src/HOL/Tools/Nitpick/nitpick_hol.ML
author blanchet
Fri Mar 18 10:17:37 2011 +0100 (2011-03-18)
changeset 41994 c567c860caf6
parent 41898 55d981e1232a
child 42361 23f352990944
permissions -rw-r--r--
always destroy constructor patterns, since this seems to be always useful
     1 (*  Title:      HOL/Tools/Nitpick/nitpick_hol.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2008, 2009, 2010
     4 
     5 Auxiliary HOL-related functions used by Nitpick.
     6 *)
     7 
     8 signature NITPICK_HOL =
     9 sig
    10   type styp = Nitpick_Util.styp
    11   type const_table = term list Symtab.table
    12   type special_fun = (styp * int list * term list) * styp
    13   type unrolled = styp * styp
    14   type wf_cache = (styp * (bool * bool)) list
    15 
    16   type hol_context =
    17     {thy: theory,
    18      ctxt: Proof.context,
    19      max_bisim_depth: int,
    20      boxes: (typ option * bool option) list,
    21      stds: (typ option * bool) list,
    22      wfs: (styp option * bool option) list,
    23      user_axioms: bool option,
    24      debug: bool,
    25      whacks: term list,
    26      binary_ints: bool option,
    27      destroy_constrs: bool,
    28      specialize: bool,
    29      star_linear_preds: bool,
    30      total_consts: bool option,
    31      needs: term list option,
    32      tac_timeout: Time.time option,
    33      evals: term list,
    34      case_names: (string * int) list,
    35      def_tables: const_table * const_table,
    36      nondef_table: const_table,
    37      user_nondefs: term list,
    38      simp_table: const_table Unsynchronized.ref,
    39      psimp_table: const_table,
    40      choice_spec_table: const_table,
    41      intro_table: const_table,
    42      ground_thm_table: term list Inttab.table,
    43      ersatz_table: (string * string) list,
    44      skolems: (string * string list) list Unsynchronized.ref,
    45      special_funs: special_fun list Unsynchronized.ref,
    46      unrolled_preds: unrolled list Unsynchronized.ref,
    47      wf_cache: wf_cache Unsynchronized.ref,
    48      constr_cache: (typ * styp list) list Unsynchronized.ref}
    49 
    50   datatype fixpoint_kind = Lfp | Gfp | NoFp
    51   datatype boxability =
    52     InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
    53 
    54   val name_sep : string
    55   val numeral_prefix : string
    56   val base_prefix : string
    57   val step_prefix : string
    58   val unrolled_prefix : string
    59   val ubfp_prefix : string
    60   val lbfp_prefix : string
    61   val quot_normal_prefix : string
    62   val skolem_prefix : string
    63   val special_prefix : string
    64   val uncurry_prefix : string
    65   val eval_prefix : string
    66   val iter_var_prefix : string
    67   val strip_first_name_sep : string -> string * string
    68   val original_name : string -> string
    69   val abs_var : indexname * typ -> term -> term
    70   val s_conj : term * term -> term
    71   val s_disj : term * term -> term
    72   val strip_any_connective : term -> term list * term
    73   val conjuncts_of : term -> term list
    74   val disjuncts_of : term -> term list
    75   val unarize_unbox_etc_type : typ -> typ
    76   val uniterize_unarize_unbox_etc_type : typ -> typ
    77   val string_for_type : Proof.context -> typ -> string
    78   val pretty_for_type : Proof.context -> typ -> Pretty.T
    79   val prefix_name : string -> string -> string
    80   val shortest_name : string -> string
    81   val short_name : string -> string
    82   val shorten_names_in_term : term -> term
    83   val strict_type_match : theory -> typ * typ -> bool
    84   val type_match : theory -> typ * typ -> bool
    85   val const_match : theory -> styp * styp -> bool
    86   val term_match : theory -> term * term -> bool
    87   val frac_from_term_pair : typ -> term -> term -> term
    88   val is_TFree : typ -> bool
    89   val is_fun_type : typ -> bool
    90   val is_set_type : typ -> bool
    91   val is_pair_type : typ -> bool
    92   val is_lfp_iterator_type : typ -> bool
    93   val is_gfp_iterator_type : typ -> bool
    94   val is_fp_iterator_type : typ -> bool
    95   val is_iterator_type : typ -> bool
    96   val is_boolean_type : typ -> bool
    97   val is_integer_type : typ -> bool
    98   val is_bit_type : typ -> bool
    99   val is_word_type : typ -> bool
   100   val is_integer_like_type : typ -> bool
   101   val is_record_type : typ -> bool
   102   val is_number_type : Proof.context -> typ -> bool
   103   val is_higher_order_type : typ -> bool
   104   val const_for_iterator_type : typ -> styp
   105   val strip_n_binders : int -> typ -> typ list * typ
   106   val nth_range_type : int -> typ -> typ
   107   val num_factors_in_type : typ -> int
   108   val num_binder_types : typ -> int
   109   val curried_binder_types : typ -> typ list
   110   val mk_flat_tuple : typ -> term list -> term
   111   val dest_n_tuple : int -> term -> term list
   112   val is_real_datatype : theory -> string -> bool
   113   val is_standard_datatype : theory -> (typ option * bool) list -> typ -> bool
   114   val is_codatatype : Proof.context -> typ -> bool
   115   val is_quot_type : Proof.context -> typ -> bool
   116   val is_pure_typedef : Proof.context -> typ -> bool
   117   val is_univ_typedef : Proof.context -> typ -> bool
   118   val is_datatype : Proof.context -> (typ option * bool) list -> typ -> bool
   119   val is_record_constr : styp -> bool
   120   val is_record_get : theory -> styp -> bool
   121   val is_record_update : theory -> styp -> bool
   122   val is_abs_fun : Proof.context -> styp -> bool
   123   val is_rep_fun : Proof.context -> styp -> bool
   124   val is_quot_abs_fun : Proof.context -> styp -> bool
   125   val is_quot_rep_fun : Proof.context -> styp -> bool
   126   val mate_of_rep_fun : Proof.context -> styp -> styp
   127   val is_constr_like : Proof.context -> styp -> bool
   128   val is_constr : Proof.context -> (typ option * bool) list -> styp -> bool
   129   val is_sel : string -> bool
   130   val is_sel_like_and_no_discr : string -> bool
   131   val box_type : hol_context -> boxability -> typ -> typ
   132   val binarize_nat_and_int_in_type : typ -> typ
   133   val binarize_nat_and_int_in_term : term -> term
   134   val discr_for_constr : styp -> styp
   135   val num_sels_for_constr_type : typ -> int
   136   val nth_sel_name_for_constr_name : string -> int -> string
   137   val nth_sel_for_constr : styp -> int -> styp
   138   val binarized_and_boxed_nth_sel_for_constr :
   139     hol_context -> bool -> styp -> int -> styp
   140   val sel_no_from_name : string -> int
   141   val close_form : term -> term
   142   val distinctness_formula : typ -> term list -> term
   143   val register_frac_type :
   144     string -> (string * string) list -> morphism -> Context.generic
   145     -> Context.generic
   146   val register_frac_type_global :
   147     string -> (string * string) list -> theory -> theory
   148   val unregister_frac_type :
   149     string -> morphism -> Context.generic -> Context.generic
   150   val unregister_frac_type_global : string -> theory -> theory
   151   val register_codatatype :
   152     typ -> string -> styp list -> morphism -> Context.generic -> Context.generic
   153   val register_codatatype_global :
   154     typ -> string -> styp list -> theory -> theory
   155   val unregister_codatatype :
   156     typ -> morphism -> Context.generic -> Context.generic
   157   val unregister_codatatype_global : typ -> theory -> theory
   158   val datatype_constrs : hol_context -> typ -> styp list
   159   val binarized_and_boxed_datatype_constrs :
   160     hol_context -> bool -> typ -> styp list
   161   val num_datatype_constrs : hol_context -> typ -> int
   162   val constr_name_for_sel_like : string -> string
   163   val binarized_and_boxed_constr_for_sel : hol_context -> bool -> styp -> styp
   164   val card_of_type : (typ * int) list -> typ -> int
   165   val bounded_card_of_type : int -> int -> (typ * int) list -> typ -> int
   166   val bounded_exact_card_of_type :
   167     hol_context -> typ list -> int -> int -> (typ * int) list -> typ -> int
   168   val typical_card_of_type : typ -> int
   169   val is_finite_type : hol_context -> typ -> bool
   170   val is_special_eligible_arg : bool -> typ list -> term -> bool
   171   val s_let :
   172     typ list -> string -> int -> typ -> typ -> (term -> term) -> term -> term
   173   val s_betapply : typ list -> term * term -> term
   174   val s_betapplys : typ list -> term * term list -> term
   175   val discriminate_value : hol_context -> styp -> term -> term
   176   val select_nth_constr_arg :
   177     Proof.context -> (typ option * bool) list -> styp -> term -> int -> typ
   178     -> term
   179   val construct_value :
   180     Proof.context -> (typ option * bool) list -> styp -> term list -> term
   181   val coerce_term : hol_context -> typ list -> typ -> typ -> term -> term
   182   val special_bounds : term list -> (indexname * typ) list
   183   val is_funky_typedef : Proof.context -> typ -> bool
   184   val all_axioms_of :
