src/HOL/Tools/Nitpick/nitpick_hol.ML
author blanchet
Tue Dec 07 11:56:01 2010 +0100 (2010-12-07)
changeset 41047 9f1d3fcef1ca
parent 41046 f2e94005d283
child 41049 0edd245892ed
permissions -rw-r--r--
simplified special handling of set products
     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      tac_timeout: Time.time option,
    31      evals: term list,
    32      case_names: (string * int) list,
    33      def_table: const_table,
    34      nondef_table: const_table,
    35      user_nondefs: term list,
    36      simp_table: const_table Unsynchronized.ref,
    37      psimp_table: const_table,
    38      choice_spec_table: const_table,
    39      intro_table: const_table,
    40      ground_thm_table: term list Inttab.table,
    41      ersatz_table: (string * string) list,
    42      skolems: (string * string list) list Unsynchronized.ref,
    43      special_funs: special_fun list Unsynchronized.ref,
    44      unrolled_preds: unrolled list Unsynchronized.ref,
    45      wf_cache: wf_cache Unsynchronized.ref,
    46      constr_cache: (typ * styp list) list Unsynchronized.ref}
    47 
    48   datatype fixpoint_kind = Lfp | Gfp | NoFp
    49   datatype boxability =
    50     InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
    51 
    52   val name_sep : string
    53   val numeral_prefix : string
    54   val base_prefix : string
    55   val step_prefix : string
    56   val unrolled_prefix : string
    57   val ubfp_prefix : string
    58   val lbfp_prefix : string
    59   val quot_normal_prefix : string
    60   val skolem_prefix : string
    61   val special_prefix : string
    62   val uncurry_prefix : string
    63   val eval_prefix : string
    64   val iter_var_prefix : string
    65   val strip_first_name_sep : string -> string * string
    66   val original_name : string -> string
    67   val abs_var : indexname * typ -> term -> term
    68   val is_higher_order_type : typ -> bool
    69   val s_let : string -> int -> typ -> typ -> (term -> term) -> term -> term
    70   val s_betapply : typ list -> term * term -> term
    71   val s_betapplys : typ list -> term * term list -> term
    72   val s_conj : term * term -> term
    73   val s_disj : term * term -> term
    74   val strip_any_connective : term -> term list * term
    75   val conjuncts_of : term -> term list
    76   val disjuncts_of : term -> term list
    77   val unarize_unbox_etc_type : typ -> typ
    78   val uniterize_unarize_unbox_etc_type : typ -> typ
    79   val string_for_type : Proof.context -> typ -> string
    80   val pretty_for_type : Proof.context -> typ -> Pretty.T
    81   val prefix_name : string -> string -> string
    82   val shortest_name : string -> string
    83   val short_name : string -> string
    84   val shorten_names_in_term : term -> term
    85   val strict_type_match : theory -> typ * typ -> bool
    86   val type_match : theory -> typ * typ -> bool
    87   val const_match : theory -> styp * styp -> bool
    88   val term_match : theory -> term * term -> bool
    89   val frac_from_term_pair : typ -> term -> term -> term
    90   val is_TFree : typ -> bool
    91   val is_fun_type : typ -> bool
    92   val is_set_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 const_for_iterator_type : typ -> styp
   106   val strip_n_binders : int -> typ -> typ list * typ
   107   val nth_range_type : int -> typ -> typ
   108   val num_factors_in_type : typ -> int
   109   val num_binder_types : typ -> int
   110   val curried_binder_types : typ -> typ list
   111   val mk_flat_tuple : typ -> term list -> term
   112   val dest_n_tuple : int -> term -> term list
   113   val is_real_datatype : theory -> string -> bool
   114   val is_standard_datatype : theory -> (typ option * bool) list -> typ -> bool
   115   val is_codatatype : Proof.context -> typ -> bool
   116   val is_quot_type : Proof.context -> typ -> bool
   117   val is_pure_typedef : Proof.context -> typ -> bool
   118   val is_univ_typedef : Proof.context -> typ -> bool
   119   val is_datatype : Proof.context -> (typ option * bool) list -> typ -> bool
   120   val is_record_constr : styp -> bool
   121   val is_record_get : theory -> styp -> bool
   122   val is_record_update : theory -> styp -> bool
   123   val is_abs_fun : Proof.context -> styp -> bool
   124   val is_rep_fun : Proof.context -> styp -> bool
   125   val is_quot_abs_fun : Proof.context -> styp -> bool
   126   val is_quot_rep_fun : Proof.context -> styp -> bool
   127   val mate_of_rep_fun : Proof.context -> styp -> styp
   128   val is_constr_like : Proof.context -> styp -> bool
   129   val is_constr : Proof.context -> (typ option * bool) list -> styp -> bool
   130   val is_sel : string -> bool
   131   val is_sel_like_and_no_discr : string -> bool
   132   val box_type : hol_context -> boxability -> typ -> typ
   133   val binarize_nat_and_int_in_type : typ -> typ
   134   val binarize_nat_and_int_in_term : term -> term
   135   val discr_for_constr : styp -> styp
   136   val num_sels_for_constr_type : typ -> int
   137   val nth_sel_name_for_constr_name : string -> int -> string
   138   val nth_sel_for_constr : styp -> int -> styp
   139   val binarized_and_boxed_nth_sel_for_constr :
   140     hol_context -> bool -> styp -> int -> styp
   141   val sel_no_from_name : string -> int
   142   val close_form : term -> term
   143   val distinctness_formula : typ -> term list -> term
   144   val register_frac_type :
   145     string -> (string * string) list -> morphism -> Context.generic
   146     -> Context.generic
   147   val register_frac_type_global :
   148     string -> (string * string) list -> theory -> theory
   149   val unregister_frac_type :
   150     string -> morphism -> Context.generic -> Context.generic
   151   val unregister_frac_type_global : string -> theory -> theory
   152   val register_codatatype :
   153     typ -> string -> styp list -> morphism -> Context.generic -> Context.generic
   154   val register_codatatype_global :
   155     typ -> string -> styp list -> theory -> theory
   156   val unregister_codatatype :
   157     typ -> morphism -> Context.generic -> Context.generic
   158   val unregister_codatatype_global : typ -> theory -> theory
   159   val datatype_constrs : hol_context -> typ -> styp list
   160   val binarized_and_boxed_datatype_constrs :
   161     hol_context -> bool -> typ -> styp list
   162   val num_datatype_constrs : hol_context -> typ -> int
   163   val constr_name_for_sel_like : string -> string
   164   val binarized_and_boxed_constr_for_sel : hol_context -> bool -> styp -> styp
   165   val discriminate_value : hol_context -> styp -> term -> term
   166   val select_nth_constr_arg :
   167     Proof.context -> (typ option * bool) list -> styp -> term -> int -> typ
   168     -> term
   169   val construct_value :
   170     Proof.context -> (typ option * bool) list -> styp -> term list -> term
   171   val coerce_term : hol_context -> typ list -> typ -> typ -> term -> term
   172   val card_of_type : (typ * int) list -> typ -> int
   173   val bounded_card_of_type : int -> int -> (typ * int) list -> typ -> int
   174   val bounded_exact_card_of_type :
   175     hol_context -> typ list -> int -> int -> (typ * int) list -> typ -> int
   176   val is_finite_type : hol_context -> typ -> bool
   177   val is_small_finite_type : hol_context -> typ -> bool
   178   val special_bounds : term list -> (indexname * typ) list
   179   val is_funky_typedef : Proof.context -> typ -> bool
   180   val all_axioms_of :
   181     Proof.context -> (term * term) list -> term list * term list * term list
   182   val arity_of_built_in_const :
   183     theory -> (typ option * bool) list -> styp -> int option
   184   val is_built_in_const :
   185     theory -> (typ option * bool) list -> styp -> bool
   186   val term_under_def : term -> term
   187   val case_const_names :
   188     Proof.context -> (typ option * bool) list -> (string * int) list
   189   val unfold_defs_in_term : hol_context -> term -> term
   190   val const_def_table :
   191     Proof.context -> (term * term) list -> term list -> const_table
   192   val const_nondef_table : term list -> const_table
   193   val const_simp_table : Proof.context -> (term * term) list -> const_table
   194   val const_psimp_table : Proof.context -> (term * term) list -> const_table
   195   val const_choice_spec_table :
   196     Proof.context -> (term * term) list -> const_table
   197   val inductive_intro_table :
   198     Proof.context -> (term * term) list -> const_table -> const_table
   199   val ground_theorem_table : theory -> term list Inttab.table
   200   val ersatz_table : Proof.context -> (string * string) list
   201   val add_simps : const_table Unsynchronized.ref -> string -> term list -> unit
   202   val inverse_axioms_for_rep_fun : Proof.context -> styp -> term list
   203   val optimized_typedef_axioms : Proof.context -> string * typ list -> term list
   204   val optimized_quot_type_axioms :
   205     Proof.context -> (typ option * bool) list -> string * typ list -> term list
   206   val def_of_const : theory -> const_table -> styp -> term option
   207   val fixpoint_kind_of_rhs : term -> fixpoint_kind
   208   val fixpoint_kind_of_const :
   209     theory -> const_table -> string * typ -> fixpoint_kind
   210   val is_real_inductive_pred : hol_context -> styp -> bool
   211   val is_constr_pattern_lhs : Proof.context -> term -> bool
   212   val is_constr_pattern_formula : Proof.context -> term -> bool
   213   val nondef_props_for_const :
   214     theory -> bool -> const_table -> styp -> term list
   215   val is_choice_spec_fun : hol_context -> styp -> bool
   216   val is_choice_spec_axiom : theory -> const_table -> term -> bool
   217   val is_real_equational_fun : hol_context -> styp -> bool
   218   val is_equational_fun_but_no_plain_def : hol_context -> styp -> bool
   219   val codatatype_bisim_axioms : hol_context -> typ -> term list
   220   val is_well_founded_inductive_pred : hol_context -> styp -> bool
   221   val unrolled_inductive_pred_const : hol_context -> bool -> styp -> term
   222   val equational_fun_axioms : hol_context -> styp -> term list
   223   val is_equational_fun_surely_complete : hol_context -> styp -> bool
   224   val merged_type_var_table_for_terms :
   225     theory -> term list -> (sort * string) list
   226   val merge_type_vars_in_term :
   227     theory -> bool -> (sort * string) list -> term -> term
   228   val ground_types_in_type : hol_context -> bool -> typ -> typ list
   229   val ground_types_in_terms : hol_context -> bool -> term list -> typ list
   230 end;
   231 
   232 structure Nitpick_HOL : NITPICK_HOL =
   233 struct
   234 
   235 open Nitpick_Util
   236 
   237 type const_table = term list Symtab.table
   238 type special_fun = (styp * int list * term list) * styp
   239 type unrolled = styp * styp
   240 type wf_cache = (styp * (bool * bool)) list
   241 
   242 type hol_context =
   243   {thy: theory,
   244    ctxt: Proof.context,
   245    max_bisim_depth: int,
   246    boxes: (typ option * bool option) list,
   247    stds: (typ option * bool) list,
   248    wfs: (styp option * bool option) list,
   249    user_axioms: bool option,
   250    debug: bool,
   251    whacks: term list,
   252    binary_ints: bool option,
   253    destroy_constrs: bool,
   254    specialize: bool,
   255    star_linear_preds: bool,
   256    tac_timeout: Time.time option,
   257    evals: term list,
   258    case_names: (string * int) list,
   259    def_table: const_table,
   260    nondef_table: const_table,
   261    user_nondefs: term list,
   262    simp_table: const_table Unsynchronized.ref,
   263    psimp_table: const_table,
   264    choice_spec_table: const_table,
   265    intro_table: const_table,
   266    ground_thm_table: term list Inttab.table,
   267    ersatz_table: (string * string) list,
   268    skolems: (string * string list) list Unsynchronized.ref,
   269    special_funs: special_fun list Unsynchronized.ref,
   270    unrolled_preds: unrolled list Unsynchronized.ref,
   271    wf_cache: wf_cache Unsynchronized.ref,
   272    constr_cache: (typ * styp list) list Unsynchronized.ref}
   273 
   274 datatype fixpoint_kind = Lfp | Gfp | NoFp
   275 datatype boxability =
   276   InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
   277 
   278 structure Data = Generic_Data(
   279   type T = {frac_types: (string * (string * string) list) list,
   280             codatatypes: (string * (string * styp list)) list}
   281   val empty = {frac_types = [], codatatypes = []}
   282   val extend = I
   283   fun merge ({frac_types = fs1, codatatypes = cs1},
   284              {frac_types = fs2, codatatypes = cs2}) : T =
   285     {frac_types = AList.merge (op =) (K true) (fs1, fs2),
   286      codatatypes = AList.merge (op =) (K true) (cs1, cs2)})
   287 
   288 val name_sep = "$"
   289 val numeral_prefix = nitpick_prefix ^ "num" ^ name_sep
   290 val sel_prefix = nitpick_prefix ^ "sel"
   291 val discr_prefix = nitpick_prefix ^ "is" ^ name_sep
   292 val set_prefix = nitpick_prefix ^ "set" ^ name_sep
   293 val lfp_iterator_prefix = nitpick_prefix ^ "lfpit" ^ name_sep
   294 val gfp_iterator_prefix = nitpick_prefix ^ "gfpit" ^ name_sep
   295 val unrolled_prefix = nitpick_prefix ^ "unroll" ^ name_sep
   296 val base_prefix = nitpick_prefix ^ "base" ^ name_sep
   297 val step_prefix = nitpick_prefix ^ "step" ^ name_sep
   298 val ubfp_prefix = nitpick_prefix ^ "ubfp" ^ name_sep
   299 val lbfp_prefix = nitpick_prefix ^ "lbfp" ^ name_sep
   300 val quot_normal_prefix = nitpick_prefix ^ "qn" ^ name_sep
   301 val skolem_prefix = nitpick_prefix ^ "sk"
   302 val special_prefix = nitpick_prefix ^ "sp"
   303 val uncurry_prefix = nitpick_prefix ^ "unc"
   304 val eval_prefix = nitpick_prefix ^ "eval"
   305 val iter_var_prefix = "i"
   306 
   307 (** Constant/type information and term/type manipulation **)
   308 
   309 fun sel_prefix_for j = sel_prefix ^ string_of_int j ^ name_sep
   310 fun quot_normal_name_for_type ctxt T =
   311   quot_normal_prefix ^ unyxml (Syntax.string_of_typ ctxt T)
   312 
   313 val strip_first_name_sep =
