src/HOL/Tools/Nitpick/nitpick_hol.ML
author blanchet
Thu Jan 16 16:20:17 2014 +0100 (2014-01-16)
changeset 55017 2df6ad1dbd66
parent 54816 10d48c2a3e32
child 55080 b7c41accbff2
permissions -rw-r--r--
adapted to move of Wfrec
     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,
    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_like_injective : Proof.context -> styp -> bool
   134   val is_constr : Proof.context -> (typ option * bool) list -> styp -> bool
   135   val is_sel : string -> bool
   136   val is_sel_like_and_no_discr : string -> bool
   137   val box_type : hol_context -> boxability -> typ -> typ
   138   val binarize_nat_and_int_in_type : typ -> typ
   139   val binarize_nat_and_int_in_term : term -> term
   140   val discr_for_constr : styp -> styp
   141   val num_sels_for_constr_type : typ -> int
   142   val nth_sel_name_for_constr_name : string -> int -> string
   143   val nth_sel_for_constr : styp -> int -> styp
   144   val binarized_and_boxed_nth_sel_for_constr :
   145     hol_context -> bool -> styp -> int -> styp
   146   val sel_no_from_name : string -> int
   147   val close_form : term -> term
   148   val distinctness_formula : typ -> term list -> term
   149   val register_frac_type :
   150     string -> (string * string) list -> morphism -> Context.generic
   151     -> Context.generic
   152   val register_frac_type_global :
   153     string -> (string * string) list -> theory -> theory
   154   val unregister_frac_type :
   155     string -> morphism -> Context.generic -> Context.generic
   156   val unregister_frac_type_global : string -> theory -> theory
   157   val register_ersatz :
   158     (string * string) list -> morphism -> Context.generic -> Context.generic
   159   val register_ersatz_global : (string * string) list -> theory -> theory
   160   val register_codatatype :
   161     typ -> string -> styp list -> morphism -> Context.generic -> Context.generic
   162   val register_codatatype_global :
   163     typ -> string -> styp list -> theory -> theory
   164   val unregister_codatatype :
   165     typ -> morphism -> Context.generic -> Context.generic
   166   val unregister_codatatype_global : typ -> theory -> theory
   167   val datatype_constrs : hol_context -> typ -> styp list
   168   val binarized_and_boxed_datatype_constrs :
   169     hol_context -> bool -> typ -> styp list
   170   val num_datatype_constrs : hol_context -> typ -> int
   171   val constr_name_for_sel_like : string -> string
   172   val binarized_and_boxed_constr_for_sel : hol_context -> bool -> styp -> styp
   173   val card_of_type : (typ * int) list -> typ -> int
   174   val bounded_card_of_type : int -> int -> (typ * int) list -> typ -> int
   175   val bounded_exact_card_of_type :
   176     hol_context -> typ list -> int -> int -> (typ * int) list -> typ -> int
   177   val typical_card_of_type : typ -> int
   178   val is_finite_type : hol_context -> typ -> bool
   179   val is_special_eligible_arg : bool -> typ list -> term -> bool
   180   val s_let :
   181     typ list -> string -> int -> typ -> typ -> (term -> term) -> term -> term
   182   val s_betapply : typ list -> term * term -> term
   183   val s_betapplys : typ list -> term * term list -> term
   184   val discriminate_value : hol_context -> styp -> term -> term
   185   val select_nth_constr_arg :
   186     Proof.context -> (typ option * bool) list -> styp -> term -> int -> typ
   187     -> term
   188   val construct_value :
   189     Proof.context -> (typ option * bool) list -> styp -> term list -> term
   190   val coerce_term : hol_context -> typ list -> typ -> typ -> term -> term
   191   val special_bounds : term list -> (indexname * typ) list
   192   val is_funky_typedef : Proof.context -> typ -> bool
   193   val all_defs_of : theory -> (term * term) list -> term list
   194   val all_nondefs_of : Proof.context -> (term * term) list -> term list
   195   val arity_of_built_in_const :
   196     theory -> (typ option * bool) list -> styp -> int option
   197   val is_built_in_const :
   198     theory -> (typ option * bool) list -> styp -> bool
   199   val term_under_def : term -> term
   200   val case_const_names :
   201     Proof.context -> (typ option * bool) list -> (string * int) list
   202   val unfold_defs_in_term : hol_context -> term -> term
   203   val const_def_tables :
   204     Proof.context -> (term * term) list -> term list
   205     -> const_table * const_table
   206   val const_nondef_table : term list -> const_table
   207   val const_simp_table : Proof.context -> (term * term) list -> const_table
   208   val const_psimp_table : Proof.context -> (term * term) list -> const_table
   209   val const_choice_spec_table :
   210     Proof.context -> (term * term) list -> const_table
   211   val inductive_intro_table :
   212     Proof.context -> (term * term) list -> const_table * const_table
   213     -> const_table
   214   val ground_theorem_table : theory -> term list Inttab.table
   215   val ersatz_table : Proof.context -> (string * string) list
   216   val add_simps : const_table Unsynchronized.ref -> string -> term list -> unit
   217   val inverse_axioms_for_rep_fun : Proof.context -> styp -> term list
   218   val optimized_typedef_axioms : Proof.context -> string * typ list -> term list
   219   val optimized_quot_type_axioms :
   220     Proof.context -> (typ option * bool) list -> string * typ list -> term list
   221   val def_of_const : theory -> const_table * const_table -> styp -> term option
   222   val fixpoint_kind_of_rhs : term -> fixpoint_kind
   223   val fixpoint_kind_of_const :
   224     theory -> const_table * const_table -> string * typ -> fixpoint_kind
   225   val is_real_inductive_pred : hol_context -> styp -> bool
   226   val is_constr_pattern : Proof.context -> term -> bool
   227   val is_constr_pattern_lhs : Proof.context -> term -> bool
   228   val is_constr_pattern_formula : Proof.context -> term -> bool
   229   val nondef_props_for_const :
   230     theory -> bool -> const_table -> styp -> term list
   231   val is_choice_spec_fun : hol_context -> styp -> bool
   232   val is_choice_spec_axiom : theory -> const_table -> term -> bool
   233   val is_real_equational_fun : hol_context -> styp -> bool
   234   val is_equational_fun_but_no_plain_def : hol_context -> styp -> bool
   235   val codatatype_bisim_axioms : hol_context -> typ -> term list
   236   val is_well_founded_inductive_pred : hol_context -> styp -> bool
   237   val unrolled_inductive_pred_const : hol_context -> bool -> styp -> term
   238   val equational_fun_axioms : hol_context -> styp -> term list
   239   val is_equational_fun_surely_complete : hol_context -> styp -> bool
   240   val merged_type_var_table_for_terms :
   241     theory -> term list -> (sort * string) list
   242   val merge_type_vars_in_term :
   243     theory -> bool -> (sort * string) list -> term -> term
   244   val ground_types_in_type : hol_context -> bool -> typ -> typ list
   245   val ground_types_in_terms : hol_context -> bool -> term list -> typ list
   246 end;
   247 
   248 structure Nitpick_HOL : NITPICK_HOL =
   249 struct
   250 
   251 open Nitpick_Util
   252 
   253 type const_table = term list Symtab.table
   254 type special_fun = (styp * int list * term list) * styp
   255 type unrolled = styp * styp
   256 type wf_cache = (styp * (bool * bool)) list
   257 
   258 type hol_context =
   259   {thy: theory,
   260    ctxt: Proof.context,
   261    max_bisim_depth: int,
   262    boxes: (typ option * bool option) list,
   263    stds: (typ option * bool) list,
   264    wfs: (styp option * bool option) list,
   265    user_axioms: bool option,
   266    debug: bool,
   267    whacks: term list,
   268    binary_ints: bool option,
   269    destroy_constrs: bool,
   270    specialize: bool,
   271    star_linear_preds: bool,
   272    total_consts: bool option,
   273    needs: term list option,
   274    tac_timeout: Time.time,
   275    evals: term list,
   276    case_names: (string * int) list,
   277    def_tables: const_table * const_table,
   278    nondef_table: const_table,
   279    nondefs: term list,
   280    simp_table: const_table Unsynchronized.ref,
   281    psimp_table: const_table,
   282    choice_spec_table: const_table,
   283    intro_table: const_table,
   284    ground_thm_table: term list Inttab.table,
   285    ersatz_table: (string * string) list,
   286    skolems: (string * string list) list Unsynchronized.ref,
   287    special_funs: special_fun list Unsynchronized.ref,
   288    unrolled_preds: unrolled list Unsynchronized.ref,
   289    wf_cache: wf_cache Unsynchronized.ref,
   290    constr_cache: (typ * styp list) list Unsynchronized.ref}
   291 
   292 datatype fixpoint_kind = Lfp | Gfp | NoFp
   293 datatype boxability =
   294   InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
   295 
   296 structure Data = Generic_Data
   297 (
   298   type T = {frac_types: (string * (string * string) list) list,
   299             ersatz_table: (string * string) list,
   300             codatatypes: (string * (string * styp list)) list}
   301   val empty = {frac_types = [], ersatz_table = [], codatatypes = []}
   302   val extend = I
   303   fun merge ({frac_types = fs1, ersatz_table = et1, codatatypes = cs1},
   304              {frac_types = fs2, ersatz_table = et2, codatatypes = cs2}) : T =
   305     {frac_types = AList.merge (op =) (K true) (fs1, fs2),
   306      ersatz_table = AList.merge (op =) (K true) (et1, et2),
   307      codatatypes = AList.merge (op =) (K true) (cs1, cs2)}
   308 )
   309 
   310 val name_sep = "$"
   311 val numeral_prefix = nitpick_prefix ^ "num" ^ name_sep
   312 val sel_prefix = nitpick_prefix ^ "sel"
   313 val discr_prefix = nitpick_prefix ^ "is" ^ 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 relcomp}, 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 Nitpick.Frac}, 0),
   400    (@{const_name Nitpick.norm_frac}, 0)]
   401 val built_in_nat_consts =
   402   [(@{const_name Suc}, 0),
   403    (@{const_name nat}, 0),
   404    (@{const_name Nitpick.nat_gcd}, 0),
   405    (@{const_name Nitpick.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    prop_of_Rep: thm, set_name: string option, Abs_inverse: thm option,
   563    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 Nitpick.Abs_Frac},
   569           Rep_name = @{const_name Nitpick.Rep_Frac},
   570           prop_of_Rep = @{prop "Nitpick.Rep_Frac x \<in> Collect Nitpick.Frac"}
   571                         |> Logic.varify_global,
   572           Abs_inverse = NONE, Rep_inverse = NONE}
   573   else case Typedef.get_info ctxt s of
   574     (* When several entries are returned, it shouldn't matter much which one
   575        we take (according to Florian Haftmann). *)
   576     (* The "Logic.varifyT_global" calls are a temporary hack because these
   577        types's type variables sometimes clash with locally fixed type variables.