   185     Proof.context -> (term * term) list -> term list * term list * term list
   186   val arity_of_built_in_const :
   187     theory -> (typ option * bool) list -> styp -> int option
   188   val is_built_in_const :
   189     theory -> (typ option * bool) list -> styp -> bool
   190   val term_under_def : term -> term
   191   val case_const_names :
   192     Proof.context -> (typ option * bool) list -> (string * int) list
   193   val unfold_defs_in_term : hol_context -> term -> term
   194   val const_def_tables :
   195     Proof.context -> (term * term) list -> term list
   196     -> const_table * const_table
   197   val const_nondef_table : term list -> const_table
   198   val const_simp_table : Proof.context -> (term * term) list -> const_table
   199   val const_psimp_table : Proof.context -> (term * term) list -> const_table
   200   val const_choice_spec_table :
   201     Proof.context -> (term * term) list -> const_table
   202   val inductive_intro_table :
   203     Proof.context -> (term * term) list -> const_table * const_table
   204     -> const_table
   205   val ground_theorem_table : theory -> term list Inttab.table
   206   val ersatz_table : Proof.context -> (string * string) list
   207   val add_simps : const_table Unsynchronized.ref -> string -> term list -> unit
   208   val inverse_axioms_for_rep_fun : Proof.context -> styp -> term list
   209   val optimized_typedef_axioms : Proof.context -> string * typ list -> term list
   210   val optimized_quot_type_axioms :
   211     Proof.context -> (typ option * bool) list -> string * typ list -> term list
   212   val def_of_const : theory -> const_table * const_table -> styp -> term option
   213   val fixpoint_kind_of_rhs : term -> fixpoint_kind
   214   val fixpoint_kind_of_const :
   215     theory -> const_table * const_table -> string * typ -> fixpoint_kind
   216   val is_real_inductive_pred : hol_context -> styp -> bool
   217   val is_constr_pattern : Proof.context -> term -> bool
   218   val is_constr_pattern_lhs : Proof.context -> term -> bool
   219   val is_constr_pattern_formula : Proof.context -> term -> bool
   220   val nondef_props_for_const :
   221     theory -> bool -> const_table -> styp -> term list
   222   val is_choice_spec_fun : hol_context -> styp -> bool
   223   val is_choice_spec_axiom : theory -> const_table -> term -> bool
   224   val is_real_equational_fun : hol_context -> styp -> bool
   225   val is_equational_fun_but_no_plain_def : hol_context -> styp -> bool
   226   val codatatype_bisim_axioms : hol_context -> typ -> term list
   227   val is_well_founded_inductive_pred : hol_context -> styp -> bool
   228   val unrolled_inductive_pred_const : hol_context -> bool -> styp -> term
   229   val equational_fun_axioms : hol_context -> styp -> term list
   230   val is_equational_fun_surely_complete : hol_context -> styp -> bool
   231   val merged_type_var_table_for_terms :
   232     theory -> term list -> (sort * string) list
   233   val merge_type_vars_in_term :
   234     theory -> bool -> (sort * string) list -> term -> term
   235   val ground_types_in_type : hol_context -> bool -> typ -> typ list
   236   val ground_types_in_terms : hol_context -> bool -> term list -> typ list
   237 end;
   238 
   239 structure Nitpick_HOL : NITPICK_HOL =
   240 struct
   241 
   242 open Nitpick_Util
   243 
   244 type const_table = term list Symtab.table
   245 type special_fun = (styp * int list * term list) * styp
   246 type unrolled = styp * styp
   247 type wf_cache = (styp * (bool * bool)) list
   248 
   249 type hol_context =
   250   {thy: theory,
   251    ctxt: Proof.context,
   252    max_bisim_depth: int,
   253    boxes: (typ option * bool option) list,
   254    stds: (typ option * bool) list,
   255    wfs: (styp option * bool option) list,
   256    user_axioms: bool option,
   257    debug: bool,
   258    whacks: term list,
   259    binary_ints: bool option,
   260    destroy_constrs: bool,
   261    specialize: bool,
   262    star_linear_preds: bool,
   263    total_consts: bool option,
   264    needs: term list option,
   265    tac_timeout: Time.time option,
   266    evals: term list,
   267    case_names: (string * int) list,
   268    def_tables: const_table * const_table,
   269    nondef_table: const_table,
   270    user_nondefs: term list,
   271    simp_table: const_table Unsynchronized.ref,
   272    psimp_table: const_table,
   273    choice_spec_table: const_table,
   274    intro_table: const_table,
   275    ground_thm_table: term list Inttab.table,
   276    ersatz_table: (string * string) list,
   277    skolems: (string * string list) list Unsynchronized.ref,
   278    special_funs: special_fun list Unsynchronized.ref,
   279    unrolled_preds: unrolled list Unsynchronized.ref,
   280    wf_cache: wf_cache Unsynchronized.ref,
   281    constr_cache: (typ * styp list) list Unsynchronized.ref}
   282 
   283 datatype fixpoint_kind = Lfp | Gfp | NoFp
   284 datatype boxability =
   285   InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2
   286 
   287 structure Data = Generic_Data
   288 (
   289   type T = {frac_types: (string * (string * string) list) list,
   290             codatatypes: (string * (string * styp list)) list}
   291   val empty = {frac_types = [], codatatypes = []}
   292   val extend = I
   293   fun merge ({frac_types = fs1, codatatypes = cs1},
   294              {frac_types = fs2, codatatypes = cs2}) : T =
   295     {frac_types = AList.merge (op =) (K true) (fs1, fs2),
   296      codatatypes = AList.merge (op =) (K true) (cs1, cs2)}
   297 )
   298 
   299 val name_sep = "$"
   300 val numeral_prefix = nitpick_prefix ^ "num" ^ name_sep
   301 val sel_prefix = nitpick_prefix ^ "sel"
   302 val discr_prefix = nitpick_prefix ^ "is" ^ name_sep
   303 val set_prefix = nitpick_prefix ^ "set" ^ name_sep
   304 val lfp_iterator_prefix = nitpick_prefix ^ "lfpit" ^ name_sep
   305 val gfp_iterator_prefix = nitpick_prefix ^ "gfpit" ^ name_sep
   306 val unrolled_prefix = nitpick_prefix ^ "unroll" ^ name_sep
   307 val base_prefix = nitpick_prefix ^ "base" ^ name_sep
   308 val step_prefix = nitpick_prefix ^ "step" ^ name_sep
   309 val ubfp_prefix = nitpick_prefix ^ "ubfp" ^ name_sep
   310 val lbfp_prefix = nitpick_prefix ^ "lbfp" ^ name_sep
   311 val quot_normal_prefix = nitpick_prefix ^ "qn" ^ name_sep
   312 val skolem_prefix = nitpick_prefix ^ "sk"
   313 val special_prefix = nitpick_prefix ^ "sp"
   314 val uncurry_prefix = nitpick_prefix ^ "unc"
   315 val eval_prefix = nitpick_prefix ^ "eval"
   316 val iter_var_prefix = "i"
   317 
   318 (** Constant/type information and term/type manipulation **)
   319 
   320 fun sel_prefix_for j = sel_prefix ^ string_of_int j ^ name_sep
   321 fun quot_normal_name_for_type ctxt T =
   322   quot_normal_prefix ^ unyxml (Syntax.string_of_typ ctxt T)
   323 
   324 val strip_first_name_sep =
   325   Substring.full #> Substring.position name_sep ##> Substring.triml 1
   326   #> pairself Substring.string
   327 fun original_name s =
   328   if String.isPrefix nitpick_prefix s then
   329     case strip_first_name_sep s of (s1, "") => s1 | (_, s2) => original_name s2
   330   else
   331     s
   332 
   333 fun s_conj (t1, @{const True}) = t1
   334   | s_conj (@{const True}, t2) = t2
   335   | s_conj (t1, t2) =
   336     if t1 = @{const False} orelse t2 = @{const False} then @{const False}
   337     else HOLogic.mk_conj (t1, t2)
   338 fun s_disj (t1, @{const False}) = t1
   339   | s_disj (@{const False}, t2) = t2
   340   | s_disj (t1, t2) =
   341     if t1 = @{const True} orelse t2 = @{const True} then @{const True}
   342     else HOLogic.mk_disj (t1, t2)
   343 
   344 fun strip_connective conn_t (t as (t0 $ t1 $ t2)) =
   345     if t0 = conn_t then strip_connective t0 t2 @ strip_connective t0 t1 else [t]
   346   | strip_connective _ t = [t]
   347 fun strip_any_connective (t as (t0 $ _ $ _)) =
   348     if t0 = @{const HOL.conj} orelse t0 = @{const HOL.disj} then
   349       (strip_connective t0 t, t0)
   350     else
   351       ([t], @{const Not})
   352   | strip_any_connective t = ([t], @{const Not})
   353 val conjuncts_of = strip_connective @{const HOL.conj}
   354 val disjuncts_of = strip_connective @{const HOL.disj}
   355 
   356 (* When you add constants to these lists, make sure to handle them in
   357    "Nitpick_Nut.nut_from_term", and perhaps in "Nitpick_Mono.consider_term" as
   358    well. *)
   359 val built_in_consts =
   360   [(@{const_name all}, 1),
   361    (@{const_name "=="}, 2),
   362    (@{const_name "==>"}, 2),
   363    (@{const_name Pure.conjunction}, 2),
   364    (@{const_name Trueprop}, 1),
   365    (@{const_name Not}, 1),
   366    (@{const_name False}, 0),
   367    (@{const_name True}, 0),
   368    (@{const_name All}, 1),
   369    (@{const_name Ex}, 1),
   370    (@{const_name HOL.eq}, 1),
   371    (@{const_name HOL.conj}, 2),
   372    (@{const_name HOL.disj}, 2),
   373    (@{const_name HOL.implies}, 2),
   374    (@{const_name If}, 3),
   375    (@{const_name Let}, 2),
   376    (@{const_name Pair}, 2),
   377    (@{const_name fst}, 1),
   378    (@{const_name snd}, 1),
   379    (@{const_name Id}, 0),
   380    (@{const_name converse}, 1),
   381    (@{const_name trancl}, 1),
