lookup_quotients ctxt s)
   764     in
   765       instantiate_type thy qtyp T rtyp
   766     end
   767   | rep_type_for_quot_type _ T =
   768     raise TYPE ("Nitpick_HOL.rep_type_for_quot_type", [T], [])
   769 fun equiv_relation_for_quot_type thy (Type (s, Ts)) =
   770     let
   771       val {qtyp, equiv_rel, equiv_thm, ...} =
   772         the (Quotient_Info.lookup_quotients thy s)
   773       val partial =
   774         case prop_of equiv_thm of
   775           @{const Trueprop} $ (Const (@{const_name equivp}, _) $ _) => false
   776         | @{const Trueprop} $ (Const (@{const_name part_equivp}, _) $ _) => true
   777         | _ => raise NOT_SUPPORTED "Ill-formed quotient type equivalence \
   778                                    \relation theorem"
   779       val Ts' = qtyp |> dest_Type |> snd
   780     in (subst_atomic_types (Ts' ~~ Ts) equiv_rel, partial) end
   781   | equiv_relation_for_quot_type _ T =
   782     raise TYPE ("Nitpick_HOL.equiv_relation_for_quot_type", [T], [])
   783 
   784 fun is_coconstr ctxt (s, T) =
   785   case body_type T of
   786     co_T as Type (co_s, _) =>
   787     let val {codatatypes, ...} = Data.get (Context.Proof ctxt) in
   788       exists (fn (s', T') => s = s' andalso repair_constr_type ctxt co_T T' = T)
   789              (AList.lookup (op =) codatatypes co_s |> Option.map snd |> these)
   790     end
   791   | _ => false
   792 fun is_constr_like ctxt (s, T) =
   793   member (op =) [@{const_name FunBox}, @{const_name PairBox},
   794                  @{const_name Quot}, @{const_name Zero_Rep},
   795                  @{const_name Suc_Rep}] s orelse
   796   let
   797     val thy = Proof_Context.theory_of ctxt
   798     val (x as (_, T)) = (s, unarize_unbox_etc_type T)
   799   in
   800     is_real_constr thy x orelse is_record_constr x orelse
   801     (is_abs_fun ctxt x andalso is_pure_typedef ctxt (range_type T)) orelse
   802     is_coconstr ctxt x
   803   end
   804 fun is_stale_constr ctxt (x as (_, T)) =
   805   is_codatatype ctxt (body_type T) andalso is_constr_like ctxt x andalso
   806   not (is_coconstr ctxt x)
   807 fun is_constr ctxt stds (x as (_, T)) =
   808   let val thy = Proof_Context.theory_of ctxt in
   809     is_constr_like ctxt x andalso
   810     not (is_basic_datatype thy stds
   811                          (fst (dest_Type (unarize_type (body_type T))))) andalso
   812     not (is_stale_constr ctxt x)
   813   end
   814 val is_sel = String.isPrefix discr_prefix orf String.isPrefix sel_prefix
   815 val is_sel_like_and_no_discr =
   816   String.isPrefix sel_prefix orf
   817   (member (op =) [@{const_name fst}, @{const_name snd}])
   818 
   819 fun in_fun_lhs_for InConstr = InSel
   820   | in_fun_lhs_for _ = InFunLHS
   821 fun in_fun_rhs_for InConstr = InConstr
   822   | in_fun_rhs_for InSel = InSel
   823   | in_fun_rhs_for InFunRHS1 = InFunRHS2
   824   | in_fun_rhs_for _ = InFunRHS1
   825 
   826 fun is_boxing_worth_it (hol_ctxt : hol_context) boxy T =
   827   case T of
   828     Type (@{type_name fun}, _) =>
   829     (boxy = InPair orelse boxy = InFunLHS) andalso
   830     not (is_boolean_type (body_type T))
   831   | Type (@{type_name prod}, Ts) =>
   832     boxy = InPair orelse boxy = InFunRHS1 orelse boxy = InFunRHS2 orelse
   833     ((boxy = InExpr orelse boxy = InFunLHS) andalso
   834      exists (is_boxing_worth_it hol_ctxt InPair)
   835             (map (box_type hol_ctxt InPair) Ts))
   836   | _ => false
   837 and should_box_type (hol_ctxt as {thy, boxes, ...}) boxy z =
   838   case triple_lookup (type_match thy) boxes (Type z) of
   839     SOME (SOME box_me) => box_me
   840   | _ => is_boxing_worth_it hol_ctxt boxy (Type z)
   841 and box_type hol_ctxt boxy T =
   842   case T of
   843     Type (z as (@{type_name fun}, [T1, T2])) =>
   844     if boxy <> InConstr andalso boxy <> InSel andalso
   845        should_box_type hol_ctxt boxy z then
   846       Type (@{type_name fun_box},
   847             [box_type hol_ctxt InFunLHS T1, box_type hol_ctxt InFunRHS1 T2])
   848     else
   849       box_type hol_ctxt (in_fun_lhs_for boxy) T1
   850       --> box_type hol_ctxt (in_fun_rhs_for boxy) T2
   851   | Type (z as (@{type_name prod}, Ts)) =>
   852     if boxy <> InConstr andalso boxy <> InSel
   853        andalso should_box_type hol_ctxt boxy z then
   854       Type (@{type_name pair_box}, map (box_type hol_ctxt InSel) Ts)
   855     else
   856       Type (@{type_name prod},
   857             map (box_type hol_ctxt
   858                           (if boxy = InConstr orelse boxy = InSel then boxy
   859                            else InPair)) Ts)
   860   | _ => T
   861 
   862 fun binarize_nat_and_int_in_type @{typ nat} = @{typ "unsigned_bit word"}
   863   | binarize_nat_and_int_in_type @{typ int} = @{typ "signed_bit word"}
   864   | binarize_nat_and_int_in_type (Type (s, Ts)) =
   865     Type (s, map binarize_nat_and_int_in_type Ts)
   866   | binarize_nat_and_int_in_type T = T
   867 val binarize_nat_and_int_in_term = map_types binarize_nat_and_int_in_type
   868 
   869 fun discr_for_constr (s, T) = (discr_prefix ^ s, body_type T --> bool_T)
   870 
   871 fun num_sels_for_constr_type T = length (maybe_curried_binder_types T)
   872 fun nth_sel_name_for_constr_name s n =
   873   if s = @{const_name Pair} then
   874     if n = 0 then @{const_name fst} else @{const_name snd}
   875   else
   876     sel_prefix_for n ^ s
   877 fun nth_sel_for_constr x ~1 = discr_for_constr x
   878   | nth_sel_for_constr (s, T) n =
   879     (nth_sel_name_for_constr_name s n,
   880      body_type T --> nth (maybe_curried_binder_types T) n)
   881 fun binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize =
   882   apsnd ((binarize ? binarize_nat_and_int_in_type) o box_type hol_ctxt InSel)
   883   oo nth_sel_for_constr
   884 
   885 fun sel_no_from_name s =
   886   if String.isPrefix discr_prefix s then
   887     ~1
   888   else if String.isPrefix sel_prefix s then
   889     s |> unprefix sel_prefix |> Int.fromString |> the
   890   else if s = @{const_name snd} then
   891     1
   892   else
   893     0
   894 
   895 val close_form =
   896   let
   897     fun close_up zs zs' =
   898       fold (fn (z as ((s, _), T)) => fn t' =>
   899                Logic.all_const T $ Abs (s, T, abstract_over (Var z, t')))
   900            (take (length zs' - length zs) zs')
   901     fun aux zs (@{const "==>"} $ t1 $ t2) =
   902         let val zs' = Term.add_vars t1 zs in
   903           close_up zs zs' (Logic.mk_implies (t1, aux zs' t2))
   904         end
   905       | aux zs t = close_up zs (Term.add_vars t zs) t
   906   in aux [] end
   907 
   908 fun distinctness_formula T =
   909   all_distinct_unordered_pairs_of
   910   #> map (fn (t1, t2) => @{const Not} $ (HOLogic.eq_const T $ t1 $ t2))
   911   #> List.foldr (s_conj o swap) @{const True}
   912 
   913 fun zero_const T = Const (@{const_name zero_class.zero}, T)
   914 fun suc_const T = Const (@{const_name Suc}, T --> T)
   915 
   916 fun uncached_datatype_constrs ({thy, ctxt, stds, ...} : hol_context)
   917                               (T as Type (s, Ts)) =
   918     (case AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   919                        s of
   920        SOME (_, xs' as (_ :: _)) => map (apsnd (repair_constr_type ctxt T)) xs'
   921      | _ =>
   922        if is_datatype ctxt stds T then
   923          case Datatype.get_info thy s of
   924            SOME {index, descr, ...} =>
   925            let
   926              val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the
   927            in
   928              map (apsnd (fn Us =>
   929                             map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))
   930                  constrs
   931            end
   932          | NONE =>
   933            if is_record_type T then
   934              let
   935                val s' = unsuffix Record.ext_typeN s ^ Record.extN
   936                val T' = (Record.get_extT_fields thy T
   937                         |> apsnd single |> uncurry append |> map snd) ---> T
   938              in [(s', T')] end
   939            else if is_real_quot_type ctxt T then
   940              [(@{const_name Quot}, rep_type_for_quot_type ctxt T --> T)]
   941            else case typedef_info ctxt s of
   942              SOME {abs_type, rep_type, Abs_name, ...} =>
   943              [(Abs_name,
   944                varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
   945            | NONE =>
   946              if T = @{typ ind} then
   947                [dest_Const @{const Zero_Rep}, dest_Const @{const Suc_Rep}]
   948              else
   949                []
   950        else
   951          [])
   952   | uncached_datatype_constrs _ _ = []
   953 fun datatype_constrs (hol_ctxt as {constr_cache, ...}) T =
   954   case AList.lookup (op =) (!constr_cache) T of
   955     SOME xs => xs
   956   | NONE =>
   957     let val xs = uncached_datatype_constrs hol_ctxt T in
   958       (Unsynchronized.change constr_cache (cons (T, xs)); xs)
   959     end
   960 fun binarized_and_boxed_datatype_constrs hol_ctxt binarize =
   961   map (apsnd ((binarize ? binarize_nat_and_int_in_type)
   962               o box_type hol_ctxt InConstr)) o datatype_constrs hol_ctxt
   963 val num_datatype_constrs = length oo datatype_constrs
   964 
   965 fun constr_name_for_sel_like @{const_name fst} = @{const_name Pair}
   966   | constr_name_for_sel_like @{const_name snd} = @{const_name Pair}
   967   | constr_name_for_sel_like s' = original_name s'
   968 fun binarized_and_boxed_constr_for_sel hol_ctxt binarize (s', T') =