   578        Remove these calls once "Typedef" is fully localized. *)
   579     ({abs_type, rep_type, Abs_name, Rep_name, ...},
   580      {Rep, Abs_inverse, Rep_inverse, ...}) :: _ =>
   581     SOME {abs_type = Logic.varifyT_global abs_type,
   582           rep_type = Logic.varifyT_global rep_type, Abs_name = Abs_name,
   583           Rep_name = Rep_name, prop_of_Rep = prop_of Rep,
   584           Abs_inverse = SOME Abs_inverse, 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 "natural" 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}, @{type_name natural}, @{type_name integer}] s orelse
   596   (s = @{type_name nat} andalso is_standard_datatype thy stds nat_T)
   597 
   598 fun repair_constr_type thy body_T' T =
   599   varify_and_instantiate_type_global thy (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 thy = Context.theory_of generic
   633     val {frac_types, ersatz_table, codatatypes} = Data.get generic
   634     val constr_xs = map (apsnd (repair_constr_type thy co_T)) constr_xs
   635     val (co_s, co_Ts) = dest_Type co_T
   636     val _ =
   637       if forall is_TFree co_Ts andalso not (has_duplicates (op =) co_Ts) andalso
   638          co_s <> @{type_name fun} andalso
   639          not (is_basic_datatype thy [(NONE, true)] co_s) then
   640         ()
   641       else
   642         raise TYPE ("Nitpick_HOL.register_codatatype_generic", [co_T], [])
   643     val codatatypes = AList.update (op =) (co_s, (case_name, constr_xs))
   644                                    codatatypes
   645   in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
   646                codatatypes = codatatypes} generic end
   647 (* TODO: Consider morphism. *)
   648 fun register_codatatype co_T case_name constr_xs (_ : morphism) =
   649   register_codatatype_generic co_T case_name constr_xs
   650 val register_codatatype_global =
   651   Context.theory_map ooo register_codatatype_generic
   652 
   653 fun unregister_codatatype_generic co_T = register_codatatype_generic co_T "" []
   654 (* TODO: Consider morphism. *)
   655 fun unregister_codatatype co_T (_ : morphism) =
   656   unregister_codatatype_generic co_T
   657 val unregister_codatatype_global =
   658   Context.theory_map o unregister_codatatype_generic
   659 
   660 fun is_codatatype ctxt (Type (s, _)) =
   661     s |> AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   662       |> Option.map snd |> these |> null |> not
   663   | is_codatatype _ _ = false
   664 fun is_registered_type ctxt T = is_frac_type ctxt T orelse is_codatatype ctxt T
   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_registered_type ctxt T) andalso
   670   T <> @{typ int}
   671 fun is_pure_typedef ctxt (T as Type (s, _)) =
   672     let val thy = Proof_Context.theory_of ctxt in
   673       is_frac_type ctxt T orelse
   674       (is_typedef ctxt s andalso
   675        not (is_real_datatype thy s orelse is_real_quot_type ctxt T orelse
   676             is_codatatype ctxt T orelse is_record_type T orelse
   677             is_integer_like_type T))
   678     end
   679   | is_pure_typedef _ _ = false
   680 fun is_univ_typedef ctxt (Type (s, _)) =
   681     (case typedef_info ctxt s of
   682        SOME {prop_of_Rep, ...} =>
   683        let
   684          val t_opt =
   685            try (snd o HOLogic.dest_mem o HOLogic.dest_Trueprop) prop_of_Rep
   686        in
   687          case t_opt of
   688            SOME (Const (@{const_name top}, _)) => true
   689            (* "Multiset.multiset" *)
   690          | SOME (Const (@{const_name Collect}, _)
   691                  $ Abs (_, _, Const (@{const_name finite}, _) $ _)) => true
   692            (* "FinFun.finfun" *)
   693          | SOME (Const (@{const_name Collect}, _) $ Abs (_, _,
   694                      Const (@{const_name Ex}, _) $ Abs (_, _,
   695                          Const (@{const_name finite}, _) $ _))) => true
   696          | _ => false
   697        end
   698      | NONE => false)
   699   | is_univ_typedef _ _ = false
   700 fun is_datatype ctxt stds (T as Type (s, _)) =
   701     let val thy = Proof_Context.theory_of ctxt in
   702       (is_typedef ctxt s orelse is_registered_type ctxt T orelse
   703        T = @{typ ind} orelse is_real_quot_type ctxt T) andalso
   704       not (is_basic_datatype thy stds s)
   705     end
   706   | is_datatype _ _ _ = false
   707 
   708 fun all_record_fields thy T =
   709   let val (recs, more) = Record.get_extT_fields thy T in
   710     recs @ more :: all_record_fields thy (snd more)
   711   end
   712   handle TYPE _ => []
   713 fun is_record_constr (s, T) =
   714   String.isSuffix Record.extN s andalso
   715   let val dataT = body_type T in
   716     is_record_type dataT andalso
   717     s = unsuffix Record.ext_typeN (fst (dest_Type dataT)) ^ Record.extN
   718   end
   719 val num_record_fields = Integer.add 1 o length o fst oo Record.get_extT_fields
   720 fun no_of_record_field thy s T1 =
   721   find_index (curry (op =) s o fst)
   722              (Record.get_extT_fields thy T1 ||> single |> op @)
   723 fun is_record_get thy (s, Type (@{type_name fun}, [T1, _])) =
   724     exists (curry (op =) s o fst) (all_record_fields thy T1)
   725   | is_record_get _ _ = false
   726 fun is_record_update thy (s, T) =
   727   String.isSuffix Record.updateN s andalso
   728   exists (curry (op =) (unsuffix Record.updateN s) o fst)
   729          (all_record_fields thy (body_type T))
   730   handle TYPE _ => false
   731 fun is_abs_fun ctxt (s, Type (@{type_name fun}, [_, Type (s', _)])) =
   732     (case typedef_info ctxt s' of
   733        SOME {Abs_name, ...} => s = Abs_name
   734      | NONE => false)
   735   | is_abs_fun _ _ = false
   736 fun is_rep_fun ctxt (s, Type (@{type_name fun}, [Type (s', _), _])) =
   737     (case typedef_info ctxt s' of
   738        SOME {Rep_name, ...} => s = Rep_name
   739      | NONE => false)
   740   | is_rep_fun _ _ = false
   741 fun is_quot_abs_fun ctxt (x as (_, Type (@{type_name fun},
   742                                          [_, abs_T as Type (s', _)]))) =
   743     try (Quotient_Term.absrep_const_chk ctxt Quotient_Term.AbsF) s'
   744     = SOME (Const x) andalso not (is_registered_type ctxt abs_T)
   745   | is_quot_abs_fun _ _ = false
   746 fun is_quot_rep_fun ctxt (s, Type (@{type_name fun},
   747                                    [abs_T as Type (abs_s, _), _])) =
   748     (case try (Quotient_Term.absrep_const_chk ctxt Quotient_Term.RepF) abs_s of
   749        SOME (Const (s', _)) =>
   750        s = s' andalso not (is_registered_type ctxt abs_T)
   751      | _ => 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   let val thy = Proof_Context.theory_of ctxt in
   786     case body_type T of
   787       co_T as Type (co_s, _) =>
   788       let val {codatatypes, ...} = Data.get (Context.Proof ctxt) in
   789         exists (fn (s', T') => s = s' andalso repair_constr_type thy co_T T' = T)
   790                (AList.lookup (op =) codatatypes co_s |> Option.map snd |> these)
   791       end
   792     | _ => false
   793   end
   794 fun is_constr_like ctxt (s, T) =
   795   member (op =) [@{const_name FunBox}, @{const_name PairBox},
   796                  @{const_name Quot}, @{const_name Zero_Rep},
   797                  @{const_name Suc_Rep}] s orelse
   798   let
   799     val thy = Proof_Context.theory_of ctxt
   800     val (x as (_, T)) = (s, unarize_unbox_etc_type T)
   801   in
   802     is_real_constr thy x orelse is_record_constr x orelse
   803     (is_abs_fun ctxt x andalso is_pure_typedef ctxt (range_type T)) orelse
   804     is_coconstr ctxt x
   805   end
   806 fun is_constr_like_injective ctxt (s, T) =
   807   is_constr_like ctxt (s, T) andalso
   808   let val (x as (_, T)) = (s, unarize_unbox_etc_type T) in
   809     not (is_abs_fun ctxt x) orelse is_univ_typedef ctxt (range_type T)
   810   end
   811 fun is_stale_constr ctxt (x as (s, T)) =
   812   is_registered_type ctxt (body_type T) andalso is_constr_like ctxt x andalso
   813   not (s = @{const_name Nitpick.Abs_Frac} orelse is_coconstr ctxt x)
   814 fun is_constr ctxt stds (x as (_, T)) =
   815   let val thy = Proof_Context.theory_of ctxt in
   816     is_constr_like ctxt x andalso
   817     not (is_basic_datatype thy stds
   818                          (fst (dest_Type (unarize_type (body_type T))))) andalso
   819     not (is_stale_constr ctxt x)
   820   end
   821 val is_sel = String.isPrefix discr_prefix orf String.isPrefix sel_prefix
   822 val is_sel_like_and_no_discr =
   823   String.isPrefix sel_prefix orf
   824   (member (op =) [@{const_name fst}, @{const_name snd}])
   825 
   826 fun in_fun_lhs_for InConstr = InSel
   827   | in_fun_lhs_for _ = InFunLHS
   828 fun in_fun_rhs_for InConstr = InConstr
   829   | in_fun_rhs_for InSel = InSel
   830   | in_fun_rhs_for InFunRHS1 = InFunRHS2
   831   | in_fun_rhs_for _ = InFunRHS1
   832 
   833 fun is_boxing_worth_it (hol_ctxt : hol_context) boxy T =
   834   case T of
   835     Type (@{type_name fun}, _) =>
   836     (boxy = InPair orelse boxy = InFunLHS) andalso
   837     not (is_boolean_type (body_type T))
   838   | Type (@{type_name prod}, Ts) =>
   839     boxy = InPair orelse boxy = InFunRHS1 orelse boxy = InFunRHS2 orelse
   840     ((boxy = InExpr orelse boxy = InFunLHS) andalso
   841      exists (is_boxing_worth_it hol_ctxt InPair)
   842             (map (box_type hol_ctxt InPair) Ts))
   843   | _ => false
   844 and should_box_type (hol_ctxt as {thy, boxes, ...}) boxy z =
   845   case triple_lookup (type_match thy) boxes (Type z) of
   846     SOME (SOME box_me) => box_me
   847   | _ => is_boxing_worth_it hol_ctxt boxy (Type z)
   848 and box_type hol_ctxt boxy T =
   849   case T of
   850     Type (z as (@{type_name fun}, [T1, T2])) =>
   851     if boxy <> InConstr andalso boxy <> InSel andalso
   852        should_box_type hol_ctxt boxy z then
   853       Type (@{type_name fun_box},
   854             [box_type hol_ctxt InFunLHS T1, box_type hol_ctxt InFunRHS1 T2])
   855     else
   856       box_type hol_ctxt (in_fun_lhs_for boxy) T1
   857       --> box_type hol_ctxt (in_fun_rhs_for boxy) T2
   858   | Type (z as (@{type_name prod}, Ts)) =>
   859     if boxy <> InConstr andalso boxy <> InSel
   860        andalso should_box_type hol_ctxt boxy z then
   861       Type (@{type_name pair_box}, map (box_type hol_ctxt InSel) Ts)
   862     else
   863       Type (@{type_name prod},
   864             map (box_type hol_ctxt
   865                           (if boxy = InConstr orelse boxy = InSel then boxy
   866                            else InPair)) Ts)
   867   | _ => T
   868 
   869 fun binarize_nat_and_int_in_type @{typ nat} = @{typ "unsigned_bit word"}
   870   | binarize_nat_and_int_in_type @{typ int} = @{typ "signed_bit word"}
   871   | binarize_nat_and_int_in_type (Type (s, Ts)) =
   872     Type (s, map binarize_nat_and_int_in_type Ts)
   873   | binarize_nat_and_int_in_type T = T
   874 val binarize_nat_and_int_in_term = map_types binarize_nat_and_int_in_type
   875 
   876 fun discr_for_constr (s, T) = (discr_prefix ^ s, body_type T --> bool_T)
   877 
   878 fun num_sels_for_constr_type T = length (maybe_curried_binder_types T)
   879 fun nth_sel_name_for_constr_name s n =
   880   if s = @{const_name Pair} then
   881     if n = 0 then @{const_name fst} else @{const_name snd}
   882   else
   883     sel_prefix_for n ^ s
   884 fun nth_sel_for_constr x ~1 = discr_for_constr x
   885   | nth_sel_for_constr (s, T) n =