   382    (@{const_name rel_comp}, 2),
   383    (@{const_name finite}, 1),
   384    (@{const_name unknown}, 0),
   385    (@{const_name is_unknown}, 1),
   386    (@{const_name safe_The}, 1),
   387    (@{const_name Frac}, 0),
   388    (@{const_name norm_frac}, 0)]
   389 val built_in_nat_consts =
   390   [(@{const_name Suc}, 0),
   391    (@{const_name nat}, 0),
   392    (@{const_name nat_gcd}, 0),
   393    (@{const_name nat_lcm}, 0)]
   394 val built_in_typed_consts =
   395   [((@{const_name zero_class.zero}, int_T), 0),
   396    ((@{const_name one_class.one}, int_T), 0),
   397    ((@{const_name plus_class.plus}, int_T --> int_T --> int_T), 0),
   398    ((@{const_name minus_class.minus}, int_T --> int_T --> int_T), 0),
   399    ((@{const_name times_class.times}, int_T --> int_T --> int_T), 0),
   400    ((@{const_name div_class.div}, int_T --> int_T --> int_T), 0),
   401    ((@{const_name uminus_class.uminus}, int_T --> int_T), 0),
   402    ((@{const_name ord_class.less}, int_T --> int_T --> bool_T), 2),
   403    ((@{const_name ord_class.less_eq}, int_T --> int_T --> bool_T), 2)]
   404 val built_in_typed_nat_consts =
   405   [((@{const_name zero_class.zero}, nat_T), 0),
   406    ((@{const_name one_class.one}, nat_T), 0),
   407    ((@{const_name plus_class.plus}, nat_T --> nat_T --> nat_T), 0),
   408    ((@{const_name minus_class.minus}, nat_T --> nat_T --> nat_T), 0),
   409    ((@{const_name times_class.times}, nat_T --> nat_T --> nat_T), 0),
   410    ((@{const_name div_class.div}, nat_T --> nat_T --> nat_T), 0),
   411    ((@{const_name ord_class.less}, nat_T --> nat_T --> bool_T), 2),
   412    ((@{const_name ord_class.less_eq}, nat_T --> nat_T --> bool_T), 2),
   413    ((@{const_name of_nat}, nat_T --> int_T), 0)]
   414 val built_in_set_consts =
   415   [(@{const_name ord_class.less_eq}, 2)]
   416 
   417 fun unarize_type @{typ "unsigned_bit word"} = nat_T
   418   | unarize_type @{typ "signed_bit word"} = int_T
   419   | unarize_type (Type (s, Ts as _ :: _)) = Type (s, map unarize_type Ts)
   420   | unarize_type T = T
   421 fun unarize_unbox_etc_type (Type (@{type_name fun_box}, Ts)) =
   422     unarize_unbox_etc_type (Type (@{type_name fun}, Ts))
   423   | unarize_unbox_etc_type (Type (@{type_name pair_box}, Ts)) =
   424     Type (@{type_name prod}, map unarize_unbox_etc_type Ts)
   425   | unarize_unbox_etc_type @{typ "unsigned_bit word"} = nat_T
   426   | unarize_unbox_etc_type @{typ "signed_bit word"} = int_T
   427   | unarize_unbox_etc_type (Type (s, Ts as _ :: _)) =
   428     Type (s, map unarize_unbox_etc_type Ts)
   429   | unarize_unbox_etc_type T = T
   430 fun uniterize_type (Type (s, Ts as _ :: _)) = Type (s, map uniterize_type Ts)
   431   | uniterize_type @{typ bisim_iterator} = nat_T
   432   | uniterize_type T = T
   433 val uniterize_unarize_unbox_etc_type = uniterize_type o unarize_unbox_etc_type
   434 
   435 fun string_for_type ctxt = Syntax.string_of_typ ctxt o unarize_unbox_etc_type
   436 fun pretty_for_type ctxt = Syntax.pretty_typ ctxt o unarize_unbox_etc_type
   437 
   438 val prefix_name = Long_Name.qualify o Long_Name.base_name
   439 fun shortest_name s = List.last (space_explode "." s) handle List.Empty => ""
   440 val prefix_abs_vars = Term.map_abs_vars o prefix_name
   441 fun short_name s =
   442   case space_explode name_sep s of
   443     [_] => s |> String.isPrefix nitpick_prefix s ? unprefix nitpick_prefix
   444   | ss => map shortest_name ss |> space_implode "_"
   445 fun shorten_names_in_type (Type (s, Ts)) =
   446     Type (short_name s, map shorten_names_in_type Ts)
   447   | shorten_names_in_type T = T
   448 val shorten_names_in_term =
   449   map_aterms (fn Const (s, T) => Const (short_name s, T) | t => t)
   450   #> map_types shorten_names_in_type
   451 
   452 fun strict_type_match thy (T1, T2) =
   453   (Sign.typ_match thy (T2, T1) Vartab.empty; true)
   454   handle Type.TYPE_MATCH => false
   455 fun type_match thy = strict_type_match thy o pairself unarize_unbox_etc_type
   456 fun const_match thy ((s1, T1), (s2, T2)) =
   457   s1 = s2 andalso type_match thy (T1, T2)
   458 fun term_match thy (Const x1, Const x2) = const_match thy (x1, x2)
   459   | term_match thy (Free (s1, T1), Free (s2, T2)) =
   460     const_match thy ((shortest_name s1, T1), (shortest_name s2, T2))
   461   | term_match _ (t1, t2) = t1 aconv t2
   462 
   463 fun frac_from_term_pair T t1 t2 =
   464   case snd (HOLogic.dest_number t1) of
   465     0 => HOLogic.mk_number T 0
   466   | n1 => case snd (HOLogic.dest_number t2) of
   467             1 => HOLogic.mk_number T n1
   468           | n2 => Const (@{const_name divide}, T --> T --> T)
   469                   $ HOLogic.mk_number T n1 $ HOLogic.mk_number T n2
   470 
   471 fun is_TFree (TFree _) = true
   472   | is_TFree _ = false
   473 fun is_fun_type (Type (@{type_name fun}, _)) = true
   474   | is_fun_type _ = false
   475 fun is_set_type (Type (@{type_name fun}, [_, @{typ bool}])) = true
   476   | is_set_type _ = false
   477 fun is_pair_type (Type (@{type_name prod}, _)) = true
   478   | is_pair_type _ = false
   479 fun is_lfp_iterator_type (Type (s, _)) = String.isPrefix lfp_iterator_prefix s
   480   | is_lfp_iterator_type _ = false
   481 fun is_gfp_iterator_type (Type (s, _)) = String.isPrefix gfp_iterator_prefix s
   482   | is_gfp_iterator_type _ = false
   483 val is_fp_iterator_type = is_lfp_iterator_type orf is_gfp_iterator_type
   484 fun is_iterator_type T =
   485   (T = @{typ bisim_iterator} orelse is_fp_iterator_type T)
   486 fun is_boolean_type T = (T = prop_T orelse T = bool_T)
   487 fun is_integer_type T = (T = nat_T orelse T = int_T)
   488 fun is_bit_type T = (T = @{typ unsigned_bit} orelse T = @{typ signed_bit})
   489 fun is_word_type (Type (@{type_name word}, _)) = true
   490   | is_word_type _ = false
   491 val is_integer_like_type = is_iterator_type orf is_integer_type orf is_word_type
   492 val is_record_type = not o null o Record.dest_recTs
   493 fun is_frac_type ctxt (Type (s, [])) =
   494     s |> AList.lookup (op =) (#frac_types (Data.get (Context.Proof ctxt)))
   495       |> these |> null |> not
   496   | is_frac_type _ _ = false
   497 fun is_number_type ctxt = is_integer_like_type orf is_frac_type ctxt
   498 fun is_higher_order_type (Type (@{type_name fun}, _)) = true
   499   | is_higher_order_type (Type (_, Ts)) = exists is_higher_order_type Ts
   500   | is_higher_order_type _ = false
   501 
   502 fun iterator_type_for_const gfp (s, T) =
   503   Type ((if gfp then gfp_iterator_prefix else lfp_iterator_prefix) ^ s,
   504         binder_types T)
   505 fun const_for_iterator_type (Type (s, Ts)) =
   506     (strip_first_name_sep s |> snd, Ts ---> bool_T)
   507   | const_for_iterator_type T =
   508     raise TYPE ("Nitpick_HOL.const_for_iterator_type", [T], [])
   509 
   510 fun strip_n_binders 0 T = ([], T)
   511   | strip_n_binders n (Type (@{type_name fun}, [T1, T2])) =
   512     strip_n_binders (n - 1) T2 |>> cons T1
   513   | strip_n_binders n (Type (@{type_name fun_box}, Ts)) =
   514     strip_n_binders n (Type (@{type_name fun}, Ts))
   515   | strip_n_binders _ T = raise TYPE ("Nitpick_HOL.strip_n_binders", [T], [])
   516 val nth_range_type = snd oo strip_n_binders
   517 
   518 fun num_factors_in_type (Type (@{type_name prod}, [T1, T2])) =
   519     fold (Integer.add o num_factors_in_type) [T1, T2] 0
   520   | num_factors_in_type _ = 1
   521 fun num_binder_types (Type (@{type_name fun}, [_, T2])) =
   522     1 + num_binder_types T2
   523   | num_binder_types _ = 0
   524 val curried_binder_types = maps HOLogic.flatten_tupleT o binder_types
   525 fun maybe_curried_binder_types T =
   526   (if is_pair_type (body_type T) then binder_types else curried_binder_types) T
   527 
   528 fun mk_flat_tuple _ [t] = t
   529   | mk_flat_tuple (Type (@{type_name prod}, [T1, T2])) (t :: ts) =
   530     HOLogic.pair_const T1 T2 $ t $ (mk_flat_tuple T2 ts)
   531   | mk_flat_tuple T ts = raise TYPE ("Nitpick_HOL.mk_flat_tuple", [T], ts)
   532 fun dest_n_tuple 1 t = [t]
   533   | dest_n_tuple n t = HOLogic.dest_prod t ||> dest_n_tuple (n - 1) |> op ::
   534 
   535 type typedef_info =
   536   {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string,
   537    set_def: thm option, prop_of_Rep: thm, set_name: string,
   538    Abs_inverse: thm option, Rep_inverse: thm option}
   539 
   540 fun typedef_info ctxt s =
   541   if is_frac_type ctxt (Type (s, [])) then
   542     SOME {abs_type = Type (s, []), rep_type = @{typ "int * int"},
   543           Abs_name = @{const_name Abs_Frac}, Rep_name = @{const_name Rep_Frac},
   544           set_def = NONE, prop_of_Rep = @{prop "Rep_Frac x \<in> Frac"}
   545                           |> Logic.varify_global,
   546           set_name = @{const_name Frac}, Abs_inverse = NONE, Rep_inverse = NONE}
   547   else case Typedef.get_info ctxt s of
   548     (* When several entries are returned, it shouldn't matter much which one
   549        we take (according to Florian Haftmann). *)
   550     (* The "Logic.varifyT_global" calls are a temporary hack because these
   551        types's type variables sometimes clash with locally fixed type variables.