   314   Substring.full #> Substring.position name_sep ##> Substring.triml 1
   315   #> pairself Substring.string
   316 fun original_name s =
   317   if String.isPrefix nitpick_prefix s then
   318     case strip_first_name_sep s of (s1, "") => s1 | (_, s2) => original_name s2
   319   else
   320     s
   321 
   322 fun is_higher_order_type (Type (@{type_name fun}, _)) = true
   323   | is_higher_order_type (Type (_, Ts)) = exists is_higher_order_type Ts
   324   | is_higher_order_type _ = false
   325 
   326 fun abs_var ((s, j), T) body = Abs (s, T, abstract_over (Var ((s, j), T), body))
   327 
   328 fun let_var s = (nitpick_prefix ^ s, 999)
   329 val let_inline_threshold = 20
   330 
   331 fun s_let s n abs_T body_T f t =
   332   if (n - 1) * (size_of_term t - 1) <= let_inline_threshold orelse
   333      is_higher_order_type abs_T then
   334     f t
   335   else
   336     let val z = (let_var s, abs_T) in
   337       Const (@{const_name Let}, abs_T --> (abs_T --> body_T) --> body_T)
   338       $ t $ abs_var z (incr_boundvars 1 (f (Var z)))
   339     end
   340 
   341 fun loose_bvar1_count (Bound i, k) = if i = k then 1 else 0
   342   | loose_bvar1_count (t1 $ t2, k) =
   343     loose_bvar1_count (t1, k) + loose_bvar1_count (t2, k)
   344   | loose_bvar1_count (Abs (_, _, t), k) = loose_bvar1_count (t, k + 1)
   345   | loose_bvar1_count _ = 0
   346 
   347 fun s_betapply _ (Const (@{const_name If}, _) $ @{const True} $ t1', _) = t1'
   348   | s_betapply _ (Const (@{const_name If}, _) $ @{const False} $ _, t2) = t2
   349   | s_betapply Ts (Const (@{const_name Let},
   350                           Type (_, [bound_T, Type (_, [_, body_T])]))
   351                    $ t12 $ Abs (s, T, t13'), t2) =
   352     let val body_T' = range_type body_T in
   353       Const (@{const_name Let}, bound_T --> (bound_T --> body_T') --> body_T')
   354       $ t12 $ Abs (s, T, s_betapply (T :: Ts) (t13', incr_boundvars 1 t2))
   355     end
   356   | s_betapply Ts (t1 as Abs (s1, T1, t1'), t2) =
   357     (s_let s1 (loose_bvar1_count (t1', 0)) T1 (fastype_of1 (T1 :: Ts, t1'))
   358               (curry betapply t1) t2
   359      handle TERM _ => betapply (t1, t2)) (* FIXME: fix all uses *)
   360   | s_betapply _ (t1, t2) = t1 $ t2
   361 fun s_betapplys Ts = Library.foldl (s_betapply Ts)
   362 
   363 fun s_beta_norm Ts t =
   364   let
   365     fun aux _ (Var _) = raise Same.SAME
   366       | aux Ts (Abs (s, T, t')) = Abs (s, T, aux (T :: Ts) t')
   367       | aux Ts ((t1 as Abs _) $ t2) =
   368         Same.commit (aux Ts) (s_betapply Ts (t1, t2))
   369       | aux Ts (t1 $ t2) =
   370         ((case aux Ts t1 of
   371            t1 as Abs _ => Same.commit (aux Ts) (s_betapply Ts (t1, t2))
   372          | t1 => t1 $ Same.commit (aux Ts) t2)
   373         handle Same.SAME => t1 $ aux Ts t2)
   374       | aux _ _ = raise Same.SAME
   375   in aux Ts t handle Same.SAME => t end
   376 
   377 fun s_conj (t1, @{const True}) = t1
   378   | s_conj (@{const True}, t2) = t2
   379   | s_conj (t1, t2) =
   380     if t1 = @{const False} orelse t2 = @{const False} then @{const False}
   381     else HOLogic.mk_conj (t1, t2)
   382 fun s_disj (t1, @{const False}) = t1
   383   | s_disj (@{const False}, t2) = t2
   384   | s_disj (t1, t2) =
   385     if t1 = @{const True} orelse t2 = @{const True} then @{const True}
   386     else HOLogic.mk_disj (t1, t2)
   387 
   388 fun strip_connective conn_t (t as (t0 $ t1 $ t2)) =
   389     if t0 = conn_t then strip_connective t0 t2 @ strip_connective t0 t1 else [t]
   390   | strip_connective _ t = [t]
   391 fun strip_any_connective (t as (t0 $ _ $ _)) =
   392     if t0 = @{const HOL.conj} orelse t0 = @{const HOL.disj} then
   393       (strip_connective t0 t, t0)
   394     else
   395       ([t], @{const Not})
   396   | strip_any_connective t = ([t], @{const Not})
   397 val conjuncts_of = strip_connective @{const HOL.conj}
   398 val disjuncts_of = strip_connective @{const HOL.disj}
   399 
   400 (* When you add constants to these lists, make sure to handle them in
   401    "Nitpick_Nut.nut_from_term", and perhaps in "Nitpick_Mono.consider_term" as
   402    well. *)
   403 val built_in_consts =
   404   [(@{const_name all}, 1),
   405    (@{const_name "=="}, 2),
   406    (@{const_name "==>"}, 2),
   407    (@{const_name Pure.conjunction}, 2),
   408    (@{const_name Trueprop}, 1),
   409    (@{const_name Not}, 1),
   410    (@{const_name False}, 0),
   411    (@{const_name True}, 0),
   412    (@{const_name All}, 1),
   413    (@{const_name Ex}, 1),
   414    (@{const_name HOL.eq}, 1),
   415    (@{const_name HOL.conj}, 2),
   416    (@{const_name HOL.disj}, 2),
   417    (@{const_name HOL.implies}, 2),
   418    (@{const_name If}, 3),
   419    (@{const_name Let}, 2),
   420    (@{const_name Pair}, 2),
   421    (@{const_name fst}, 1),
   422    (@{const_name snd}, 1),
   423    (@{const_name Id}, 0),
   424    (@{const_name converse}, 1),
   425    (@{const_name trancl}, 1),
   426    (@{const_name rel_comp}, 2),
   427    (@{const_name image}, 2),
   428    (@{const_name finite}, 1),
   429    (@{const_name unknown}, 0),
   430    (@{const_name is_unknown}, 1),
   431    (@{const_name safe_The}, 1),
   432    (@{const_name Frac}, 0),
   433    (@{const_name norm_frac}, 0)]
   434 val built_in_nat_consts =
   435   [(@{const_name Suc}, 0),
   436    (@{const_name nat}, 0),
   437    (@{const_name nat_gcd}, 0),
   438    (@{const_name nat_lcm}, 0)]
   439 val built_in_typed_consts =
   440   [((@{const_name zero_class.zero}, int_T), 0),
   441    ((@{const_name one_class.one}, int_T), 0),
   442    ((@{const_name plus_class.plus}, int_T --> int_T --> int_T), 0),
   443    ((@{const_name minus_class.minus}, int_T --> int_T --> int_T), 0),
   444    ((@{const_name times_class.times}, int_T --> int_T --> int_T), 0),
   445    ((@{const_name div_class.div}, int_T --> int_T --> int_T), 0),
   446    ((@{const_name uminus_class.uminus}, int_T --> int_T), 0),
   447    ((@{const_name ord_class.less}, int_T --> int_T --> bool_T), 2),
   448    ((@{const_name ord_class.less_eq}, int_T --> int_T --> bool_T), 2)]
   449 val built_in_typed_nat_consts =
   450   [((@{const_name zero_class.zero}, nat_T), 0),
   451    ((@{const_name one_class.one}, nat_T), 0),
   452    ((@{const_name plus_class.plus}, nat_T --> nat_T --> nat_T), 0),
   453    ((@{const_name minus_class.minus}, nat_T --> nat_T --> nat_T), 0),
   454    ((@{const_name times_class.times}, nat_T --> nat_T --> nat_T), 0),
   455    ((@{const_name div_class.div}, nat_T --> nat_T --> nat_T), 0),
   456    ((@{const_name ord_class.less}, nat_T --> nat_T --> bool_T), 2),
   457    ((@{const_name ord_class.less_eq}, nat_T --> nat_T --> bool_T), 2),
   458    ((@{const_name of_nat}, nat_T --> int_T), 0)]
   459 val built_in_set_consts =
   460   [(@{const_name ord_class.less_eq}, 2)]
   461 
   462 fun unarize_type @{typ "unsigned_bit word"} = nat_T
   463   | unarize_type @{typ "signed_bit word"} = int_T
   464   | unarize_type (Type (s, Ts as _ :: _)) = Type (s, map unarize_type Ts)
   465   | unarize_type T = T
   466 fun unarize_unbox_etc_type (Type (@{type_name fin_fun}, Ts)) =
   467     unarize_unbox_etc_type (Type (@{type_name fun}, Ts))
   468   | unarize_unbox_etc_type (Type (@{type_name fun_box}, Ts)) =
   469     unarize_unbox_etc_type (Type (@{type_name fun}, Ts))
   470   | unarize_unbox_etc_type (Type (@{type_name pair_box}, Ts)) =
   471     Type (@{type_name prod}, map unarize_unbox_etc_type Ts)
   472   | unarize_unbox_etc_type @{typ "unsigned_bit word"} = nat_T
   473   | unarize_unbox_etc_type @{typ "signed_bit word"} = int_T
   474   | unarize_unbox_etc_type (Type (s, Ts as _ :: _)) =
   475     Type (s, map unarize_unbox_etc_type Ts)
   476   | unarize_unbox_etc_type T = T
   477 fun uniterize_type (Type (s, Ts as _ :: _)) = Type (s, map uniterize_type Ts)
   478   | uniterize_type @{typ bisim_iterator} = nat_T
   479   | uniterize_type T = T
   480 val uniterize_unarize_unbox_etc_type = uniterize_type o unarize_unbox_etc_type
   481 
   482 fun string_for_type ctxt = Syntax.string_of_typ ctxt o unarize_unbox_etc_type
   483 fun pretty_for_type ctxt = Syntax.pretty_typ ctxt o unarize_unbox_etc_type
   484 
   485 val prefix_name = Long_Name.qualify o Long_Name.base_name
   486 fun shortest_name s = List.last (space_explode "." s) handle List.Empty => ""
   487 val prefix_abs_vars = Term.map_abs_vars o prefix_name
   488 fun short_name s =
   489   case space_explode name_sep s of
   490     [_] => s |> String.isPrefix nitpick_prefix s ? unprefix nitpick_prefix
   491   | ss => map shortest_name ss |> space_implode "_"
   492 fun shorten_names_in_type (Type (s, Ts)) =
   493     Type (short_name s, map shorten_names_in_type Ts)
   494   | shorten_names_in_type T = T
   495 val shorten_names_in_term =
   496   map_aterms (fn Const (s, T) => Const (short_name s, T) | t => t)
   497   #> map_types shorten_names_in_type
   498 
   499 fun strict_type_match thy (T1, T2) =
   500   (Sign.typ_match thy (T2, T1) Vartab.empty; true)
   501   handle Type.TYPE_MATCH => false
   502 fun type_match thy = strict_type_match thy o pairself unarize_unbox_etc_type
   503 fun const_match thy ((s1, T1), (s2, T2)) =
   504   s1 = s2 andalso type_match thy (T1, T2)
   505 fun term_match thy (Const x1, Const x2) = const_match thy (x1, x2)
   506   | term_match thy (Free (s1, T1), Free (s2, T2)) =
   507     const_match thy ((shortest_name s1, T1), (shortest_name s2, T2))
   508   | term_match _ (t1, t2) = t1 aconv t2
   509 
   510 fun frac_from_term_pair T t1 t2 =
   511   case snd (HOLogic.dest_number t1) of
   512     0 => HOLogic.mk_number T 0
   513   | n1 => case snd (HOLogic.dest_number t2) of
   514             1 => HOLogic.mk_number T n1
   515           | n2 => Const (@{const_name divide}, T --> T --> T)
   516                   $ HOLogic.mk_number T n1 $ HOLogic.mk_number T n2
   517 
   518 fun is_TFree (TFree _) = true
   519   | is_TFree _ = false
   520 fun is_fun_type (Type (@{type_name fun}, _)) = true
   521   | is_fun_type _ = false
   522 fun is_set_type (Type (@{type_name fun}, [_, @{typ bool}])) = true
   523   | is_set_type _ = false
   524 fun is_pair_type (Type (@{type_name prod}, _)) = true
   525   | is_pair_type _ = false
   526 fun is_lfp_iterator_type (Type (s, _)) = String.isPrefix lfp_iterator_prefix s
   527   | is_lfp_iterator_type _ = false
   528 fun is_gfp_iterator_type (Type (s, _)) = String.isPrefix gfp_iterator_prefix s
   529   | is_gfp_iterator_type _ = false
   530 val is_fp_iterator_type = is_lfp_iterator_type orf is_gfp_iterator_type
   531 fun is_iterator_type T =
   532   (T = @{typ bisim_iterator} orelse is_fp_iterator_type T)
   533 fun is_boolean_type T = (T = prop_T orelse T = bool_T)
   534 fun is_integer_type T = (T = nat_T orelse T = int_T)
   535 fun is_bit_type T = (T = @{typ unsigned_bit} orelse T = @{typ signed_bit})
   536 fun is_word_type (Type (@{type_name word}, _)) = true
   537   | is_word_type _ = false
   538 val is_integer_like_type = is_iterator_type orf is_integer_type orf is_word_type
   539 val is_record_type = not o null o Record.dest_recTs
   540 fun is_frac_type ctxt (Type (s, [])) =
   541     s |> AList.lookup (op =) (#frac_types (Data.get (Context.Proof ctxt)))
   542       |> these |> null |> not
   543   | is_frac_type _ _ = false
   544 fun is_number_type ctxt = is_integer_like_type orf is_frac_type ctxt
   545 
   546 fun iterator_type_for_const gfp (s, T) =
   547   Type ((if gfp then gfp_iterator_prefix else lfp_iterator_prefix) ^ s,
   548         binder_types T)
   549 fun const_for_iterator_type (Type (s, Ts)) =
   550     (strip_first_name_sep s |> snd, Ts ---> bool_T)
   551   | const_for_iterator_type T =
   552     raise TYPE ("Nitpick_HOL.const_for_iterator_type", [T], [])
   553 
   554 fun strip_n_binders 0 T = ([], T)
   555   | strip_n_binders n (Type (@{type_name fun}, [T1, T2])) =
   556     strip_n_binders (n - 1) T2 |>> cons T1
   557   | strip_n_binders n (Type (@{type_name fun_box}, Ts)) =
   558     strip_n_binders n (Type (@{type_name fun}, Ts))
   559   | strip_n_binders _ T = raise TYPE ("Nitpick_HOL.strip_n_binders", [T], [])
   560 val nth_range_type = snd oo strip_n_binders
   561 
   562 fun num_factors_in_type (Type (@{type_name prod}, [T1, T2])) =
   563     fold (Integer.add o num_factors_in_type) [T1, T2] 0
   564   | num_factors_in_type _ = 1
   565 fun num_binder_types (Type (@{type_name fun}, [_, T2])) =
   566     1 + num_binder_types T2
   567   | num_binder_types _ = 0
   568 val curried_binder_types = maps HOLogic.flatten_tupleT o binder_types
   569 fun maybe_curried_binder_types T =
   570   (if is_pair_type (body_type T) then binder_types else curried_binder_types) T
   571 
   572 fun mk_flat_tuple _ [t] = t
   573   | mk_flat_tuple (Type (@{type_name prod}, [T1, T2])) (t :: ts) =
   574     HOLogic.pair_const T1 T2 $ t $ (mk_flat_tuple T2 ts)
   575   | mk_flat_tuple T ts = raise TYPE ("Nitpick_HOL.mk_flat_tuple", [T], ts)
   576 fun dest_n_tuple 1 t = [t]
   577   | dest_n_tuple n t = HOLogic.dest_prod t ||> dest_n_tuple (n - 1) |> op ::
   578 
   579 type typedef_info =
   580   {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string,
   581    set_def: thm option, prop_of_Rep: thm, set_name: string,
   582    Abs_inverse: thm option, Rep_inverse: thm option}
   583 
   584 fun typedef_info ctxt s =
   585   if is_frac_type ctxt (Type (s, [])) then
   586     SOME {abs_type = Type (s, []), rep_type = @{typ "int * int"},
   587           Abs_name = @{const_name Abs_Frac}, Rep_name = @{const_name Rep_Frac},
   588           set_def = NONE, prop_of_Rep = @{prop "Rep_Frac x \<in> Frac"}
   589                           |> Logic.varify_global,
   590           set_name = @{const_name Frac}, Abs_inverse = NONE, Rep_inverse = NONE}
   591   else case Typedef.get_info ctxt s of
   592     (* When several entries are returned, it shouldn't matter much which one
   593        we take (according to Florian Haftmann). *)
   594     (* The "Logic.varifyT_global" calls are a temporary hack because these
   595        types's type variables sometimes clash with locally fixed type variables.