   969   let val s = constr_name_for_sel_like s' in
   970     AList.lookup (op =)
   971         (binarized_and_boxed_datatype_constrs hol_ctxt binarize (domain_type T'))
   972         s
   973     |> the |> pair s
   974   end
   975 
   976 fun card_of_type assigns (Type (@{type_name fun}, [T1, T2])) =
   977     reasonable_power (card_of_type assigns T2) (card_of_type assigns T1)
   978   | card_of_type assigns (Type (@{type_name prod}, [T1, T2])) =
   979     card_of_type assigns T1 * card_of_type assigns T2
   980   | card_of_type assigns (Type (@{type_name set}, [T'])) =
   981     reasonable_power 2 (card_of_type assigns T')
   982   | card_of_type _ (Type (@{type_name itself}, _)) = 1
   983   | card_of_type _ @{typ prop} = 2
   984   | card_of_type _ @{typ bool} = 2
   985   | card_of_type assigns T =
   986     case AList.lookup (op =) assigns T of
   987       SOME k => k
   988     | NONE => if T = @{typ bisim_iterator} then 0
   989               else raise TYPE ("Nitpick_HOL.card_of_type", [T], [])
   990 
   991 fun bounded_card_of_type max default_card assigns
   992                          (Type (@{type_name fun}, [T1, T2])) =
   993     let
   994       val k1 = bounded_card_of_type max default_card assigns T1
   995       val k2 = bounded_card_of_type max default_card assigns T2
   996     in
   997       if k1 = max orelse k2 = max then max
   998       else Int.min (max, reasonable_power k2 k1)
   999     end
  1000   | bounded_card_of_type max default_card assigns
  1001                          (Type (@{type_name prod}, [T1, T2])) =
  1002     let
  1003       val k1 = bounded_card_of_type max default_card assigns T1
  1004       val k2 = bounded_card_of_type max default_card assigns T2
  1005     in if k1 = max orelse k2 = max then max else Int.min (max, k1 * k2) end
  1006   | bounded_card_of_type max default_card assigns
  1007                          (Type (@{type_name set}, [T'])) =
  1008     bounded_card_of_type max default_card assigns (T' --> bool_T)
  1009   | bounded_card_of_type max default_card assigns T =
  1010     Int.min (max, if default_card = ~1 then
  1011                     card_of_type assigns T
  1012                   else
  1013                     card_of_type assigns T
  1014                     handle TYPE ("Nitpick_HOL.card_of_type", _, _) =>
  1015                            default_card)
  1016 
  1017 (* Similar to "ATP_Util.tiny_card_of_type". *)
  1018 fun bounded_exact_card_of_type hol_ctxt finitizable_dataTs max default_card
  1019                                assigns T =
  1020   let
  1021     fun aux avoid T =
  1022       (if member (op =) avoid T then
  1023          0
  1024        else if member (op =) finitizable_dataTs T then
  1025          raise SAME ()
  1026        else case T of
  1027          Type (@{type_name fun}, [T1, T2]) =>
  1028          (case (aux avoid T1, aux avoid T2) of
  1029             (_, 1) => 1
  1030           | (0, _) => 0
  1031           | (_, 0) => 0
  1032           | (k1, k2) =>
  1033             if k1 >= max orelse k2 >= max then max
  1034             else Int.min (max, reasonable_power k2 k1))
  1035        | Type (@{type_name prod}, [T1, T2]) =>
  1036          (case (aux avoid T1, aux avoid T2) of
  1037             (0, _) => 0
  1038           | (_, 0) => 0
  1039           | (k1, k2) =>
  1040             if k1 >= max orelse k2 >= max then max
  1041             else Int.min (max, k1 * k2))
  1042        | Type (@{type_name set}, [T']) => aux avoid (T' --> bool_T)
  1043        | Type (@{type_name itself}, _) => 1
  1044        | @{typ prop} => 2
  1045        | @{typ bool} => 2
  1046        | Type _ =>
  1047          (case datatype_constrs hol_ctxt T of
  1048             [] => if is_integer_type T orelse is_bit_type T then 0
  1049                   else raise SAME ()
  1050           | constrs =>
  1051             let
  1052               val constr_cards =
  1053                 map (Integer.prod o map (aux (T :: avoid)) o binder_types o snd)
  1054                     constrs
  1055             in
  1056               if exists (curry (op =) 0) constr_cards then 0
  1057               else Int.min (max, Integer.sum constr_cards)
  1058             end)
  1059        | _ => raise SAME ())
  1060       handle SAME () =>
  1061              AList.lookup (op =) assigns T |> the_default default_card
  1062   in Int.min (max, aux [] T) end
  1063 
  1064 val typical_atomic_card = 4
  1065 val typical_card_of_type = bounded_card_of_type 16777217 typical_atomic_card []
  1066 
  1067 fun is_finite_type hol_ctxt T =
  1068   bounded_exact_card_of_type hol_ctxt [] 1 2 [] T > 0
  1069 
  1070 fun is_special_eligible_arg strict Ts t =
  1071   case map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t) of
  1072     [] => true
  1073   | bad_Ts =>
  1074     let
  1075       val bad_Ts_cost =
  1076         if strict then fold (curry (op *) o typical_card_of_type) bad_Ts 1
  1077         else fold (Integer.max o typical_card_of_type) bad_Ts 0
  1078       val T_cost = typical_card_of_type (fastype_of1 (Ts, t))
  1079     in (bad_Ts_cost, T_cost) |> (if strict then op < else op <=) end
  1080 
  1081 fun abs_var ((s, j), T) body = Abs (s, T, abstract_over (Var ((s, j), T), body))
  1082 
  1083 fun let_var s = (nitpick_prefix ^ s, 999)
  1084 val let_inline_threshold = 20
  1085 
  1086 fun s_let Ts s n abs_T body_T f t =
  1087   if (n - 1) * (size_of_term t - 1) <= let_inline_threshold orelse
  1088      is_special_eligible_arg false Ts t then
  1089     f t
  1090   else
  1091     let val z = (let_var s, abs_T) in
  1092       Const (@{const_name Let}, abs_T --> (abs_T --> body_T) --> body_T)
  1093       $ t $ abs_var z (incr_boundvars 1 (f (Var z)))
  1094     end
  1095 
  1096 fun loose_bvar1_count (Bound i, k) = if i = k then 1 else 0
  1097   | loose_bvar1_count (t1 $ t2, k) =
  1098     loose_bvar1_count (t1, k) + loose_bvar1_count (t2, k)
  1099   | loose_bvar1_count (Abs (_, _, t), k) = loose_bvar1_count (t, k + 1)
  1100   | loose_bvar1_count _ = 0
  1101 
  1102 fun s_betapply _ (t1 as Const (@{const_name "=="}, _) $ t1', t2) =
  1103     if t1' aconv t2 then @{prop True} else t1 $ t2
  1104   | s_betapply _ (t1 as Const (@{const_name HOL.eq}, _) $ t1', t2) =
  1105     if t1' aconv t2 then @{term True} else t1 $ t2
  1106   | s_betapply _ (Const (@{const_name If}, _) $ @{const True} $ t1', _) = t1'
  1107   | s_betapply _ (Const (@{const_name If}, _) $ @{const False} $ _, t2) = t2
  1108   | s_betapply Ts (Const (@{const_name Let},
  1109                           Type (_, [bound_T, Type (_, [_, body_T])]))
  1110                    $ t12 $ Abs (s, T, t13'), t2) =
  1111     let val body_T' = range_type body_T in
  1112       Const (@{const_name Let}, bound_T --> (bound_T --> body_T') --> body_T')
  1113       $ t12 $ Abs (s, T, s_betapply (T :: Ts) (t13', incr_boundvars 1 t2))
  1114     end
  1115   | s_betapply Ts (t1 as Abs (s1, T1, t1'), t2) =
  1116     (s_let Ts s1 (loose_bvar1_count (t1', 0)) T1 (fastype_of1 (T1 :: Ts, t1'))
  1117            (curry betapply t1) t2
  1118      (* FIXME: fix all "s_betapply []" calls *)
  1119      handle TERM _ => betapply (t1, t2)
  1120           | General.Subscript => betapply (t1, t2))
  1121   | s_betapply _ (t1, t2) = t1 $ t2
  1122 fun s_betapplys Ts = Library.foldl (s_betapply Ts)
  1123 
  1124 fun s_beta_norm Ts t =
  1125   let
  1126     fun aux _ (Var _) = raise Same.SAME
  1127       | aux Ts (Abs (s, T, t')) = Abs (s, T, aux (T :: Ts) t')
  1128       | aux Ts ((t1 as Abs _) $ t2) =
  1129         Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1130       | aux Ts (t1 $ t2) =
  1131         ((case aux Ts t1 of
  1132            t1 as Abs _ => Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1133          | t1 => t1 $ Same.commit (aux Ts) t2)
  1134         handle Same.SAME => t1 $ aux Ts t2)
  1135       | aux _ _ = raise Same.SAME
  1136   in aux Ts t handle Same.SAME => t end
  1137 
  1138 fun discr_term_for_constr hol_ctxt (x as (s, T)) =
  1139   let val dataT = body_type T in
  1140     if s = @{const_name Suc} then
  1141       Abs (Name.uu, dataT,
  1142            @{const Not} $ HOLogic.mk_eq (zero_const dataT, Bound 0))
  1143     else if num_datatype_constrs hol_ctxt dataT >= 2 then
  1144       Const (discr_for_constr x)
  1145     else
  1146       Abs (Name.uu, dataT, @{const True})
  1147   end
  1148 fun discriminate_value (hol_ctxt as {ctxt, ...}) x t =
  1149   case head_of t of
  1150     Const x' =>
  1151     if x = x' then @{const True}
  1152     else if is_constr_like ctxt x' then @{const False}
  1153     else s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1154   | _ => s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1155 
  1156 fun nth_arg_sel_term_for_constr thy stds (x as (s, T)) n =
  1157   let val (arg_Ts, dataT) = strip_type T in
  1158     if dataT = nat_T andalso is_standard_datatype thy stds nat_T then
  1159       @{term "%n::nat. n - 1"}
  1160     else if is_pair_type dataT then
  1161       Const (nth_sel_for_constr x n)
  1162     else
  1163       let
  1164         fun aux m (Type (@{type_name prod}, [T1, T2])) =
  1165             let
  1166               val (m, t1) = aux m T1
  1167               val (m, t2) = aux m T2
  1168             in (m, HOLogic.mk_prod (t1, t2)) end
  1169           | aux m T =
  1170             (m + 1, Const (nth_sel_name_for_constr_name s m, dataT --> T)
  1171                     $ Bound 0)
  1172         val m = fold (Integer.add o num_factors_in_type)
  1173                      (List.take (arg_Ts, n)) 0
  1174       in Abs ("x", dataT, aux m (nth arg_Ts n) |> snd) end
  1175   end