   552        Remove these calls once "Typedef" is fully localized. *)
   553     ({abs_type, rep_type, Abs_name, Rep_name, ...},
   554      {set_def, Rep, Abs_inverse, Rep_inverse, ...}) :: _ =>
   555     SOME {abs_type = Logic.varifyT_global abs_type,
   556           rep_type = Logic.varifyT_global rep_type, Abs_name = Abs_name,
   557           Rep_name = Rep_name, set_def = set_def, prop_of_Rep = prop_of Rep,
   558           set_name = set_prefix ^ s, Abs_inverse = SOME Abs_inverse,
   559           Rep_inverse = SOME Rep_inverse}
   560   | _ => NONE
   561 
   562 val is_typedef = is_some oo typedef_info
   563 val is_real_datatype = is_some oo Datatype.get_info
   564 fun is_standard_datatype thy = the oo triple_lookup (type_match thy)
   565 
   566 (* FIXME: Use antiquotation for "code_numeral" below or detect "rep_datatype",
   567    e.g., by adding a field to "Datatype_Aux.info". *)
   568 fun is_basic_datatype thy stds s =
   569   member (op =) [@{type_name prod}, @{type_name bool}, @{type_name int},
   570                  "Code_Numeral.code_numeral"] s orelse
   571   (s = @{type_name nat} andalso is_standard_datatype thy stds nat_T)
   572 
   573 (* TODO: use "Term_Subst.instantiateT" instead? *)
   574 fun instantiate_type thy T1 T1' T2 =
   575   Same.commit (Envir.subst_type_same
   576                    (Sign.typ_match thy (T1, T1') Vartab.empty)) T2
   577   handle Type.TYPE_MATCH =>
   578          raise TYPE ("Nitpick_HOL.instantiate_type", [T1, T1'], [])
   579 fun varify_and_instantiate_type ctxt T1 T1' T2 =
   580   let val thy = ProofContext.theory_of ctxt in
   581     instantiate_type thy (varify_type ctxt T1) T1' (varify_type ctxt T2)
   582   end
   583 
   584 fun repair_constr_type ctxt body_T' T =
   585   varify_and_instantiate_type ctxt (body_type T) body_T' T
   586 
   587 fun register_frac_type_generic frac_s ersaetze generic =
   588   let
   589     val {frac_types, codatatypes} = Data.get generic
   590     val frac_types = AList.update (op =) (frac_s, ersaetze) frac_types
   591   in Data.put {frac_types = frac_types, codatatypes = codatatypes} generic end
   592 (* TODO: Consider morphism. *)
   593 fun register_frac_type frac_s ersaetze (_ : morphism) =
   594   register_frac_type_generic frac_s ersaetze
   595 val register_frac_type_global = Context.theory_map oo register_frac_type_generic
   596 
   597 fun unregister_frac_type_generic frac_s = register_frac_type_generic frac_s []
   598 (* TODO: Consider morphism. *)
   599 fun unregister_frac_type frac_s (_ : morphism) =
   600   unregister_frac_type_generic frac_s
   601 val unregister_frac_type_global =
   602   Context.theory_map o unregister_frac_type_generic
   603 
   604 fun register_codatatype_generic co_T case_name constr_xs generic =
   605   let
   606     val ctxt = Context.proof_of generic
   607     val thy = Context.theory_of generic
   608     val {frac_types, codatatypes} = Data.get generic
   609     val constr_xs = map (apsnd (repair_constr_type ctxt co_T)) constr_xs
   610     val (co_s, co_Ts) = dest_Type co_T
   611     val _ =
   612       if forall is_TFree co_Ts andalso not (has_duplicates (op =) co_Ts) andalso
   613          co_s <> @{type_name fun} andalso
   614          not (is_basic_datatype thy [(NONE, true)] co_s) then
   615         ()
   616       else
   617         raise TYPE ("Nitpick_HOL.register_codatatype_generic", [co_T], [])
   618     val codatatypes = AList.update (op =) (co_s, (case_name, constr_xs))
   619                                    codatatypes
   620   in Data.put {frac_types = frac_types, codatatypes = codatatypes} generic end
   621 (* TODO: Consider morphism. *)
   622 fun register_codatatype co_T case_name constr_xs (_ : morphism) =
   623   register_codatatype_generic co_T case_name constr_xs
   624 val register_codatatype_global =
   625   Context.theory_map ooo register_codatatype_generic
   626 
   627 fun unregister_codatatype_generic co_T = register_codatatype_generic co_T "" []
   628 (* TODO: Consider morphism. *)
   629 fun unregister_codatatype co_T (_ : morphism) =
   630   unregister_codatatype_generic co_T
   631 val unregister_codatatype_global =
   632   Context.theory_map o unregister_codatatype_generic
   633 
   634 fun is_codatatype ctxt (Type (s, _)) =
   635     s |> AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   636       |> Option.map snd |> these |> null |> not
   637   | is_codatatype _ _ = false
   638 fun is_real_quot_type thy (Type (s, _)) =
   639     is_some (Quotient_Info.quotdata_lookup_raw thy s)
   640   | is_real_quot_type _ _ = false
   641 fun is_quot_type ctxt T =
   642   let val thy = ProofContext.theory_of ctxt in
   643     is_real_quot_type thy T andalso not (is_codatatype ctxt T)
   644   end
   645 fun is_pure_typedef ctxt (T as Type (s, _)) =
   646     let val thy = ProofContext.theory_of ctxt in
   647       is_typedef ctxt s andalso
   648       not (is_real_datatype thy s orelse is_real_quot_type thy T orelse
   649            is_codatatype ctxt T orelse is_record_type T orelse
   650            is_integer_like_type T)
   651     end
   652   | is_pure_typedef _ _ = false
   653 fun is_univ_typedef ctxt (Type (s, _)) =
   654     (case typedef_info ctxt s of
   655        SOME {set_def, prop_of_Rep, ...} =>
   656        let
   657          val t_opt =
   658            case set_def of
   659              SOME thm => try (snd o Logic.dest_equals o prop_of) thm
   660            | NONE => try (snd o HOLogic.dest_mem o HOLogic.dest_Trueprop)
   661                          prop_of_Rep
   662        in
   663          case t_opt of
   664            SOME (Const (@{const_name top}, _)) => true
   665            (* "Multiset.multiset" *)
   666          | SOME (Const (@{const_name Collect}, _)
   667                  $ Abs (_, _, Const (@{const_name finite}, _) $ _)) => true
   668            (* "FinFun.finfun" *)
   669          | SOME (Const (@{const_name Collect}, _) $ Abs (_, _,
   670                      Const (@{const_name Ex}, _) $ Abs (_, _,
   671                          Const (@{const_name finite}, _) $ _))) => true
   672          | _ => false
   673        end
   674      | NONE => false)
   675   | is_univ_typedef _ _ = false
   676 fun is_datatype ctxt stds (T as Type (s, _)) =
   677     let val thy = ProofContext.theory_of ctxt in
   678       (is_typedef ctxt s orelse is_codatatype ctxt T orelse
   679        T = @{typ ind} orelse is_real_quot_type thy T) andalso
   680       not (is_basic_datatype thy stds s)
   681     end
   682   | is_datatype _ _ _ = false
   683 
   684 fun all_record_fields thy T =
   685   let val (recs, more) = Record.get_extT_fields thy T in
   686     recs @ more :: all_record_fields thy (snd more)
   687   end
   688   handle TYPE _ => []
   689 fun is_record_constr (s, T) =
   690   String.isSuffix Record.extN s andalso
   691   let val dataT = body_type T in
   692     is_record_type dataT andalso
   693     s = unsuffix Record.ext_typeN (fst (dest_Type dataT)) ^ Record.extN
   694   end
   695 val num_record_fields = Integer.add 1 o length o fst oo Record.get_extT_fields
   696 fun no_of_record_field thy s T1 =
   697   find_index (curry (op =) s o fst)
   698              (Record.get_extT_fields thy T1 ||> single |> op @)
   699 fun is_record_get thy (s, Type (@{type_name fun}, [T1, _])) =
   700     exists (curry (op =) s o fst) (all_record_fields thy T1)
   701   | is_record_get _ _ = false
   702 fun is_record_update thy (s, T) =
   703   String.isSuffix Record.updateN s andalso
   704   exists (curry (op =) (unsuffix Record.updateN s) o fst)
   705          (all_record_fields thy (body_type T))
   706   handle TYPE _ => false
   707 fun is_abs_fun ctxt (s, Type (@{type_name fun}, [_, Type (s', _)])) =
   708     (case typedef_info ctxt s' of
   709        SOME {Abs_name, ...} => s = Abs_name
   710      | NONE => false)
   711   | is_abs_fun _ _ = false
   712 fun is_rep_fun ctxt (s, Type (@{type_name fun}, [Type (s', _), _])) =
   713     (case typedef_info ctxt s' of
   714        SOME {Rep_name, ...} => s = Rep_name
   715      | NONE => false)
   716   | is_rep_fun _ _ = false
   717 fun is_quot_abs_fun ctxt (x as (_, Type (@{type_name fun},
   718                                          [_, abs_T as Type (s', _)]))) =
   719     try (Quotient_Term.absrep_const_chk Quotient_Term.AbsF ctxt) s'
   720     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   721   | is_quot_abs_fun _ _ = false
   722 fun is_quot_rep_fun ctxt (x as (_, Type (@{type_name fun},
   723                                          [abs_T as Type (s', _), _]))) =