   596        Remove these calls once "Typedef" is fully localized. *)
   597     ({abs_type, rep_type, Abs_name, Rep_name, ...},
   598      {set_def, Rep, Abs_inverse, Rep_inverse, ...}) :: _ =>
   599     SOME {abs_type = Logic.varifyT_global abs_type,
   600           rep_type = Logic.varifyT_global rep_type, Abs_name = Abs_name,
   601           Rep_name = Rep_name, set_def = set_def, prop_of_Rep = prop_of Rep,
   602           set_name = set_prefix ^ s, Abs_inverse = SOME Abs_inverse,
   603           Rep_inverse = SOME Rep_inverse}
   604   | _ => NONE
   605 
   606 val is_typedef = is_some oo typedef_info
   607 val is_real_datatype = is_some oo Datatype.get_info
   608 fun is_standard_datatype thy = the oo triple_lookup (type_match thy)
   609 
   610 (* FIXME: Use antiquotation for "code_numeral" below or detect "rep_datatype",
   611    e.g., by adding a field to "Datatype_Aux.info". *)
   612 fun is_basic_datatype thy stds s =
   613   member (op =) [@{type_name prod}, @{type_name bool}, @{type_name int},
   614                  "Code_Numeral.code_numeral"] s orelse
   615   (s = @{type_name nat} andalso is_standard_datatype thy stds nat_T)
   616 
   617 (* TODO: use "Term_Subst.instantiateT" instead? *)
   618 fun instantiate_type thy T1 T1' T2 =
   619   Same.commit (Envir.subst_type_same
   620                    (Sign.typ_match thy (T1, T1') Vartab.empty)) T2
   621   handle Type.TYPE_MATCH =>
   622          raise TYPE ("Nitpick_HOL.instantiate_type", [T1, T1'], [])
   623 fun varify_and_instantiate_type ctxt T1 T1' T2 =
   624   let val thy = ProofContext.theory_of ctxt in
   625     instantiate_type thy (varify_type ctxt T1) T1' (varify_type ctxt T2)
   626   end
   627 
   628 fun repair_constr_type ctxt body_T' T =
   629   varify_and_instantiate_type ctxt (body_type T) body_T' T
   630 
   631 fun register_frac_type_generic frac_s ersaetze generic =
   632   let
   633     val {frac_types, codatatypes} = Data.get generic
   634     val frac_types = AList.update (op =) (frac_s, ersaetze) frac_types
   635   in Data.put {frac_types = frac_types, codatatypes = codatatypes} generic end
   636 (* TODO: Consider morphism. *)
   637 fun register_frac_type frac_s ersaetze (_ : morphism) =
   638   register_frac_type_generic frac_s ersaetze
   639 val register_frac_type_global = Context.theory_map oo register_frac_type_generic
   640 
   641 fun unregister_frac_type_generic frac_s = register_frac_type_generic frac_s []
   642 (* TODO: Consider morphism. *)
   643 fun unregister_frac_type frac_s (_ : morphism) =
   644   unregister_frac_type_generic frac_s
   645 val unregister_frac_type_global =
   646   Context.theory_map o unregister_frac_type_generic
   647 
   648 fun register_codatatype_generic co_T case_name constr_xs generic =
   649   let
   650     val ctxt = Context.proof_of generic
   651     val thy = Context.theory_of generic
   652     val {frac_types, codatatypes} = Data.get generic
   653     val constr_xs = map (apsnd (repair_constr_type ctxt co_T)) constr_xs
   654     val (co_s, co_Ts) = dest_Type co_T
   655     val _ =
   656       if forall is_TFree co_Ts andalso not (has_duplicates (op =) co_Ts) andalso
   657          co_s <> @{type_name fun} andalso
   658          not (is_basic_datatype thy [(NONE, true)] co_s) then
   659         ()
   660       else
   661         raise TYPE ("Nitpick_HOL.register_codatatype_generic", [co_T], [])
   662     val codatatypes = AList.update (op =) (co_s, (case_name, constr_xs))
   663                                    codatatypes
   664   in Data.put {frac_types = frac_types, codatatypes = codatatypes} generic end
   665 (* TODO: Consider morphism. *)
   666 fun register_codatatype co_T case_name constr_xs (_ : morphism) =
   667   register_codatatype_generic co_T case_name constr_xs
   668 val register_codatatype_global =
   669   Context.theory_map ooo register_codatatype_generic
   670 
   671 fun unregister_codatatype_generic co_T = register_codatatype_generic co_T "" []
   672 (* TODO: Consider morphism. *)
   673 fun unregister_codatatype co_T (_ : morphism) =
   674   unregister_codatatype_generic co_T
   675 val unregister_codatatype_global =
   676   Context.theory_map o unregister_codatatype_generic
   677 
   678 fun is_codatatype ctxt (Type (s, _)) =
   679     s |> AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   680       |> Option.map snd |> these |> null |> not
   681   | is_codatatype _ _ = false
   682 fun is_real_quot_type thy (Type (s, _)) =
   683     is_some (Quotient_Info.quotdata_lookup_raw thy s)
   684   | is_real_quot_type _ _ = false
   685 fun is_quot_type ctxt T =
   686   let val thy = ProofContext.theory_of ctxt in
   687     is_real_quot_type thy T andalso not (is_codatatype ctxt T)
   688   end
   689 fun is_pure_typedef ctxt (T as Type (s, _)) =
   690     let val thy = ProofContext.theory_of ctxt in
   691       is_typedef ctxt s andalso
   692       not (is_real_datatype thy s orelse is_real_quot_type thy T orelse
   693            is_codatatype ctxt T orelse is_record_type T orelse
   694            is_integer_like_type T)
   695     end
   696   | is_pure_typedef _ _ = false
   697 fun is_univ_typedef ctxt (Type (s, _)) =
   698     (case typedef_info ctxt s of
   699        SOME {set_def, prop_of_Rep, ...} =>
   700        let
   701          val t_opt =
   702            case set_def of
   703              SOME thm => try (snd o Logic.dest_equals o prop_of) thm
   704            | NONE => try (snd o HOLogic.dest_mem o HOLogic.dest_Trueprop)
   705                          prop_of_Rep
   706        in
   707          case t_opt of
   708            SOME (Const (@{const_name top}, _)) => true
   709            (* "Multiset.multiset" *)
   710          | SOME (Const (@{const_name Collect}, _)
   711                  $ Abs (_, _, Const (@{const_name finite}, _) $ _)) => true
   712            (* "FinFun.finfun" *)
   713          | SOME (Const (@{const_name Collect}, _) $ Abs (_, _,
   714                      Const (@{const_name Ex}, _) $ Abs (_, _,
   715                          Const (@{const_name finite}, _) $ _))) => true
   716          | _ => false
   717        end
   718      | NONE => false)
   719   | is_univ_typedef _ _ = false
   720 fun is_datatype ctxt stds (T as Type (s, _)) =
   721     let val thy = ProofContext.theory_of ctxt in
   722       (is_typedef ctxt s orelse is_codatatype ctxt T orelse
   723        T = @{typ ind} orelse is_real_quot_type thy T) andalso
   724       not (is_basic_datatype thy stds s)
   725     end
   726   | is_datatype _ _ _ = false
   727 
   728 fun all_record_fields thy T =
   729   let val (recs, more) = Record.get_extT_fields thy T in
   730     recs @ more :: all_record_fields thy (snd more)
   731   end
   732   handle TYPE _ => []
   733 fun is_record_constr (s, T) =
   734   String.isSuffix Record.extN s andalso
   735   let val dataT = body_type T in
   736     is_record_type dataT andalso
   737     s = unsuffix Record.ext_typeN (fst (dest_Type dataT)) ^ Record.extN
   738   end
   739 val num_record_fields = Integer.add 1 o length o fst oo Record.get_extT_fields
   740 fun no_of_record_field thy s T1 =
   741   find_index (curry (op =) s o fst)
   742              (Record.get_extT_fields thy T1 ||> single |> op @)
   743 fun is_record_get thy (s, Type (@{type_name fun}, [T1, _])) =
   744     exists (curry (op =) s o fst) (all_record_fields thy T1)
   745   | is_record_get _ _ = false
   746 fun is_record_update thy (s, T) =
   747   String.isSuffix Record.updateN s andalso
   748   exists (curry (op =) (unsuffix Record.updateN s) o fst)
   749          (all_record_fields thy (body_type T))
   750   handle TYPE _ => false
   751 fun is_abs_fun ctxt (s, Type (@{type_name fun}, [_, Type (s', _)])) =
   752     (case typedef_info ctxt s' of
   753        SOME {Abs_name, ...} => s = Abs_name
   754      | NONE => false)
   755   | is_abs_fun _ _ = false
   756 fun is_rep_fun ctxt (s, Type (@{type_name fun}, [Type (s', _), _])) =
   757     (case typedef_info ctxt s' of
   758        SOME {Rep_name, ...} => s = Rep_name
   759      | NONE => false)
   760   | is_rep_fun _ _ = false
   761 fun is_quot_abs_fun ctxt (x as (_, Type (@{type_name fun},
   762                                          [_, abs_T as Type (s', _)]))) =
   763     try (Quotient_Term.absrep_const_chk Quotient_Term.AbsF ctxt) s'
   764     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   765   | is_quot_abs_fun _ _ = false
   766 fun is_quot_rep_fun ctxt (x as (_, Type (@{type_name fun},
   767                                          [abs_T as Type (s', _), _]))) =
   768     try (Quotient_Term.absrep_const_chk Quotient_Term.RepF ctxt) s'
   769     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   770   | is_quot_rep_fun _ _ = false
   771 
   772 fun mate_of_rep_fun ctxt (x as (_, Type (@{type_name fun},
   773                                          [T1 as Type (s', _), T2]))) =
   774     (case typedef_info ctxt s' of
   775        SOME {Abs_name, ...} => (Abs_name, Type (@{type_name fun}, [T2, T1]))
   776      | NONE => raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x]))
   777   | mate_of_rep_fun _ x = raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x])
   778 fun rep_type_for_quot_type thy (T as Type (s, _)) =
   779     let val {qtyp, rtyp, ...} = Quotient_Info.quotdata_lookup thy s in
   780       instantiate_type thy qtyp T rtyp
   781     end
   782   | rep_type_for_quot_type _ T =
   783     raise TYPE ("Nitpick_HOL.rep_type_for_quot_type", [T], [])
   784 fun equiv_relation_for_quot_type thy (Type (s, Ts)) =
   785     let
   786       val {qtyp, equiv_rel, equiv_thm, ...} =
   787         Quotient_Info.quotdata_lookup thy s
   788       val partial =
   789         case prop_of equiv_thm of
   790           @{const Trueprop} $ (Const (@{const_name equivp}, _) $ _) => false
   791         | @{const Trueprop} $ (Const (@{const_name part_equivp}, _) $ _) => true
   792         | _ => raise NOT_SUPPORTED "Ill-formed quotient type equivalence \
   793                                    \relation theorem"
   794       val Ts' = qtyp |> dest_Type |> snd
   795     in (subst_atomic_types (Ts' ~~ Ts) equiv_rel, partial) end
   796   | equiv_relation_for_quot_type _ T =
   797     raise TYPE ("Nitpick_HOL.equiv_relation_for_quot_type", [T], [])
   798 
   799 fun is_coconstr ctxt (s, T) =
   800   case body_type T of
   801     co_T as Type (co_s, _) =>
   802     let val {codatatypes, ...} = Data.get (Context.Proof ctxt) in
   803       exists (fn (s', T') => s = s' andalso repair_constr_type ctxt co_T T' = T)
   804              (AList.lookup (op =) codatatypes co_s |> Option.map snd |> these)
   805     end
   806   | _ => false
   807 fun is_constr_like ctxt (s, T) =
   808   member (op =) [@{const_name FinFun}, @{const_name FunBox},
   809                  @{const_name PairBox}, @{const_name Quot},
   810                  @{const_name Zero_Rep}, @{const_name Suc_Rep}] s orelse
   811   let
   812     val thy = ProofContext.theory_of ctxt
   813     val (x as (_, T)) = (s, unarize_unbox_etc_type T)
   814   in
   815     is_real_constr thy x orelse is_record_constr x orelse
   816     (is_abs_fun ctxt x andalso is_pure_typedef ctxt (range_type T)) orelse
   817     is_coconstr ctxt x
   818   end
   819 fun is_stale_constr ctxt (x as (_, T)) =
   820   is_codatatype ctxt (body_type T) andalso is_constr_like ctxt x andalso
   821   not (is_coconstr ctxt x)
   822 fun is_constr ctxt stds (x as (_, T)) =
   823   let val thy = ProofContext.theory_of ctxt in
   824     is_constr_like ctxt x andalso
   825     not (is_basic_datatype thy stds
   826                          (fst (dest_Type (unarize_type (body_type T))))) andalso
   827     not (is_stale_constr ctxt x)
   828   end
   829 val is_sel = String.isPrefix discr_prefix orf String.isPrefix sel_prefix
   830 val is_sel_like_and_no_discr =
   831   String.isPrefix sel_prefix orf
   832   (member (op =) [@{const_name fst}, @{const_name snd}])
   833 
   834 fun in_fun_lhs_for InConstr = InSel
   835   | in_fun_lhs_for _ = InFunLHS
   836 fun in_fun_rhs_for InConstr = InConstr
   837   | in_fun_rhs_for InSel = InSel
   838   | in_fun_rhs_for InFunRHS1 = InFunRHS2
   839   | in_fun_rhs_for _ = InFunRHS1
   840 
   841 fun is_boxing_worth_it (hol_ctxt : hol_context) boxy T =
   842   case T of
   843     Type (@{type_name fun}, _) =>
   844     (boxy = InPair orelse boxy = InFunLHS) andalso
   845     not (is_boolean_type (body_type T))
   846   | Type (@{type_name prod}, Ts) =>
   847     boxy = InPair orelse boxy = InFunRHS1 orelse boxy = InFunRHS2 orelse
   848     ((boxy = InExpr orelse boxy = InFunLHS) andalso
   849      exists (is_boxing_worth_it hol_ctxt InPair)
   850             (map (box_type hol_ctxt InPair) Ts))
   851   | _ => false
   852 and should_box_type (hol_ctxt as {thy, boxes, ...}) boxy z =
   853   case triple_lookup (type_match thy) boxes (Type z) of
   854     SOME (SOME box_me) => box_me
   855   | _ => is_boxing_worth_it hol_ctxt boxy (Type z)
   856 and box_type hol_ctxt boxy T =
   857   case T of
   858     Type (z as (@{type_name fun}, [T1, T2])) =>
   859     if boxy <> InConstr andalso boxy <> InSel andalso
   860        should_box_type hol_ctxt boxy z then
   861       Type (@{type_name fun_box},
   862             [box_type hol_ctxt InFunLHS T1, box_type hol_ctxt InFunRHS1 T2])
   863     else
   864       box_type hol_ctxt (in_fun_lhs_for boxy) T1
   865       --> box_type hol_ctxt (in_fun_rhs_for boxy) T2
   866   | Type (z as (@{type_name prod}, Ts)) =>
   867     if boxy <> InConstr andalso boxy <> InSel
   868        andalso should_box_type hol_ctxt boxy z then
   869       Type (@{type_name pair_box}, map (box_type hol_ctxt InSel) Ts)
   870     else
   871       Type (@{type_name prod},
   872             map (box_type hol_ctxt
   873                           (if boxy = InConstr orelse boxy = InSel then boxy
   874                            else InPair)) Ts)
   875   | _ => T
   876 
   877 fun binarize_nat_and_int_in_type @{typ nat} = @{typ "unsigned_bit word"}
   878   | binarize_nat_and_int_in_type @{typ int} = @{typ "signed_bit word"}
   879   | binarize_nat_and_int_in_type (Type (s, Ts)) =
   880     Type (s, map binarize_nat_and_int_in_type Ts)
   881   | binarize_nat_and_int_in_type T = T
   882 val binarize_nat_and_int_in_term = map_types binarize_nat_and_int_in_type
   883 
   884 fun discr_for_constr (s, T) = (discr_prefix ^ s, body_type T --> bool_T)
   885 
   886 fun num_sels_for_constr_type T = length (maybe_curried_binder_types T)
   887 fun nth_sel_name_for_constr_name s n =
   888   if s = @{const_name Pair} then
   889     if n = 0 then @{const_name fst} else @{const_name snd}
   890   else
   891     sel_prefix_for n ^ s
   892 fun nth_sel_for_constr x ~1 = discr_for_constr x
   893   | nth_sel_for_constr (s, T) n =
   894     (nth_sel_name_for_constr_name s n,
   895      body_type T --> nth (maybe_curried_binder_types T) n)
   896 fun binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize =
   897   apsnd ((binarize ? binarize_nat_and_int_in_type) o box_type hol_ctxt InSel)
   898   oo nth_sel_for_constr
   899 
   900 fun sel_no_from_name s =
   901   if String.isPrefix discr_prefix s then
   902     ~1
   903   else if String.isPrefix sel_prefix s then
   904     s |> unprefix sel_prefix |> Int.fromString |> the
   905   else if s = @{const_name snd} then
   906     1
   907   else
   908     0
   909 
   910 val close_form =
   911   let
   912     fun close_up zs zs' =
   913       fold (fn (z as ((s, _), T)) => fn t' =>
   914                Term.all T $ Abs (s, T, abstract_over (Var z, t')))
   915            (take (length zs' - length zs) zs')
   916     fun aux zs (@{const "==>"} $ t1 $ t2) =