   886     (nth_sel_name_for_constr_name s n,
   887      body_type T --> nth (maybe_curried_binder_types T) n)
   888 fun binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize =
   889   apsnd ((binarize ? binarize_nat_and_int_in_type) o box_type hol_ctxt InSel)
   890   oo nth_sel_for_constr
   891 
   892 fun sel_no_from_name s =
   893   if String.isPrefix discr_prefix s then
   894     ~1
   895   else if String.isPrefix sel_prefix s then
   896     s |> unprefix sel_prefix |> Int.fromString |> the
   897   else if s = @{const_name snd} then
   898     1
   899   else
   900     0
   901 
   902 val close_form =
   903   let
   904     fun close_up zs zs' =
   905       fold (fn (z as ((s, _), T)) => fn t' =>
   906                Logic.all_const T $ Abs (s, T, abstract_over (Var z, t')))
   907            (take (length zs' - length zs) zs')
   908     fun aux zs (@{const "==>"} $ t1 $ t2) =
   909         let val zs' = Term.add_vars t1 zs in
   910           close_up zs zs' (Logic.mk_implies (t1, aux zs' t2))
   911         end
   912       | aux zs t = close_up zs (Term.add_vars t zs) t
   913   in aux [] end
   914 
   915 fun distinctness_formula T =
   916   all_distinct_unordered_pairs_of
   917   #> map (fn (t1, t2) => @{const Not} $ (HOLogic.eq_const T $ t1 $ t2))
   918   #> List.foldr (s_conj o swap) @{const True}
   919 
   920 fun zero_const T = Const (@{const_name zero_class.zero}, T)
   921 fun suc_const T = Const (@{const_name Suc}, T --> T)
   922 
   923 fun uncached_datatype_constrs ({thy, ctxt, stds, ...} : hol_context)
   924                               (T as Type (s, Ts)) =
   925     (case AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   926                        s of
   927        SOME (_, xs' as (_ :: _)) => map (apsnd (repair_constr_type thy T)) xs'
   928      | _ =>
   929        if is_frac_type ctxt T then
   930          case typedef_info ctxt s of
   931            SOME {abs_type, rep_type, Abs_name, ...} =>
   932            [(Abs_name,
   933              varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
   934          | NONE => [] (* impossible *)
   935        else if is_datatype ctxt stds T then
   936          case Datatype.get_info thy s of
   937            SOME {index, descr, ...} =>
   938            let
   939              val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the
   940            in
   941              map (apsnd (fn Us =>
   942                             map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))
   943                  constrs
   944            end
   945          | NONE =>
   946            if is_record_type T then
   947              let
   948                val s' = unsuffix Record.ext_typeN s ^ Record.extN
   949                val T' = (Record.get_extT_fields thy T
   950                         |> apsnd single |> uncurry append |> map snd) ---> T
   951              in [(s', T')] end
   952            else if is_real_quot_type ctxt T then
   953              [(@{const_name Quot}, rep_type_for_quot_type ctxt T --> T)]
   954            else case typedef_info ctxt s of
   955              SOME {abs_type, rep_type, Abs_name, ...} =>
   956              [(Abs_name,
   957                varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
   958            | NONE =>
   959              if T = @{typ ind} then
   960                [dest_Const @{const Zero_Rep}, dest_Const @{const Suc_Rep}]
   961              else
   962                []
   963        else
   964          [])
   965   | uncached_datatype_constrs _ _ = []
   966 fun datatype_constrs (hol_ctxt as {constr_cache, ...}) T =
   967   case AList.lookup (op =) (!constr_cache) T of
   968     SOME xs => xs
   969   | NONE =>
   970     let val xs = uncached_datatype_constrs hol_ctxt T in
   971       (Unsynchronized.change constr_cache (cons (T, xs)); xs)
   972     end
   973 fun binarized_and_boxed_datatype_constrs hol_ctxt binarize =
   974   map (apsnd ((binarize ? binarize_nat_and_int_in_type)
   975               o box_type hol_ctxt InConstr)) o datatype_constrs hol_ctxt
   976 val num_datatype_constrs = length oo datatype_constrs
   977 
   978 fun constr_name_for_sel_like @{const_name fst} = @{const_name Pair}
   979   | constr_name_for_sel_like @{const_name snd} = @{const_name Pair}
   980   | constr_name_for_sel_like s' = original_name s'
   981 fun binarized_and_boxed_constr_for_sel hol_ctxt binarize (s', T') =
   982   let val s = constr_name_for_sel_like s' in
   983     AList.lookup (op =)
   984         (binarized_and_boxed_datatype_constrs hol_ctxt binarize (domain_type T'))
   985         s
   986     |> the |> pair s
   987   end
   988 
   989 fun card_of_type assigns (Type (@{type_name fun}, [T1, T2])) =
   990     reasonable_power (card_of_type assigns T2) (card_of_type assigns T1)
   991   | card_of_type assigns (Type (@{type_name prod}, [T1, T2])) =
   992     card_of_type assigns T1 * card_of_type assigns T2
   993   | card_of_type assigns (Type (@{type_name set}, [T'])) =
   994     reasonable_power 2 (card_of_type assigns T')
   995   | card_of_type _ (Type (@{type_name itself}, _)) = 1
   996   | card_of_type _ @{typ prop} = 2
   997   | card_of_type _ @{typ bool} = 2
   998   | card_of_type assigns T =
   999     case AList.lookup (op =) assigns T of
  1000       SOME k => k
  1001     | NONE => if T = @{typ bisim_iterator} then 0
  1002               else raise TYPE ("Nitpick_HOL.card_of_type", [T], [])
  1003 
  1004 fun bounded_card_of_type max default_card assigns
  1005                          (Type (@{type_name fun}, [T1, T2])) =
  1006     let
  1007       val k1 = bounded_card_of_type max default_card assigns T1
  1008       val k2 = bounded_card_of_type max default_card assigns T2
  1009     in
  1010       if k1 = max orelse k2 = max then max
  1011       else Int.min (max, reasonable_power k2 k1)
  1012       handle TOO_LARGE _ => max
  1013     end
  1014   | bounded_card_of_type max default_card assigns
  1015                          (Type (@{type_name prod}, [T1, T2])) =
  1016     let
  1017       val k1 = bounded_card_of_type max default_card assigns T1
  1018       val k2 = bounded_card_of_type max default_card assigns T2
  1019     in if k1 = max orelse k2 = max then max else Int.min (max, k1 * k2) end
  1020   | bounded_card_of_type max default_card assigns
  1021                          (Type (@{type_name set}, [T'])) =
  1022     bounded_card_of_type max default_card assigns (T' --> bool_T)
  1023   | bounded_card_of_type max default_card assigns T =
  1024     Int.min (max, if default_card = ~1 then
  1025                     card_of_type assigns T
  1026                   else
  1027                     card_of_type assigns T
  1028                     handle TYPE ("Nitpick_HOL.card_of_type", _, _) =>
  1029                            default_card)
  1030 
  1031 (* Similar to "ATP_Util.tiny_card_of_type". *)
  1032 fun bounded_exact_card_of_type hol_ctxt finitizable_dataTs max default_card
  1033                                assigns T =
  1034   let
  1035     fun aux avoid T =
  1036       (if member (op =) avoid T then
  1037          0
  1038        else if member (op =) finitizable_dataTs T then
  1039          raise SAME ()
  1040        else case T of
  1041          Type (@{type_name fun}, [T1, T2]) =>
  1042          (case (aux avoid T1, aux avoid T2) of
  1043             (_, 1) => 1
  1044           | (0, _) => 0
  1045           | (_, 0) => 0
  1046           | (k1, k2) =>
  1047             if k1 >= max orelse k2 >= max then max
  1048             else Int.min (max, reasonable_power k2 k1))
  1049        | Type (@{type_name prod}, [T1, T2]) =>
  1050          (case (aux avoid T1, aux avoid T2) of
  1051             (0, _) => 0
  1052           | (_, 0) => 0
  1053           | (k1, k2) =>
  1054             if k1 >= max orelse k2 >= max then max
  1055             else Int.min (max, k1 * k2))
  1056        | Type (@{type_name set}, [T']) => aux avoid (T' --> bool_T)
  1057        | Type (@{type_name itself}, _) => 1
  1058        | @{typ prop} => 2
  1059        | @{typ bool} => 2
  1060        | Type _ =>
  1061          (case datatype_constrs hol_ctxt T of
  1062             [] => if is_integer_type T orelse is_bit_type T then 0
  1063                   else raise SAME ()
  1064           | constrs =>
  1065             let
  1066               val constr_cards =
  1067                 map (Integer.prod o map (aux (T :: avoid)) o binder_types o snd)
  1068                     constrs
  1069             in
  1070               if exists (curry (op =) 0) constr_cards then 0
  1071               else Int.min (max, Integer.sum constr_cards)
  1072             end)
  1073        | _ => raise SAME ())
  1074       handle SAME () =>
  1075              AList.lookup (op =) assigns T |> the_default default_card
  1076   in Int.min (max, aux [] T) end
  1077 
  1078 val typical_atomic_card = 4
  1079 val typical_card_of_type = bounded_card_of_type 16777217 typical_atomic_card []
  1080 
  1081 fun is_finite_type hol_ctxt T =
  1082   bounded_exact_card_of_type hol_ctxt [] 1 2 [] T > 0
  1083 
  1084 fun is_special_eligible_arg strict Ts t =
  1085   case map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t) of
  1086     [] => true
  1087   | bad_Ts =>
  1088     let
  1089       val bad_Ts_cost =
  1090         if strict then fold (curry (op *) o typical_card_of_type) bad_Ts 1
  1091         else fold (Integer.max o typical_card_of_type) bad_Ts 0
  1092       val T_cost = typical_card_of_type (fastype_of1 (Ts, t))
  1093     in (bad_Ts_cost, T_cost) |> (if strict then op < else op <=) end
  1094 
  1095 fun abs_var ((s, j), T) body = Abs (s, T, abstract_over (Var ((s, j), T), body))
  1096 
  1097 fun let_var s = (nitpick_prefix ^ s, 999)
  1098 val let_inline_threshold = 20
  1099 
  1100 fun s_let Ts s n abs_T body_T f t =
  1101   if (n - 1) * (size_of_term t - 1) <= let_inline_threshold orelse
  1102      is_special_eligible_arg false Ts t then
  1103     f t
  1104   else
  1105     let val z = (let_var s, abs_T) in
  1106       Const (@{const_name Let}, abs_T --> (abs_T --> body_T) --> body_T)
  1107       $ t $ abs_var z (incr_boundvars 1 (f (Var z)))
  1108     end
  1109 
  1110 fun loose_bvar1_count (Bound i, k) = if i = k then 1 else 0
  1111   | loose_bvar1_count (t1 $ t2, k) =
  1112     loose_bvar1_count (t1, k) + loose_bvar1_count (t2, k)
  1113   | loose_bvar1_count (Abs (_, _, t), k) = loose_bvar1_count (t, k + 1)
  1114   | loose_bvar1_count _ = 0
  1115 
  1116 fun s_betapply _ (t1 as Const (@{const_name "=="}, _) $ t1', t2) =
  1117     if t1' aconv t2 then @{prop True} else t1 $ t2
  1118   | s_betapply _ (t1 as Const (@{const_name HOL.eq}, _) $ t1', t2) =
  1119     if t1' aconv t2 then @{term True} else t1 $ t2
  1120   | s_betapply _ (Const (@{const_name If}, _) $ @{const True} $ t1', _) = t1'
  1121   | s_betapply _ (Const (@{const_name If}, _) $ @{const False} $ _, t2) = t2
  1122   | s_betapply Ts (Const (@{const_name Let},
  1123                           Type (_, [bound_T, Type (_, [_, body_T])]))
  1124                    $ t12 $ Abs (s, T, t13'), t2) =
  1125     let val body_T' = range_type body_T in
  1126       Const (@{const_name Let}, bound_T --> (bound_T --> body_T') --> body_T')
  1127       $ t12 $ Abs (s, T, s_betapply (T :: Ts) (t13', incr_boundvars 1 t2))
  1128     end
  1129   | s_betapply Ts (t1 as Abs (s1, T1, t1'), t2) =
  1130     (s_let Ts s1 (loose_bvar1_count (t1', 0)) T1 (fastype_of1 (T1 :: Ts, t1'))
  1131            (curry betapply t1) t2