   724     try (Quotient_Term.absrep_const_chk Quotient_Term.RepF ctxt) s'
   725     = SOME (Const x) andalso not (is_codatatype ctxt abs_T)
   726   | is_quot_rep_fun _ _ = false
   727 
   728 fun mate_of_rep_fun ctxt (x as (_, Type (@{type_name fun},
   729                                          [T1 as Type (s', _), T2]))) =
   730     (case typedef_info ctxt s' of
   731        SOME {Abs_name, ...} => (Abs_name, Type (@{type_name fun}, [T2, T1]))
   732      | NONE => raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x]))
   733   | mate_of_rep_fun _ x = raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x])
   734 fun rep_type_for_quot_type thy (T as Type (s, _)) =
   735     let val {qtyp, rtyp, ...} = Quotient_Info.quotdata_lookup thy s in
   736       instantiate_type thy qtyp T rtyp
   737     end
   738   | rep_type_for_quot_type _ T =
   739     raise TYPE ("Nitpick_HOL.rep_type_for_quot_type", [T], [])
   740 fun equiv_relation_for_quot_type thy (Type (s, Ts)) =
   741     let
   742       val {qtyp, equiv_rel, equiv_thm, ...} =
   743         Quotient_Info.quotdata_lookup thy s
   744       val partial =
   745         case prop_of equiv_thm of
   746           @{const Trueprop} $ (Const (@{const_name equivp}, _) $ _) => false
   747         | @{const Trueprop} $ (Const (@{const_name part_equivp}, _) $ _) => true
   748         | _ => raise NOT_SUPPORTED "Ill-formed quotient type equivalence \
   749                                    \relation theorem"
   750       val Ts' = qtyp |> dest_Type |> snd
   751     in (subst_atomic_types (Ts' ~~ Ts) equiv_rel, partial) end
   752   | equiv_relation_for_quot_type _ T =
   753     raise TYPE ("Nitpick_HOL.equiv_relation_for_quot_type", [T], [])
   754 
   755 fun is_coconstr ctxt (s, T) =
   756   case body_type T of
   757     co_T as Type (co_s, _) =>
   758     let val {codatatypes, ...} = Data.get (Context.Proof ctxt) in
   759       exists (fn (s', T') => s = s' andalso repair_constr_type ctxt co_T T' = T)
   760              (AList.lookup (op =) codatatypes co_s |> Option.map snd |> these)
   761     end
   762   | _ => false
   763 fun is_constr_like ctxt (s, T) =
   764   member (op =) [@{const_name FunBox}, @{const_name PairBox},
   765                  @{const_name Quot}, @{const_name Zero_Rep},
   766                  @{const_name Suc_Rep}] s orelse
   767   let
   768     val thy = ProofContext.theory_of ctxt
   769     val (x as (_, T)) = (s, unarize_unbox_etc_type T)
   770   in
   771     is_real_constr thy x orelse is_record_constr x orelse
   772     (is_abs_fun ctxt x andalso is_pure_typedef ctxt (range_type T)) orelse
   773     is_coconstr ctxt x
   774   end
   775 fun is_stale_constr ctxt (x as (_, T)) =
   776   is_codatatype ctxt (body_type T) andalso is_constr_like ctxt x andalso
   777   not (is_coconstr ctxt x)
   778 fun is_constr ctxt stds (x as (_, T)) =
   779   let val thy = ProofContext.theory_of ctxt in
   780     is_constr_like ctxt x andalso
   781     not (is_basic_datatype thy stds
   782                          (fst (dest_Type (unarize_type (body_type T))))) andalso
   783     not (is_stale_constr ctxt x)
   784   end
   785 val is_sel = String.isPrefix discr_prefix orf String.isPrefix sel_prefix
   786 val is_sel_like_and_no_discr =
   787   String.isPrefix sel_prefix orf
   788   (member (op =) [@{const_name fst}, @{const_name snd}])
   789 
   790 fun in_fun_lhs_for InConstr = InSel
   791   | in_fun_lhs_for _ = InFunLHS
   792 fun in_fun_rhs_for InConstr = InConstr
   793   | in_fun_rhs_for InSel = InSel
   794   | in_fun_rhs_for InFunRHS1 = InFunRHS2
   795   | in_fun_rhs_for _ = InFunRHS1
   796 
   797 fun is_boxing_worth_it (hol_ctxt : hol_context) boxy T =
   798   case T of
   799     Type (@{type_name fun}, _) =>
   800     (boxy = InPair orelse boxy = InFunLHS) andalso
   801     not (is_boolean_type (body_type T))
   802   | Type (@{type_name prod}, Ts) =>
   803     boxy = InPair orelse boxy = InFunRHS1 orelse boxy = InFunRHS2 orelse
   804     ((boxy = InExpr orelse boxy = InFunLHS) andalso
   805      exists (is_boxing_worth_it hol_ctxt InPair)
   806             (map (box_type hol_ctxt InPair) Ts))
   807   | _ => false
   808 and should_box_type (hol_ctxt as {thy, boxes, ...}) boxy z =
   809   case triple_lookup (type_match thy) boxes (Type z) of
   810     SOME (SOME box_me) => box_me
   811   | _ => is_boxing_worth_it hol_ctxt boxy (Type z)
   812 and box_type hol_ctxt boxy T =
   813   case T of
   814     Type (z as (@{type_name fun}, [T1, T2])) =>
   815     if boxy <> InConstr andalso boxy <> InSel andalso
   816        should_box_type hol_ctxt boxy z then
   817       Type (@{type_name fun_box},
   818             [box_type hol_ctxt InFunLHS T1, box_type hol_ctxt InFunRHS1 T2])
   819     else
   820       box_type hol_ctxt (in_fun_lhs_for boxy) T1
   821       --> box_type hol_ctxt (in_fun_rhs_for boxy) T2
   822   | Type (z as (@{type_name prod}, Ts)) =>
   823     if boxy <> InConstr andalso boxy <> InSel
   824        andalso should_box_type hol_ctxt boxy z then
   825       Type (@{type_name pair_box}, map (box_type hol_ctxt InSel) Ts)
   826     else
   827       Type (@{type_name prod},
   828             map (box_type hol_ctxt
   829                           (if boxy = InConstr orelse boxy = InSel then boxy
   830                            else InPair)) Ts)
   831   | _ => T
   832 
   833 fun binarize_nat_and_int_in_type @{typ nat} = @{typ "unsigned_bit word"}
   834   | binarize_nat_and_int_in_type @{typ int} = @{typ "signed_bit word"}
   835   | binarize_nat_and_int_in_type (Type (s, Ts)) =
   836     Type (s, map binarize_nat_and_int_in_type Ts)
   837   | binarize_nat_and_int_in_type T = T
   838 val binarize_nat_and_int_in_term = map_types binarize_nat_and_int_in_type
   839 
   840 fun discr_for_constr (s, T) = (discr_prefix ^ s, body_type T --> bool_T)
   841 
   842 fun num_sels_for_constr_type T = length (maybe_curried_binder_types T)
   843 fun nth_sel_name_for_constr_name s n =
   844   if s = @{const_name Pair} then
   845     if n = 0 then @{const_name fst} else @{const_name snd}
   846   else
   847     sel_prefix_for n ^ s
   848 fun nth_sel_for_constr x ~1 = discr_for_constr x
   849   | nth_sel_for_constr (s, T) n =
   850     (nth_sel_name_for_constr_name s n,
   851      body_type T --> nth (maybe_curried_binder_types T) n)
   852 fun binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize =
   853   apsnd ((binarize ? binarize_nat_and_int_in_type) o box_type hol_ctxt InSel)
   854   oo nth_sel_for_constr
   855 
   856 fun sel_no_from_name s =
   857   if String.isPrefix discr_prefix s then
   858     ~1
   859   else if String.isPrefix sel_prefix s then
   860     s |> unprefix sel_prefix |> Int.fromString |> the
   861   else if s = @{const_name snd} then
   862     1
   863   else
   864     0
   865 
   866 val close_form =
   867   let
   868     fun close_up zs zs' =
   869       fold (fn (z as ((s, _), T)) => fn t' =>
   870                Term.all T $ Abs (s, T, abstract_over (Var z, t')))
   871            (take (length zs' - length zs) zs')
   872     fun aux zs (@{const "==>"} $ t1 $ t2) =
   873         let val zs' = Term.add_vars t1 zs in
   874           close_up zs zs' (Logic.mk_implies (t1, aux zs' t2))
   875         end
   876       | aux zs t = close_up zs (Term.add_vars t zs) t
   877   in aux [] end
   878 
   879 fun distinctness_formula T =
   880   all_distinct_unordered_pairs_of
   881   #> map (fn (t1, t2) => @{const Not} $ (HOLogic.eq_const T $ t1 $ t2))
   882   #> List.foldr (s_conj o swap) @{const True}
   883 
   884 fun zero_const T = Const (@{const_name zero_class.zero}, T)
   885 fun suc_const T = Const (@{const_name Suc}, T --> T)
   886 
   887 fun uncached_datatype_constrs ({thy, ctxt, stds, ...} : hol_context)
   888                               (T as Type (s, Ts)) =
   889     (case AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
   890                        s of
   891        SOME (_, xs' as (_ :: _)) => map (apsnd (repair_constr_type ctxt T)) xs'
   892      | _ =>
   893        if is_datatype ctxt stds T then
   894          case Datatype.get_info thy s of
   895            SOME {index, descr, ...} =>
   896            let
   897              val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the
   898            in
   899              map (apsnd (fn Us =>
   900                             map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))
   901                  constrs
   902            end
   903          | NONE =>