  1176 fun select_nth_constr_arg ctxt stds x t n res_T =
  1177   let val thy = Proof_Context.theory_of ctxt in
  1178     (case strip_comb t of
  1179        (Const x', args) =>
  1180        if x = x' then nth args n
  1181        else if is_constr_like ctxt x' then Const (@{const_name unknown}, res_T)
  1182        else raise SAME ()
  1183      | _ => raise SAME())
  1184     handle SAME () =>
  1185            s_betapply [] (nth_arg_sel_term_for_constr thy stds x n, t)
  1186   end
  1187 
  1188 fun construct_value _ _ x [] = Const x
  1189   | construct_value ctxt stds (x as (s, _)) args =
  1190     let val args = map Envir.eta_contract args in
  1191       case hd args of
  1192         Const (s', _) $ t =>
  1193         if is_sel_like_and_no_discr s' andalso
  1194            constr_name_for_sel_like s' = s andalso
  1195            forall (fn (n, t') =>
  1196                       select_nth_constr_arg ctxt stds x t n dummyT = t')
  1197                   (index_seq 0 (length args) ~~ args) then
  1198           t
  1199         else
  1200           list_comb (Const x, args)
  1201       | _ => list_comb (Const x, args)
  1202     end
  1203 
  1204 fun constr_expand (hol_ctxt as {ctxt, stds, ...}) T t =
  1205   (case head_of t of
  1206      Const x => if is_constr_like ctxt x then t else raise SAME ()
  1207    | _ => raise SAME ())
  1208   handle SAME () =>
  1209          let
  1210            val x' as (_, T') =
  1211              if is_pair_type T then
  1212                let val (T1, T2) = HOLogic.dest_prodT T in
  1213                  (@{const_name Pair}, T1 --> T2 --> T)
  1214                end
  1215              else
  1216                datatype_constrs hol_ctxt T |> hd
  1217            val arg_Ts = binder_types T'
  1218          in
  1219            list_comb (Const x', map2 (select_nth_constr_arg ctxt stds x' t)
  1220                                      (index_seq 0 (length arg_Ts)) arg_Ts)
  1221          end
  1222 
  1223 fun coerce_bound_no f j t =
  1224   case t of
  1225     t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2
  1226   | Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')
  1227   | Bound j' => if j' = j then f t else t
  1228   | _ => t
  1229 fun coerce_bound_0_in_term hol_ctxt new_T old_T =
  1230   old_T <> new_T ? coerce_bound_no (coerce_term hol_ctxt [new_T] old_T new_T) 0
  1231 and coerce_term (hol_ctxt as {ctxt, stds, ...}) Ts new_T old_T t =
  1232   if old_T = new_T then
  1233     t
  1234   else
  1235     case (new_T, old_T) of
  1236       (Type (new_s, new_Ts as [new_T1, new_T2]),
  1237        Type (@{type_name fun}, [old_T1, old_T2])) =>
  1238       (case eta_expand Ts t 1 of
  1239          Abs (s, _, t') =>
  1240          Abs (s, new_T1,
  1241               t' |> coerce_bound_0_in_term hol_ctxt new_T1 old_T1
  1242                  |> coerce_term hol_ctxt (new_T1 :: Ts) new_T2 old_T2)
  1243          |> Envir.eta_contract
  1244          |> new_s <> @{type_name fun}
  1245             ? construct_value ctxt stds
  1246                   (@{const_name FunBox},
  1247                    Type (@{type_name fun}, new_Ts) --> new_T)
  1248               o single
  1249        | t' => raise TERM ("Nitpick_HOL.coerce_term", [t']))
  1250     | (Type (new_s, new_Ts as [new_T1, new_T2]),
  1251        Type (old_s, old_Ts as [old_T1, old_T2])) =>
  1252       if old_s = @{type_name fun_box} orelse
  1253          old_s = @{type_name pair_box} orelse old_s = @{type_name prod} then
  1254         case constr_expand hol_ctxt old_T t of
  1255           Const (old_s, _) $ t1 =>
  1256           if new_s = @{type_name fun} then
  1257             coerce_term hol_ctxt Ts new_T (Type (@{type_name fun}, old_Ts)) t1
  1258           else
  1259             construct_value ctxt stds
  1260                 (old_s, Type (@{type_name fun}, new_Ts) --> new_T)
  1261                 [coerce_term hol_ctxt Ts (Type (@{type_name fun}, new_Ts))
  1262                              (Type (@{type_name fun}, old_Ts)) t1]
  1263         | Const _ $ t1 $ t2 =>
  1264           construct_value ctxt stds
  1265               (if new_s = @{type_name prod} then @{const_name Pair}
  1266                else @{const_name PairBox}, new_Ts ---> new_T)
  1267               (map3 (coerce_term hol_ctxt Ts) [new_T1, new_T2] [old_T1, old_T2]
  1268                     [t1, t2])
  1269         | t' => raise TERM ("Nitpick_HOL.coerce_term", [t'])
  1270       else
  1271         raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1272     | _ => raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1273 
  1274 fun is_ground_term (t1 $ t2) = is_ground_term t1 andalso is_ground_term t2
  1275   | is_ground_term (Const _) = true
  1276   | is_ground_term _ = false
  1277 
  1278 fun special_bounds ts =
  1279   fold Term.add_vars ts [] |> sort (Term_Ord.fast_indexname_ord o pairself fst)
  1280 
  1281 (* FIXME: detect "rep_datatype"? *)
  1282 fun is_funky_typedef_name ctxt s =
  1283   member (op =) [@{type_name unit}, @{type_name prod}, @{type_name set},
  1284                  @{type_name Sum_Type.sum}, @{type_name int}] s orelse
  1285   is_frac_type ctxt (Type (s, []))
  1286 fun is_funky_typedef ctxt (Type (s, _)) = is_funky_typedef_name ctxt s
  1287   | is_funky_typedef _ _ = false
  1288 fun is_typedef_axiom ctxt boring (@{const "==>"} $ _ $ t2) =
  1289     is_typedef_axiom ctxt boring t2
  1290   | is_typedef_axiom ctxt boring
  1291         (@{const Trueprop} $ (Const (@{const_name Typedef.type_definition}, _)
  1292          $ Const (_, Type (@{type_name fun}, [Type (s, _), _]))
  1293          $ Const _ $ _)) =
  1294     boring <> is_funky_typedef_name ctxt s andalso is_typedef ctxt s
  1295   | is_typedef_axiom _ _ _ = false
  1296 
  1297 fun all_defs_of thy subst =
  1298   let
  1299     val def_names =
  1300       thy |> Theory.defs_of
  1301           |> Defs.all_specifications_of
  1302           |> maps snd |> map_filter #def
  1303           |> Ord_List.make fast_string_ord
  1304   in
  1305     Theory.nodes_of thy
  1306     |> maps Thm.axioms_of
  1307     |> map (apsnd (subst_atomic subst o prop_of))
  1308     |> sort (fast_string_ord o pairself fst)
  1309     |> Ord_List.inter (fast_string_ord o apsnd fst) def_names
  1310     |> map snd
  1311   end
  1312 
  1313 (* Ideally we would check against "Complex_Main", not "Refute", but any theory
  1314    will do as long as it contains all the "axioms" and "axiomatization"
  1315    commands. *)
  1316 fun is_built_in_theory thy = Theory.subthy (thy, @{theory Refute})
  1317 
  1318 fun all_nondefs_of ctxt subst =
  1319   ctxt |> Spec_Rules.get
  1320        |> filter (curry (op =) Spec_Rules.Unknown o fst)
  1321        |> maps (snd o snd)
  1322        |> filter_out (is_built_in_theory o theory_of_thm)
  1323        |> map (subst_atomic subst o prop_of)
  1324 
  1325 fun arity_of_built_in_const thy stds (s, T) =
  1326   if s = @{const_name If} then
  1327     if nth_range_type 3 T = @{typ bool} then NONE else SOME 3
  1328   else
  1329     let val std_nats = is_standard_datatype thy stds nat_T in
  1330       case AList.lookup (op =)
  1331                     (built_in_consts
  1332                      |> std_nats ? append built_in_nat_consts) s of
  1333         SOME n => SOME n
  1334       | NONE =>
  1335         case AList.lookup (op =)
  1336                  (built_in_typed_consts
  1337                   |> std_nats ? append built_in_typed_nat_consts)
  1338                  (s, unarize_type T) of
  1339           SOME n => SOME n
  1340         | NONE =>
  1341           case s of
  1342             @{const_name zero_class.zero} =>
  1343             if is_iterator_type T then SOME 0 else NONE
  1344           | @{const_name Suc} =>
  1345             if is_iterator_type (domain_type T) then SOME 0 else NONE
  1346           | _ => if is_fun_type T andalso is_set_like_type (domain_type T) then
  1347                    AList.lookup (op =) built_in_set_like_consts s
  1348                  else
  1349                    NONE
  1350     end
  1351 val is_built_in_const = is_some ooo arity_of_built_in_const
  1352 
  1353 (* This function is designed to work for both real definition axioms and
  1354    simplification rules (equational specifications). *)
  1355 fun term_under_def t =
  1356   case t of
  1357     @{const "==>"} $ _ $ t2 => term_under_def t2
  1358   | Const (@{const_name "=="}, _) $ t1 $ _ => term_under_def t1
  1359   | @{const Trueprop} $ t1 => term_under_def t1
  1360   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => term_under_def t1
  1361   | Abs (_, _, t') => term_under_def t'
  1362   | t1 $ _ => term_under_def t1
  1363   | _ => t
  1364 
  1365 (* Here we crucially rely on "specialize_type" performing a preorder traversal
  1366    of the term, without which the wrong occurrence of a constant could be
  1367    matched in the face of overloading. *)
  1368 fun def_props_for_const thy stds table (x as (s, _)) =
  1369   if is_built_in_const thy stds x then
  1370     []
  1371   else
  1372     these (Symtab.lookup table s)
  1373     |> map_filter (try (specialize_type thy x))
  1374     |> filter (curry (op =) (Const x) o term_under_def)
  1375 
  1376 fun normalized_rhs_of t =
  1377   let
  1378     fun aux (v as Var _) (SOME t) = SOME (lambda v t)
  1379       | aux (c as Const (@{const_name TYPE}, _)) (SOME t) = SOME (lambda c t)
  1380       | aux _ _ = NONE
  1381     val (lhs, rhs) =
  1382       case t of
  1383         Const (@{const_name "=="}, _) $ t1 $ t2 => (t1, t2)
  1384       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ t2) =>
  1385         (t1, t2)