  1132      (* FIXME: fix all "s_betapply []" calls *)
  1133      handle TERM _ => betapply (t1, t2)
  1134           | General.Subscript => betapply (t1, t2))
  1135   | s_betapply _ (t1, t2) = t1 $ t2
  1136 fun s_betapplys Ts = Library.foldl (s_betapply Ts)
  1137 
  1138 fun s_beta_norm Ts t =
  1139   let
  1140     fun aux _ (Var _) = raise Same.SAME
  1141       | aux Ts (Abs (s, T, t')) = Abs (s, T, aux (T :: Ts) t')
  1142       | aux Ts ((t1 as Abs _) $ t2) =
  1143         Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1144       | aux Ts (t1 $ t2) =
  1145         ((case aux Ts t1 of
  1146            t1 as Abs _ => Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1147          | t1 => t1 $ Same.commit (aux Ts) t2)
  1148         handle Same.SAME => t1 $ aux Ts t2)
  1149       | aux _ _ = raise Same.SAME
  1150   in aux Ts t handle Same.SAME => t end
  1151 
  1152 fun discr_term_for_constr hol_ctxt (x as (s, T)) =
  1153   let val dataT = body_type T in
  1154     if s = @{const_name Suc} then
  1155       Abs (Name.uu, dataT,
  1156            @{const Not} $ HOLogic.mk_eq (zero_const dataT, Bound 0))
  1157     else if num_datatype_constrs hol_ctxt dataT >= 2 then
  1158       Const (discr_for_constr x)
  1159     else
  1160       Abs (Name.uu, dataT, @{const True})
  1161   end
  1162 fun discriminate_value (hol_ctxt as {ctxt, ...}) x t =
  1163   case head_of t of
  1164     Const x' =>
  1165     if x = x' then @{const True}
  1166     else if is_constr_like ctxt x' then @{const False}
  1167     else s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1168   | _ => s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1169 
  1170 fun nth_arg_sel_term_for_constr thy stds (x as (s, T)) n =
  1171   let val (arg_Ts, dataT) = strip_type T in
  1172     if dataT = nat_T andalso is_standard_datatype thy stds nat_T then
  1173       @{term "%n::nat. n - 1"}
  1174     else if is_pair_type dataT then
  1175       Const (nth_sel_for_constr x n)
  1176     else
  1177       let
  1178         fun aux m (Type (@{type_name prod}, [T1, T2])) =
  1179             let
  1180               val (m, t1) = aux m T1
  1181               val (m, t2) = aux m T2
  1182             in (m, HOLogic.mk_prod (t1, t2)) end
  1183           | aux m T =
  1184             (m + 1, Const (nth_sel_name_for_constr_name s m, dataT --> T)
  1185                     $ Bound 0)
  1186         val m = fold (Integer.add o num_factors_in_type)
  1187                      (List.take (arg_Ts, n)) 0
  1188       in Abs ("x", dataT, aux m (nth arg_Ts n) |> snd) end
  1189   end
  1190 fun select_nth_constr_arg ctxt stds x t n res_T =
  1191   let val thy = Proof_Context.theory_of ctxt in
  1192     (case strip_comb t of
  1193        (Const x', args) =>
  1194        if x = x' then
  1195           if is_constr_like_injective ctxt x then nth args n else raise SAME ()
  1196        else if is_constr_like ctxt x' then
  1197          Const (@{const_name unknown}, res_T)
  1198        else
  1199          raise SAME ()
  1200      | _ => raise SAME())
  1201     handle SAME () =>
  1202            s_betapply [] (nth_arg_sel_term_for_constr thy stds x n, t)
  1203   end
  1204 
  1205 fun construct_value _ _ x [] = Const x
  1206   | construct_value ctxt stds (x as (s, _)) args =
  1207     let val args = map Envir.eta_contract args in
  1208       case hd args of
  1209         Const (s', _) $ t =>
  1210         if is_sel_like_and_no_discr s' andalso
  1211            constr_name_for_sel_like s' = s andalso
  1212            forall (fn (n, t') =>
  1213                       select_nth_constr_arg ctxt stds x t n dummyT = t')
  1214                   (index_seq 0 (length args) ~~ args) then
  1215           t
  1216         else
  1217           list_comb (Const x, args)
  1218       | _ => list_comb (Const x, args)
  1219     end
  1220 
  1221 fun constr_expand (hol_ctxt as {ctxt, stds, ...}) T t =
  1222   (case head_of t of
  1223      Const x => if is_constr_like ctxt x then t else raise SAME ()
  1224    | _ => raise SAME ())
  1225   handle SAME () =>
  1226          let
  1227            val x' as (_, T') =
  1228              if is_pair_type T then
  1229                let val (T1, T2) = HOLogic.dest_prodT T in
  1230                  (@{const_name Pair}, T1 --> T2 --> T)
  1231                end
  1232              else
  1233                datatype_constrs hol_ctxt T |> hd
  1234            val arg_Ts = binder_types T'
  1235          in
  1236            list_comb (Const x', map2 (select_nth_constr_arg ctxt stds x' t)
  1237                                      (index_seq 0 (length arg_Ts)) arg_Ts)
  1238          end
  1239 
  1240 fun coerce_bound_no f j t =
  1241   case t of
  1242     t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2
  1243   | Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')
  1244   | Bound j' => if j' = j then f t else t
  1245   | _ => t
  1246 fun coerce_bound_0_in_term hol_ctxt new_T old_T =
  1247   old_T <> new_T ? coerce_bound_no (coerce_term hol_ctxt [new_T] old_T new_T) 0
  1248 and coerce_term (hol_ctxt as {ctxt, stds, ...}) Ts new_T old_T t =
  1249   if old_T = new_T then
  1250     t
  1251   else
  1252     case (new_T, old_T) of
  1253       (Type (new_s, new_Ts as [new_T1, new_T2]),
  1254        Type (@{type_name fun}, [old_T1, old_T2])) =>
  1255       (case eta_expand Ts t 1 of
  1256          Abs (s, _, t') =>
  1257          Abs (s, new_T1,
  1258               t' |> coerce_bound_0_in_term hol_ctxt new_T1 old_T1
  1259                  |> coerce_term hol_ctxt (new_T1 :: Ts) new_T2 old_T2)
  1260          |> Envir.eta_contract
  1261          |> new_s <> @{type_name fun}
  1262             ? construct_value ctxt stds
  1263                   (@{const_name FunBox},
  1264                    Type (@{type_name fun}, new_Ts) --> new_T)
  1265               o single
  1266        | t' => raise TERM ("Nitpick_HOL.coerce_term", [t']))
  1267     | (Type (new_s, new_Ts as [new_T1, new_T2]),
  1268        Type (old_s, old_Ts as [old_T1, old_T2])) =>
  1269       if old_s = @{type_name fun_box} orelse
  1270          old_s = @{type_name pair_box} orelse old_s = @{type_name prod} then
  1271         case constr_expand hol_ctxt old_T t of
  1272           Const (old_s, _) $ t1 =>
  1273           if new_s = @{type_name fun} then
  1274             coerce_term hol_ctxt Ts new_T (Type (@{type_name fun}, old_Ts)) t1
  1275           else
  1276             construct_value ctxt stds
  1277                 (old_s, Type (@{type_name fun}, new_Ts) --> new_T)
  1278                 [coerce_term hol_ctxt Ts (Type (@{type_name fun}, new_Ts))
  1279                              (Type (@{type_name fun}, old_Ts)) t1]
  1280         | Const _ $ t1 $ t2 =>
  1281           construct_value ctxt stds
  1282               (if new_s = @{type_name prod} then @{const_name Pair}
  1283                else @{const_name PairBox}, new_Ts ---> new_T)
  1284               (map3 (coerce_term hol_ctxt Ts) [new_T1, new_T2] [old_T1, old_T2]
  1285                     [t1, t2])
  1286         | t' => raise TERM ("Nitpick_HOL.coerce_term", [t'])
  1287       else
  1288         raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1289     | _ => raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1290 
  1291 fun is_ground_term (t1 $ t2) = is_ground_term t1 andalso is_ground_term t2
  1292   | is_ground_term (Const _) = true
  1293   | is_ground_term _ = false
  1294 
  1295 fun special_bounds ts =
  1296   fold Term.add_vars ts [] |> sort (Term_Ord.fast_indexname_ord o pairself fst)
  1297 
  1298 (* FIXME: detect "rep_datatype"? *)
  1299 fun is_funky_typedef_name ctxt s =
  1300   member (op =) [@{type_name unit}, @{type_name prod}, @{type_name set},
  1301                  @{type_name Sum_Type.sum}, @{type_name int}] s orelse
  1302   is_frac_type ctxt (Type (s, []))
  1303 fun is_funky_typedef ctxt (Type (s, _)) = is_funky_typedef_name ctxt s
  1304   | is_funky_typedef _ _ = false
  1305 
  1306 fun all_defs_of thy subst =
  1307   let
  1308     val def_names =
  1309       thy |> Theory.defs_of
  1310           |> Defs.all_specifications_of
  1311           |> maps snd |> map_filter #def
  1312           |> Ord_List.make fast_string_ord
  1313   in
  1314     Theory.nodes_of thy
  1315     |> maps Thm.axioms_of
  1316     |> map (apsnd (subst_atomic subst o prop_of))
  1317     |> sort (fast_string_ord o pairself fst)
  1318     |> Ord_List.inter (fast_string_ord o apsnd fst) def_names
  1319     |> map snd
  1320   end
  1321 
  1322 (* Ideally we would check against "Complex_Main", not "Hilbert_Choice", but any
  1323    theory will do as long as it contains all the "axioms" and "axiomatization"
  1324    commands. *)
  1325 fun is_built_in_theory thy = Theory.subthy (thy, @{theory Hilbert_Choice})
  1326 
  1327 fun all_nondefs_of ctxt subst =
  1328   ctxt |> Spec_Rules.get
  1329        |> filter (curry (op =) Spec_Rules.Unknown o fst)
  1330        |> maps (snd o snd)
  1331        |> filter_out (is_built_in_theory o theory_of_thm)
  1332        |> map (subst_atomic subst o prop_of)
  1333 
  1334 fun arity_of_built_in_const thy stds (s, T) =
  1335   if s = @{const_name If} then
  1336     if nth_range_type 3 T = @{typ bool} then NONE else SOME 3
  1337   else
  1338     let val std_nats = is_standard_datatype thy stds nat_T in
  1339       case AList.lookup (op =)
  1340                     (built_in_consts
  1341                      |> std_nats ? append built_in_nat_consts) s of
  1342         SOME n => SOME n
  1343       | NONE =>
  1344         case AList.lookup (op =)
  1345                  (built_in_typed_consts
  1346                   |> std_nats ? append built_in_typed_nat_consts)
  1347                  (s, unarize_type T) of
  1348           SOME n => SOME n
  1349         | NONE =>
  1350           case s of
  1351             @{const_name zero_class.zero} =>
  1352             if is_iterator_type T then SOME 0 else NONE
  1353           | @{const_name Suc} =>
  1354             if is_iterator_type (domain_type T) then SOME 0 else NONE
  1355           | _ => if is_fun_type T andalso is_set_like_type (domain_type T) then
  1356                    AList.lookup (op =) built_in_set_like_consts s
  1357                  else
  1358                    NONE
  1359     end
  1360 val is_built_in_const = is_some ooo arity_of_built_in_const
  1361 
  1362 (* This function is designed to work for both real definition axioms and
  1363    simplification rules (equational specifications). *)
  1364 fun term_under_def t =
  1365   case t of
  1366     @{const "==>"} $ _ $ t2 => term_under_def t2
  1367   | Const (@{const_name "=="}, _) $ t1 $ _ => term_under_def t1
  1368   | @{const Trueprop} $ t1 => term_under_def t1
  1369   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => term_under_def t1
  1370   | Abs (_, _, t') => term_under_def t'
  1371   | t1 $ _ => term_under_def t1
  1372   | _ => t
  1373 
  1374 (* Here we crucially rely on "specialize_type" performing a preorder traversal
  1375    of the term, without which the wrong occurrence of a constant could be
  1376    matched in the face of overloading. *)
  1377 fun def_props_for_const thy stds table (x as (s, _)) =
  1378   if is_built_in_const thy stds x then
  1379     []
  1380   else
  1381     these (Symtab.lookup table s)
  1382     |> map_filter (try (specialize_type thy x))
  1383     |> filter (curry (op =) (Const x) o term_under_def)
  1384 
  1385 fun normalized_rhs_of t =
  1386   let