   904            if is_record_type T then
   905              let
   906                val s' = unsuffix Record.ext_typeN s ^ Record.extN
   907                val T' = (Record.get_extT_fields thy T
   908                         |> apsnd single |> uncurry append |> map snd) ---> T
   909              in [(s', T')] end
   910            else if is_real_quot_type thy T then
   911              [(@{const_name Quot}, rep_type_for_quot_type thy T --> T)]
   912            else case typedef_info ctxt s of
   913              SOME {abs_type, rep_type, Abs_name, ...} =>
   914              [(Abs_name,
   915                varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
   916            | NONE =>
   917              if T = @{typ ind} then
   918                [dest_Const @{const Zero_Rep}, dest_Const @{const Suc_Rep}]
   919              else
   920                []
   921        else
   922          [])
   923   | uncached_datatype_constrs _ _ = []
   924 fun datatype_constrs (hol_ctxt as {constr_cache, ...}) T =
   925   case AList.lookup (op =) (!constr_cache) T of
   926     SOME xs => xs
   927   | NONE =>
   928     let val xs = uncached_datatype_constrs hol_ctxt T in
   929       (Unsynchronized.change constr_cache (cons (T, xs)); xs)
   930     end
   931 fun binarized_and_boxed_datatype_constrs hol_ctxt binarize =
   932   map (apsnd ((binarize ? binarize_nat_and_int_in_type)
   933               o box_type hol_ctxt InConstr)) o datatype_constrs hol_ctxt
   934 val num_datatype_constrs = length oo datatype_constrs
   935 
   936 fun constr_name_for_sel_like @{const_name fst} = @{const_name Pair}
   937   | constr_name_for_sel_like @{const_name snd} = @{const_name Pair}
   938   | constr_name_for_sel_like s' = original_name s'
   939 fun binarized_and_boxed_constr_for_sel hol_ctxt binarize (s', T') =
   940   let val s = constr_name_for_sel_like s' in
   941     AList.lookup (op =)
   942         (binarized_and_boxed_datatype_constrs hol_ctxt binarize (domain_type T'))
   943         s
   944     |> the |> pair s
   945   end
   946 
   947 fun card_of_type assigns (Type (@{type_name fun}, [T1, T2])) =
   948     reasonable_power (card_of_type assigns T2) (card_of_type assigns T1)
   949   | card_of_type assigns (Type (@{type_name prod}, [T1, T2])) =
   950     card_of_type assigns T1 * card_of_type assigns T2
   951   | card_of_type _ (Type (@{type_name itself}, _)) = 1
   952   | card_of_type _ @{typ prop} = 2
   953   | card_of_type _ @{typ bool} = 2
   954   | card_of_type assigns T =
   955     case AList.lookup (op =) assigns T of
   956       SOME k => k
   957     | NONE => if T = @{typ bisim_iterator} then 0
   958               else raise TYPE ("Nitpick_HOL.card_of_type", [T], [])
   959 
   960 fun bounded_card_of_type max default_card assigns
   961                          (Type (@{type_name fun}, [T1, T2])) =
   962     let
   963       val k1 = bounded_card_of_type max default_card assigns T1
   964       val k2 = bounded_card_of_type max default_card assigns T2
   965     in
   966       if k1 = max orelse k2 = max then max
   967       else Int.min (max, reasonable_power k2 k1)
   968     end
   969   | bounded_card_of_type max default_card assigns
   970                          (Type (@{type_name prod}, [T1, T2])) =
   971     let
   972       val k1 = bounded_card_of_type max default_card assigns T1
   973       val k2 = bounded_card_of_type max default_card assigns T2
   974     in if k1 = max orelse k2 = max then max else Int.min (max, k1 * k2) end
   975   | bounded_card_of_type max default_card assigns T =
   976     Int.min (max, if default_card = ~1 then
   977                     card_of_type assigns T
   978                   else
   979                     card_of_type assigns T
   980                     handle TYPE ("Nitpick_HOL.card_of_type", _, _) =>
   981                            default_card)
   982 
   983 fun bounded_exact_card_of_type hol_ctxt finitizable_dataTs max default_card
   984                                assigns T =
   985   let
   986     fun aux avoid T =
   987       (if member (op =) avoid T then
   988          0
   989        else if member (op =) finitizable_dataTs T then
   990          raise SAME ()
   991        else case T of
   992          Type (@{type_name fun}, [T1, T2]) =>
   993          let
   994            val k1 = aux avoid T1
   995            val k2 = aux avoid T2
   996          in
   997            if k1 = 0 orelse k2 = 0 then 0
   998            else if k1 >= max orelse k2 >= max then max
   999            else Int.min (max, reasonable_power k2 k1)
  1000          end
  1001        | Type (@{type_name prod}, [T1, T2]) =>
  1002          let
  1003            val k1 = aux avoid T1
  1004            val k2 = aux avoid T2
  1005          in
  1006            if k1 = 0 orelse k2 = 0 then 0
  1007            else if k1 >= max orelse k2 >= max then max
  1008            else Int.min (max, k1 * k2)
  1009          end
  1010        | Type (@{type_name itself}, _) => 1
  1011        | @{typ prop} => 2
  1012        | @{typ bool} => 2
  1013        | Type _ =>
  1014          (case datatype_constrs hol_ctxt T of
  1015             [] => if is_integer_type T orelse is_bit_type T then 0
  1016                   else raise SAME ()
  1017           | constrs =>
  1018             let
  1019               val constr_cards =
  1020                 map (Integer.prod o map (aux (T :: avoid)) o binder_types o snd)
  1021                     constrs
  1022             in
  1023               if exists (curry (op =) 0) constr_cards then 0
  1024               else Integer.sum constr_cards
  1025             end)
  1026        | _ => raise SAME ())
  1027       handle SAME () =>
  1028              AList.lookup (op =) assigns T |> the_default default_card
  1029   in Int.min (max, aux [] T) end
  1030 
  1031 val typical_atomic_card = 4
  1032 val typical_card_of_type = bounded_card_of_type 16777217 typical_atomic_card []
  1033 
  1034 fun is_finite_type hol_ctxt T =
  1035   bounded_exact_card_of_type hol_ctxt [] 1 2 [] T > 0
  1036 
  1037 fun is_special_eligible_arg strict Ts t =
  1038   case map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t) of
  1039     [] => true
  1040   | bad_Ts =>
  1041     let
  1042       val bad_Ts_cost =
  1043         if strict then fold (curry (op *) o typical_card_of_type) bad_Ts 1
  1044         else fold (Integer.max o typical_card_of_type) bad_Ts 0
  1045       val T_cost = typical_card_of_type (fastype_of1 (Ts, t))
  1046     in (bad_Ts_cost, T_cost) |> (if strict then op < else op <=) end
  1047 
  1048 fun abs_var ((s, j), T) body = Abs (s, T, abstract_over (Var ((s, j), T), body))
  1049 
  1050 fun let_var s = (nitpick_prefix ^ s, 999)
  1051 val let_inline_threshold = 20
  1052 
  1053 fun s_let Ts s n abs_T body_T f t =
  1054   if (n - 1) * (size_of_term t - 1) <= let_inline_threshold orelse
  1055      is_special_eligible_arg false Ts t then
  1056     f t
  1057   else
  1058     let val z = (let_var s, abs_T) in
  1059       Const (@{const_name Let}, abs_T --> (abs_T --> body_T) --> body_T)
  1060       $ t $ abs_var z (incr_boundvars 1 (f (Var z)))
  1061     end
  1062 
  1063 fun loose_bvar1_count (Bound i, k) = if i = k then 1 else 0
  1064   | loose_bvar1_count (t1 $ t2, k) =
  1065     loose_bvar1_count (t1, k) + loose_bvar1_count (t2, k)
  1066   | loose_bvar1_count (Abs (_, _, t), k) = loose_bvar1_count (t, k + 1)
  1067   | loose_bvar1_count _ = 0
  1068 
  1069 fun s_betapply _ (Const (@{const_name If}, _) $ @{const True} $ t1', _) = t1'
  1070   | s_betapply _ (Const (@{const_name If}, _) $ @{const False} $ _, t2) = t2
  1071   | s_betapply Ts (Const (@{const_name Let},
  1072                           Type (_, [bound_T, Type (_, [_, body_T])]))
  1073                    $ t12 $ Abs (s, T, t13'), t2) =
  1074     let val body_T' = range_type body_T in
  1075       Const (@{const_name Let}, bound_T --> (bound_T --> body_T') --> body_T')
  1076       $ t12 $ Abs (s, T, s_betapply (T :: Ts) (t13', incr_boundvars 1 t2))
  1077     end
  1078   | s_betapply Ts (t1 as Abs (s1, T1, t1'), t2) =
  1079     (s_let Ts s1 (loose_bvar1_count (t1', 0)) T1 (fastype_of1 (T1 :: Ts, t1'))
  1080               (curry betapply t1) t2
  1081      handle TERM _ => betapply (t1, t2)) (* FIXME: fix all uses *)
  1082   | s_betapply _ (t1, t2) = t1 $ t2
  1083 fun s_betapplys Ts = Library.foldl (s_betapply Ts)
  1084 
  1085 fun s_beta_norm Ts t =
  1086   let
  1087     fun aux _ (Var _) = raise Same.SAME
  1088       | aux Ts (Abs (s, T, t')) = Abs (s, T, aux (T :: Ts) t')