   917         let val zs' = Term.add_vars t1 zs in
   918           close_up zs zs' (Logic.mk_implies (t1, aux zs' t2))
   919         end
   920       | aux zs t = close_up zs (Term.add_vars t zs) t
   921   in aux [] end
   922 
   923 fun distinctness_formula T =
   924   all_distinct_unordered_pairs_of
   925   #> map (fn (t1, t2) => @{const Not} $ (HOLogic.eq_const T $ t1 $ t2))
   926   #> List.foldr (s_conj o swap) @{const True}
   927 
   928 fun zero_const T = Const (@{const_name zero_class.zero}, T)
   929 fun suc_const T = Const (@{const_name Suc}, T --> T)
   930 
   931 fun uncached_datatype_constrs ({thy, ctxt, stds, ...} : hol_context)
   932                               (T as Type (s, Ts)) =
   933     (case AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   934                        s of
   935        SOME (_, xs' as (_ :: _)) => map (apsnd (repair_constr_type ctxt T)) xs'
   936      | _ =>
   937        if is_datatype ctxt stds T then
   938          case Datatype.get_info thy s of
   939            SOME {index, descr, ...} =>
   940            let
   941              val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the
   942            in
   943              map (apsnd (fn Us =>
   944                             map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))
   945                  constrs
   946            end
   947          | NONE =>
   948            if is_record_type T then
   949              let
   950                val s' = unsuffix Record.ext_typeN s ^ Record.extN
   951                val T' = (Record.get_extT_fields thy T
   952                         |> apsnd single |> uncurry append |> map snd) ---> T
   953              in [(s', T')] end
   954            else if is_real_quot_type thy T then
   955              [(@{const_name Quot}, rep_type_for_quot_type thy T --> T)]
   956            else case typedef_info ctxt s of
   957              SOME {abs_type, rep_type, Abs_name, ...} =>
   958              [(Abs_name,
   959                varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
   960            | NONE =>
   961              if T = @{typ ind} then
   962                [dest_Const @{const Zero_Rep}, dest_Const @{const Suc_Rep}]
   963              else
   964                []
   965        else
   966          [])
   967   | uncached_datatype_constrs _ _ = []
   968 fun datatype_constrs (hol_ctxt as {constr_cache, ...}) T =
   969   case AList.lookup (op =) (!constr_cache) T of
   970     SOME xs => xs
   971   | NONE =>
   972     let val xs = uncached_datatype_constrs hol_ctxt T in
   973       (Unsynchronized.change constr_cache (cons (T, xs)); xs)
   974     end
   975 fun binarized_and_boxed_datatype_constrs hol_ctxt binarize =
   976   map (apsnd ((binarize ? binarize_nat_and_int_in_type)
   977               o box_type hol_ctxt InConstr)) o datatype_constrs hol_ctxt
   978 val num_datatype_constrs = length oo datatype_constrs
   979 
   980 fun constr_name_for_sel_like @{const_name fst} = @{const_name Pair}
   981   | constr_name_for_sel_like @{const_name snd} = @{const_name Pair}
   982   | constr_name_for_sel_like s' = original_name s'
   983 fun binarized_and_boxed_constr_for_sel hol_ctxt binarize (s', T') =
   984   let val s = constr_name_for_sel_like s' in
   985     AList.lookup (op =)
   986         (binarized_and_boxed_datatype_constrs hol_ctxt binarize (domain_type T'))
   987         s
   988     |> the |> pair s
   989   end
   990 
   991 fun discr_term_for_constr hol_ctxt (x as (s, T)) =
   992   let val dataT = body_type T in
   993     if s = @{const_name Suc} then
   994       Abs (Name.uu, dataT,
   995            @{const Not} $ HOLogic.mk_eq (zero_const dataT, Bound 0))
   996     else if num_datatype_constrs hol_ctxt dataT >= 2 then
   997       Const (discr_for_constr x)
   998     else
   999       Abs (Name.uu, dataT, @{const True})
  1000   end
  1001 fun discriminate_value (hol_ctxt as {ctxt, ...}) x t =
  1002   case head_of t of
  1003     Const x' =>
  1004     if x = x' then @{const True}
  1005     else if is_constr_like ctxt x' then @{const False}
  1006     else s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1007   | _ => s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1008 
  1009 fun nth_arg_sel_term_for_constr thy stds (x as (s, T)) n =
  1010   let val (arg_Ts, dataT) = strip_type T in
  1011     if dataT = nat_T andalso is_standard_datatype thy stds nat_T then
  1012       @{term "%n::nat. n - 1"}
  1013     else if is_pair_type dataT then
  1014       Const (nth_sel_for_constr x n)
  1015     else
  1016       let
  1017         fun aux m (Type (@{type_name prod}, [T1, T2])) =
  1018             let
  1019               val (m, t1) = aux m T1
  1020               val (m, t2) = aux m T2
  1021             in (m, HOLogic.mk_prod (t1, t2)) end
  1022           | aux m T =
  1023             (m + 1, Const (nth_sel_name_for_constr_name s m, dataT --> T)
  1024                     $ Bound 0)
  1025         val m = fold (Integer.add o num_factors_in_type)
  1026                      (List.take (arg_Ts, n)) 0
  1027       in Abs ("x", dataT, aux m (nth arg_Ts n) |> snd) end
  1028   end
  1029 fun select_nth_constr_arg ctxt stds x t n res_T =
  1030   let val thy = ProofContext.theory_of ctxt in
  1031     (case strip_comb t of
  1032        (Const x', args) =>
  1033        if x = x' then nth args n
  1034        else if is_constr_like ctxt x' then Const (@{const_name unknown}, res_T)
  1035        else raise SAME ()
  1036      | _ => raise SAME())
  1037     handle SAME () =>
  1038            s_betapply [] (nth_arg_sel_term_for_constr thy stds x n, t)
  1039   end
  1040 
  1041 fun construct_value _ _ x [] = Const x
  1042   | construct_value ctxt stds (x as (s, _)) args =
  1043     let val args = map Envir.eta_contract args in
  1044       case hd args of
  1045         Const (s', _) $ t =>
  1046         if is_sel_like_and_no_discr s' andalso
  1047            constr_name_for_sel_like s' = s andalso
  1048            forall (fn (n, t') =>
  1049                       select_nth_constr_arg ctxt stds x t n dummyT = t')
  1050                   (index_seq 0 (length args) ~~ args) then
  1051           t
  1052         else
  1053           list_comb (Const x, args)
  1054       | _ => list_comb (Const x, args)
  1055     end
  1056 
  1057 fun constr_expand (hol_ctxt as {ctxt, stds, ...}) T t =
  1058   (case head_of t of
  1059      Const x => if is_constr_like ctxt x then t else raise SAME ()
  1060    | _ => raise SAME ())
  1061   handle SAME () =>
  1062          let
  1063            val x' as (_, T') =
  1064              if is_pair_type T then
  1065                let val (T1, T2) = HOLogic.dest_prodT T in
  1066                  (@{const_name Pair}, T1 --> T2 --> T)
  1067                end
  1068              else
  1069                datatype_constrs hol_ctxt T |> hd
  1070            val arg_Ts = binder_types T'
  1071          in
  1072            list_comb (Const x', map2 (select_nth_constr_arg ctxt stds x' t)
  1073                                      (index_seq 0 (length arg_Ts)) arg_Ts)
  1074          end
  1075 
  1076 fun coerce_bound_no f j t =
  1077   case t of
  1078     t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2
  1079   | Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')
  1080   | Bound j' => if j' = j then f t else t
  1081   | _ => t
  1082 fun coerce_bound_0_in_term hol_ctxt new_T old_T =
  1083   old_T <> new_T ? coerce_bound_no (coerce_term hol_ctxt [new_T] old_T new_T) 0
  1084 and coerce_term (hol_ctxt as {ctxt, stds, ...}) Ts new_T old_T t =
  1085   if old_T = new_T then
  1086     t
  1087   else
  1088     case (new_T, old_T) of
  1089       (Type (new_s, new_Ts as [new_T1, new_T2]),
  1090        Type (@{type_name fun}, [old_T1, old_T2])) =>
  1091       (case eta_expand Ts t 1 of
  1092          Abs (s, _, t') =>
  1093          Abs (s, new_T1,
  1094               t' |> coerce_bound_0_in_term hol_ctxt new_T1 old_T1
  1095                  |> coerce_term hol_ctxt (new_T1 :: Ts) new_T2 old_T2)
  1096          |> Envir.eta_contract
  1097          |> new_s <> @{type_name fun}
  1098             ? construct_value ctxt stds
  1099                   (if new_s = @{type_name fin_fun} then @{const_name FinFun}
  1100                    else @{const_name FunBox},
  1101                    Type (@{type_name fun}, new_Ts) --> new_T)
  1102               o single
  1103        | t' => raise TERM ("Nitpick_HOL.coerce_term", [t']))
  1104     | (Type (new_s, new_Ts as [new_T1, new_T2]),
  1105        Type (old_s, old_Ts as [old_T1, old_T2])) =>
  1106       if old_s = @{type_name fin_fun} orelse old_s = @{type_name fun_box} orelse
  1107          old_s = @{type_name pair_box} orelse old_s = @{type_name prod} then
  1108         case constr_expand hol_ctxt old_T t of
  1109           Const (old_s, _) $ t1 =>
  1110           if new_s = @{type_name fun} then
  1111             coerce_term hol_ctxt Ts new_T (Type (@{type_name fun}, old_Ts)) t1
  1112           else
  1113             construct_value ctxt stds
  1114                 (old_s, Type (@{type_name fun}, new_Ts) --> new_T)
  1115                 [coerce_term hol_ctxt Ts (Type (@{type_name fun}, new_Ts))
  1116                              (Type (@{type_name fun}, old_Ts)) t1]
  1117         | Const _ $ t1 $ t2 =>
  1118           construct_value ctxt stds
  1119               (if new_s = @{type_name prod} then @{const_name Pair}
  1120                else @{const_name PairBox}, new_Ts ---> new_T)
  1121               (map3 (coerce_term hol_ctxt Ts) [new_T1, new_T2] [old_T1, old_T2]
  1122                     [t1, t2])
  1123         | t' => raise TERM ("Nitpick_HOL.coerce_term", [t'])
  1124       else
  1125         raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1126     | _ => raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1127 
  1128 fun card_of_type assigns (Type (@{type_name fun}, [T1, T2])) =
  1129     reasonable_power (card_of_type assigns T2) (card_of_type assigns T1)
  1130   | card_of_type assigns (Type (@{type_name prod}, [T1, T2])) =
  1131     card_of_type assigns T1 * card_of_type assigns T2
  1132   | card_of_type _ (Type (@{type_name itself}, _)) = 1
  1133   | card_of_type _ @{typ prop} = 2
  1134   | card_of_type _ @{typ bool} = 2
  1135   | card_of_type assigns T =
  1136     case AList.lookup (op =) assigns T of
  1137       SOME k => k
  1138     | NONE => if T = @{typ bisim_iterator} then 0
  1139               else raise TYPE ("Nitpick_HOL.card_of_type", [T], [])
  1140 fun bounded_card_of_type max default_card assigns
  1141                          (Type (@{type_name fun}, [T1, T2])) =
  1142     let
  1143       val k1 = bounded_card_of_type max default_card assigns T1
  1144       val k2 = bounded_card_of_type max default_card assigns T2
  1145     in
  1146       if k1 = max orelse k2 = max then max
  1147       else Int.min (max, reasonable_power k2 k1)
  1148     end
  1149   | bounded_card_of_type max default_card assigns
  1150                          (Type (@{type_name prod}, [T1, T2])) =
  1151     let
  1152       val k1 = bounded_card_of_type max default_card assigns T1
  1153       val k2 = bounded_card_of_type max default_card assigns T2
  1154     in if k1 = max orelse k2 = max then max else Int.min (max, k1 * k2) end
  1155   | bounded_card_of_type max default_card assigns T =
  1156     Int.min (max, if default_card = ~1 then
  1157                     card_of_type assigns T
  1158                   else
  1159                     card_of_type assigns T
  1160                     handle TYPE ("Nitpick_HOL.card_of_type", _, _) =>
  1161                            default_card)
  1162 fun bounded_exact_card_of_type hol_ctxt finitizable_dataTs max default_card
  1163                                assigns T =
  1164   let
  1165     fun aux avoid T =
  1166       (if member (op =) avoid T then
  1167          0
  1168        else if member (op =) finitizable_dataTs T then
  1169          raise SAME ()
  1170        else case T of
  1171          Type (@{type_name fun}, [T1, T2]) =>
  1172          let
  1173            val k1 = aux avoid T1
  1174            val k2 = aux avoid T2
  1175          in
  1176            if k1 = 0 orelse k2 = 0 then 0
  1177            else if k1 >= max orelse k2 >= max then max
  1178            else Int.min (max, reasonable_power k2 k1)
  1179          end
  1180        | Type (@{type_name prod}, [T1, T2]) =>
  1181          let
  1182            val k1 = aux avoid T1
  1183            val k2 = aux avoid T2
  1184          in
  1185            if k1 = 0 orelse k2 = 0 then 0
  1186            else if k1 >= max orelse k2 >= max then max
  1187            else Int.min (max, k1 * k2)
  1188          end
  1189        | Type (@{type_name itself}, _) => 1
  1190        | @{typ prop} => 2
  1191        | @{typ bool} => 2
  1192        | Type _ =>
  1193          (case datatype_constrs hol_ctxt T of
  1194             [] => if is_integer_type T orelse is_bit_type T then 0
  1195                   else raise SAME ()
  1196           | constrs =>
  1197             let
  1198               val constr_cards =
  1199                 map (Integer.prod o map (aux (T :: avoid)) o binder_types o snd)
  1200                     constrs
  1201             in
  1202               if exists (curry (op =) 0) constr_cards then 0
  1203               else Integer.sum constr_cards
  1204             end)
  1205        | _ => raise SAME ())
  1206       handle SAME () =>
  1207              AList.lookup (op =) assigns T |> the_default default_card
  1208   in Int.min (max, aux [] T) end
  1209 
  1210 val small_type_max_card = 5
  1211 
  1212 fun is_finite_type hol_ctxt T =
  1213   bounded_exact_card_of_type hol_ctxt [] 1 2 [] T > 0
  1214 fun is_small_finite_type hol_ctxt T =
  1215   let val n = bounded_exact_card_of_type hol_ctxt [] 1 2 [] T in
  1216     n > 0 andalso n <= small_type_max_card
  1217   end
  1218 
  1219 fun is_ground_term (t1 $ t2) = is_ground_term t1 andalso is_ground_term t2
  1220   | is_ground_term (Const _) = true
  1221   | is_ground_term _ = false
  1222 
  1223 fun hashw_term (t1 $ t2) = hashw (hashw_term t1, hashw_term t2)
  1224   | hashw_term (Const (s, _)) = hashw_string (s, 0w0)
  1225   | hashw_term _ = 0w0
  1226 val hash_term = Word.toInt o hashw_term
  1227 
  1228 fun special_bounds ts =
  1229   fold Term.add_vars ts [] |> sort (Term_Ord.fast_indexname_ord o pairself fst)
  1230 
  1231 (* FIXME: detect "rep_datatype"? *)
  1232 fun is_funky_typedef_name ctxt s =
  1233   member (op =) [@{type_name unit}, @{type_name prod},
  1234                  @{type_name Sum_Type.sum}, @{type_name int}] s orelse
  1235   is_frac_type ctxt (Type (s, []))
  1236 fun is_funky_typedef ctxt (Type (s, _)) = is_funky_typedef_name ctxt s
  1237   | is_funky_typedef _ _ = false
  1238 fun is_arity_type_axiom (Const (@{const_name HOL.type_class}, _)
  1239                          $ Const (@{const_name TYPE}, _)) = true
  1240   | is_arity_type_axiom _ = false
  1241 fun is_typedef_axiom ctxt boring (@{const "==>"} $ _ $ t2) =
  1242     is_typedef_axiom ctxt boring t2
  1243   | is_typedef_axiom ctxt boring
  1244         (@{const Trueprop} $ (Const (@{const_name Typedef.type_definition}, _)
  1245          $ Const (_, Type (@{type_name fun}, [Type (s, _), _]))
  1246          $ Const _ $ _)) =
  1247     boring <> is_funky_typedef_name ctxt s andalso is_typedef ctxt s
  1248   | is_typedef_axiom _ _ _ = false
  1249 val is_class_axiom =
  1250   Logic.strip_horn #> swap #> op :: #> forall (can Logic.dest_of_class)
  1251 
  1252 (* Distinguishes between (1) constant definition axioms, (2) type arity and
  1253    typedef axioms, and (3) other axioms, and returns the pair ((1), (3)).