  1387     fun aux (v as Var _) (SOME t) = SOME (lambda v t)
  1388       | aux (c as Const (@{const_name TYPE}, _)) (SOME t) = SOME (lambda c t)
  1389       | aux _ _ = NONE
  1390     val (lhs, rhs) =
  1391       case t of
  1392         Const (@{const_name "=="}, _) $ t1 $ t2 => (t1, t2)
  1393       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ t2) =>
  1394         (t1, t2)
  1395       | _ => raise TERM ("Nitpick_HOL.normalized_rhs_of", [t])
  1396     val args = strip_comb lhs |> snd
  1397   in fold_rev aux args (SOME rhs) end
  1398 
  1399 fun get_def_of_const thy table (x as (s, _)) =
  1400   x |> def_props_for_const thy [(NONE, false)] table |> List.last
  1401     |> normalized_rhs_of |> Option.map (prefix_abs_vars s)
  1402   handle List.Empty => NONE
  1403 
  1404 fun def_of_const_ext thy (unfold_table, fallback_table) (x as (s, _)) =
  1405   if is_built_in_const thy [(NONE, false)] x orelse original_name s <> s then
  1406     NONE
  1407   else case get_def_of_const thy unfold_table x of
  1408     SOME def => SOME (true, def)
  1409   | NONE => get_def_of_const thy fallback_table x |> Option.map (pair false)
  1410 
  1411 val def_of_const = Option.map snd ooo def_of_const_ext
  1412 
  1413 fun fixpoint_kind_of_rhs (Abs (_, _, t)) = fixpoint_kind_of_rhs t
  1414   | fixpoint_kind_of_rhs (Const (@{const_name lfp}, _) $ Abs _) = Lfp
  1415   | fixpoint_kind_of_rhs (Const (@{const_name gfp}, _) $ Abs _) = Gfp
  1416   | fixpoint_kind_of_rhs _ = NoFp
  1417 
  1418 fun is_mutually_inductive_pred_def thy table t =
  1419   let
  1420     fun is_good_arg (Bound _) = true
  1421       | is_good_arg (Const (s, _)) =
  1422         s = @{const_name True} orelse s = @{const_name False} orelse
  1423         s = @{const_name undefined}
  1424       | is_good_arg _ = false
  1425   in
  1426     case t |> strip_abs_body |> strip_comb of
  1427       (Const x, ts as (_ :: _)) =>
  1428       (case def_of_const thy table x of
  1429          SOME t' => fixpoint_kind_of_rhs t' <> NoFp andalso
  1430                     forall is_good_arg ts
  1431        | NONE => false)
  1432     | _ => false
  1433   end
  1434 fun unfold_mutually_inductive_preds thy table =
  1435   map_aterms (fn t as Const x =>
  1436                  (case def_of_const thy table x of
  1437                     SOME t' =>
  1438                     let val t' = Envir.eta_contract t' in
  1439                       if is_mutually_inductive_pred_def thy table t' then t'
  1440                       else t
  1441                     end
  1442                  | NONE => t)
  1443                | t => t)
  1444 
  1445 fun case_const_names ctxt stds =
  1446   let val thy = Proof_Context.theory_of ctxt in
  1447     Symtab.fold (fn (dtype_s, {index, descr, case_name, ...}) =>
  1448                     if is_basic_datatype thy stds dtype_s then
  1449                       I
  1450                     else
  1451                       cons (case_name, AList.lookup (op =) descr index
  1452                                        |> the |> #3 |> length))
  1453                 (Datatype.get_all thy) [] @
  1454     map (apsnd length o snd) (#codatatypes (Data.get (Context.Proof ctxt)))
  1455   end
  1456 
  1457 fun fixpoint_kind_of_const thy table x =
  1458   if is_built_in_const thy [(NONE, false)] x then NoFp
  1459   else fixpoint_kind_of_rhs (the (def_of_const thy table x))
  1460   handle Option.Option => NoFp
  1461 
  1462 fun is_real_inductive_pred ({thy, stds, def_tables, intro_table, ...}
  1463                             : hol_context) x =
  1464   fixpoint_kind_of_const thy def_tables x <> NoFp andalso
  1465   not (null (def_props_for_const thy stds intro_table x))
  1466 fun is_inductive_pred hol_ctxt (x as (s, _)) =
  1467   is_real_inductive_pred hol_ctxt x orelse String.isPrefix ubfp_prefix s orelse
  1468   String.isPrefix lbfp_prefix s
  1469 
  1470 fun lhs_of_equation t =
  1471   case t of
  1472     Const (@{const_name all}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1473   | Const (@{const_name "=="}, _) $ t1 $ _ => SOME t1
  1474   | @{const "==>"} $ _ $ t2 => lhs_of_equation t2
  1475   | @{const Trueprop} $ t1 => lhs_of_equation t1
  1476   | Const (@{const_name All}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1477   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => SOME t1
  1478   | @{const HOL.implies} $ _ $ t2 => lhs_of_equation t2
  1479   | _ => NONE
  1480 fun is_constr_pattern _ (Bound _) = true
  1481   | is_constr_pattern _ (Var _) = true
  1482   | is_constr_pattern ctxt t =
  1483     case strip_comb t of
  1484       (Const x, args) =>
  1485       is_constr_like ctxt x andalso forall (is_constr_pattern ctxt) args
  1486     | _ => false
  1487 fun is_constr_pattern_lhs ctxt t =
  1488   forall (is_constr_pattern ctxt) (snd (strip_comb t))
  1489 fun is_constr_pattern_formula ctxt t =
  1490   case lhs_of_equation t of
  1491     SOME t' => is_constr_pattern_lhs ctxt t'
  1492   | NONE => false
  1493 
  1494 (* Similar to "specialize_type" but returns all matches rather than only the
  1495    first (preorder) match. *)
  1496 fun multi_specialize_type thy slack (s, T) t =
  1497   let
  1498     fun aux (Const (s', T')) ys =
  1499         if s = s' then
  1500           ys |> (if AList.defined (op =) ys T' then
  1501                    I
  1502                  else
  1503                    cons (T', monomorphic_term (Sign.typ_match thy (T', T)
  1504                                                               Vartab.empty) t)
  1505                    handle Type.TYPE_MATCH => I
  1506                         | TERM _ =>
  1507                           if slack then
  1508                             I
  1509                           else
  1510                             raise NOT_SUPPORTED
  1511                                       ("too much polymorphism in axiom \"" ^
  1512                                        Syntax.string_of_term_global thy t ^
  1513                                        "\" involving " ^ quote s))
  1514         else
  1515           ys
  1516       | aux _ ys = ys
  1517   in map snd (fold_aterms aux t []) end
  1518 fun nondef_props_for_const thy slack table (x as (s, _)) =
  1519   these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)
  1520 
  1521 fun unvarify_term (t1 $ t2) = unvarify_term t1 $ unvarify_term t2
  1522   | unvarify_term (Var ((s, 0), T)) = Free (s, T)
  1523   | unvarify_term (Abs (s, T, t')) = Abs (s, T, unvarify_term t')
  1524   | unvarify_term t = t
  1525 fun axiom_for_choice_spec thy =
  1526   unvarify_term
  1527   #> Object_Logic.atomize_term thy
  1528   #> Choice_Specification.close_form
  1529   #> HOLogic.mk_Trueprop
  1530 fun is_choice_spec_fun ({thy, def_tables, nondef_table, choice_spec_table, ...}
  1531                         : hol_context) x =
  1532   case nondef_props_for_const thy true choice_spec_table x of
  1533     [] => false
  1534   | ts => case def_of_const thy def_tables x of
  1535             SOME (Const (@{const_name Eps}, _) $ _) => true
  1536           | SOME _ => false
  1537           | NONE =>
  1538             let val ts' = nondef_props_for_const thy true nondef_table x in
  1539               length ts' = length ts andalso
  1540               forall (fn t =>
  1541                          exists (curry (op aconv) (axiom_for_choice_spec thy t))
  1542                                 ts') ts
  1543             end
  1544 
  1545 fun is_choice_spec_axiom thy choice_spec_table t =
  1546   Symtab.exists (fn (_, ts) =>
  1547                     exists (curry (op aconv) t o axiom_for_choice_spec thy) ts)
  1548                 choice_spec_table
  1549 
  1550 fun is_real_equational_fun ({thy, stds, simp_table, psimp_table, ...}
  1551                             : hol_context) x =
  1552   exists (fn table => not (null (def_props_for_const thy stds table x)))
  1553          [!simp_table, psimp_table]
  1554 fun is_equational_fun_but_no_plain_def hol_ctxt =
  1555   is_real_equational_fun hol_ctxt orf is_inductive_pred hol_ctxt
  1556 
  1557 (** Constant unfolding **)
  1558 
  1559 fun constr_case_body ctxt stds Ts (func_t, (x as (_, T))) =
  1560   let val arg_Ts = binder_types T in
  1561     s_betapplys Ts (func_t, map2 (select_nth_constr_arg ctxt stds x (Bound 0))
  1562                                  (index_seq 0 (length arg_Ts)) arg_Ts)
  1563   end
  1564 fun add_constr_case res_T (body_t, guard_t) res_t =
  1565   if res_T = bool_T then
  1566     s_conj (HOLogic.mk_imp (guard_t, body_t), res_t)
  1567   else
  1568     Const (@{const_name If}, bool_T --> res_T --> res_T --> res_T)
  1569     $ guard_t $ body_t $ res_t
  1570 fun optimized_case_def (hol_ctxt as {ctxt, stds, ...}) Ts dataT res_T func_ts =
  1571   let
  1572     val xs = datatype_constrs hol_ctxt dataT
  1573     val cases =
  1574       func_ts ~~ xs
  1575       |> map (fn (func_t, x) =>
  1576                  (constr_case_body ctxt stds (dataT :: Ts)
  1577                                    (incr_boundvars 1 func_t, x),
  1578                   discriminate_value hol_ctxt x (Bound 0)))
  1579       |> AList.group (op aconv)
  1580       |> map (apsnd (List.foldl s_disj @{const False}))
  1581       |> sort (int_ord o pairself (size_of_term o snd))
  1582       |> rev
  1583   in
  1584     if res_T = bool_T then
  1585       if forall (member (op =) [@{const False}, @{const True}] o fst) cases then
  1586         case cases of
  1587           [(body_t, _)] => body_t
  1588         | [_, (@{const True}, head_t2)] => head_t2
  1589         | [_, (@{const False}, head_t2)] => @{const Not} $ head_t2
  1590         | _ => raise BAD ("Nitpick_HOL.optimized_case_def", "impossible cases")
  1591       else
  1592         @{const True} |> fold_rev (add_constr_case res_T) cases
  1593     else
  1594       fst (hd cases) |> fold_rev (add_constr_case res_T) (tl cases)
  1595   end
  1596   |> absdummy dataT
  1597 
  1598 fun optimized_record_get (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T res_T t =
  1599   let val constr_x = hd (datatype_constrs hol_ctxt rec_T) in
  1600     case no_of_record_field thy s rec_T of
  1601       ~1 => (case rec_T of
  1602                Type (_, Ts as _ :: _) =>
  1603                let
  1604                  val rec_T' = List.last Ts
  1605                  val j = num_record_fields thy rec_T - 1
  1606                in
  1607                  select_nth_constr_arg ctxt stds constr_x t j res_T
  1608                  |> optimized_record_get hol_ctxt s rec_T' res_T
  1609                end
  1610              | _ => raise TYPE ("Nitpick_HOL.optimized_record_get", [rec_T],
  1611                                 []))
  1612     | j => select_nth_constr_arg ctxt stds constr_x t j res_T
  1613   end
  1614 fun optimized_record_update (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T fun_t
  1615                             rec_t =
  1616   let
  1617     val constr_x as (_, constr_T) = hd (datatype_constrs hol_ctxt rec_T)
  1618     val Ts = binder_types constr_T
  1619     val n = length Ts
  1620     val special_j = no_of_record_field thy s rec_T
  1621     val ts =
  1622       map2 (fn j => fn T =>
  1623                let val t = select_nth_constr_arg ctxt stds constr_x rec_t j T in
  1624                  if j = special_j then
  1625                    s_betapply [] (fun_t, t)
  1626                  else if j = n - 1 andalso special_j = ~1 then
  1627                    optimized_record_update hol_ctxt s