  1386       | _ => raise TERM ("Nitpick_HOL.normalized_rhs_of", [t])
  1387     val args = strip_comb lhs |> snd
  1388   in fold_rev aux args (SOME rhs) end
  1389 
  1390 fun get_def_of_const thy table (x as (s, _)) =
  1391   x |> def_props_for_const thy [(NONE, false)] table |> List.last
  1392     |> normalized_rhs_of |> Option.map (prefix_abs_vars s)
  1393   handle List.Empty => NONE
  1394 
  1395 fun def_of_const_ext thy (unfold_table, fallback_table) (x as (s, _)) =
  1396   if is_built_in_const thy [(NONE, false)] x orelse original_name s <> s then
  1397     NONE
  1398   else case get_def_of_const thy unfold_table x of
  1399     SOME def => SOME (true, def)
  1400   | NONE => get_def_of_const thy fallback_table x |> Option.map (pair false)
  1401 
  1402 val def_of_const = Option.map snd ooo def_of_const_ext
  1403 
  1404 fun fixpoint_kind_of_rhs (Abs (_, _, t)) = fixpoint_kind_of_rhs t
  1405   | fixpoint_kind_of_rhs (Const (@{const_name lfp}, _) $ Abs _) = Lfp
  1406   | fixpoint_kind_of_rhs (Const (@{const_name gfp}, _) $ Abs _) = Gfp
  1407   | fixpoint_kind_of_rhs _ = NoFp
  1408 
  1409 fun is_mutually_inductive_pred_def thy table t =
  1410   let
  1411     fun is_good_arg (Bound _) = true
  1412       | is_good_arg (Const (s, _)) =
  1413         s = @{const_name True} orelse s = @{const_name False} orelse
  1414         s = @{const_name undefined}
  1415       | is_good_arg _ = false
  1416   in
  1417     case t |> strip_abs_body |> strip_comb of
  1418       (Const x, ts as (_ :: _)) =>
  1419       (case def_of_const thy table x of
  1420          SOME t' => fixpoint_kind_of_rhs t' <> NoFp andalso
  1421                     forall is_good_arg ts
  1422        | NONE => false)
  1423     | _ => false
  1424   end
  1425 fun unfold_mutually_inductive_preds thy table =
  1426   map_aterms (fn t as Const x =>
  1427                  (case def_of_const thy table x of
  1428                     SOME t' =>
  1429                     let val t' = Envir.eta_contract t' in
  1430                       if is_mutually_inductive_pred_def thy table t' then t'
  1431                       else t
  1432                     end
  1433                  | NONE => t)
  1434                | t => t)
  1435 
  1436 fun case_const_names ctxt stds =
  1437   let val thy = Proof_Context.theory_of ctxt in
  1438     Symtab.fold (fn (dtype_s, {index, descr, case_name, ...}) =>
  1439                     if is_basic_datatype thy stds dtype_s then
  1440                       I
  1441                     else
  1442                       cons (case_name, AList.lookup (op =) descr index
  1443                                        |> the |> #3 |> length))
  1444                 (Datatype.get_all thy) [] @
  1445     map (apsnd length o snd) (#codatatypes (Data.get (Context.Proof ctxt)))
  1446   end
  1447 
  1448 fun fixpoint_kind_of_const thy table x =
  1449   if is_built_in_const thy [(NONE, false)] x then NoFp
  1450   else fixpoint_kind_of_rhs (the (def_of_const thy table x))
  1451   handle Option.Option => NoFp
  1452 
  1453 fun is_real_inductive_pred ({thy, stds, def_tables, intro_table, ...}
  1454                             : hol_context) x =
  1455   fixpoint_kind_of_const thy def_tables x <> NoFp andalso
  1456   not (null (def_props_for_const thy stds intro_table x))
  1457 fun is_inductive_pred hol_ctxt (x as (s, _)) =
  1458   is_real_inductive_pred hol_ctxt x orelse String.isPrefix ubfp_prefix s orelse
  1459   String.isPrefix lbfp_prefix s
  1460 
  1461 fun lhs_of_equation t =
  1462   case t of
  1463     Const (@{const_name all}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1464   | Const (@{const_name "=="}, _) $ t1 $ _ => SOME t1
  1465   | @{const "==>"} $ _ $ t2 => lhs_of_equation t2
  1466   | @{const Trueprop} $ t1 => lhs_of_equation t1
  1467   | Const (@{const_name All}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1468   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => SOME t1
  1469   | @{const HOL.implies} $ _ $ t2 => lhs_of_equation t2
  1470   | _ => NONE
  1471 fun is_constr_pattern _ (Bound _) = true
  1472   | is_constr_pattern _ (Var _) = true
  1473   | is_constr_pattern ctxt t =
  1474     case strip_comb t of
  1475       (Const x, args) =>
  1476       is_constr_like ctxt x andalso forall (is_constr_pattern ctxt) args
  1477     | _ => false
  1478 fun is_constr_pattern_lhs ctxt t =
  1479   forall (is_constr_pattern ctxt) (snd (strip_comb t))
  1480 fun is_constr_pattern_formula ctxt t =
  1481   case lhs_of_equation t of
  1482     SOME t' => is_constr_pattern_lhs ctxt t'
  1483   | NONE => false
  1484 
  1485 (* Similar to "specialize_type" but returns all matches rather than only the
  1486    first (preorder) match. *)
  1487 fun multi_specialize_type thy slack (s, T) t =
  1488   let
  1489     fun aux (Const (s', T')) ys =
  1490         if s = s' then
  1491           ys |> (if AList.defined (op =) ys T' then
  1492                    I
  1493                  else
  1494                    cons (T', monomorphic_term (Sign.typ_match thy (T', T)
  1495                                                               Vartab.empty) t)
  1496                    handle Type.TYPE_MATCH => I
  1497                         | TERM _ =>
  1498                           if slack then
  1499                             I
  1500                           else
  1501                             raise NOT_SUPPORTED
  1502                                       ("too much polymorphism in axiom \"" ^
  1503                                        Syntax.string_of_term_global thy t ^
  1504                                        "\" involving " ^ quote s))
  1505         else
  1506           ys
  1507       | aux _ ys = ys
  1508   in map snd (fold_aterms aux t []) end
  1509 fun nondef_props_for_const thy slack table (x as (s, _)) =
  1510   these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)
  1511 
  1512 fun unvarify_term (t1 $ t2) = unvarify_term t1 $ unvarify_term t2
  1513   | unvarify_term (Var ((s, 0), T)) = Free (s, T)
  1514   | unvarify_term (Abs (s, T, t')) = Abs (s, T, unvarify_term t')
  1515   | unvarify_term t = t
  1516 fun axiom_for_choice_spec thy =
  1517   unvarify_term
  1518   #> Object_Logic.atomize_term thy
  1519   #> Choice_Specification.close_form
  1520   #> HOLogic.mk_Trueprop
  1521 fun is_choice_spec_fun ({thy, def_tables, nondef_table, choice_spec_table, ...}
  1522                         : hol_context) x =
  1523   case nondef_props_for_const thy true choice_spec_table x of
  1524     [] => false
  1525   | ts => case def_of_const thy def_tables x of
  1526             SOME (Const (@{const_name Eps}, _) $ _) => true
  1527           | SOME _ => false
  1528           | NONE =>
  1529             let val ts' = nondef_props_for_const thy true nondef_table x in
  1530               length ts' = length ts andalso
  1531               forall (fn t =>
  1532                          exists (curry (op aconv) (axiom_for_choice_spec thy t))
  1533                                 ts') ts
  1534             end
  1535 
  1536 fun is_choice_spec_axiom thy choice_spec_table t =
  1537   Symtab.exists (fn (_, ts) =>
  1538                     exists (curry (op aconv) t o axiom_for_choice_spec thy) ts)
  1539                 choice_spec_table
  1540 
  1541 fun is_real_equational_fun ({thy, stds, simp_table, psimp_table, ...}
  1542                             : hol_context) x =
  1543   exists (fn table => not (null (def_props_for_const thy stds table x)))
  1544          [!simp_table, psimp_table]
  1545 fun is_equational_fun_but_no_plain_def hol_ctxt =
  1546   is_real_equational_fun hol_ctxt orf is_inductive_pred hol_ctxt
  1547 
  1548 (** Constant unfolding **)
  1549 
  1550 fun constr_case_body ctxt stds Ts (func_t, (x as (_, T))) =
  1551   let val arg_Ts = binder_types T in
  1552     s_betapplys Ts (func_t, map2 (select_nth_constr_arg ctxt stds x (Bound 0))
  1553                                  (index_seq 0 (length arg_Ts)) arg_Ts)
  1554   end
  1555 fun add_constr_case res_T (body_t, guard_t) res_t =
  1556   if res_T = bool_T then
  1557     s_conj (HOLogic.mk_imp (guard_t, body_t), res_t)
  1558   else
  1559     Const (@{const_name If}, bool_T --> res_T --> res_T --> res_T)
  1560     $ guard_t $ body_t $ res_t
  1561 fun optimized_case_def (hol_ctxt as {ctxt, stds, ...}) Ts dataT res_T func_ts =
  1562   let
  1563     val xs = datatype_constrs hol_ctxt dataT
  1564     val cases =
  1565       func_ts ~~ xs
  1566       |> map (fn (func_t, x) =>
  1567                  (constr_case_body ctxt stds (dataT :: Ts)
  1568                                    (incr_boundvars 1 func_t, x),
  1569                   discriminate_value hol_ctxt x (Bound 0)))
  1570       |> AList.group (op aconv)
  1571       |> map (apsnd (List.foldl s_disj @{const False}))
  1572       |> sort (int_ord o pairself (size_of_term o snd))
  1573       |> rev
  1574   in
  1575     if res_T = bool_T then
  1576       if forall (member (op =) [@{const False}, @{const True}] o fst) cases then
  1577         case cases of
  1578           [(body_t, _)] => body_t
  1579         | [_, (@{const True}, head_t2)] => head_t2
  1580         | [_, (@{const False}, head_t2)] => @{const Not} $ head_t2
  1581         | _ => raise BAD ("Nitpick_HOL.optimized_case_def", "impossible cases")
  1582       else
  1583         @{const True} |> fold_rev (add_constr_case res_T) cases
  1584     else
  1585       fst (hd cases) |> fold_rev (add_constr_case res_T) (tl cases)
  1586   end
  1587   |> absdummy dataT
  1588 
  1589 fun optimized_record_get (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T res_T t =