  1254    Typedef axioms are uninteresting to Nitpick, because it can retrieve them
  1255    using "typedef_info". *)
  1256 fun partition_axioms_by_definitionality ctxt axioms def_names =
  1257   let
  1258     val axioms = sort (fast_string_ord o pairself fst) axioms
  1259     val defs = Ord_List.inter (fast_string_ord o apsnd fst) def_names axioms
  1260     val nondefs =
  1261       Ord_List.subtract (fast_string_ord o apsnd fst) def_names axioms
  1262       |> filter_out ((is_arity_type_axiom orf is_typedef_axiom ctxt true) o snd)
  1263   in pairself (map snd) (defs, nondefs) end
  1264 
  1265 (* Ideally we would check against "Complex_Main", not "Refute", but any theory
  1266    will do as long as it contains all the "axioms" and "axiomatization"
  1267    commands. *)
  1268 fun is_built_in_theory thy = Theory.subthy (thy, @{theory Refute})
  1269 
  1270 val is_trivial_definition =
  1271   the_default false o try (op aconv o Logic.dest_equals)
  1272 val is_plain_definition =
  1273   let
  1274     fun do_lhs t1 =
  1275       case strip_comb t1 of
  1276         (Const _, args) =>
  1277         forall is_Var args andalso not (has_duplicates (op =) args)
  1278       | _ => false
  1279     fun do_eq (Const (@{const_name "=="}, _) $ t1 $ _) = do_lhs t1
  1280       | do_eq (@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)) =
  1281         do_lhs t1
  1282       | do_eq _ = false
  1283   in do_eq end
  1284 
  1285 fun all_axioms_of ctxt subst =
  1286   let
  1287     val thy = ProofContext.theory_of ctxt
  1288     val axioms_of_thys =
  1289       maps Thm.axioms_of
  1290       #> map (apsnd (subst_atomic subst o prop_of))
  1291       #> filter_out (is_class_axiom o snd)
  1292     val specs = Defs.all_specifications_of (Theory.defs_of thy)
  1293     val def_names = specs |> maps snd |> map_filter #def
  1294                     |> Ord_List.make fast_string_ord
  1295     val thys = thy :: Theory.ancestors_of thy
  1296     val (built_in_thys, user_thys) = List.partition is_built_in_theory thys
  1297     val built_in_axioms = axioms_of_thys built_in_thys
  1298     val user_axioms = axioms_of_thys user_thys
  1299     val (built_in_defs, built_in_nondefs) =
  1300       partition_axioms_by_definitionality ctxt built_in_axioms def_names
  1301       ||> filter (is_typedef_axiom ctxt false)
  1302     val (user_defs, user_nondefs) =
  1303       partition_axioms_by_definitionality ctxt user_axioms def_names
  1304     val (built_in_nondefs, user_nondefs) =
  1305       List.partition (is_typedef_axiom ctxt false) user_nondefs
  1306       |>> append built_in_nondefs
  1307     val defs =
  1308       (thy |> Global_Theory.all_thms_of
  1309            |> filter (curry (op =) Thm.definitionK o Thm.get_kind o snd)
  1310            |> map (prop_of o snd)
  1311            |> filter_out is_trivial_definition
  1312            |> filter is_plain_definition) @
  1313       user_defs @ built_in_defs
  1314   in (defs, built_in_nondefs, user_nondefs) end
  1315 
  1316 fun arity_of_built_in_const thy stds (s, T) =
  1317   if s = @{const_name If} then
  1318     if nth_range_type 3 T = @{typ bool} then NONE else SOME 3
  1319   else
  1320     let val std_nats = is_standard_datatype thy stds nat_T in
  1321       case AList.lookup (op =)
  1322                     (built_in_consts
  1323                      |> std_nats ? append built_in_nat_consts) s of
  1324         SOME n => SOME n
  1325       | NONE =>
  1326         case AList.lookup (op =)
  1327                  (built_in_typed_consts
  1328                   |> std_nats ? append built_in_typed_nat_consts)
  1329                  (s, unarize_type T) of
  1330           SOME n => SOME n
  1331         | NONE =>
  1332           case s of
  1333             @{const_name zero_class.zero} =>
  1334             if is_iterator_type T then SOME 0 else NONE
  1335           | @{const_name Suc} =>
  1336             if is_iterator_type (domain_type T) then SOME 0 else NONE
  1337           | _ => if is_fun_type T andalso is_set_type (domain_type T) then
  1338                    AList.lookup (op =) built_in_set_consts s
  1339                  else
  1340                    NONE
  1341     end
  1342 val is_built_in_const = is_some ooo arity_of_built_in_const
  1343 
  1344 (* This function is designed to work for both real definition axioms and
  1345    simplification rules (equational specifications). *)
  1346 fun term_under_def t =
  1347   case t of
  1348     @{const "==>"} $ _ $ t2 => term_under_def t2
  1349   | Const (@{const_name "=="}, _) $ t1 $ _ => term_under_def t1
  1350   | @{const Trueprop} $ t1 => term_under_def t1
  1351   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => term_under_def t1
  1352   | Abs (_, _, t') => term_under_def t'
  1353   | t1 $ _ => term_under_def t1
  1354   | _ => t
  1355 
  1356 (* Here we crucially rely on "specialize_type" performing a preorder traversal
  1357    of the term, without which the wrong occurrence of a constant could be
  1358    matched in the face of overloading. *)
  1359 fun def_props_for_const thy stds table (x as (s, _)) =
  1360   if is_built_in_const thy stds x then
  1361     []
  1362   else
  1363     these (Symtab.lookup table s)
  1364     |> map_filter (try (specialize_type thy x))
  1365     |> filter (curry (op =) (Const x) o term_under_def)
  1366 
  1367 fun normalized_rhs_of t =
  1368   let
  1369     fun aux (v as Var _) (SOME t) = SOME (lambda v t)
  1370       | aux (c as Const (@{const_name TYPE}, _)) (SOME t) = SOME (lambda c t)
  1371       | aux _ _ = NONE
  1372     val (lhs, rhs) =
  1373       case t of
  1374         Const (@{const_name "=="}, _) $ t1 $ t2 => (t1, t2)
  1375       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ t2) =>
  1376         (t1, t2)
  1377       | _ => raise TERM ("Nitpick_HOL.normalized_rhs_of", [t])
  1378     val args = strip_comb lhs |> snd
  1379   in fold_rev aux args (SOME rhs) end
  1380 
  1381 fun def_of_const thy table (x as (s, _)) =
  1382   if is_built_in_const thy [(NONE, false)] x orelse
  1383      original_name s <> s then
  1384     NONE
  1385   else
  1386     x |> def_props_for_const thy [(NONE, false)] table |> List.last
  1387       |> normalized_rhs_of |> Option.map (prefix_abs_vars s)
  1388     handle List.Empty => NONE
  1389 
  1390 fun fixpoint_kind_of_rhs (Abs (_, _, t)) = fixpoint_kind_of_rhs t
  1391   | fixpoint_kind_of_rhs (Const (@{const_name lfp}, _) $ Abs _) = Lfp
  1392   | fixpoint_kind_of_rhs (Const (@{const_name gfp}, _) $ Abs _) = Gfp
  1393   | fixpoint_kind_of_rhs _ = NoFp
  1394 
  1395 fun is_mutually_inductive_pred_def thy table t =
  1396   let
  1397     fun is_good_arg (Bound _) = true
  1398       | is_good_arg (Const (s, _)) =
  1399         s = @{const_name True} orelse s = @{const_name False} orelse
  1400         s = @{const_name undefined}
  1401       | is_good_arg _ = false
  1402   in
  1403     case t |> strip_abs_body |> strip_comb of
  1404       (Const x, ts as (_ :: _)) =>
  1405       (case def_of_const thy table x of
  1406          SOME t' => fixpoint_kind_of_rhs t' <> NoFp andalso
  1407                     forall is_good_arg ts
  1408        | NONE => false)
  1409     | _ => false
  1410   end
  1411 fun unfold_mutually_inductive_preds thy table =
  1412   map_aterms (fn t as Const x =>
  1413                  (case def_of_const thy table x of
  1414                     SOME t' =>
  1415                     let val t' = Envir.eta_contract t' in
  1416                       if is_mutually_inductive_pred_def thy table t' then t'
  1417                       else t
  1418                     end
  1419                  | NONE => t)
  1420                | t => t)
  1421 
  1422 fun case_const_names ctxt stds =
  1423   let val thy = ProofContext.theory_of ctxt in
  1424     Symtab.fold (fn (dtype_s, {index, descr, case_name, ...}) =>
  1425                     if is_basic_datatype thy stds dtype_s then
  1426                       I
  1427                     else
  1428                       cons (case_name, AList.lookup (op =) descr index
  1429                                        |> the |> #3 |> length))
  1430                 (Datatype.get_all thy) [] @
  1431     map (apsnd length o snd) (#codatatypes (Data.get (Context.Proof ctxt)))
  1432   end
  1433 
  1434 fun fixpoint_kind_of_const thy table x =
  1435   if is_built_in_const thy [(NONE, false)] x then NoFp
  1436   else fixpoint_kind_of_rhs (the (def_of_const thy table x))
  1437   handle Option.Option => NoFp
  1438 
  1439 fun is_real_inductive_pred ({thy, stds, def_table, intro_table, ...}
  1440                             : hol_context) x =
  1441   fixpoint_kind_of_const thy def_table x <> NoFp andalso
  1442   not (null (def_props_for_const thy stds intro_table x))
  1443 fun is_inductive_pred hol_ctxt (x as (s, _)) =
  1444   is_real_inductive_pred hol_ctxt x orelse String.isPrefix ubfp_prefix s orelse
  1445   String.isPrefix lbfp_prefix s
  1446 
  1447 fun lhs_of_equation t =
  1448   case t of
  1449     Const (@{const_name all}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1450   | Const (@{const_name "=="}, _) $ t1 $ _ => SOME t1
  1451   | @{const "==>"} $ _ $ t2 => lhs_of_equation t2
  1452   | @{const Trueprop} $ t1 => lhs_of_equation t1
  1453   | Const (@{const_name All}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1454   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => SOME t1
  1455   | @{const HOL.implies} $ _ $ t2 => lhs_of_equation t2
  1456   | _ => NONE
  1457 fun is_constr_pattern _ (Bound _) = true
  1458   | is_constr_pattern _ (Var _) = true
  1459   | is_constr_pattern ctxt t =
  1460     case strip_comb t of
  1461       (Const x, args) =>
  1462       is_constr_like ctxt x andalso forall (is_constr_pattern ctxt) args
  1463     | _ => false
  1464 fun is_constr_pattern_lhs ctxt t =
  1465   forall (is_constr_pattern ctxt) (snd (strip_comb t))
  1466 fun is_constr_pattern_formula ctxt t =
  1467   case lhs_of_equation t of
  1468     SOME t' => is_constr_pattern_lhs ctxt t'
  1469   | NONE => false
  1470 
  1471 (* Similar to "specialize_type" but returns all matches rather than only the
  1472    first (preorder) match. *)
  1473 fun multi_specialize_type thy slack (s, T) t =
  1474   let
  1475     fun aux (Const (s', T')) ys =
  1476         if s = s' then
  1477           ys |> (if AList.defined (op =) ys T' then
  1478                    I
  1479                  else
  1480                    cons (T', monomorphic_term (Sign.typ_match thy (T', T)
  1481                                                               Vartab.empty) t)
  1482                    handle Type.TYPE_MATCH => I
  1483                         | TERM _ =>
  1484                           if slack then
  1485                             I
  1486                           else
  1487                             raise NOT_SUPPORTED
  1488                                       ("too much polymorphism in axiom \"" ^
  1489                                        Syntax.string_of_term_global thy t ^
  1490                                        "\" involving " ^ quote s))
  1491         else
  1492           ys
  1493       | aux _ ys = ys
  1494   in map snd (fold_aterms aux t []) end
  1495 fun nondef_props_for_const thy slack table (x as (s, _)) =
  1496   these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)
  1497 
  1498 fun unvarify_term (t1 $ t2) = unvarify_term t1 $ unvarify_term t2
  1499   | unvarify_term (Var ((s, 0), T)) = Free (s, T)
  1500   | unvarify_term (Abs (s, T, t')) = Abs (s, T, unvarify_term t')
  1501   | unvarify_term t = t
  1502 fun axiom_for_choice_spec thy =
  1503   unvarify_term
  1504   #> Object_Logic.atomize_term thy
  1505   #> Choice_Specification.close_form
  1506   #> HOLogic.mk_Trueprop
  1507 fun is_choice_spec_fun ({thy, def_table, nondef_table, choice_spec_table, ...}
  1508                         : hol_context) x =
  1509   case nondef_props_for_const thy true choice_spec_table x of
  1510     [] => false
  1511   | ts => case def_of_const thy def_table x of
  1512             SOME (Const (@{const_name Eps}, _) $ _) => true
  1513           | SOME _ => false
  1514           | NONE =>
  1515             let val ts' = nondef_props_for_const thy true nondef_table x in
  1516               length ts' = length ts andalso
  1517               forall (fn t =>
  1518                          exists (curry (op aconv) (axiom_for_choice_spec thy t))
  1519                                 ts') ts
  1520             end
  1521 
  1522 fun is_choice_spec_axiom thy choice_spec_table t =
  1523   Symtab.exists (fn (_, ts) =>
  1524                     exists (curry (op aconv) t o axiom_for_choice_spec thy) ts)
  1525                 choice_spec_table
  1526 
  1527 fun is_real_equational_fun ({thy, stds, simp_table, psimp_table, ...}
  1528                             : hol_context) x =
  1529   exists (fn table => not (null (def_props_for_const thy stds table x)))
  1530          [!simp_table, psimp_table]
  1531 fun is_equational_fun_but_no_plain_def hol_ctxt =