  1628                        (rec_T |> dest_Type |> snd |> List.last) fun_t t
  1629                  else
  1630                    t
  1631                end) (index_seq 0 n) Ts
  1632   in list_comb (Const constr_x, ts) end
  1633 
  1634 (* Prevents divergence in case of cyclic or infinite definition dependencies. *)
  1635 val unfold_max_depth = 255
  1636 
  1637 (* Inline definitions or define as an equational constant? Booleans tend to
  1638    benefit more from inlining, due to the polarity analysis. (However, if
  1639    "total_consts" is set, the polarity analysis is likely not to be so
  1640    crucial.) *)
  1641 val def_inline_threshold_for_booleans = 60
  1642 val def_inline_threshold_for_non_booleans = 20
  1643 
  1644 fun unfold_defs_in_term
  1645         (hol_ctxt as {thy, ctxt, stds, whacks, total_consts, case_names,
  1646                       def_tables, ground_thm_table, ersatz_table, ...}) =
  1647   let
  1648     fun do_numeral depth Ts mult T some_t0 t1 t2 =
  1649       (if is_number_type ctxt T then
  1650          let
  1651            val j = mult * HOLogic.dest_num t2
  1652          in
  1653            if j = 1 then
  1654              raise SAME ()
  1655            else
  1656              let
  1657                val s = numeral_prefix ^ signed_string_of_int j
  1658              in
  1659                if is_integer_like_type T then
  1660                  if is_standard_datatype thy stds T then Const (s, T)
  1661                  else funpow j (curry (op $) (suc_const T)) (zero_const T)
  1662                else
  1663                  do_term depth Ts (Const (@{const_name of_int}, int_T --> T)
  1664                                    $ Const (s, int_T))
  1665              end
  1666          end
  1667          handle TERM _ => raise SAME ()
  1668        else
  1669          raise SAME ())
  1670       handle SAME () => (case some_t0 of NONE => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2)
  1671          | SOME t0 => s_betapply [] (do_term depth Ts t0, s_betapply [] (do_term depth Ts t1, do_term depth Ts t2)))
  1672     and do_term depth Ts t =
  1673       case t of
  1674         (t0 as Const (@{const_name uminus}, _) $ ((t1 as Const (@{const_name numeral},
  1675                       Type (@{type_name fun}, [_, ran_T]))) $ t2)) =>
  1676         do_numeral depth Ts ~1 ran_T (SOME t0) t1 t2
  1677       | (t1 as Const (@{const_name numeral},
  1678                       Type (@{type_name fun}, [_, ran_T]))) $ t2 =>
  1679         do_numeral depth Ts 1 ran_T NONE t1 t2
  1680       | Const (@{const_name refl_on}, T) $ Const (@{const_name top}, _) $ t2 =>
  1681         do_const depth Ts t (@{const_name refl'}, range_type T) [t2]
  1682       | (t0 as Const (@{const_name Sigma}, Type (_, [T1, Type (_, [T2, T3])])))
  1683         $ t1 $ (t2 as Abs (_, _, t2')) =>
  1684         if loose_bvar1 (t2', 0) then
  1685           s_betapplys Ts (do_term depth Ts t0, map (do_term depth Ts) [t1, t2])
  1686         else
  1687           do_term depth Ts
  1688                   (Const (@{const_name prod}, T1 --> range_type T2 --> T3)
  1689                    $ t1 $ incr_boundvars ~1 t2')
  1690       | Const (x as (@{const_name distinct},
  1691                Type (@{type_name fun}, [Type (@{type_name list}, [T']), _])))
  1692         $ (t1 as _ $ _) =>
  1693         (t1 |> HOLogic.dest_list |> distinctness_formula T'
  1694          handle TERM _ => do_const depth Ts t x [t1])
  1695       | Const (x as (@{const_name If}, _)) $ t1 $ t2 $ t3 =>
  1696         if is_ground_term t1 andalso
  1697            exists (Pattern.matches thy o rpair t1)
  1698                   (Inttab.lookup_list ground_thm_table (hash_term t1)) then
  1699           do_term depth Ts t2
  1700         else
  1701           do_const depth Ts t x [t1, t2, t3]
  1702       | Const (@{const_name Let}, _) $ t1 $ t2 =>
  1703         s_betapply Ts (pairself (do_term depth Ts) (t2, t1))
  1704       | Const x => do_const depth Ts t x []
  1705       | t1 $ t2 =>
  1706         (case strip_comb t of
  1707            (Const x, ts) => do_const depth Ts t x ts
  1708          | _ => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2))
  1709       | Bound _ => t
  1710       | Abs (s, T, body) => Abs (s, T, do_term depth (T :: Ts) body)
  1711       | _ => if member (term_match thy) whacks t then
  1712                Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1713              else
  1714                t
  1715     and select_nth_constr_arg_with_args _ _ (x as (_, T)) [] n res_T =
  1716         (Abs (Name.uu, body_type T,
  1717               select_nth_constr_arg ctxt stds x (Bound 0) n res_T), [])
  1718       | select_nth_constr_arg_with_args depth Ts x (t :: ts) n res_T =
  1719         (select_nth_constr_arg ctxt stds x (do_term depth Ts t) n res_T, ts)
  1720     and quot_rep_of depth Ts abs_T rep_T ts =
  1721       select_nth_constr_arg_with_args depth Ts
  1722           (@{const_name Quot}, rep_T --> abs_T) ts 0 rep_T
  1723     and do_const depth Ts t (x as (s, T)) ts =
  1724       if member (term_match thy) whacks (Const x) then
  1725         Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1726       else case AList.lookup (op =) ersatz_table s of
  1727         SOME s' =>
  1728         do_const (depth + 1) Ts (list_comb (Const (s', T), ts)) (s', T) ts
  1729       | NONE =>
  1730         let
  1731           fun def_inline_threshold () =
  1732             if is_boolean_type (body_type T) andalso
  1733                total_consts <> SOME true then
  1734               def_inline_threshold_for_booleans
  1735             else
  1736               def_inline_threshold_for_non_booleans
  1737           val (const, ts) =
  1738             if is_built_in_const thy stds x then
  1739               (Const x, ts)
  1740             else case AList.lookup (op =) case_names s of
  1741               SOME n =>
  1742               if length ts < n then
  1743                 (do_term depth Ts (eta_expand Ts t (n - length ts)), [])
  1744               else
  1745                 let
  1746                   val (dataT, res_T) = nth_range_type n T
  1747                                        |> pairf domain_type range_type
  1748                 in
  1749                   (optimized_case_def hol_ctxt Ts dataT res_T
  1750                                       (map (do_term depth Ts) (take n ts)),
  1751                    drop n ts)
  1752                 end
  1753             | _ =>
  1754               if is_constr ctxt stds x then
  1755                 (Const x, ts)
  1756               else if is_stale_constr ctxt x then
  1757                 raise NOT_SUPPORTED ("(non-co)constructors of codatatypes \
  1758                                      \(\"" ^ s ^ "\")")
  1759               else if is_quot_abs_fun ctxt x then
  1760                 let
  1761                   val rep_T = domain_type T
  1762                   val abs_T = range_type T
  1763                 in
  1764                   (Abs (Name.uu, rep_T,
  1765                         Const (@{const_name Quot}, rep_T --> abs_T)
  1766                                $ (Const (quot_normal_name_for_type ctxt abs_T,
  1767                                          rep_T --> rep_T) $ Bound 0)), ts)
  1768                 end
  1769               else if is_quot_rep_fun ctxt x then
  1770                 quot_rep_of depth Ts (domain_type T) (range_type T) ts
  1771               else if is_record_get thy x then
  1772                 case length ts of
  1773                   0 => (do_term depth Ts (eta_expand Ts t 1), [])
  1774                 | _ => (optimized_record_get hol_ctxt s (domain_type T)
  1775                             (range_type T) (do_term depth Ts (hd ts)), tl ts)
  1776               else if is_record_update thy x then
  1777                 case length ts of
  1778                   2 => (optimized_record_update hol_ctxt
  1779                             (unsuffix Record.updateN s) (nth_range_type 2 T)
  1780                             (do_term depth Ts (hd ts))
  1781                             (do_term depth Ts (nth ts 1)), [])
  1782                 | n => (do_term depth Ts (eta_expand Ts t (2 - n)), [])
  1783               else if is_abs_fun ctxt x andalso
  1784                       is_quot_type ctxt (range_type T) then
  1785                 let
  1786                   val abs_T = range_type T
  1787                   val rep_T = elem_type (domain_type T)
  1788                   val eps_fun = Const (@{const_name Eps},
  1789                                        (rep_T --> bool_T) --> rep_T)
  1790                   val normal_fun =
  1791                     Const (quot_normal_name_for_type ctxt abs_T,
  1792                            rep_T --> rep_T)
  1793                   val abs_fun = Const (@{const_name Quot}, rep_T --> abs_T)
  1794                   val pred =
  1795                     Abs (Name.uu, rep_T,
  1796                          Const (@{const_name Set.member},
  1797                                 rep_T --> domain_type T --> bool_T)
  1798                          $ Bound 0 $ Bound 1)
  1799                 in
  1800                   (Abs (Name.uu, HOLogic.mk_setT rep_T,
  1801                         abs_fun $ (normal_fun $ (eps_fun $ pred)))
  1802                    |> do_term (depth + 1) Ts, ts)
  1803                 end
  1804               else if is_rep_fun ctxt x then
  1805                 let val x' = mate_of_rep_fun ctxt x in
  1806                   if is_constr ctxt stds x' then
  1807                     select_nth_constr_arg_with_args depth Ts x' ts 0
  1808                                                     (range_type T)
  1809                   else if is_quot_type ctxt (domain_type T) then
  1810                     let
  1811                       val abs_T = domain_type T
  1812                       val rep_T = elem_type (range_type T)
  1813                       val (rep_fun, _) = quot_rep_of depth Ts abs_T rep_T []
  1814                       val (equiv_rel, _) =
  1815                         equiv_relation_for_quot_type ctxt abs_T
  1816                     in
  1817                       (Abs (Name.uu, abs_T,
  1818                             HOLogic.Collect_const rep_T
  1819                             $ (equiv_rel $ (rep_fun $ Bound 0))),
  1820                        ts)
  1821                     end
  1822                   else
  1823                     (Const x, ts)
  1824                 end
  1825               else if is_equational_fun_but_no_plain_def hol_ctxt x orelse
  1826                       is_choice_spec_fun hol_ctxt x then
  1827                 (Const x, ts)
  1828               else case def_of_const_ext thy def_tables x of
  1829                 SOME (unfold, def) =>
  1830                 if depth > unfold_max_depth then
  1831                   raise TOO_LARGE ("Nitpick_HOL.unfold_defs_in_term",
  1832                                    "too many nested definitions (" ^
  1833                                    string_of_int depth ^ ") while expanding " ^
  1834                                    quote s)
  1835                 else if s = @{const_name wfrec'} then
  1836                   (do_term (depth + 1) Ts (s_betapplys Ts (def, ts)), [])
  1837                 else if not unfold andalso
  1838                      size_of_term def > def_inline_threshold () then
  1839                   (Const x, ts)
  1840                 else
  1841                   (do_term (depth + 1) Ts def, ts)