  1590   let val constr_x = hd (datatype_constrs hol_ctxt rec_T) in
  1591     case no_of_record_field thy s rec_T of
  1592       ~1 => (case rec_T of
  1593                Type (_, Ts as _ :: _) =>
  1594                let
  1595                  val rec_T' = List.last Ts
  1596                  val j = num_record_fields thy rec_T - 1
  1597                in
  1598                  select_nth_constr_arg ctxt stds constr_x t j res_T
  1599                  |> optimized_record_get hol_ctxt s rec_T' res_T
  1600                end
  1601              | _ => raise TYPE ("Nitpick_HOL.optimized_record_get", [rec_T],
  1602                                 []))
  1603     | j => select_nth_constr_arg ctxt stds constr_x t j res_T
  1604   end
  1605 fun optimized_record_update (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T fun_t
  1606                             rec_t =
  1607   let
  1608     val constr_x as (_, constr_T) = hd (datatype_constrs hol_ctxt rec_T)
  1609     val Ts = binder_types constr_T
  1610     val n = length Ts
  1611     val special_j = no_of_record_field thy s rec_T
  1612     val ts =
  1613       map2 (fn j => fn T =>
  1614                let val t = select_nth_constr_arg ctxt stds constr_x rec_t j T in
  1615                  if j = special_j then
  1616                    s_betapply [] (fun_t, t)
  1617                  else if j = n - 1 andalso special_j = ~1 then
  1618                    optimized_record_update hol_ctxt s
  1619                        (rec_T |> dest_Type |> snd |> List.last) fun_t t
  1620                  else
  1621                    t
  1622                end) (index_seq 0 n) Ts
  1623   in list_comb (Const constr_x, ts) end
  1624 
  1625 (* Prevents divergence in case of cyclic or infinite definition dependencies. *)
  1626 val unfold_max_depth = 255
  1627 
  1628 (* Inline definitions or define as an equational constant? Booleans tend to
  1629    benefit more from inlining, due to the polarity analysis. (However, if
  1630    "total_consts" is set, the polarity analysis is likely not to be so
  1631    crucial.) *)
  1632 val def_inline_threshold_for_booleans = 60
  1633 val def_inline_threshold_for_non_booleans = 20
  1634 
  1635 fun unfold_defs_in_term
  1636         (hol_ctxt as {thy, ctxt, stds, whacks, total_consts, case_names,
  1637                       def_tables, ground_thm_table, ersatz_table, ...}) =
  1638   let
  1639     fun do_term depth Ts t =
  1640       case t of
  1641         (t0 as Const (@{const_name Num.numeral_class.numeral},
  1642                       Type (@{type_name fun}, [_, ran_T]))) $ t1 =>
  1643         ((if is_number_type ctxt ran_T then
  1644             let
  1645               val j = t1 |> HOLogic.dest_num
  1646                          |> ran_T = nat_T ? Integer.max 0
  1647               val s = numeral_prefix ^ signed_string_of_int j
  1648             in
  1649               if is_integer_like_type ran_T then
  1650                 if is_standard_datatype thy stds ran_T then
  1651                   Const (s, ran_T)
  1652                 else
  1653                   funpow j (curry (op $) (suc_const ran_T)) (zero_const ran_T)
  1654               else
  1655                 do_term depth Ts (Const (@{const_name of_int}, int_T --> ran_T)
  1656                                   $ Const (s, int_T))
  1657             end
  1658             handle TERM _ => raise SAME ()
  1659           else
  1660             raise SAME ())
  1661          handle SAME () =>
  1662                 s_betapply [] (do_term depth Ts t0, do_term depth Ts t1))
  1663       | Const (@{const_name refl_on}, T) $ Const (@{const_name top}, _) $ t2 =>
  1664         do_const depth Ts t (@{const_name refl'}, range_type T) [t2]
  1665       | (t0 as Const (@{const_name Sigma}, Type (_, [T1, Type (_, [T2, T3])])))
  1666         $ t1 $ (t2 as Abs (_, _, t2')) =>
  1667         if loose_bvar1 (t2', 0) then
  1668           s_betapplys Ts (do_term depth Ts t0, map (do_term depth Ts) [t1, t2])
  1669         else
  1670           do_term depth Ts
  1671                   (Const (@{const_name prod}, T1 --> range_type T2 --> T3)
  1672                    $ t1 $ incr_boundvars ~1 t2')
  1673       | Const (x as (@{const_name distinct},
  1674                Type (@{type_name fun}, [Type (@{type_name list}, [T']), _])))
  1675         $ (t1 as _ $ _) =>
  1676         (t1 |> HOLogic.dest_list |> distinctness_formula T'
  1677          handle TERM _ => do_const depth Ts t x [t1])
  1678       | Const (x as (@{const_name If}, _)) $ t1 $ t2 $ t3 =>
  1679         if is_ground_term t1 andalso
  1680            exists (Pattern.matches thy o rpair t1)
  1681                   (Inttab.lookup_list ground_thm_table (hash_term t1)) then
  1682           do_term depth Ts t2
  1683         else
  1684           do_const depth Ts t x [t1, t2, t3]
  1685       | Const (@{const_name Let}, _) $ t1 $ t2 =>
  1686         s_betapply Ts (pairself (do_term depth Ts) (t2, t1))
  1687       | Const x => do_const depth Ts t x []
  1688       | t1 $ t2 =>
  1689         (case strip_comb t of
  1690            (Const x, ts) => do_const depth Ts t x ts
  1691          | _ => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2))
  1692       | Bound _ => t
  1693       | Abs (s, T, body) => Abs (s, T, do_term depth (T :: Ts) body)
  1694       | _ => if member (term_match thy) whacks t then
  1695                Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1696              else
  1697                t
  1698     and select_nth_constr_arg_with_args _ _ (x as (_, T)) [] n res_T =
  1699         (Abs (Name.uu, body_type T,
  1700               select_nth_constr_arg ctxt stds x (Bound 0) n res_T), [])
  1701       | select_nth_constr_arg_with_args depth Ts x (t :: ts) n res_T =
  1702         (select_nth_constr_arg ctxt stds x (do_term depth Ts t) n res_T, ts)
  1703     and quot_rep_of depth Ts abs_T rep_T ts =
  1704       select_nth_constr_arg_with_args depth Ts
  1705           (@{const_name Quot}, rep_T --> abs_T) ts 0 rep_T
  1706     and do_const depth Ts t (x as (s, T)) ts =
  1707       if member (term_match thy) whacks (Const x) then
  1708         Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1709       else case AList.lookup (op =) ersatz_table s of
  1710         SOME s' =>
  1711         do_const (depth + 1) Ts (list_comb (Const (s', T), ts)) (s', T) ts
  1712       | NONE =>
  1713         let
  1714           fun def_inline_threshold () =
  1715             if is_boolean_type (nth_range_type (length ts) T) andalso
  1716                total_consts <> SOME true then
  1717               def_inline_threshold_for_booleans
  1718             else
  1719               def_inline_threshold_for_non_booleans
  1720           val (const, ts) =
  1721             if is_built_in_const thy stds x then
  1722               (Const x, ts)
  1723             else case AList.lookup (op =) case_names s of
  1724               SOME n =>
  1725               if length ts < n then
  1726                 (do_term depth Ts (eta_expand Ts t (n - length ts)), [])
  1727               else
  1728                 let
  1729                   val (dataT, res_T) = nth_range_type n T
  1730                                        |> pairf domain_type range_type
  1731                 in
  1732                   (optimized_case_def hol_ctxt Ts dataT res_T
  1733                                       (map (do_term depth Ts) (take n ts)),
  1734                    drop n ts)
  1735                 end
  1736             | _ =>
  1737               if is_constr ctxt stds x then
  1738                 (Const x, ts)
  1739               else if is_stale_constr ctxt x then
  1740                 raise NOT_SUPPORTED ("(non-co)constructors of codatatypes \
  1741                                      \(\"" ^ s ^ "\")")
  1742               else if is_quot_abs_fun ctxt x then
  1743                 let
  1744                   val rep_T = domain_type T
  1745                   val abs_T = range_type T
  1746                 in
  1747                   (Abs (Name.uu, rep_T,
  1748                         Const (@{const_name Quot}, rep_T --> abs_T)
  1749                                $ (Const (quot_normal_name_for_type ctxt abs_T,
  1750                                          rep_T --> rep_T) $ Bound 0)), ts)
  1751                 end
  1752               else if is_quot_rep_fun ctxt x then
  1753                 quot_rep_of depth Ts (domain_type T) (range_type T) ts
  1754               else if is_record_get thy x then
  1755                 case length ts of
  1756                   0 => (do_term depth Ts (eta_expand Ts t 1), [])
  1757                 | _ => (optimized_record_get hol_ctxt s (domain_type T)
  1758                             (range_type T) (do_term depth Ts (hd ts)), tl ts)
  1759               else if is_record_update thy x then
  1760                 case length ts of
  1761                   2 => (optimized_record_update hol_ctxt
  1762                             (unsuffix Record.updateN s) (nth_range_type 2 T)
  1763                             (do_term depth Ts (hd ts))
  1764                             (do_term depth Ts (nth ts 1)), [])
  1765                 | n => (do_term depth Ts (eta_expand Ts t (2 - n)), [])
  1766               else if is_abs_fun ctxt x andalso
  1767                       is_quot_type ctxt (range_type T) then
  1768                 let
  1769                   val abs_T = range_type T
  1770                   val rep_T = elem_type (domain_type T)
  1771                   val eps_fun = Const (@{const_name Eps},
  1772                                        (rep_T --> bool_T) --> rep_T)