  1532   is_real_equational_fun hol_ctxt orf is_inductive_pred hol_ctxt
  1533 
  1534 (** Constant unfolding **)
  1535 
  1536 fun constr_case_body ctxt stds (func_t, (x as (_, T))) =
  1537   let val arg_Ts = binder_types T in
  1538     s_betapplys [] (func_t, map2 (select_nth_constr_arg ctxt stds x (Bound 0))
  1539                                  (index_seq 0 (length arg_Ts)) arg_Ts)
  1540   end
  1541 fun add_constr_case res_T (body_t, guard_t) res_t =
  1542   if res_T = bool_T then
  1543     s_conj (HOLogic.mk_imp (guard_t, body_t), res_t)
  1544   else
  1545     Const (@{const_name If}, bool_T --> res_T --> res_T --> res_T)
  1546     $ guard_t $ body_t $ res_t
  1547 fun optimized_case_def (hol_ctxt as {ctxt, stds, ...}) dataT res_T func_ts =
  1548   let
  1549     val xs = datatype_constrs hol_ctxt dataT
  1550     val cases =
  1551       func_ts ~~ xs
  1552       |> map (fn (func_t, x) =>
  1553                  (constr_case_body ctxt stds (incr_boundvars 1 func_t, x),
  1554                   discriminate_value hol_ctxt x (Bound 0)))
  1555       |> AList.group (op aconv)
  1556       |> map (apsnd (List.foldl s_disj @{const False}))
  1557       |> sort (int_ord o pairself (size_of_term o snd))
  1558       |> rev
  1559   in
  1560     if res_T = bool_T then
  1561       if forall (member (op =) [@{const False}, @{const True}] o fst) cases then
  1562         case cases of
  1563           [(body_t, _)] => body_t
  1564         | [_, (@{const True}, head_t2)] => head_t2
  1565         | [_, (@{const False}, head_t2)] => @{const Not} $ head_t2
  1566         | _ => raise BAD ("Nitpick_HOL.optimized_case_def", "impossible cases")
  1567       else
  1568         @{const True} |> fold_rev (add_constr_case res_T) cases
  1569     else
  1570       fst (hd cases) |> fold_rev (add_constr_case res_T) (tl cases)
  1571   end
  1572   |> curry absdummy dataT
  1573 
  1574 fun optimized_record_get (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T res_T t =
  1575   let val constr_x = hd (datatype_constrs hol_ctxt rec_T) in
  1576     case no_of_record_field thy s rec_T of
  1577       ~1 => (case rec_T of
  1578                Type (_, Ts as _ :: _) =>
  1579                let
  1580                  val rec_T' = List.last Ts
  1581                  val j = num_record_fields thy rec_T - 1
  1582                in
  1583                  select_nth_constr_arg ctxt stds constr_x t j res_T
  1584                  |> optimized_record_get hol_ctxt s rec_T' res_T
  1585                end
  1586              | _ => raise TYPE ("Nitpick_HOL.optimized_record_get", [rec_T],
  1587                                 []))
  1588     | j => select_nth_constr_arg ctxt stds constr_x t j res_T
  1589   end
  1590 fun optimized_record_update (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T fun_t
  1591                             rec_t =
  1592   let
  1593     val constr_x as (_, constr_T) = hd (datatype_constrs hol_ctxt rec_T)
  1594     val Ts = binder_types constr_T
  1595     val n = length Ts
  1596     val special_j = no_of_record_field thy s rec_T
  1597     val ts =
  1598       map2 (fn j => fn T =>
  1599                let val t = select_nth_constr_arg ctxt stds constr_x rec_t j T in
  1600                  if j = special_j then
  1601                    s_betapply [] (fun_t, t)
  1602                  else if j = n - 1 andalso special_j = ~1 then
  1603                    optimized_record_update hol_ctxt s
  1604                        (rec_T |> dest_Type |> snd |> List.last) fun_t t
  1605                  else
  1606                    t
  1607                end) (index_seq 0 n) Ts
  1608   in list_comb (Const constr_x, ts) end
  1609 
  1610 (* Prevents divergence in case of cyclic or infinite definition dependencies. *)
  1611 val unfold_max_depth = 255
  1612 
  1613 (* Inline definitions or define as an equational constant? Booleans tend to
  1614    benefit more from inlining, due to the polarity analysis. *)
  1615 val def_inline_threshold_for_booleans = 50
  1616 val def_inline_threshold_for_non_booleans = 20
  1617 
  1618 fun unfold_defs_in_term
  1619         (hol_ctxt as {thy, ctxt, stds, whacks, case_names, def_table,
  1620                       ground_thm_table, ersatz_table, ...}) =
  1621   let
  1622     fun do_term depth Ts t =
  1623       case t of
  1624         (t0 as Const (@{const_name Int.number_class.number_of},
  1625                       Type (@{type_name fun}, [_, ran_T]))) $ t1 =>
  1626         ((if is_number_type ctxt ran_T then
  1627             let
  1628               val j = t1 |> HOLogic.dest_numeral
  1629                          |> ran_T = nat_T ? Integer.max 0
  1630               val s = numeral_prefix ^ signed_string_of_int j
  1631             in
  1632               if is_integer_like_type ran_T then
  1633                 if is_standard_datatype thy stds ran_T then
  1634                   Const (s, ran_T)
  1635                 else
  1636                   funpow j (curry (op $) (suc_const ran_T)) (zero_const ran_T)
  1637               else
  1638                 do_term depth Ts (Const (@{const_name of_int}, int_T --> ran_T)
  1639                                   $ Const (s, int_T))
  1640             end
  1641             handle TERM _ => raise SAME ()
  1642           else
  1643             raise SAME ())
  1644          handle SAME () =>
  1645                 s_betapply [] (do_term depth Ts t0, do_term depth Ts t1))
  1646       | Const (@{const_name refl_on}, T) $ Const (@{const_name top}, _) $ t2 =>
  1647         do_const depth Ts t (@{const_name refl'}, range_type T) [t2]
  1648       | (t0 as Const (@{const_name Sigma}, Type (_, [T1, Type (_, [T2, T3])])))
  1649         $ t1 $ (t2 as Abs (_, _, t2')) =>
  1650         if loose_bvar1 (t2', 0) then
  1651           s_betapplys Ts (do_term depth Ts t0, map (do_term depth Ts) [t1, t2])
  1652         else
  1653           do_term depth Ts
  1654                   (Const (@{const_name prod}, T1 --> range_type T2 --> T3)
  1655                    $ t1 $ incr_boundvars ~1 t2')
  1656       | Const (x as (@{const_name distinct},
  1657                Type (@{type_name fun}, [Type (@{type_name list}, [T']), _])))
  1658         $ (t1 as _ $ _) =>
  1659         (t1 |> HOLogic.dest_list |> distinctness_formula T'
  1660          handle TERM _ => do_const depth Ts t x [t1])
  1661       | Const (x as (@{const_name If}, _)) $ t1 $ t2 $ t3 =>
  1662         if is_ground_term t1 andalso
  1663            exists (Pattern.matches thy o rpair t1)
  1664                   (Inttab.lookup_list ground_thm_table (hash_term t1)) then
  1665           do_term depth Ts t2
  1666         else
  1667           do_const depth Ts t x [t1, t2, t3]
  1668       | Const (@{const_name Let}, _) $ t1 $ t2 =>
  1669         s_betapply Ts (pairself (do_term depth Ts) (t2, t1))
  1670       | Const x => do_const depth Ts t x []
  1671       | t1 $ t2 =>
  1672         (case strip_comb t of
  1673            (Const x, ts) => do_const depth Ts t x ts
  1674          | _ => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2))
  1675       | Bound _ => t
  1676       | Abs (s, T, body) => Abs (s, T, do_term depth (T :: Ts) body)
  1677       | _ => if member (term_match thy) whacks t then
  1678                Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1679              else
  1680                t
  1681     and select_nth_constr_arg_with_args _ _ (x as (_, T)) [] n res_T =
  1682         (Abs (Name.uu, body_type T,
  1683               select_nth_constr_arg ctxt stds x (Bound 0) n res_T), [])
  1684       | select_nth_constr_arg_with_args depth Ts x (t :: ts) n res_T =
  1685         (select_nth_constr_arg ctxt stds x (do_term depth Ts t) n res_T, ts)
  1686     and quot_rep_of depth Ts abs_T rep_T ts =
  1687       select_nth_constr_arg_with_args depth Ts
  1688           (@{const_name Quot}, rep_T --> abs_T) ts 0 rep_T
  1689     and do_const depth Ts t (x as (s, T)) ts =
  1690       if member (term_match thy) whacks (Const x) then
  1691         Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1692       else case AList.lookup (op =) ersatz_table s of
  1693         SOME s' =>
  1694         do_const (depth + 1) Ts (list_comb (Const (s', T), ts)) (s', T) ts
  1695       | NONE =>
  1696         let
  1697           fun def_inline_threshold () =
  1698             if is_boolean_type (nth_range_type (length ts) T) then
  1699               def_inline_threshold_for_booleans
  1700             else
  1701               def_inline_threshold_for_non_booleans
  1702           val (const, ts) =
  1703             if is_built_in_const thy stds x then
  1704               (Const x, ts)
  1705             else case AList.lookup (op =) case_names s of
  1706               SOME n =>
  1707               if length ts < n then
  1708                 (do_term depth Ts (eta_expand Ts t (n - length ts)), [])
  1709               else
  1710                 let
  1711                   val (dataT, res_T) = nth_range_type n T
  1712                                        |> pairf domain_type range_type
  1713                 in
  1714                   (optimized_case_def hol_ctxt dataT res_T
  1715                                       (map (do_term depth Ts) (take n ts)),
  1716                    drop n ts)
  1717                 end
  1718             | _ =>
  1719               if is_constr ctxt stds x then
  1720                 (Const x, ts)
  1721               else if is_stale_constr ctxt x then
  1722                 raise NOT_SUPPORTED ("(non-co)constructors of codatatypes \
  1723                                      \(\"" ^ s ^ "\")")
  1724               else if is_quot_abs_fun ctxt x then
  1725                 let
  1726                   val rep_T = domain_type T
  1727                   val abs_T = range_type T
  1728                 in
  1729                   (Abs (Name.uu, rep_T,
  1730                         Const (@{const_name Quot}, rep_T --> abs_T)
  1731                                $ (Const (quot_normal_name_for_type ctxt abs_T,
  1732                                          rep_T --> rep_T) $ Bound 0)), ts)
  1733                 end
  1734               else if is_quot_rep_fun ctxt x then
  1735                 quot_rep_of depth Ts (domain_type T) (range_type T) ts
  1736               else if is_record_get thy x then
  1737                 case length ts of
  1738                   0 => (do_term depth Ts (eta_expand Ts t 1), [])
  1739                 | _ => (optimized_record_get hol_ctxt s (domain_type T)
  1740                             (range_type T) (do_term depth Ts (hd ts)), tl ts)
  1741               else if is_record_update thy x then
  1742                 case length ts of
  1743                   2 => (optimized_record_update hol_ctxt
  1744                             (unsuffix Record.updateN s) (nth_range_type 2 T)
  1745                             (do_term depth Ts (hd ts))
  1746                             (do_term depth Ts (nth ts 1)), [])
  1747                 | n => (do_term depth Ts (eta_expand Ts t (2 - n)), [])
  1748               else if is_abs_fun ctxt x andalso
  1749                       is_quot_type ctxt (range_type T) then
  1750                 let
  1751                   val abs_T = range_type T
  1752                   val rep_T = domain_type (domain_type T)
  1753                   val eps_fun = Const (@{const_name Eps},
  1754                                        (rep_T --> bool_T) --> rep_T)
  1755                   val normal_fun =
  1756                     Const (quot_normal_name_for_type ctxt abs_T,
  1757                            rep_T --> rep_T)
  1758                   val abs_fun = Const (@{const_name Quot}, rep_T --> abs_T)
  1759                 in
  1760                   (Abs (Name.uu, rep_T --> bool_T,
  1761                         abs_fun $ (normal_fun $ (eps_fun $ Bound 0)))
  1762                    |> do_term (depth + 1) Ts, ts)
  1763                 end
  1764               else if is_rep_fun ctxt x then
  1765                 let val x' = mate_of_rep_fun ctxt x in
  1766                   if is_constr ctxt stds x' then
  1767                     select_nth_constr_arg_with_args depth Ts x' ts 0
  1768                                                     (range_type T)
  1769                   else if is_quot_type ctxt (domain_type T) then
  1770                     let
  1771                       val abs_T = domain_type T
  1772                       val rep_T = domain_type (range_type T)
  1773                       val (rep_fun, _) = quot_rep_of depth Ts abs_T rep_T []
  1774                       val (equiv_rel, _) =
  1775                         equiv_relation_for_quot_type thy abs_T
  1776                     in
  1777                       (Abs (Name.uu, abs_T, equiv_rel $ (rep_fun $ Bound 0)),
  1778                        ts)
  1779                     end
  1780                   else
  1781                     (Const x, ts)
  1782                 end
  1783               else if is_equational_fun_but_no_plain_def hol_ctxt x orelse