  1842               | NONE => (Const x, ts)
  1843         in
  1844           s_betapplys Ts (const, map (do_term depth Ts) ts)
  1845           |> s_beta_norm Ts
  1846         end
  1847   in do_term 0 [] end
  1848 
  1849 (** Axiom extraction/generation **)
  1850 
  1851 fun extensional_equal j T t1 t2 =
  1852   if is_fun_type T then
  1853     let
  1854       val dom_T = pseudo_domain_type T
  1855       val ran_T = pseudo_range_type T
  1856       val var_t = Var (("x", j), dom_T)
  1857     in
  1858       extensional_equal (j + 1) ran_T (betapply (t1, var_t))
  1859                         (betapply (t2, var_t))
  1860     end
  1861   else
  1862     Const (@{const_name HOL.eq}, T --> T --> bool_T) $ t1 $ t2
  1863 
  1864 (* FIXME: needed? *)
  1865 fun equationalize_term ctxt tag t =
  1866   let
  1867     val j = maxidx_of_term t + 1
  1868     val (prems, concl) = Logic.strip_horn t
  1869   in
  1870     Logic.list_implies (prems,
  1871         case concl of
  1872           @{const Trueprop} $ (Const (@{const_name HOL.eq}, Type (_, [T, _]))
  1873                                $ t1 $ t2) =>
  1874           @{const Trueprop} $ extensional_equal j T t1 t2
  1875         | @{const Trueprop} $ t' =>
  1876           @{const Trueprop} $ HOLogic.mk_eq (t', @{const True})
  1877         | Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
  1878           @{const Trueprop} $ extensional_equal j T t1 t2
  1879         | _ => (warning ("Ignoring " ^ quote tag ^ " for non-equation " ^
  1880                          quote (Syntax.string_of_term ctxt t) ^ ".");
  1881                 raise SAME ()))
  1882     |> SOME
  1883   end
  1884   handle SAME () => NONE
  1885 
  1886 fun pair_for_prop t =
  1887   case term_under_def t of
  1888     Const (s, _) => (s, t)
  1889   | t' => raise TERM ("Nitpick_HOL.pair_for_prop", [t, t'])
  1890 
  1891 fun def_table_for get ctxt subst =
  1892   ctxt |> get |> map (pair_for_prop o subst_atomic subst)
  1893        |> AList.group (op =) |> Symtab.make
  1894 
  1895 fun const_def_tables ctxt subst ts =
  1896   (def_table_for (map prop_of o Nitpick_Unfolds.get) ctxt subst,
  1897    fold (fn (s, t) => Symtab.map_default (s, []) (cons t))
  1898         (map pair_for_prop ts) Symtab.empty)
  1899 
  1900 fun paired_with_consts t = map (rpair t) (Term.add_const_names t [])
  1901 fun const_nondef_table ts =
  1902   fold (append o paired_with_consts) ts [] |> AList.group (op =) |> Symtab.make
  1903 
  1904 fun const_simp_table ctxt =
  1905   def_table_for (map_filter (equationalize_term ctxt "nitpick_simp" o prop_of)
  1906                  o Nitpick_Simps.get) ctxt
  1907 fun const_psimp_table ctxt =
  1908   def_table_for (map_filter (equationalize_term ctxt "nitpick_psimp" o prop_of)
  1909                  o Nitpick_Psimps.get) ctxt
  1910 
  1911 fun const_choice_spec_table ctxt subst =
  1912   map (subst_atomic subst o prop_of) (Nitpick_Choice_Specs.get ctxt)
  1913   |> const_nondef_table
  1914 
  1915 fun inductive_intro_table ctxt subst def_tables =
  1916   let val thy = Proof_Context.theory_of ctxt in
  1917     def_table_for
  1918         (maps (map (unfold_mutually_inductive_preds thy def_tables o prop_of)
  1919                o snd o snd)
  1920          o filter (fn (cat, _) => cat = Spec_Rules.Inductive orelse
  1921                                   cat = Spec_Rules.Co_Inductive)
  1922          o Spec_Rules.get) ctxt subst
  1923   end
  1924 
  1925 fun ground_theorem_table thy =
  1926   fold ((fn @{const Trueprop} $ t1 =>
  1927             is_ground_term t1 ? Inttab.map_default (hash_term t1, []) (cons t1)
  1928           | _ => I) o prop_of o snd) (Global_Theory.all_thms_of thy) Inttab.empty
  1929 
  1930 fun ersatz_table ctxt =
  1931  #ersatz_table (Data.get (Context.Proof ctxt))
  1932  |> fold (append o snd) (#frac_types (Data.get (Context.Proof ctxt)))
  1933 
  1934 fun add_simps simp_table s eqs =
  1935   Unsynchronized.change simp_table
  1936       (Symtab.update (s, eqs @ these (Symtab.lookup (!simp_table) s)))
  1937 
  1938 fun inverse_axioms_for_rep_fun ctxt (x as (_, T)) =
  1939   let
  1940     val thy = Proof_Context.theory_of ctxt
  1941     val abs_T = domain_type T
  1942   in
  1943     typedef_info ctxt (fst (dest_Type abs_T)) |> the
  1944     |> pairf #Abs_inverse #Rep_inverse
  1945     |> pairself (specialize_type thy x o prop_of o the)
  1946     ||> single |> op ::
  1947   end
  1948 fun optimized_typedef_axioms ctxt (abs_z as (abs_s, _)) =
  1949   let
  1950     val thy = Proof_Context.theory_of ctxt
  1951     val abs_T = Type abs_z
  1952   in
  1953     if is_univ_typedef ctxt abs_T then
  1954       []
  1955     else case typedef_info ctxt abs_s of
  1956       SOME {abs_type, rep_type, Rep_name, prop_of_Rep, ...} =>
  1957       let
  1958         val rep_T = varify_and_instantiate_type ctxt abs_type abs_T rep_type
  1959         val rep_t = Const (Rep_name, abs_T --> rep_T)
  1960         val set_t =
  1961           prop_of_Rep |> HOLogic.dest_Trueprop
  1962                       |> specialize_type thy (dest_Const rep_t)
  1963                       |> HOLogic.dest_mem |> snd
  1964       in
  1965         [HOLogic.all_const abs_T
  1966              $ Abs (Name.uu, abs_T, HOLogic.mk_mem (rep_t $ Bound 0, set_t))
  1967          |> HOLogic.mk_Trueprop]
  1968       end
  1969     | NONE => []
  1970   end
  1971 fun optimized_quot_type_axioms ctxt stds abs_z =
  1972   let
  1973     val abs_T = Type abs_z
  1974     val rep_T = rep_type_for_quot_type ctxt abs_T
  1975     val (equiv_rel, partial) = equiv_relation_for_quot_type ctxt abs_T
  1976     val a_var = Var (("a", 0), abs_T)
  1977     val x_var = Var (("x", 0), rep_T)
  1978     val y_var = Var (("y", 0), rep_T)
  1979     val x = (@{const_name Quot}, rep_T --> abs_T)
  1980     val sel_a_t = select_nth_constr_arg ctxt stds x a_var 0 rep_T
  1981     val normal_fun =
  1982       Const (quot_normal_name_for_type ctxt abs_T, rep_T --> rep_T)
  1983     val normal_x = normal_fun $ x_var
  1984     val normal_y = normal_fun $ y_var
  1985     val is_unknown_t = Const (@{const_name is_unknown}, rep_T --> bool_T)
  1986   in
  1987     [Logic.mk_equals (normal_fun $ sel_a_t, sel_a_t),
  1988      Logic.list_implies
  1989          ([@{const Not} $ (is_unknown_t $ normal_x),
  1990            @{const Not} $ (is_unknown_t $ normal_y),
  1991            equiv_rel $ x_var $ y_var] |> map HOLogic.mk_Trueprop,
  1992            Logic.mk_equals (normal_x, normal_y)),
  1993      Logic.list_implies
  1994          ([HOLogic.mk_Trueprop (@{const Not} $ (is_unknown_t $ normal_x)),
  1995            HOLogic.mk_Trueprop (@{const Not} $ HOLogic.mk_eq (normal_x, x_var))],
  1996           HOLogic.mk_Trueprop (equiv_rel $ x_var $ normal_x))]
  1997     |> partial ? cons (HOLogic.mk_Trueprop (equiv_rel $ sel_a_t $ sel_a_t))
  1998   end
  1999 
  2000 fun codatatype_bisim_axioms (hol_ctxt as {ctxt, stds, ...}) T =
  2001   let
  2002     val xs = datatype_constrs hol_ctxt T
  2003     val pred_T = T --> bool_T
  2004     val iter_T = @{typ bisim_iterator}
  2005     val bisim_max = @{const bisim_iterator_max}
  2006     val n_var = Var (("n", 0), iter_T)
  2007     val n_var_minus_1 =
  2008       Const (@{const_name safe_The}, (iter_T --> bool_T) --> iter_T)
  2009       $ Abs ("m", iter_T, HOLogic.eq_const iter_T
  2010                           $ (suc_const iter_T $ Bound 0) $ n_var)
  2011     val x_var = Var (("x", 0), T)
  2012     val y_var = Var (("y", 0), T)
  2013     fun bisim_const T = Const (@{const_name bisim}, [iter_T, T, T] ---> bool_T)
  2014     fun nth_sub_bisim x n nth_T =
  2015       (if is_codatatype ctxt nth_T then bisim_const nth_T $ n_var_minus_1
  2016        else HOLogic.eq_const nth_T)
  2017       $ select_nth_constr_arg ctxt stds x x_var n nth_T
  2018       $ select_nth_constr_arg ctxt stds x y_var n nth_T
  2019     fun case_func (x as (_, T)) =
  2020       let
  2021         val arg_Ts = binder_types T
  2022         val core_t =
  2023           discriminate_value hol_ctxt x y_var ::
  2024           map2 (nth_sub_bisim x) (index_seq 0 (length arg_Ts)) arg_Ts
  2025           |> foldr1 s_conj
  2026       in fold_rev absdummy arg_Ts core_t end
  2027   in
  2028     [HOLogic.mk_imp
  2029        (HOLogic.mk_disj (HOLogic.eq_const iter_T $ n_var $ zero_const iter_T,
  2030             s_betapply [] (optimized_case_def hol_ctxt [] T bool_T
  2031                                               (map case_func xs), x_var)),
  2032         bisim_const T $ n_var $ x_var $ y_var),
  2033      HOLogic.eq_const pred_T $ (bisim_const T $ bisim_max $ x_var)
  2034      $ Abs (Name.uu, T, HOLogic.mk_eq (x_var, Bound 0))]
  2035     |> map HOLogic.mk_Trueprop
  2036   end
  2037 
  2038 exception NO_TRIPLE of unit
  2039 
  2040 fun triple_for_intro_rule thy x t =
  2041   let
  2042     val prems = Logic.strip_imp_prems t |> map (Object_Logic.atomize_term thy)
  2043     val concl = Logic.strip_imp_concl t |> Object_Logic.atomize_term thy
  2044     val (main, side) = List.partition (exists_Const (curry (op =) x)) prems
  2045     val is_good_head = curry (op =) (Const x) o head_of
  2046   in
  2047     if forall is_good_head main then (side, main, concl) else raise NO_TRIPLE ()
  2048   end
  2049 
  2050 val tuple_for_args = HOLogic.mk_tuple o snd o strip_comb
  2051 fun wf_constraint_for rel side concl main =
  2052   let
  2053     val core = HOLogic.mk_mem (HOLogic.mk_prod
  2054                                (pairself tuple_for_args (main, concl)), Var rel)
  2055     val t = List.foldl HOLogic.mk_imp core side
  2056     val vars = filter_out (curry (op =) rel) (Term.add_vars t [])
  2057   in
  2058     Library.foldl (fn (t', ((x, j), T)) =>
  2059                       HOLogic.all_const T
  2060                       $ Abs (x, T, abstract_over (Var ((x, j), T), t')))
  2061                   (t, vars)
  2062   end
  2063 fun wf_constraint_for_triple rel (side, main, concl) =
  2064   map (wf_constraint_for rel side concl) main |> foldr1 s_conj
  2065 
  2066 fun terminates_by ctxt timeout goal tac =
  2067   can (SINGLE (Classical.safe_tac ctxt) #> the
  2068        #> SINGLE (DETERM_TIMEOUT timeout (tac ctxt (auto_tac ctxt)))
  2069        #> the #> Goal.finish ctxt) goal
  2070 
  2071 val max_cached_wfs = 50
  2072 val cached_timeout = Synchronized.var "Nitpick_HOL.cached_timeout" Time.zeroTime
  2073 val cached_wf_props =
  2074   Synchronized.var "Nitpick_HOL.cached_wf_props" ([] : (term * bool) list)
  2075 
  2076 val termination_tacs = [Lexicographic_Order.lex_order_tac true,
  2077                         ScnpReconstruct.sizechange_tac]
  2078 
  2079 fun uncached_is_well_founded_inductive_pred
  2080         ({thy, ctxt, stds, debug, tac_timeout, intro_table, ...} : hol_context)
  2081         (x as (_, T)) =
  2082   case def_props_for_const thy stds intro_table x of
  2083     [] => raise TERM ("Nitpick_HOL.uncached_is_well_founded_inductive",
  2084                       [Const x])
  2085   | intro_ts =>
  2086     (case map (triple_for_intro_rule thy x) intro_ts
  2087           |> filter_out (null o #2) of
  2088        [] => true
  2089      | triples =>
  2090        let
  2091          val binders_T = HOLogic.mk_tupleT (binder_types T)
  2092          val rel_T = HOLogic.mk_setT (HOLogic.mk_prodT (binders_T, binders_T))
  2093          val j = fold Integer.max (map maxidx_of_term intro_ts) 0 + 1
  2094          val rel = (("R", j), rel_T)
  2095          val prop = Const (@{const_name wf}, rel_T --> bool_T) $ Var rel ::