  1773                   val normal_fun =
  1774                     Const (quot_normal_name_for_type ctxt abs_T,
  1775                            rep_T --> rep_T)
  1776                   val abs_fun = Const (@{const_name Quot}, rep_T --> abs_T)
  1777                   val pred =
  1778                     Abs (Name.uu, rep_T,
  1779                          Const (@{const_name Set.member},
  1780                                 rep_T --> domain_type T --> bool_T)
  1781                          $ Bound 0 $ Bound 1)
  1782                 in
  1783                   (Abs (Name.uu, HOLogic.mk_setT rep_T,
  1784                         abs_fun $ (normal_fun $ (eps_fun $ pred)))
  1785                    |> do_term (depth + 1) Ts, ts)
  1786                 end
  1787               else if is_rep_fun ctxt x then
  1788                 let val x' = mate_of_rep_fun ctxt x in
  1789                   if is_constr ctxt stds x' then
  1790                     select_nth_constr_arg_with_args depth Ts x' ts 0
  1791                                                     (range_type T)
  1792                   else if is_quot_type ctxt (domain_type T) then
  1793                     let
  1794                       val abs_T = domain_type T
  1795                       val rep_T = elem_type (range_type T)
  1796                       val (rep_fun, _) = quot_rep_of depth Ts abs_T rep_T []
  1797                       val (equiv_rel, _) =
  1798                         equiv_relation_for_quot_type ctxt abs_T
  1799                     in
  1800                       (Abs (Name.uu, abs_T,
  1801                             HOLogic.Collect_const rep_T
  1802                             $ (equiv_rel $ (rep_fun $ Bound 0))),
  1803                        ts)
  1804                     end
  1805                   else
  1806                     (Const x, ts)
  1807                 end
  1808               else if is_equational_fun_but_no_plain_def hol_ctxt x orelse
  1809                       is_choice_spec_fun hol_ctxt x then
  1810                 (Const x, ts)
  1811               else case def_of_const_ext thy def_tables x of
  1812                 SOME (unfold, def) =>
  1813                 if depth > unfold_max_depth then
  1814                   raise TOO_LARGE ("Nitpick_HOL.unfold_defs_in_term",
  1815                                    "too many nested definitions (" ^
  1816                                    string_of_int depth ^ ") while expanding " ^
  1817                                    quote s)
  1818                 else if s = "Wfrec.wfrec'" (* FIXME unchecked! *) then
  1819                   (do_term (depth + 1) Ts (s_betapplys Ts (def, ts)), [])
  1820                 else if not unfold andalso
  1821                      size_of_term def > def_inline_threshold () then
  1822                   (Const x, ts)
  1823                 else
  1824                   (do_term (depth + 1) Ts def, ts)
  1825               | NONE => (Const x, ts)
  1826         in
  1827           s_betapplys Ts (const, map (do_term depth Ts) ts)
  1828           |> s_beta_norm Ts
  1829         end
  1830   in do_term 0 [] end
  1831 
  1832 (** Axiom extraction/generation **)
  1833 
  1834 fun extensional_equal j T t1 t2 =
  1835   if is_fun_type T then
  1836     let
  1837       val dom_T = pseudo_domain_type T
  1838       val ran_T = pseudo_range_type T
  1839       val var_t = Var (("x", j), dom_T)
  1840     in
  1841       extensional_equal (j + 1) ran_T (betapply (t1, var_t))
  1842                         (betapply (t2, var_t))
  1843     end
  1844   else
  1845     Const (@{const_name HOL.eq}, T --> T --> bool_T) $ t1 $ t2
  1846 
  1847 (* FIXME: needed? *)
  1848 fun equationalize_term ctxt tag t =
  1849   let
  1850     val j = maxidx_of_term t + 1
  1851     val (prems, concl) = Logic.strip_horn t
  1852   in
  1853     Logic.list_implies (prems,
  1854         case concl of
  1855           @{const Trueprop} $ (Const (@{const_name HOL.eq}, Type (_, [T, _]))
  1856                                $ t1 $ t2) =>
  1857           @{const Trueprop} $ extensional_equal j T t1 t2
  1858         | @{const Trueprop} $ t' =>
  1859           @{const Trueprop} $ HOLogic.mk_eq (t', @{const True})
  1860         | Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
  1861           @{const Trueprop} $ extensional_equal j T t1 t2
  1862         | _ => (warning ("Ignoring " ^ quote tag ^ " for non-equation " ^
  1863                          quote (Syntax.string_of_term ctxt t) ^ ".");
  1864                 raise SAME ()))
  1865     |> SOME
  1866   end
  1867   handle SAME () => NONE
  1868 
  1869 fun pair_for_prop t =
  1870   case term_under_def t of
  1871     Const (s, _) => (s, t)
  1872   | t' => raise TERM ("Nitpick_HOL.pair_for_prop", [t, t'])
  1873 
  1874 fun def_table_for get ctxt subst =
  1875   ctxt |> get |> map (pair_for_prop o subst_atomic subst)
  1876        |> AList.group (op =) |> Symtab.make
  1877 
  1878 fun const_def_tables ctxt subst ts =
  1879   (def_table_for (map prop_of o Nitpick_Unfolds.get) ctxt subst,
  1880    fold (fn (s, t) => Symtab.map_default (s, []) (cons t))
  1881         (map pair_for_prop ts) Symtab.empty)
  1882 
  1883 fun paired_with_consts t = map (rpair t) (Term.add_const_names t [])
  1884 fun const_nondef_table ts =
  1885   fold (append o paired_with_consts) ts [] |> AList.group (op =) |> Symtab.make
  1886 
  1887 fun const_simp_table ctxt =
  1888   def_table_for (map_filter (equationalize_term ctxt "nitpick_simp" o prop_of)
  1889                  o Nitpick_Simps.get) ctxt
  1890 fun const_psimp_table ctxt =
  1891   def_table_for (map_filter (equationalize_term ctxt "nitpick_psimp" o prop_of)
  1892                  o Nitpick_Psimps.get) ctxt
  1893 
  1894 fun const_choice_spec_table ctxt subst =
  1895   map (subst_atomic subst o prop_of) (Nitpick_Choice_Specs.get ctxt)
  1896   |> const_nondef_table
  1897 
  1898 fun inductive_intro_table ctxt subst def_tables =
  1899   let val thy = Proof_Context.theory_of ctxt in
  1900     def_table_for
  1901         (maps (map (unfold_mutually_inductive_preds thy def_tables o prop_of)
  1902                o snd o snd)
  1903          o filter (fn (cat, _) => cat = Spec_Rules.Inductive orelse
  1904                                   cat = Spec_Rules.Co_Inductive)
  1905          o Spec_Rules.get) ctxt subst
  1906   end
  1907 
  1908 fun ground_theorem_table thy =
  1909   fold ((fn @{const Trueprop} $ t1 =>
  1910             is_ground_term t1 ? Inttab.map_default (hash_term t1, []) (cons t1)
  1911           | _ => I) o prop_of o snd) (Global_Theory.all_thms_of thy) Inttab.empty
  1912 
  1913 fun ersatz_table ctxt =
  1914  #ersatz_table (Data.get (Context.Proof ctxt))
  1915  |> fold (append o snd) (#frac_types (Data.get (Context.Proof ctxt)))
  1916 
  1917 fun add_simps simp_table s eqs =
  1918   Unsynchronized.change simp_table
  1919       (Symtab.update (s, eqs @ these (Symtab.lookup (!simp_table) s)))
  1920 
  1921 fun inverse_axioms_for_rep_fun ctxt (x as (_, T)) =
  1922   let
  1923     val thy = Proof_Context.theory_of ctxt
  1924     val abs_T = domain_type T
  1925   in
  1926     typedef_info ctxt (fst (dest_Type abs_T)) |> the
  1927     |> pairf #Abs_inverse #Rep_inverse
  1928     |> pairself (specialize_type thy x o prop_of o the)
  1929     ||> single |> op ::
  1930   end
  1931 fun optimized_typedef_axioms ctxt (abs_z as (abs_s, _)) =
  1932   let
  1933     val thy = Proof_Context.theory_of ctxt
  1934     val abs_T = Type abs_z
  1935   in
  1936     if is_univ_typedef ctxt abs_T then
  1937       []
  1938     else case typedef_info ctxt abs_s of
  1939       SOME {abs_type, rep_type, Rep_name, prop_of_Rep, set_name, ...} =>
  1940       let
  1941         val rep_T = varify_and_instantiate_type ctxt abs_type abs_T rep_type
  1942         val rep_t = Const (Rep_name, abs_T --> rep_T)
  1943         val set_t = Const (set_name, HOLogic.mk_setT rep_T)
  1944         val set_t' =
  1945           prop_of_Rep |> HOLogic.dest_Trueprop
  1946                       |> specialize_type thy (dest_Const rep_t)
  1947                       |> HOLogic.dest_mem |> snd
  1948       in
  1949         [HOLogic.all_const abs_T
  1950          $ Abs (Name.uu, abs_T, HOLogic.mk_mem (rep_t $ Bound 0, set_t))]
  1951         |> set_t <> set_t' ? cons (HOLogic.mk_eq (set_t, set_t'))
  1952         |> map HOLogic.mk_Trueprop
  1953       end
  1954     | NONE => []
  1955   end
  1956 fun optimized_quot_type_axioms ctxt stds abs_z =
  1957   let
  1958     val abs_T = Type abs_z
  1959     val rep_T = rep_type_for_quot_type ctxt abs_T
  1960     val (equiv_rel, partial) = equiv_relation_for_quot_type ctxt abs_T
  1961     val a_var = Var (("a", 0), abs_T)
  1962     val x_var = Var (("x", 0), rep_T)
  1963     val y_var = Var (("y", 0), rep_T)
  1964     val x = (@{const_name Quot}, rep_T --> abs_T)
  1965     val sel_a_t = select_nth_constr_arg ctxt stds x a_var 0 rep_T
  1966     val normal_fun =
  1967       Const (quot_normal_name_for_type ctxt abs_T, rep_T --> rep_T)
  1968     val normal_x = normal_fun $ x_var
  1969     val normal_y = normal_fun $ y_var
  1970     val is_unknown_t = Const (@{const_name is_unknown}, rep_T --> bool_T)
  1971   in
  1972     [Logic.mk_equals (normal_fun $ sel_a_t, sel_a_t),
  1973      Logic.list_implies
  1974          ([@{const Not} $ (is_unknown_t $ normal_x),
  1975            @{const Not} $ (is_unknown_t $ normal_y),
  1976            equiv_rel $ x_var $ y_var] |> map HOLogic.mk_Trueprop,
  1977            Logic.mk_equals (normal_x, normal_y)),
  1978      Logic.list_implies
  1979          ([HOLogic.mk_Trueprop (@{const Not} $ (is_unknown_t $ normal_x)),