  1089       | aux Ts ((t1 as Abs _) $ t2) =
  1090         Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1091       | aux Ts (t1 $ t2) =
  1092         ((case aux Ts t1 of
  1093            t1 as Abs _ => Same.commit (aux Ts) (s_betapply Ts (t1, t2))
  1094          | t1 => t1 $ Same.commit (aux Ts) t2)
  1095         handle Same.SAME => t1 $ aux Ts t2)
  1096       | aux _ _ = raise Same.SAME
  1097   in aux Ts t handle Same.SAME => t end
  1098 
  1099 fun discr_term_for_constr hol_ctxt (x as (s, T)) =
  1100   let val dataT = body_type T in
  1101     if s = @{const_name Suc} then
  1102       Abs (Name.uu, dataT,
  1103            @{const Not} $ HOLogic.mk_eq (zero_const dataT, Bound 0))
  1104     else if num_datatype_constrs hol_ctxt dataT >= 2 then
  1105       Const (discr_for_constr x)
  1106     else
  1107       Abs (Name.uu, dataT, @{const True})
  1108   end
  1109 fun discriminate_value (hol_ctxt as {ctxt, ...}) x t =
  1110   case head_of t of
  1111     Const x' =>
  1112     if x = x' then @{const True}
  1113     else if is_constr_like ctxt x' then @{const False}
  1114     else s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1115   | _ => s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  1116 
  1117 fun nth_arg_sel_term_for_constr thy stds (x as (s, T)) n =
  1118   let val (arg_Ts, dataT) = strip_type T in
  1119     if dataT = nat_T andalso is_standard_datatype thy stds nat_T then
  1120       @{term "%n::nat. n - 1"}
  1121     else if is_pair_type dataT then
  1122       Const (nth_sel_for_constr x n)
  1123     else
  1124       let
  1125         fun aux m (Type (@{type_name prod}, [T1, T2])) =
  1126             let
  1127               val (m, t1) = aux m T1
  1128               val (m, t2) = aux m T2
  1129             in (m, HOLogic.mk_prod (t1, t2)) end
  1130           | aux m T =
  1131             (m + 1, Const (nth_sel_name_for_constr_name s m, dataT --> T)
  1132                     $ Bound 0)
  1133         val m = fold (Integer.add o num_factors_in_type)
  1134                      (List.take (arg_Ts, n)) 0
  1135       in Abs ("x", dataT, aux m (nth arg_Ts n) |> snd) end
  1136   end
  1137 fun select_nth_constr_arg ctxt stds x t n res_T =
  1138   let val thy = ProofContext.theory_of ctxt in
  1139     (case strip_comb t of
  1140        (Const x', args) =>
  1141        if x = x' then nth args n
  1142        else if is_constr_like ctxt x' then Const (@{const_name unknown}, res_T)
  1143        else raise SAME ()
  1144      | _ => raise SAME())
  1145     handle SAME () =>
  1146            s_betapply [] (nth_arg_sel_term_for_constr thy stds x n, t)
  1147   end
  1148 
  1149 fun construct_value _ _ x [] = Const x
  1150   | construct_value ctxt stds (x as (s, _)) args =
  1151     let val args = map Envir.eta_contract args in
  1152       case hd args of
  1153         Const (s', _) $ t =>
  1154         if is_sel_like_and_no_discr s' andalso
  1155            constr_name_for_sel_like s' = s andalso
  1156            forall (fn (n, t') =>
  1157                       select_nth_constr_arg ctxt stds x t n dummyT = t')
  1158                   (index_seq 0 (length args) ~~ args) then
  1159           t
  1160         else
  1161           list_comb (Const x, args)
  1162       | _ => list_comb (Const x, args)
  1163     end
  1164 
  1165 fun constr_expand (hol_ctxt as {ctxt, stds, ...}) T t =
  1166   (case head_of t of
  1167      Const x => if is_constr_like ctxt x then t else raise SAME ()
  1168    | _ => raise SAME ())
  1169   handle SAME () =>
  1170          let
  1171            val x' as (_, T') =
  1172              if is_pair_type T then
  1173                let val (T1, T2) = HOLogic.dest_prodT T in
  1174                  (@{const_name Pair}, T1 --> T2 --> T)
  1175                end
  1176              else
  1177                datatype_constrs hol_ctxt T |> hd
  1178            val arg_Ts = binder_types T'
  1179          in
  1180            list_comb (Const x', map2 (select_nth_constr_arg ctxt stds x' t)
  1181                                      (index_seq 0 (length arg_Ts)) arg_Ts)
  1182          end
  1183 
  1184 fun coerce_bound_no f j t =
  1185   case t of
  1186     t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2
  1187   | Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')
  1188   | Bound j' => if j' = j then f t else t
  1189   | _ => t
  1190 fun coerce_bound_0_in_term hol_ctxt new_T old_T =
  1191   old_T <> new_T ? coerce_bound_no (coerce_term hol_ctxt [new_T] old_T new_T) 0
  1192 and coerce_term (hol_ctxt as {ctxt, stds, ...}) Ts new_T old_T t =
  1193   if old_T = new_T then
  1194     t
  1195   else
  1196     case (new_T, old_T) of
  1197       (Type (new_s, new_Ts as [new_T1, new_T2]),
  1198        Type (@{type_name fun}, [old_T1, old_T2])) =>
  1199       (case eta_expand Ts t 1 of
  1200          Abs (s, _, t') =>
  1201          Abs (s, new_T1,
  1202               t' |> coerce_bound_0_in_term hol_ctxt new_T1 old_T1
  1203                  |> coerce_term hol_ctxt (new_T1 :: Ts) new_T2 old_T2)
  1204          |> Envir.eta_contract
  1205          |> new_s <> @{type_name fun}
  1206             ? construct_value ctxt stds
  1207                   (@{const_name FunBox},
  1208                    Type (@{type_name fun}, new_Ts) --> new_T)
  1209               o single
  1210        | t' => raise TERM ("Nitpick_HOL.coerce_term", [t']))
  1211     | (Type (new_s, new_Ts as [new_T1, new_T2]),
  1212        Type (old_s, old_Ts as [old_T1, old_T2])) =>
  1213       if old_s = @{type_name fun_box} orelse
  1214          old_s = @{type_name pair_box} orelse old_s = @{type_name prod} then
  1215         case constr_expand hol_ctxt old_T t of
  1216           Const (old_s, _) $ t1 =>
  1217           if new_s = @{type_name fun} then
  1218             coerce_term hol_ctxt Ts new_T (Type (@{type_name fun}, old_Ts)) t1
  1219           else
  1220             construct_value ctxt stds
  1221                 (old_s, Type (@{type_name fun}, new_Ts) --> new_T)
  1222                 [coerce_term hol_ctxt Ts (Type (@{type_name fun}, new_Ts))
  1223                              (Type (@{type_name fun}, old_Ts)) t1]
  1224         | Const _ $ t1 $ t2 =>
  1225           construct_value ctxt stds
  1226               (if new_s = @{type_name prod} then @{const_name Pair}
  1227                else @{const_name PairBox}, new_Ts ---> new_T)
  1228               (map3 (coerce_term hol_ctxt Ts) [new_T1, new_T2] [old_T1, old_T2]
  1229                     [t1, t2])
  1230         | t' => raise TERM ("Nitpick_HOL.coerce_term", [t'])
  1231       else
  1232         raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1233     | _ => raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
  1234 
  1235 fun is_ground_term (t1 $ t2) = is_ground_term t1 andalso is_ground_term t2
  1236   | is_ground_term (Const _) = true
  1237   | is_ground_term _ = false
  1238 
  1239 fun hashw_term (t1 $ t2) = hashw (hashw_term t1, hashw_term t2)
  1240   | hashw_term (Const (s, _)) = hashw_string (s, 0w0)
  1241   | hashw_term _ = 0w0
  1242 val hash_term = Word.toInt o hashw_term
  1243 
  1244 fun special_bounds ts =
  1245   fold Term.add_vars ts [] |> sort (Term_Ord.fast_indexname_ord o pairself fst)
  1246 
  1247 (* FIXME: detect "rep_datatype"? *)
  1248 fun is_funky_typedef_name ctxt s =
  1249   member (op =) [@{type_name unit}, @{type_name prod},
  1250                  @{type_name Sum_Type.sum}, @{type_name int}] s orelse
  1251   is_frac_type ctxt (Type (s, []))
  1252 fun is_funky_typedef ctxt (Type (s, _)) = is_funky_typedef_name ctxt s
  1253   | is_funky_typedef _ _ = false
  1254 fun is_arity_type_axiom (Const (@{const_name HOL.type_class}, _)
  1255                          $ Const (@{const_name TYPE}, _)) = true
  1256   | is_arity_type_axiom _ = false
  1257 fun is_typedef_axiom ctxt boring (@{const "==>"} $ _ $ t2) =
  1258     is_typedef_axiom ctxt boring t2
  1259   | is_typedef_axiom ctxt boring
  1260         (@{const Trueprop} $ (Const (@{const_name Typedef.type_definition}, _)
  1261          $ Const (_, Type (@{type_name fun}, [Type (s, _), _]))
  1262          $ Const _ $ _)) =
  1263     boring <> is_funky_typedef_name ctxt s andalso is_typedef ctxt s
  1264   | is_typedef_axiom _ _ _ = false
  1265 val is_class_axiom =
  1266   Logic.strip_horn #> swap #> op :: #> forall (can Logic.dest_of_class)
  1267 
  1268 (* Distinguishes between (1) constant definition axioms, (2) type arity and
  1269    typedef axioms, and (3) other axioms, and returns the pair ((1), (3)).