  1784                       is_choice_spec_fun hol_ctxt x then
  1785                 (Const x, ts)
  1786               else case def_of_const thy def_table x of
  1787                 SOME def =>
  1788                 if depth > unfold_max_depth then
  1789                   raise TOO_LARGE ("Nitpick_HOL.unfold_defs_in_term",
  1790                                    "too many nested definitions (" ^
  1791                                    string_of_int depth ^ ") while expanding " ^
  1792                                    quote s)
  1793                 else if s = @{const_name wfrec'} then
  1794                   (do_term (depth + 1) Ts (s_betapplys Ts (def, ts)), [])
  1795                 else if size_of_term def > def_inline_threshold () then
  1796                   (Const x, ts)
  1797                 else
  1798                   (do_term (depth + 1) Ts def, ts)
  1799               | NONE => (Const x, ts)
  1800         in
  1801           s_betapplys Ts (const, map (do_term depth Ts) ts)
  1802           |> s_beta_norm Ts
  1803         end
  1804   in do_term 0 [] end
  1805 
  1806 (** Axiom extraction/generation **)
  1807 
  1808 fun extensional_equal j (Type (@{type_name fun}, [dom_T, ran_T])) t1 t2 =
  1809     let val var_t = Var (("x", j), dom_T) in
  1810       extensional_equal (j + 1) ran_T (betapply (t1, var_t))
  1811                         (betapply (t2, var_t))
  1812     end
  1813   | extensional_equal _ T t1 t2 =
  1814     Const (@{const_name HOL.eq}, T --> T --> bool_T) $ t1 $ t2
  1815 
  1816 fun equationalize_term ctxt tag t =
  1817   let
  1818     val j = maxidx_of_term t + 1
  1819     val (prems, concl) = Logic.strip_horn t
  1820   in
  1821     Logic.list_implies (prems,
  1822         case concl of
  1823           @{const Trueprop} $ (Const (@{const_name HOL.eq}, Type (_, [T, _]))
  1824                                $ t1 $ t2) =>
  1825           @{const Trueprop} $ extensional_equal j T t1 t2
  1826         | @{const Trueprop} $ t' =>
  1827           @{const Trueprop} $ HOLogic.mk_eq (t', @{const True})
  1828         | Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
  1829           @{const Trueprop} $ extensional_equal j T t1 t2
  1830         | _ => (warning ("Ignoring " ^ quote tag ^ " for non-equation" ^
  1831                          quote (Syntax.string_of_term ctxt t) ^ ".");
  1832                 raise SAME ()))
  1833     |> SOME
  1834   end
  1835   handle SAME () => NONE
  1836 
  1837 fun pair_for_prop t =
  1838   case term_under_def t of
  1839     Const (s, _) => (s, t)
  1840   | t' => raise TERM ("Nitpick_HOL.pair_for_prop", [t, t'])
  1841 
  1842 fun def_table_for get ctxt subst =
  1843   ctxt |> get |> map (pair_for_prop o subst_atomic subst)
  1844        |> AList.group (op =) |> Symtab.make
  1845 
  1846 fun const_def_table ctxt subst ts =
  1847   def_table_for (map prop_of o Nitpick_Defs.get) ctxt subst
  1848   |> fold (fn (s, t) => Symtab.map_default (s, []) (cons t))
  1849           (map pair_for_prop ts)
  1850 
  1851 fun paired_with_consts t = map (rpair t) (Term.add_const_names t [])
  1852 fun const_nondef_table ts =
  1853   fold (append o paired_with_consts) ts [] |> AList.group (op =) |> Symtab.make
  1854 
  1855 fun const_simp_table ctxt =
  1856   def_table_for (map_filter (equationalize_term ctxt "nitpick_simp" o prop_of)
  1857                  o Nitpick_Simps.get) ctxt
  1858 fun const_psimp_table ctxt =
  1859   def_table_for (map_filter (equationalize_term ctxt "nitpick_psimp" o prop_of)
  1860                  o Nitpick_Psimps.get) ctxt
  1861 
  1862 fun const_choice_spec_table ctxt subst =
  1863   map (subst_atomic subst o prop_of) (Nitpick_Choice_Specs.get ctxt)
  1864   |> const_nondef_table
  1865 
  1866 fun inductive_intro_table ctxt subst def_table =
  1867   let val thy = ProofContext.theory_of ctxt in
  1868     def_table_for
  1869         (maps (map (unfold_mutually_inductive_preds thy def_table o prop_of)
  1870                o snd o snd)
  1871          o filter (fn (cat, _) => cat = Spec_Rules.Inductive orelse
  1872                                   cat = Spec_Rules.Co_Inductive)
  1873          o Spec_Rules.get) ctxt subst
  1874   end
  1875 
  1876 fun ground_theorem_table thy =
  1877   fold ((fn @{const Trueprop} $ t1 =>
  1878             is_ground_term t1 ? Inttab.map_default (hash_term t1, []) (cons t1)
  1879           | _ => I) o prop_of o snd) (Global_Theory.all_thms_of thy) Inttab.empty
  1880 
  1881 (* TODO: Move to "Nitpick.thy" *)
  1882 val basic_ersatz_table =
  1883   [(@{const_name card}, @{const_name card'}),
  1884    (@{const_name setsum}, @{const_name setsum'}),
  1885    (@{const_name fold_graph}, @{const_name fold_graph'}),
  1886    (@{const_name wf}, @{const_name wf'}),
  1887    (@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
  1888    (@{const_name wfrec}, @{const_name wfrec'})]
  1889 
  1890 fun ersatz_table ctxt =
  1891  basic_ersatz_table
  1892  |> fold (append o snd) (#frac_types (Data.get (Context.Proof ctxt)))
  1893 
  1894 fun add_simps simp_table s eqs =
  1895   Unsynchronized.change simp_table
  1896       (Symtab.update (s, eqs @ these (Symtab.lookup (!simp_table) s)))
  1897 
  1898 fun inverse_axioms_for_rep_fun ctxt (x as (_, T)) =
  1899   let
  1900     val thy = ProofContext.theory_of ctxt
  1901     val abs_T = domain_type T
  1902   in
  1903     typedef_info ctxt (fst (dest_Type abs_T)) |> the
  1904     |> pairf #Abs_inverse #Rep_inverse
  1905     |> pairself (specialize_type thy x o prop_of o the)
  1906     ||> single |> op ::
  1907   end
  1908 fun optimized_typedef_axioms ctxt (abs_z as (abs_s, _)) =
  1909   let
  1910     val thy = ProofContext.theory_of ctxt
  1911     val abs_T = Type abs_z
  1912   in
  1913     if is_univ_typedef ctxt abs_T then
  1914       []
  1915     else case typedef_info ctxt abs_s of
  1916       SOME {abs_type, rep_type, Rep_name, prop_of_Rep, set_name, ...} =>
  1917       let
  1918         val rep_T = varify_and_instantiate_type ctxt abs_type abs_T rep_type
  1919         val rep_t = Const (Rep_name, abs_T --> rep_T)
  1920         val set_t = Const (set_name, rep_T --> bool_T)
  1921         val set_t' =
  1922           prop_of_Rep |> HOLogic.dest_Trueprop
  1923                       |> specialize_type thy (dest_Const rep_t)
  1924                       |> HOLogic.dest_mem |> snd
  1925       in
  1926         [HOLogic.all_const abs_T
  1927          $ Abs (Name.uu, abs_T, set_t $ (rep_t $ Bound 0))]
  1928         |> set_t <> set_t' ? cons (HOLogic.mk_eq (set_t, set_t'))
  1929         |> map HOLogic.mk_Trueprop
  1930       end
  1931     | NONE => []
  1932   end
  1933 fun optimized_quot_type_axioms ctxt stds abs_z =
  1934   let
  1935     val thy = ProofContext.theory_of ctxt
  1936     val abs_T = Type abs_z
  1937     val rep_T = rep_type_for_quot_type thy abs_T
  1938     val (equiv_rel, partial) = equiv_relation_for_quot_type thy abs_T
  1939     val a_var = Var (("a", 0), abs_T)
  1940     val x_var = Var (("x", 0), rep_T)
  1941     val y_var = Var (("y", 0), rep_T)
  1942     val x = (@{const_name Quot}, rep_T --> abs_T)
  1943     val sel_a_t = select_nth_constr_arg ctxt stds x a_var 0 rep_T
  1944     val normal_fun =
  1945       Const (quot_normal_name_for_type ctxt abs_T, rep_T --> rep_T)
  1946     val normal_x = normal_fun $ x_var
  1947     val normal_y = normal_fun $ y_var
  1948     val is_unknown_t = Const (@{const_name is_unknown}, rep_T --> bool_T)
  1949   in
  1950     [Logic.mk_equals (normal_fun $ sel_a_t, sel_a_t),
  1951      Logic.list_implies
  1952          ([@{const Not} $ (is_unknown_t $ normal_x),
  1953            @{const Not} $ (is_unknown_t $ normal_y),
  1954            equiv_rel $ x_var $ y_var] |> map HOLogic.mk_Trueprop,
  1955            Logic.mk_equals (normal_x, normal_y)),
  1956      Logic.list_implies
  1957          ([HOLogic.mk_Trueprop (@{const Not} $ (is_unknown_t $ normal_x)),
  1958            HOLogic.mk_Trueprop (@{const Not} $ HOLogic.mk_eq (normal_x, x_var))],
  1959           HOLogic.mk_Trueprop (equiv_rel $ x_var $ normal_x))]
  1960     |> partial ? cons (HOLogic.mk_Trueprop (equiv_rel $ sel_a_t $ sel_a_t))
  1961   end
  1962 
  1963 fun codatatype_bisim_axioms (hol_ctxt as {ctxt, stds, ...}) T =
  1964   let
  1965     val xs = datatype_constrs hol_ctxt T
  1966     val set_T = T --> bool_T
  1967     val iter_T = @{typ bisim_iterator}
  1968     val bisim_max = @{const bisim_iterator_max}
  1969     val n_var = Var (("n", 0), iter_T)
  1970     val n_var_minus_1 =
  1971       Const (@{const_name safe_The}, (iter_T --> bool_T) --> iter_T)
  1972       $ Abs ("m", iter_T, HOLogic.eq_const iter_T
  1973                           $ (suc_const iter_T $ Bound 0) $ n_var)
  1974     val x_var = Var (("x", 0), T)
  1975     val y_var = Var (("y", 0), T)
  1976     fun bisim_const T =
  1977       Const (@{const_name bisim}, iter_T --> T --> T --> bool_T)
  1978     fun nth_sub_bisim x n nth_T =
  1979       (if is_codatatype ctxt nth_T then bisim_const nth_T $ n_var_minus_1
  1980        else HOLogic.eq_const nth_T)
  1981       $ select_nth_constr_arg ctxt stds x x_var n nth_T
  1982       $ select_nth_constr_arg ctxt stds x y_var n nth_T
  1983     fun case_func (x as (_, T)) =
  1984       let
  1985         val arg_Ts = binder_types T
  1986         val core_t =
  1987           discriminate_value hol_ctxt x y_var ::
  1988           map2 (nth_sub_bisim x) (index_seq 0 (length arg_Ts)) arg_Ts
  1989           |> foldr1 s_conj
  1990       in List.foldr absdummy core_t arg_Ts end
  1991   in
  1992     [HOLogic.mk_imp
  1993        (HOLogic.mk_disj (HOLogic.eq_const iter_T $ n_var $ zero_const iter_T,
  1994             s_betapply [] (optimized_case_def hol_ctxt T bool_T
  1995                                               (map case_func xs), x_var)),
  1996         bisim_const T $ n_var $ x_var $ y_var),
  1997      HOLogic.eq_const set_T $ (bisim_const T $ bisim_max $ x_var)
  1998      $ (Const (@{const_name insert}, T --> set_T --> set_T)
  1999         $ x_var $ Const (@{const_name bot_class.bot}, set_T))]
  2000     |> map HOLogic.mk_Trueprop
  2001   end
  2002 
  2003 exception NO_TRIPLE of unit
  2004 
  2005 fun triple_for_intro_rule thy x t =
  2006   let
  2007     val prems = Logic.strip_imp_prems t |> map (Object_Logic.atomize_term thy)
  2008     val concl = Logic.strip_imp_concl t |> Object_Logic.atomize_term thy
  2009     val (main, side) = List.partition (exists_Const (curry (op =) x)) prems
  2010     val is_good_head = curry (op =) (Const x) o head_of
  2011   in
  2012     if forall is_good_head main then (side, main, concl) else raise NO_TRIPLE ()
  2013   end
  2014 
  2015 val tuple_for_args = HOLogic.mk_tuple o snd o strip_comb
  2016 fun wf_constraint_for rel side concl main =
  2017   let
  2018     val core = HOLogic.mk_mem (HOLogic.mk_prod
  2019                                (pairself tuple_for_args (main, concl)), Var rel)
  2020     val t = List.foldl HOLogic.mk_imp core side
  2021     val vars = filter_out (curry (op =) rel) (Term.add_vars t [])
  2022   in
  2023     Library.foldl (fn (t', ((x, j), T)) =>
  2024                       HOLogic.all_const T
  2025                       $ Abs (x, T, abstract_over (Var ((x, j), T), t')))
  2026                   (t, vars)
  2027   end
  2028 fun wf_constraint_for_triple rel (side, main, concl) =
  2029   map (wf_constraint_for rel side concl) main |> foldr1 s_conj
  2030 
  2031 fun terminates_by ctxt timeout goal tac =
  2032   can (SINGLE (Classical.safe_tac (claset_of ctxt)) #> the
  2033        #> SINGLE (DETERM_TIMEOUT timeout
  2034                                  (tac ctxt (auto_tac (clasimpset_of ctxt))))
  2035        #> the #> Goal.finish ctxt) goal
  2036 
  2037 val max_cached_wfs = 50
  2038 val cached_timeout =
  2039   Synchronized.var "Nitpick_HOL.cached_timeout" (SOME Time.zeroTime)
  2040 val cached_wf_props =
  2041   Synchronized.var "Nitpick_HOL.cached_wf_props" ([] : (term * bool) list)
  2042 
  2043 val termination_tacs = [Lexicographic_Order.lex_order_tac true,
  2044                         ScnpReconstruct.sizechange_tac]
  2045 
  2046 fun uncached_is_well_founded_inductive_pred
  2047         ({thy, ctxt, stds, debug, tac_timeout, intro_table, ...} : hol_context)
  2048         (x as (_, T)) =
  2049   case def_props_for_const thy stds intro_table x of
  2050     [] => raise TERM ("Nitpick_HOL.uncached_is_well_founded_inductive",
  2051                       [Const x])
  2052   | intro_ts =>
  2053     (case map (triple_for_intro_rule thy x) intro_ts
  2054           |> filter_out (null o #2) of
  2055        [] => true
  2056      | triples =>
  2057        let
  2058          val binders_T = HOLogic.mk_tupleT (binder_types T)
  2059          val rel_T = HOLogic.mk_prodT (binders_T, binders_T) --> bool_T
  2060          val j = fold Integer.max (map maxidx_of_term intro_ts) 0 + 1
  2061          val rel = (("R", j), rel_T)
  2062          val prop = Const (@{const_name wf}, rel_T --> bool_T) $ Var rel ::