  1980            HOLogic.mk_Trueprop (@{const Not} $ HOLogic.mk_eq (normal_x, x_var))],
  1981           HOLogic.mk_Trueprop (equiv_rel $ x_var $ normal_x))]
  1982     |> partial ? cons (HOLogic.mk_Trueprop (equiv_rel $ sel_a_t $ sel_a_t))
  1983   end
  1984 
  1985 fun codatatype_bisim_axioms (hol_ctxt as {ctxt, stds, ...}) T =
  1986   let
  1987     val xs = datatype_constrs hol_ctxt T
  1988     val pred_T = T --> bool_T
  1989     val iter_T = @{typ bisim_iterator}
  1990     val bisim_max = @{const bisim_iterator_max}
  1991     val n_var = Var (("n", 0), iter_T)
  1992     val n_var_minus_1 =
  1993       Const (@{const_name safe_The}, (iter_T --> bool_T) --> iter_T)
  1994       $ Abs ("m", iter_T, HOLogic.eq_const iter_T
  1995                           $ (suc_const iter_T $ Bound 0) $ n_var)
  1996     val x_var = Var (("x", 0), T)
  1997     val y_var = Var (("y", 0), T)
  1998     fun bisim_const T = Const (@{const_name bisim}, [iter_T, T, T] ---> bool_T)
  1999     fun nth_sub_bisim x n nth_T =
  2000       (if is_codatatype ctxt nth_T then bisim_const nth_T $ n_var_minus_1
  2001        else HOLogic.eq_const nth_T)
  2002       $ select_nth_constr_arg ctxt stds x x_var n nth_T
  2003       $ select_nth_constr_arg ctxt stds x y_var n nth_T
  2004     fun case_func (x as (_, T)) =
  2005       let
  2006         val arg_Ts = binder_types T
  2007         val core_t =
  2008           discriminate_value hol_ctxt x y_var ::
  2009           map2 (nth_sub_bisim x) (index_seq 0 (length arg_Ts)) arg_Ts
  2010           |> foldr1 s_conj
  2011       in fold_rev absdummy arg_Ts core_t end
  2012   in
  2013     [HOLogic.mk_imp
  2014        (HOLogic.mk_disj (HOLogic.eq_const iter_T $ n_var $ zero_const iter_T,
  2015             s_betapply [] (optimized_case_def hol_ctxt [] T bool_T
  2016                                               (map case_func xs), x_var)),
  2017         bisim_const T $ n_var $ x_var $ y_var),
  2018      HOLogic.eq_const pred_T $ (bisim_const T $ bisim_max $ x_var)
  2019      $ Abs (Name.uu, T, HOLogic.mk_eq (x_var, Bound 0))]
  2020     |> map HOLogic.mk_Trueprop
  2021   end
  2022 
  2023 exception NO_TRIPLE of unit
  2024 
  2025 fun triple_for_intro_rule thy x t =
  2026   let
  2027     val prems = Logic.strip_imp_prems t |> map (Object_Logic.atomize_term thy)
  2028     val concl = Logic.strip_imp_concl t |> Object_Logic.atomize_term thy
  2029     val (main, side) = List.partition (exists_Const (curry (op =) x)) prems
  2030     val is_good_head = curry (op =) (Const x) o head_of
  2031   in
  2032     if forall is_good_head main then (side, main, concl) else raise NO_TRIPLE ()
  2033   end
  2034 
  2035 val tuple_for_args = HOLogic.mk_tuple o snd o strip_comb
  2036 fun wf_constraint_for rel side concl main =
  2037   let
  2038     val core = HOLogic.mk_mem (HOLogic.mk_prod
  2039                                (pairself tuple_for_args (main, concl)), Var rel)
  2040     val t = List.foldl HOLogic.mk_imp core side
  2041     val vars = filter_out (curry (op =) rel) (Term.add_vars t [])
  2042   in
  2043     Library.foldl (fn (t', ((x, j), T)) =>
  2044                       HOLogic.all_const T
  2045                       $ Abs (x, T, abstract_over (Var ((x, j), T), t')))
  2046                   (t, vars)
  2047   end
  2048 fun wf_constraint_for_triple rel (side, main, concl) =
  2049   map (wf_constraint_for rel side concl) main |> foldr1 s_conj
  2050 
  2051 fun terminates_by ctxt timeout goal tac =
  2052   can (SINGLE (Classical.safe_tac ctxt) #> the
  2053        #> SINGLE (DETERM_TIMEOUT timeout (tac ctxt (auto_tac ctxt)))
  2054        #> the #> Goal.finish ctxt) goal
  2055 
  2056 val max_cached_wfs = 50
  2057 val cached_timeout =
  2058   Synchronized.var "Nitpick_HOL.cached_timeout" (SOME Time.zeroTime)
  2059 val cached_wf_props =
  2060   Synchronized.var "Nitpick_HOL.cached_wf_props" ([] : (term * bool) list)
  2061 
  2062 val termination_tacs = [Lexicographic_Order.lex_order_tac true,
  2063                         ScnpReconstruct.sizechange_tac]
  2064 
  2065 fun uncached_is_well_founded_inductive_pred
  2066         ({thy, ctxt, stds, debug, tac_timeout, intro_table, ...} : hol_context)
  2067         (x as (_, T)) =
  2068   case def_props_for_const thy stds intro_table x of
  2069     [] => raise TERM ("Nitpick_HOL.uncached_is_well_founded_inductive",
  2070                       [Const x])
  2071   | intro_ts =>
  2072     (case map (triple_for_intro_rule thy x) intro_ts
  2073           |> filter_out (null o #2) of
  2074        [] => true
  2075      | triples =>
  2076        let
  2077          val binders_T = HOLogic.mk_tupleT (binder_types T)
  2078          val rel_T = HOLogic.mk_setT (HOLogic.mk_prodT (binders_T, binders_T))
  2079          val j = fold Integer.max (map maxidx_of_term intro_ts) 0 + 1
  2080          val rel = (("R", j), rel_T)
  2081          val prop = Const (@{const_name wf}, rel_T --> bool_T) $ Var rel ::
  2082                     map (wf_constraint_for_triple rel) triples
  2083                     |> foldr1 s_conj |> HOLogic.mk_Trueprop
  2084          val _ = if debug then
  2085                    Output.urgent_message ("Wellfoundedness goal: " ^
  2086                              Syntax.string_of_term ctxt prop ^ ".")
  2087                  else
  2088                    ()
  2089        in
  2090          if tac_timeout = Synchronized.value cached_timeout andalso
  2091             length (Synchronized.value cached_wf_props) < max_cached_wfs then
  2092            ()
  2093          else
  2094            (Synchronized.change cached_wf_props (K []);
  2095             Synchronized.change cached_timeout (K tac_timeout));
  2096          case AList.lookup (op =) (Synchronized.value cached_wf_props) prop of
  2097            SOME wf => wf
  2098          | NONE =>
  2099            let
  2100              val goal = prop |> cterm_of thy |> Goal.init
  2101              val wf = exists (terminates_by ctxt tac_timeout goal)
  2102                              termination_tacs
  2103            in Synchronized.change cached_wf_props (cons (prop, wf)); wf end
  2104        end)
  2105     handle List.Empty => false | NO_TRIPLE () => false
  2106 
  2107 (* The type constraint below is a workaround for a Poly/ML crash. *)
  2108 
  2109 fun is_well_founded_inductive_pred
  2110         (hol_ctxt as {thy, wfs, def_tables, wf_cache, ...} : hol_context)
  2111         (x as (s, _)) =
  2112   case triple_lookup (const_match thy) wfs x of
  2113     SOME (SOME b) => b
  2114   | _ => s = @{const_name Nats} orelse s = @{const_name fold_graph'} orelse
  2115          case AList.lookup (op =) (!wf_cache) x of
  2116            SOME (_, wf) => wf
  2117          | NONE =>
  2118            let
  2119              val gfp = (fixpoint_kind_of_const thy def_tables x = Gfp)
  2120              val wf = uncached_is_well_founded_inductive_pred hol_ctxt x
  2121            in
  2122              Unsynchronized.change wf_cache (cons (x, (gfp, wf))); wf
  2123            end
  2124 
  2125 fun ap_curry [_] _ t = t
  2126   | ap_curry arg_Ts tuple_T t =
  2127     let val n = length arg_Ts in
  2128       fold_rev (Term.abs o pair "c") arg_Ts
  2129                 (incr_boundvars n t $ mk_flat_tuple tuple_T (map Bound (n - 1 downto 0)))
  2130     end
  2131 
  2132 fun num_occs_of_bound_in_term j (t1 $ t2) =
  2133     op + (pairself (num_occs_of_bound_in_term j) (t1, t2))
  2134   | num_occs_of_bound_in_term j (Abs (_, _, t')) =
  2135     num_occs_of_bound_in_term (j + 1) t'
  2136   | num_occs_of_bound_in_term j (Bound j') = if j' = j then 1 else 0
  2137   | num_occs_of_bound_in_term _ _ = 0
  2138 
  2139 val is_linear_inductive_pred_def =
  2140   let
  2141     fun do_disjunct j (Const (@{const_name Ex}, _) $ Abs (_, _, t2)) =
  2142         do_disjunct (j + 1) t2
  2143       | do_disjunct j t =
  2144         case num_occs_of_bound_in_term j t of
  2145           0 => true
  2146         | 1 => exists (curry (op =) (Bound j) o head_of) (conjuncts_of t)
  2147         | _ => false
  2148     fun do_lfp_def (Const (@{const_name lfp}, _) $ t2) =
  2149         let val (xs, body) = strip_abs t2 in
  2150           case length xs of
  2151             1 => false
  2152           | n => forall (do_disjunct (n - 1)) (disjuncts_of body)
  2153         end
  2154       | do_lfp_def _ = false
  2155   in do_lfp_def o strip_abs_body end
  2156 
  2157 fun n_ptuple_paths 0 = []
  2158   | n_ptuple_paths 1 = []
  2159   | n_ptuple_paths n = [] :: map (cons 2) (n_ptuple_paths (n - 1))
  2160 val ap_n_split = HOLogic.mk_psplits o n_ptuple_paths
  2161 
  2162 val linear_pred_base_and_step_rhss =
  2163   let
  2164     fun aux (Const (@{const_name lfp}, _) $ t2) =
  2165         let
  2166           val (xs, body) = strip_abs t2
  2167           val arg_Ts = map snd (tl xs)
  2168           val tuple_T = HOLogic.mk_tupleT arg_Ts
  2169           val j = length arg_Ts
  2170           fun repair_rec j (Const (@{const_name Ex}, T1) $ Abs (s2, T2, t2')) =
  2171               Const (@{const_name Ex}, T1)
  2172               $ Abs (s2, T2, repair_rec (j + 1) t2')
  2173             | repair_rec j (@{const HOL.conj} $ t1 $ t2) =
  2174               @{const HOL.conj} $ repair_rec j t1 $ repair_rec j t2
  2175             | repair_rec j t =
  2176               let val (head, args) = strip_comb t in
  2177                 if head = Bound j then
  2178                   HOLogic.eq_const tuple_T $ Bound j
  2179                   $ mk_flat_tuple tuple_T args
  2180                 else
  2181                   t
  2182               end
  2183           val (nonrecs, recs) =