  2096                     map (wf_constraint_for_triple rel) triples
  2097                     |> foldr1 s_conj |> HOLogic.mk_Trueprop
  2098          val _ = if debug then
  2099                    Output.urgent_message ("Wellfoundedness goal: " ^
  2100                              Syntax.string_of_term ctxt prop ^ ".")
  2101                  else
  2102                    ()
  2103        in
  2104          if tac_timeout = Synchronized.value cached_timeout andalso
  2105             length (Synchronized.value cached_wf_props) < max_cached_wfs then
  2106            ()
  2107          else
  2108            (Synchronized.change cached_wf_props (K []);
  2109             Synchronized.change cached_timeout (K tac_timeout));
  2110          case AList.lookup (op =) (Synchronized.value cached_wf_props) prop of
  2111            SOME wf => wf
  2112          | NONE =>
  2113            let
  2114              val goal = prop |> cterm_of thy |> Goal.init
  2115              val wf = exists (terminates_by ctxt tac_timeout goal)
  2116                              termination_tacs
  2117            in Synchronized.change cached_wf_props (cons (prop, wf)); wf end
  2118        end)
  2119     handle List.Empty => false | NO_TRIPLE () => false
  2120 
  2121 (* The type constraint below is a workaround for a Poly/ML crash. *)
  2122 
  2123 fun is_well_founded_inductive_pred
  2124         (hol_ctxt as {thy, wfs, def_tables, wf_cache, ...} : hol_context)
  2125         (x as (s, _)) =
  2126   case triple_lookup (const_match thy) wfs x of
  2127     SOME (SOME b) => b
  2128   | _ => s = @{const_name Nats} orelse s = @{const_name fold_graph'} orelse
  2129          case AList.lookup (op =) (!wf_cache) x of
  2130            SOME (_, wf) => wf
  2131          | NONE =>
  2132            let
  2133              val gfp = (fixpoint_kind_of_const thy def_tables x = Gfp)
  2134              val wf = uncached_is_well_founded_inductive_pred hol_ctxt x
  2135            in
  2136              Unsynchronized.change wf_cache (cons (x, (gfp, wf))); wf
  2137            end
  2138 
  2139 fun ap_curry [_] _ t = t
  2140   | ap_curry arg_Ts tuple_T t =
  2141     let val n = length arg_Ts in
  2142       fold_rev (Term.abs o pair "c") arg_Ts
  2143                 (incr_boundvars n t $ mk_flat_tuple tuple_T (map Bound (n - 1 downto 0)))
  2144     end
  2145 
  2146 fun num_occs_of_bound_in_term j (t1 $ t2) =
  2147     op + (pairself (num_occs_of_bound_in_term j) (t1, t2))
  2148   | num_occs_of_bound_in_term j (Abs (_, _, t')) =
  2149     num_occs_of_bound_in_term (j + 1) t'
  2150   | num_occs_of_bound_in_term j (Bound j') = if j' = j then 1 else 0
  2151   | num_occs_of_bound_in_term _ _ = 0
  2152 
  2153 val is_linear_inductive_pred_def =
  2154   let
  2155     fun do_disjunct j (Const (@{const_name Ex}, _) $ Abs (_, _, t2)) =
  2156         do_disjunct (j + 1) t2
  2157       | do_disjunct j t =
  2158         case num_occs_of_bound_in_term j t of
  2159           0 => true
  2160         | 1 => exists (curry (op =) (Bound j) o head_of) (conjuncts_of t)
  2161         | _ => false
  2162     fun do_lfp_def (Const (@{const_name lfp}, _) $ t2) =
  2163         let val (xs, body) = strip_abs t2 in
  2164           case length xs of
  2165             1 => false
  2166           | n => forall (do_disjunct (n - 1)) (disjuncts_of body)
  2167         end
  2168       | do_lfp_def _ = false
  2169   in do_lfp_def o strip_abs_body end
  2170 
  2171 fun n_ptuple_paths 0 = []
  2172   | n_ptuple_paths 1 = []
  2173   | n_ptuple_paths n = [] :: map (cons 2) (n_ptuple_paths (n - 1))
  2174 val ap_n_split = HOLogic.mk_psplits o n_ptuple_paths
  2175 
  2176 val linear_pred_base_and_step_rhss =
  2177   let
  2178     fun aux (Const (@{const_name lfp}, _) $ t2) =
  2179         let
  2180           val (xs, body) = strip_abs t2
  2181           val arg_Ts = map snd (tl xs)
  2182           val tuple_T = HOLogic.mk_tupleT arg_Ts
  2183           val j = length arg_Ts
  2184           fun repair_rec j (Const (@{const_name Ex}, T1) $ Abs (s2, T2, t2')) =
  2185               Const (@{const_name Ex}, T1)
  2186               $ Abs (s2, T2, repair_rec (j + 1) t2')
  2187             | repair_rec j (@{const HOL.conj} $ t1 $ t2) =
  2188               @{const HOL.conj} $ repair_rec j t1 $ repair_rec j t2
  2189             | repair_rec j t =
  2190               let val (head, args) = strip_comb t in
  2191                 if head = Bound j then
  2192                   HOLogic.eq_const tuple_T $ Bound j
  2193                   $ mk_flat_tuple tuple_T args
  2194                 else
  2195                   t
  2196               end
  2197           val (nonrecs, recs) =
  2198             List.partition (curry (op =) 0 o num_occs_of_bound_in_term j)
  2199                            (disjuncts_of body)
  2200           val base_body = nonrecs |> List.foldl s_disj @{const False}
  2201           val step_body = recs |> map (repair_rec j)
  2202                                |> List.foldl s_disj @{const False}
  2203         in
  2204           (fold_rev Term.abs (tl xs) (incr_bv (~1, j, base_body))
  2205            |> ap_n_split (length arg_Ts) tuple_T bool_T,
  2206            Abs ("y", tuple_T, fold_rev Term.abs (tl xs) step_body
  2207                               |> ap_n_split (length arg_Ts) tuple_T bool_T))
  2208         end
  2209       | aux t =
  2210         raise TERM ("Nitpick_HOL.linear_pred_base_and_step_rhss.aux", [t])
  2211   in aux end
  2212 
  2213 fun predicatify T t =
  2214   let val set_T = HOLogic.mk_setT T in
  2215     Abs (Name.uu, T,
  2216          Const (@{const_name Set.member}, T --> set_T --> bool_T)
  2217          $ Bound 0 $ incr_boundvars 1 t)
  2218   end
  2219 
  2220 fun starred_linear_pred_const (hol_ctxt as {simp_table, ...}) (s, T) def =
  2221   let
  2222     val j = maxidx_of_term def + 1
  2223     val (outer, fp_app) = strip_abs def
  2224     val outer_bounds = map Bound (length outer - 1 downto 0)
  2225     val outer_vars = map (fn (s, T) => Var ((s, j), T)) outer
  2226     val fp_app = subst_bounds (rev outer_vars, fp_app)
  2227     val (outer_Ts, rest_T) = strip_n_binders (length outer) T
  2228     val tuple_arg_Ts = strip_type rest_T |> fst
  2229     val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
  2230     val prod_T = HOLogic.mk_prodT (tuple_T, tuple_T)
  2231     val set_T = HOLogic.mk_setT tuple_T
  2232     val rel_T = HOLogic.mk_setT prod_T
  2233     val pred_T = tuple_T --> bool_T
  2234     val curried_T = tuple_T --> pred_T
  2235     val uncurried_T = prod_T --> bool_T
  2236     val (base_rhs, step_rhs) = linear_pred_base_and_step_rhss fp_app
  2237     val base_x as (base_s, _) = (base_prefix ^ s, outer_Ts ---> pred_T)
  2238     val base_eq = HOLogic.mk_eq (list_comb (Const base_x, outer_vars), base_rhs)
  2239                   |> HOLogic.mk_Trueprop
  2240     val _ = add_simps simp_table base_s [base_eq]
  2241     val step_x as (step_s, _) = (step_prefix ^ s, outer_Ts ---> curried_T)
  2242     val step_eq = HOLogic.mk_eq (list_comb (Const step_x, outer_vars), step_rhs)
  2243                   |> HOLogic.mk_Trueprop
  2244     val _ = add_simps simp_table step_s [step_eq]
  2245     val image_const = Const (@{const_name Image}, rel_T --> set_T --> set_T)
  2246     val rtrancl_const = Const (@{const_name rtrancl}, rel_T --> rel_T)
  2247     val base_set =
  2248       HOLogic.Collect_const tuple_T $ list_comb (Const base_x, outer_bounds)
  2249     val step_set =
  2250       HOLogic.Collect_const prod_T
  2251       $ (Const (@{const_name prod_case}, curried_T --> uncurried_T)
  2252                 $ list_comb (Const step_x, outer_bounds))
  2253     val image_set =
  2254       image_const $ (rtrancl_const $ step_set) $ base_set
  2255       |> predicatify tuple_T
  2256   in
  2257     fold_rev Term.abs outer (image_set |> ap_curry tuple_arg_Ts tuple_T)
  2258     |> unfold_defs_in_term hol_ctxt
  2259   end
  2260 
  2261 fun is_good_starred_linear_pred_type (Type (@{type_name fun}, Ts)) =
  2262     forall (not o (is_fun_or_set_type orf is_pair_type)) Ts
  2263   | is_good_starred_linear_pred_type _ = false
  2264 
  2265 fun unrolled_inductive_pred_const (hol_ctxt as {thy, star_linear_preds,
  2266                                                 def_tables, simp_table, ...})
  2267                                   gfp (x as (s, T)) =
  2268   let
  2269     val iter_T = iterator_type_for_const gfp x
  2270     val x' as (s', _) = (unrolled_prefix ^ s, iter_T --> T)
  2271     val unrolled_const = Const x' $ zero_const iter_T
  2272     val def = the (def_of_const thy def_tables x)
  2273   in
  2274     if is_equational_fun_but_no_plain_def hol_ctxt x' then
  2275       unrolled_const (* already done *)
  2276     else if not gfp andalso star_linear_preds andalso
  2277          is_linear_inductive_pred_def def andalso
  2278          is_good_starred_linear_pred_type T then
  2279       starred_linear_pred_const hol_ctxt x def
  2280     else
  2281       let
  2282         val j = maxidx_of_term def + 1
  2283         val (outer, fp_app) = strip_abs def
  2284         val outer_bounds = map Bound (length outer - 1 downto 0)
  2285         val cur = Var ((iter_var_prefix, j + 1), iter_T)
  2286         val next = suc_const iter_T $ cur
  2287         val rhs =
  2288           case fp_app of
  2289             Const _ $ t =>
  2290             s_betapply [] (t, list_comb (Const x', next :: outer_bounds))
  2291           | _ => raise TERM ("Nitpick_HOL.unrolled_inductive_pred_const",
  2292                              [fp_app])
  2293         val (inner, naked_rhs) = strip_abs rhs
  2294         val all = outer @ inner
  2295         val bounds = map Bound (length all - 1 downto 0)
  2296         val vars = map (fn (s, T) => Var ((s, j), T)) all
  2297         val eq = HOLogic.mk_eq (list_comb (Const x', cur :: bounds), naked_rhs)
  2298                  |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2299         val _ = add_simps simp_table s' [eq]
  2300       in unrolled_const end
  2301   end
  2302 
  2303 fun raw_inductive_pred_axiom ({thy, def_tables, ...} : hol_context) x =
  2304   let
  2305     val def = the (def_of_const thy def_tables x)
  2306     val (outer, fp_app) = strip_abs def
  2307     val outer_bounds = map Bound (length outer - 1 downto 0)
  2308     val rhs =
  2309       case fp_app of
  2310         Const _ $ t => s_betapply [] (t, list_comb (Const x, outer_bounds))
  2311       | _ => raise TERM ("Nitpick_HOL.raw_inductive_pred_axiom", [fp_app])
  2312     val (inner, naked_rhs) = strip_abs rhs
  2313     val all = outer @ inner
  2314     val bounds = map Bound (length all - 1 downto 0)
  2315     val j = maxidx_of_term def + 1
  2316     val vars = map (fn (s, T) => Var ((s, j), T)) all
  2317   in
  2318     HOLogic.mk_eq (list_comb (Const x, bounds), naked_rhs)
  2319     |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2320   end
  2321 fun inductive_pred_axiom hol_ctxt (x as (s, T)) =
  2322   if String.isPrefix ubfp_prefix s orelse String.isPrefix lbfp_prefix s then
  2323     let val x' = (strip_first_name_sep s |> snd, T) in
  2324       raw_inductive_pred_axiom hol_ctxt x' |> subst_atomic [(Const x', Const x)]
  2325     end
  2326   else
  2327     raw_inductive_pred_axiom hol_ctxt x
  2328 
  2329 fun equational_fun_axioms (hol_ctxt as {thy, ctxt, stds, def_tables, simp_table,
  2330                                         psimp_table, ...}) x =
  2331   case def_props_for_const thy stds (!simp_table) x of
  2332     [] => (case def_props_for_const thy stds psimp_table x of
  2333              [] => (if is_inductive_pred hol_ctxt x then
  2334                       [inductive_pred_axiom hol_ctxt x]
  2335                     else case def_of_const thy def_tables x of
  2336                       SOME def =>
  2337                       @{const Trueprop} $ HOLogic.mk_eq (Const x, def)
  2338                       |> equationalize_term ctxt "" |> the |> single
  2339                     | NONE => [])
  2340            | psimps => psimps)
  2341   | simps => simps
  2342 fun is_equational_fun_surely_complete hol_ctxt x =
  2343   case equational_fun_axioms hol_ctxt x of
  2344     [@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)] =>
  2345     strip_comb t1 |> snd |> forall is_Var
  2346   | _ => false
  2347 
  2348 (** Type preprocessing **)
  2349 
  2350 fun merged_type_var_table_for_terms thy ts =
  2351   let
  2352     fun add (s, S) table =
  2353       table
  2354       |> (case AList.lookup (Sign.subsort thy o swap) table S of
  2355             SOME _ => I
  2356           | NONE =>
  2357             filter_out (fn (S', _) => Sign.subsort thy (S, S'))
  2358             #> cons (S, s))
  2359     val tfrees = [] |> fold Term.add_tfrees ts
  2360                     |> sort (string_ord o pairself fst)
  2361   in [] |> fold add tfrees |> rev end
  2362 
  2363 fun merge_type_vars_in_term thy merge_type_vars table =
  2364   merge_type_vars
  2365   ? map_types (map_atyps
  2366         (fn TFree (_, S) =>
  2367             TFree (table |> find_first (fn (S', _) => Sign.subsort thy (S', S))
  2368                          |> the |> swap)
  2369           | T => T))
  2370 
  2371 fun add_ground_types hol_ctxt binarize =
  2372   let
  2373     fun aux T accum =
  2374       case T of
  2375         Type (@{type_name fun}, Ts) => fold aux Ts accum
  2376       | Type (@{type_name prod}, Ts) => fold aux Ts accum
  2377       | Type (@{type_name set}, Ts) => fold aux Ts accum
  2378       | Type (@{type_name itself}, [T1]) => aux T1 accum
  2379       | Type (_, Ts) =>
  2380         if member (op =) (@{typ prop} :: @{typ bool} :: accum) T then
  2381           accum
  2382         else
  2383           T :: accum
  2384           |> fold aux (case binarized_and_boxed_datatype_constrs hol_ctxt
  2385                                                                  binarize T of
  2386                          [] => Ts
  2387                        | xs => map snd xs)
  2388       | _ => insert (op =) T accum
  2389   in aux end
  2390 fun ground_types_in_type hol_ctxt binarize T =
  2391   add_ground_types hol_ctxt binarize T []
  2392 fun ground_types_in_terms hol_ctxt binarize ts =
  2393   fold (fold_types (add_ground_types hol_ctxt binarize)) ts []
  2394 
  2395 end;