  2063                     map (wf_constraint_for_triple rel) triples
  2064                     |> foldr1 s_conj |> HOLogic.mk_Trueprop
  2065          val _ = if debug then
  2066                    Output.urgent_message ("Wellfoundedness goal: " ^
  2067                              Syntax.string_of_term ctxt prop ^ ".")
  2068                  else
  2069                    ()
  2070        in
  2071          if tac_timeout = Synchronized.value cached_timeout andalso
  2072             length (Synchronized.value cached_wf_props) < max_cached_wfs then
  2073            ()
  2074          else
  2075            (Synchronized.change cached_wf_props (K []);
  2076             Synchronized.change cached_timeout (K tac_timeout));
  2077          case AList.lookup (op =) (Synchronized.value cached_wf_props) prop of
  2078            SOME wf => wf
  2079          | NONE =>
  2080            let
  2081              val goal = prop |> cterm_of thy |> Goal.init
  2082              val wf = exists (terminates_by ctxt tac_timeout goal)
  2083                              termination_tacs
  2084            in Synchronized.change cached_wf_props (cons (prop, wf)); wf end
  2085        end)
  2086     handle List.Empty => false | NO_TRIPLE () => false
  2087 
  2088 (* The type constraint below is a workaround for a Poly/ML crash. *)
  2089 
  2090 fun is_well_founded_inductive_pred
  2091         (hol_ctxt as {thy, wfs, def_table, wf_cache, ...} : hol_context)
  2092         (x as (s, _)) =
  2093   case triple_lookup (const_match thy) wfs x of
  2094     SOME (SOME b) => b
  2095   | _ => s = @{const_name Nats} orelse s = @{const_name fold_graph'} orelse
  2096          case AList.lookup (op =) (!wf_cache) x of
  2097            SOME (_, wf) => wf
  2098          | NONE =>
  2099            let
  2100              val gfp = (fixpoint_kind_of_const thy def_table x = Gfp)
  2101              val wf = uncached_is_well_founded_inductive_pred hol_ctxt x
  2102            in
  2103              Unsynchronized.change wf_cache (cons (x, (gfp, wf))); wf
  2104            end
  2105 
  2106 fun ap_curry [_] _ t = t
  2107   | ap_curry arg_Ts tuple_T t =
  2108     let val n = length arg_Ts in
  2109       list_abs (map (pair "c") arg_Ts,
  2110                 incr_boundvars n t
  2111                 $ mk_flat_tuple tuple_T (map Bound (n - 1 downto 0)))
  2112     end
  2113 
  2114 fun num_occs_of_bound_in_term j (t1 $ t2) =
  2115     op + (pairself (num_occs_of_bound_in_term j) (t1, t2))
  2116   | num_occs_of_bound_in_term j (Abs (_, _, t')) =
  2117     num_occs_of_bound_in_term (j + 1) t'
  2118   | num_occs_of_bound_in_term j (Bound j') = if j' = j then 1 else 0
  2119   | num_occs_of_bound_in_term _ _ = 0
  2120 
  2121 val is_linear_inductive_pred_def =
  2122   let
  2123     fun do_disjunct j (Const (@{const_name Ex}, _) $ Abs (_, _, t2)) =
  2124         do_disjunct (j + 1) t2
  2125       | do_disjunct j t =
  2126         case num_occs_of_bound_in_term j t of
  2127           0 => true
  2128         | 1 => exists (curry (op =) (Bound j) o head_of) (conjuncts_of t)
  2129         | _ => false
  2130     fun do_lfp_def (Const (@{const_name lfp}, _) $ t2) =
  2131         let val (xs, body) = strip_abs t2 in
  2132           case length xs of
  2133             1 => false
  2134           | n => forall (do_disjunct (n - 1)) (disjuncts_of body)
  2135         end
  2136       | do_lfp_def _ = false
  2137   in do_lfp_def o strip_abs_body end
  2138 
  2139 fun n_ptuple_paths 0 = []
  2140   | n_ptuple_paths 1 = []
  2141   | n_ptuple_paths n = [] :: map (cons 2) (n_ptuple_paths (n - 1))
  2142 val ap_n_split = HOLogic.mk_psplits o n_ptuple_paths
  2143 
  2144 val linear_pred_base_and_step_rhss =
  2145   let
  2146     fun aux (Const (@{const_name lfp}, _) $ t2) =
  2147         let
  2148           val (xs, body) = strip_abs t2
  2149           val arg_Ts = map snd (tl xs)
  2150           val tuple_T = HOLogic.mk_tupleT arg_Ts
  2151           val j = length arg_Ts
  2152           fun repair_rec j (Const (@{const_name Ex}, T1) $ Abs (s2, T2, t2')) =
  2153               Const (@{const_name Ex}, T1)
  2154               $ Abs (s2, T2, repair_rec (j + 1) t2')
  2155             | repair_rec j (@{const HOL.conj} $ t1 $ t2) =
  2156               @{const HOL.conj} $ repair_rec j t1 $ repair_rec j t2
  2157             | repair_rec j t =
  2158               let val (head, args) = strip_comb t in
  2159                 if head = Bound j then
  2160                   HOLogic.eq_const tuple_T $ Bound j
  2161                   $ mk_flat_tuple tuple_T args
  2162                 else
  2163                   t
  2164               end
  2165           val (nonrecs, recs) =
  2166             List.partition (curry (op =) 0 o num_occs_of_bound_in_term j)
  2167                            (disjuncts_of body)
  2168           val base_body = nonrecs |> List.foldl s_disj @{const False}
  2169           val step_body = recs |> map (repair_rec j)
  2170                                |> List.foldl s_disj @{const False}
  2171         in
  2172           (list_abs (tl xs, incr_bv (~1, j, base_body))
  2173            |> ap_n_split (length arg_Ts) tuple_T bool_T,
  2174            Abs ("y", tuple_T, list_abs (tl xs, step_body)
  2175                               |> ap_n_split (length arg_Ts) tuple_T bool_T))
  2176         end
  2177       | aux t =
  2178         raise TERM ("Nitpick_HOL.linear_pred_base_and_step_rhss.aux", [t])
  2179   in aux end
  2180 
  2181 fun starred_linear_pred_const (hol_ctxt as {simp_table, ...}) (s, T) def =
  2182   let
  2183     val j = maxidx_of_term def + 1
  2184     val (outer, fp_app) = strip_abs def
  2185     val outer_bounds = map Bound (length outer - 1 downto 0)
  2186     val outer_vars = map (fn (s, T) => Var ((s, j), T)) outer
  2187     val fp_app = subst_bounds (rev outer_vars, fp_app)
  2188     val (outer_Ts, rest_T) = strip_n_binders (length outer) T
  2189     val tuple_arg_Ts = strip_type rest_T |> fst
  2190     val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
  2191     val set_T = tuple_T --> bool_T
  2192     val curried_T = tuple_T --> set_T
  2193     val uncurried_T = Type (@{type_name prod}, [tuple_T, tuple_T]) --> bool_T
  2194     val (base_rhs, step_rhs) = linear_pred_base_and_step_rhss fp_app
  2195     val base_x as (base_s, _) = (base_prefix ^ s, outer_Ts ---> set_T)
  2196     val base_eq = HOLogic.mk_eq (list_comb (Const base_x, outer_vars), base_rhs)
  2197                   |> HOLogic.mk_Trueprop
  2198     val _ = add_simps simp_table base_s [base_eq]
  2199     val step_x as (step_s, _) = (step_prefix ^ s, outer_Ts ---> curried_T)
  2200     val step_eq = HOLogic.mk_eq (list_comb (Const step_x, outer_vars), step_rhs)
  2201                   |> HOLogic.mk_Trueprop
  2202     val _ = add_simps simp_table step_s [step_eq]
  2203   in
  2204     list_abs (outer,
  2205               Const (@{const_name Image}, uncurried_T --> set_T --> set_T)
  2206               $ (Const (@{const_name rtrancl}, uncurried_T --> uncurried_T)
  2207                  $ (Const (@{const_name prod_case}, curried_T --> uncurried_T)
  2208                     $ list_comb (Const step_x, outer_bounds)))
  2209               $ list_comb (Const base_x, outer_bounds)
  2210               |> ap_curry tuple_arg_Ts tuple_T)
  2211     |> unfold_defs_in_term hol_ctxt
  2212   end
  2213 
  2214 fun is_good_starred_linear_pred_type (Type (@{type_name fun}, Ts)) =
  2215     forall (not o (is_fun_type orf is_pair_type)) Ts
  2216   | is_good_starred_linear_pred_type _ = false
  2217 
  2218 fun unrolled_inductive_pred_const (hol_ctxt as {thy, star_linear_preds,
  2219                                                 def_table, simp_table, ...})
  2220                                   gfp (x as (s, T)) =
  2221   let
  2222     val iter_T = iterator_type_for_const gfp x
  2223     val x' as (s', _) = (unrolled_prefix ^ s, iter_T --> T)
  2224     val unrolled_const = Const x' $ zero_const iter_T
  2225     val def = the (def_of_const thy def_table x)
  2226   in
  2227     if is_equational_fun_but_no_plain_def hol_ctxt x' then
  2228       unrolled_const (* already done *)
  2229     else if not gfp andalso star_linear_preds andalso
  2230          is_linear_inductive_pred_def def andalso
  2231          is_good_starred_linear_pred_type T then
  2232       starred_linear_pred_const hol_ctxt x def
  2233     else
  2234       let
  2235         val j = maxidx_of_term def + 1
  2236         val (outer, fp_app) = strip_abs def
  2237         val outer_bounds = map Bound (length outer - 1 downto 0)
  2238         val cur = Var ((iter_var_prefix, j + 1), iter_T)
  2239         val next = suc_const iter_T $ cur
  2240         val rhs =
  2241           case fp_app of
  2242             Const _ $ t =>
  2243             s_betapply [] (t, list_comb (Const x', next :: outer_bounds))
  2244           | _ => raise TERM ("Nitpick_HOL.unrolled_inductive_pred_const",
  2245                              [fp_app])
  2246         val (inner, naked_rhs) = strip_abs rhs
  2247         val all = outer @ inner
  2248         val bounds = map Bound (length all - 1 downto 0)
  2249         val vars = map (fn (s, T) => Var ((s, j), T)) all
  2250         val eq = HOLogic.mk_eq (list_comb (Const x', cur :: bounds), naked_rhs)
  2251                  |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2252         val _ = add_simps simp_table s' [eq]
  2253       in unrolled_const end
  2254   end
  2255 
  2256 fun raw_inductive_pred_axiom ({thy, def_table, ...} : hol_context) x =
  2257   let
  2258     val def = the (def_of_const thy def_table x)
  2259     val (outer, fp_app) = strip_abs def
  2260     val outer_bounds = map Bound (length outer - 1 downto 0)
  2261     val rhs =
  2262       case fp_app of
  2263         Const _ $ t => s_betapply [] (t, list_comb (Const x, outer_bounds))
  2264       | _ => raise TERM ("Nitpick_HOL.raw_inductive_pred_axiom", [fp_app])
  2265     val (inner, naked_rhs) = strip_abs rhs
  2266     val all = outer @ inner
  2267     val bounds = map Bound (length all - 1 downto 0)
  2268     val j = maxidx_of_term def + 1
  2269     val vars = map (fn (s, T) => Var ((s, j), T)) all
  2270   in
  2271     HOLogic.mk_eq (list_comb (Const x, bounds), naked_rhs)
  2272     |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2273   end
  2274 fun inductive_pred_axiom hol_ctxt (x as (s, T)) =
  2275   if String.isPrefix ubfp_prefix s orelse String.isPrefix lbfp_prefix s then
  2276     let val x' = (strip_first_name_sep s |> snd, T) in
  2277       raw_inductive_pred_axiom hol_ctxt x' |> subst_atomic [(Const x', Const x)]
  2278     end
  2279   else
  2280     raw_inductive_pred_axiom hol_ctxt x
  2281 
  2282 fun equational_fun_axioms (hol_ctxt as {thy, ctxt, stds, def_table, simp_table,
  2283                                         psimp_table, ...}) x =
  2284   case def_props_for_const thy stds (!simp_table) x of
  2285     [] => (case def_props_for_const thy stds psimp_table x of
  2286              [] => (if is_inductive_pred hol_ctxt x then
  2287                       [inductive_pred_axiom hol_ctxt x]
  2288                     else case def_of_const thy def_table x of
  2289                       SOME def =>
  2290                       @{const Trueprop} $ HOLogic.mk_eq (Const x, def)
  2291                       |> equationalize_term ctxt "" |> the |> single
  2292                     | NONE => [])
  2293            | psimps => psimps)
  2294   | simps => simps
  2295 fun is_equational_fun_surely_complete hol_ctxt x =
  2296   case equational_fun_axioms hol_ctxt x of
  2297     [@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)] =>
  2298     strip_comb t1 |> snd |> forall is_Var
  2299   | _ => false
  2300 
  2301 (** Type preprocessing **)
  2302 
  2303 fun merged_type_var_table_for_terms thy ts =
  2304   let
  2305     fun add (s, S) table =
  2306       table
  2307       |> (case AList.lookup (Sign.subsort thy o swap) table S of
  2308             SOME _ => I
  2309           | NONE =>
  2310             filter_out (fn (S', _) => Sign.subsort thy (S, S'))
  2311             #> cons (S, s))
  2312     val tfrees = [] |> fold Term.add_tfrees ts
  2313                     |> sort (string_ord o pairself fst)
  2314   in [] |> fold add tfrees |> rev end
  2315 
  2316 fun merge_type_vars_in_term thy merge_type_vars table =
  2317   merge_type_vars
  2318   ? map_types (map_atyps
  2319         (fn TFree (_, S) =>
  2320             TFree (table |> find_first (fn (S', _) => Sign.subsort thy (S', S))
  2321                          |> the |> swap)
  2322           | T => T))
  2323 
  2324 fun add_ground_types hol_ctxt binarize =
  2325   let
  2326     fun aux T accum =
  2327       case T of
  2328         Type (@{type_name fun}, Ts) => fold aux Ts accum
  2329       | Type (@{type_name prod}, Ts) => fold aux Ts accum
  2330       | Type (@{type_name itself}, [T1]) => aux T1 accum
  2331       | Type (_, Ts) =>
  2332         if member (op =) (@{typ prop} :: @{typ bool} :: accum) T then
  2333           accum
  2334         else
  2335           T :: accum
  2336           |> fold aux (case binarized_and_boxed_datatype_constrs hol_ctxt
  2337                                                                  binarize T of
  2338                          [] => Ts
  2339                        | xs => map snd xs)
  2340       | _ => insert (op =) T accum
  2341   in aux end
  2342 
  2343 fun ground_types_in_type hol_ctxt binarize T =
  2344   add_ground_types hol_ctxt binarize T []
  2345 fun ground_types_in_terms hol_ctxt binarize ts =
  2346   fold (fold_types (add_ground_types hol_ctxt binarize)) ts []
  2347 
  2348 end;