  2184             List.partition (curry (op =) 0 o num_occs_of_bound_in_term j)
  2185                            (disjuncts_of body)
  2186           val base_body = nonrecs |> List.foldl s_disj @{const False}
  2187           val step_body = recs |> map (repair_rec j)
  2188                                |> List.foldl s_disj @{const False}
  2189         in
  2190           (fold_rev Term.abs (tl xs) (incr_bv (~1, j, base_body))
  2191            |> ap_n_split (length arg_Ts) tuple_T bool_T,
  2192            Abs ("y", tuple_T, fold_rev Term.abs (tl xs) step_body
  2193                               |> ap_n_split (length arg_Ts) tuple_T bool_T))
  2194         end
  2195       | aux t =
  2196         raise TERM ("Nitpick_HOL.linear_pred_base_and_step_rhss.aux", [t])
  2197   in aux end
  2198 
  2199 fun predicatify T t =
  2200   let val set_T = HOLogic.mk_setT T in
  2201     Abs (Name.uu, T,
  2202          Const (@{const_name Set.member}, T --> set_T --> bool_T)
  2203          $ Bound 0 $ incr_boundvars 1 t)
  2204   end
  2205 
  2206 fun starred_linear_pred_const (hol_ctxt as {simp_table, ...}) (s, T) def =
  2207   let
  2208     val j = maxidx_of_term def + 1
  2209     val (outer, fp_app) = strip_abs def
  2210     val outer_bounds = map Bound (length outer - 1 downto 0)
  2211     val outer_vars = map (fn (s, T) => Var ((s, j), T)) outer
  2212     val fp_app = subst_bounds (rev outer_vars, fp_app)
  2213     val (outer_Ts, rest_T) = strip_n_binders (length outer) T
  2214     val tuple_arg_Ts = strip_type rest_T |> fst
  2215     val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
  2216     val prod_T = HOLogic.mk_prodT (tuple_T, tuple_T)
  2217     val set_T = HOLogic.mk_setT tuple_T
  2218     val rel_T = HOLogic.mk_setT prod_T
  2219     val pred_T = tuple_T --> bool_T
  2220     val curried_T = tuple_T --> pred_T
  2221     val uncurried_T = prod_T --> bool_T
  2222     val (base_rhs, step_rhs) = linear_pred_base_and_step_rhss fp_app
  2223     val base_x as (base_s, _) = (base_prefix ^ s, outer_Ts ---> pred_T)
  2224     val base_eq = HOLogic.mk_eq (list_comb (Const base_x, outer_vars), base_rhs)
  2225                   |> HOLogic.mk_Trueprop
  2226     val _ = add_simps simp_table base_s [base_eq]
  2227     val step_x as (step_s, _) = (step_prefix ^ s, outer_Ts ---> curried_T)
  2228     val step_eq = HOLogic.mk_eq (list_comb (Const step_x, outer_vars), step_rhs)
  2229                   |> HOLogic.mk_Trueprop
  2230     val _ = add_simps simp_table step_s [step_eq]
  2231     val image_const = Const (@{const_name Image}, rel_T --> set_T --> set_T)
  2232     val rtrancl_const = Const (@{const_name rtrancl}, rel_T --> rel_T)
  2233     val base_set =
  2234       HOLogic.Collect_const tuple_T $ list_comb (Const base_x, outer_bounds)
  2235     val step_set =
  2236       HOLogic.Collect_const prod_T
  2237       $ (Const (@{const_name prod_case}, curried_T --> uncurried_T)
  2238                 $ list_comb (Const step_x, outer_bounds))
  2239     val image_set =
  2240       image_const $ (rtrancl_const $ step_set) $ base_set
  2241       |> predicatify tuple_T
  2242   in
  2243     fold_rev Term.abs outer (image_set |> ap_curry tuple_arg_Ts tuple_T)
  2244     |> unfold_defs_in_term hol_ctxt
  2245   end
  2246 
  2247 fun is_good_starred_linear_pred_type (Type (@{type_name fun}, Ts)) =
  2248     forall (not o (is_fun_or_set_type orf is_pair_type)) Ts
  2249   | is_good_starred_linear_pred_type _ = false
  2250 
  2251 fun unrolled_inductive_pred_const (hol_ctxt as {thy, star_linear_preds,
  2252                                                 def_tables, simp_table, ...})
  2253                                   gfp (x as (s, T)) =
  2254   let
  2255     val iter_T = iterator_type_for_const gfp x
  2256     val x' as (s', _) = (unrolled_prefix ^ s, iter_T --> T)
  2257     val unrolled_const = Const x' $ zero_const iter_T
  2258     val def = the (def_of_const thy def_tables x)
  2259   in
  2260     if is_equational_fun_but_no_plain_def hol_ctxt x' then
  2261       unrolled_const (* already done *)
  2262     else if not gfp andalso star_linear_preds andalso
  2263          is_linear_inductive_pred_def def andalso
  2264          is_good_starred_linear_pred_type T then
  2265       starred_linear_pred_const hol_ctxt x def
  2266     else
  2267       let
  2268         val j = maxidx_of_term def + 1
  2269         val (outer, fp_app) = strip_abs def
  2270         val outer_bounds = map Bound (length outer - 1 downto 0)
  2271         val cur = Var ((iter_var_prefix, j + 1), iter_T)
  2272         val next = suc_const iter_T $ cur
  2273         val rhs =
  2274           case fp_app of
  2275             Const _ $ t =>
  2276             s_betapply [] (t, list_comb (Const x', next :: outer_bounds))
  2277           | _ => raise TERM ("Nitpick_HOL.unrolled_inductive_pred_const",
  2278                              [fp_app])
  2279         val (inner, naked_rhs) = strip_abs rhs
  2280         val all = outer @ inner
  2281         val bounds = map Bound (length all - 1 downto 0)
  2282         val vars = map (fn (s, T) => Var ((s, j), T)) all
  2283         val eq = HOLogic.mk_eq (list_comb (Const x', cur :: bounds), naked_rhs)
  2284                  |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2285         val _ = add_simps simp_table s' [eq]
  2286       in unrolled_const end
  2287   end
  2288 
  2289 fun raw_inductive_pred_axiom ({thy, def_tables, ...} : hol_context) x =
  2290   let
  2291     val def = the (def_of_const thy def_tables x)
  2292     val (outer, fp_app) = strip_abs def
  2293     val outer_bounds = map Bound (length outer - 1 downto 0)
  2294     val rhs =
  2295       case fp_app of
  2296         Const _ $ t => s_betapply [] (t, list_comb (Const x, outer_bounds))
  2297       | _ => raise TERM ("Nitpick_HOL.raw_inductive_pred_axiom", [fp_app])
  2298     val (inner, naked_rhs) = strip_abs rhs
  2299     val all = outer @ inner
  2300     val bounds = map Bound (length all - 1 downto 0)
  2301     val j = maxidx_of_term def + 1
  2302     val vars = map (fn (s, T) => Var ((s, j), T)) all
  2303   in
  2304     HOLogic.mk_eq (list_comb (Const x, bounds), naked_rhs)
  2305     |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2306   end
  2307 fun inductive_pred_axiom hol_ctxt (x as (s, T)) =
  2308   if String.isPrefix ubfp_prefix s orelse String.isPrefix lbfp_prefix s then
  2309     let val x' = (strip_first_name_sep s |> snd, T) in
  2310       raw_inductive_pred_axiom hol_ctxt x' |> subst_atomic [(Const x', Const x)]
  2311     end
  2312   else
  2313     raw_inductive_pred_axiom hol_ctxt x
  2314 
  2315 fun equational_fun_axioms (hol_ctxt as {thy, ctxt, stds, def_tables, simp_table,
  2316                                         psimp_table, ...}) x =
  2317   case def_props_for_const thy stds (!simp_table) x of
  2318     [] => (case def_props_for_const thy stds psimp_table x of
  2319              [] => (if is_inductive_pred hol_ctxt x then
  2320                       [inductive_pred_axiom hol_ctxt x]
  2321                     else case def_of_const thy def_tables x of
  2322                       SOME def =>
  2323                       @{const Trueprop} $ HOLogic.mk_eq (Const x, def)
  2324                       |> equationalize_term ctxt "" |> the |> single
  2325                     | NONE => [])
  2326            | psimps => psimps)
  2327   | simps => simps
  2328 fun is_equational_fun_surely_complete hol_ctxt x =
  2329   case equational_fun_axioms hol_ctxt x of
  2330     [@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)] =>
  2331     strip_comb t1 |> snd |> forall is_Var
  2332   | _ => false
  2333 
  2334 (** Type preprocessing **)
  2335 
  2336 fun merged_type_var_table_for_terms thy ts =
  2337   let
  2338     fun add (s, S) table =
  2339       table
  2340       |> (case AList.lookup (Sign.subsort thy o swap) table S of
  2341             SOME _ => I
  2342           | NONE =>
  2343             filter_out (fn (S', _) => Sign.subsort thy (S, S'))
  2344             #> cons (S, s))
  2345     val tfrees = [] |> fold Term.add_tfrees ts
  2346                     |> sort (string_ord o pairself fst)
  2347   in [] |> fold add tfrees |> rev end
  2348 
  2349 fun merge_type_vars_in_term thy merge_type_vars table =
  2350   merge_type_vars
  2351   ? map_types (map_atyps
  2352         (fn TFree (_, S) =>
  2353             TFree (table |> find_first (fn (S', _) => Sign.subsort thy (S', S))
  2354                          |> the |> swap)
  2355           | T => T))
  2356 
  2357 fun add_ground_types hol_ctxt binarize =
  2358   let
  2359     fun aux T accum =
  2360       case T of
  2361         Type (@{type_name fun}, Ts) => fold aux Ts accum
  2362       | Type (@{type_name prod}, Ts) => fold aux Ts accum
  2363       | Type (@{type_name set}, Ts) => fold aux Ts accum
  2364       | Type (@{type_name itself}, [T1]) => aux T1 accum
  2365       | Type (_, Ts) =>
  2366         if member (op =) (@{typ prop} :: @{typ bool} :: accum) T then
  2367           accum
  2368         else
  2369           T :: accum
  2370           |> fold aux (case binarized_and_boxed_datatype_constrs hol_ctxt
  2371                                                                  binarize T of
  2372                          [] => Ts
  2373                        | xs => map snd xs)
  2374       | _ => insert (op =) T accum
  2375   in aux end
  2376 fun ground_types_in_type hol_ctxt binarize T =
  2377   add_ground_types hol_ctxt binarize T []
  2378 fun ground_types_in_terms hol_ctxt binarize ts =
  2379   fold (fold_types (add_ground_types hol_ctxt binarize)) ts []
  2380 
  2381 end;