  1270    Typedef axioms are uninteresting to Nitpick, because it can retrieve them
  1271    using "typedef_info". *)
  1272 fun partition_axioms_by_definitionality ctxt axioms def_names =
  1273   let
  1274     val axioms = sort (fast_string_ord o pairself fst) axioms
  1275     val defs = Ord_List.inter (fast_string_ord o apsnd fst) def_names axioms
  1276     val nondefs =
  1277       Ord_List.subtract (fast_string_ord o apsnd fst) def_names axioms
  1278       |> filter_out ((is_arity_type_axiom orf is_typedef_axiom ctxt true) o snd)
  1279   in pairself (map snd) (defs, nondefs) end
  1280 
  1281 (* Ideally we would check against "Complex_Main", not "Refute", but any theory
  1282    will do as long as it contains all the "axioms" and "axiomatization"
  1283    commands. *)
  1284 fun is_built_in_theory thy = Theory.subthy (thy, @{theory Refute})
  1285 
  1286 val is_trivial_definition =
  1287   the_default false o try (op aconv o Logic.dest_equals)
  1288 val is_plain_definition =
  1289   let
  1290     fun do_lhs t1 =
  1291       case strip_comb t1 of
  1292         (Const _, args) =>
  1293         forall is_Var args andalso not (has_duplicates (op =) args)
  1294       | _ => false
  1295     fun do_eq (Const (@{const_name "=="}, _) $ t1 $ _) = do_lhs t1
  1296       | do_eq (@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)) =
  1297         do_lhs t1
  1298       | do_eq _ = false
  1299   in do_eq end
  1300 
  1301 fun all_axioms_of ctxt subst =
  1302   let
  1303     val thy = ProofContext.theory_of ctxt
  1304     val axioms_of_thys =
  1305       maps Thm.axioms_of
  1306       #> map (apsnd (subst_atomic subst o prop_of))
  1307       #> filter_out (is_class_axiom o snd)
  1308     val specs = Defs.all_specifications_of (Theory.defs_of thy)
  1309     val def_names = specs |> maps snd |> map_filter #def
  1310                     |> Ord_List.make fast_string_ord
  1311     val thys = thy :: Theory.ancestors_of thy
  1312     val (built_in_thys, user_thys) = List.partition is_built_in_theory thys
  1313     val built_in_axioms = axioms_of_thys built_in_thys
  1314     val user_axioms = axioms_of_thys user_thys
  1315     val (built_in_defs, built_in_nondefs) =
  1316       partition_axioms_by_definitionality ctxt built_in_axioms def_names
  1317       ||> filter (is_typedef_axiom ctxt false)
  1318     val (user_defs, user_nondefs) =
  1319       partition_axioms_by_definitionality ctxt user_axioms def_names
  1320     val (built_in_nondefs, user_nondefs) =
  1321       List.partition (is_typedef_axiom ctxt false) user_nondefs
  1322       |>> append built_in_nondefs
  1323     val defs =
  1324       (thy |> Global_Theory.all_thms_of
  1325            |> filter (curry (op =) Thm.definitionK o Thm.get_kind o snd)
  1326            |> map (prop_of o snd)
  1327            |> filter_out is_trivial_definition
  1328            |> filter is_plain_definition) @
  1329       user_defs @ built_in_defs
  1330   in (defs, built_in_nondefs, user_nondefs) end
  1331 
  1332 fun arity_of_built_in_const thy stds (s, T) =
  1333   if s = @{const_name If} then
  1334     if nth_range_type 3 T = @{typ bool} then NONE else SOME 3
  1335   else
  1336     let val std_nats = is_standard_datatype thy stds nat_T in
  1337       case AList.lookup (op =)
  1338                     (built_in_consts
  1339                      |> std_nats ? append built_in_nat_consts) s of
  1340         SOME n => SOME n
  1341       | NONE =>
  1342         case AList.lookup (op =)
  1343                  (built_in_typed_consts
  1344                   |> std_nats ? append built_in_typed_nat_consts)
  1345                  (s, unarize_type T) of
  1346           SOME n => SOME n
  1347         | NONE =>
  1348           case s of
  1349             @{const_name zero_class.zero} =>
  1350             if is_iterator_type T then SOME 0 else NONE
  1351           | @{const_name Suc} =>
  1352             if is_iterator_type (domain_type T) then SOME 0 else NONE
  1353           | _ => if is_fun_type T andalso is_set_type (domain_type T) then
  1354                    AList.lookup (op =) built_in_set_consts s
  1355                  else
  1356                    NONE
  1357     end
  1358 val is_built_in_const = is_some ooo arity_of_built_in_const
  1359 
  1360 (* This function is designed to work for both real definition axioms and
  1361    simplification rules (equational specifications). *)
  1362 fun term_under_def t =
  1363   case t of
  1364     @{const "==>"} $ _ $ t2 => term_under_def t2
  1365   | Const (@{const_name "=="}, _) $ t1 $ _ => term_under_def t1
  1366   | @{const Trueprop} $ t1 => term_under_def t1
  1367   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => term_under_def t1
  1368   | Abs (_, _, t') => term_under_def t'
  1369   | t1 $ _ => term_under_def t1
  1370   | _ => t
  1371 
  1372 (* Here we crucially rely on "specialize_type" performing a preorder traversal
  1373    of the term, without which the wrong occurrence of a constant could be
  1374    matched in the face of overloading. *)
  1375 fun def_props_for_const thy stds table (x as (s, _)) =
  1376   if is_built_in_const thy stds x then
  1377     []
  1378   else
  1379     these (Symtab.lookup table s)
  1380     |> map_filter (try (specialize_type thy x))
  1381     |> filter (curry (op =) (Const x) o term_under_def)
  1382 
  1383 fun normalized_rhs_of t =
  1384   let
  1385     fun aux (v as Var _) (SOME t) = SOME (lambda v t)
  1386       | aux (c as Const (@{const_name TYPE}, _)) (SOME t) = SOME (lambda c t)
  1387       | aux _ _ = NONE
  1388     val (lhs, rhs) =
  1389       case t of
  1390         Const (@{const_name "=="}, _) $ t1 $ t2 => (t1, t2)
  1391       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ t2) =>
  1392         (t1, t2)
  1393       | _ => raise TERM ("Nitpick_HOL.normalized_rhs_of", [t])
  1394     val args = strip_comb lhs |> snd
  1395   in fold_rev aux args (SOME rhs) end
  1396 
  1397 fun get_def_of_const thy table (x as (s, _)) =
  1398   x |> def_props_for_const thy [(NONE, false)] table |> List.last
  1399     |> normalized_rhs_of |> Option.map (prefix_abs_vars s)
  1400   handle List.Empty => NONE
  1401 
  1402 fun def_of_const_ext thy (unfold_table, fallback_table) (x as (s, _)) =
  1403   if is_built_in_const thy [(NONE, false)] x orelse original_name s <> s then
  1404     NONE
  1405   else case get_def_of_const thy unfold_table x of
  1406     SOME def => SOME (true, def)
  1407   | NONE => get_def_of_const thy fallback_table x |> Option.map (pair false)
  1408 
  1409 val def_of_const = Option.map snd ooo def_of_const_ext
  1410 
  1411 fun fixpoint_kind_of_rhs (Abs (_, _, t)) = fixpoint_kind_of_rhs t
  1412   | fixpoint_kind_of_rhs (Const (@{const_name lfp}, _) $ Abs _) = Lfp
  1413   | fixpoint_kind_of_rhs (Const (@{const_name gfp}, _) $ Abs _) = Gfp
  1414   | fixpoint_kind_of_rhs _ = NoFp
  1415 
  1416 fun is_mutually_inductive_pred_def thy table t =
  1417   let
  1418     fun is_good_arg (Bound _) = true
  1419       | is_good_arg (Const (s, _)) =
  1420         s = @{const_name True} orelse s = @{const_name False} orelse
  1421         s = @{const_name undefined}
  1422       | is_good_arg _ = false
  1423   in
  1424     case t |> strip_abs_body |> strip_comb of
  1425       (Const x, ts as (_ :: _)) =>
  1426       (case def_of_const thy table x of
  1427          SOME t' => fixpoint_kind_of_rhs t' <> NoFp andalso
  1428                     forall is_good_arg ts
  1429        | NONE => false)
  1430     | _ => false
  1431   end
  1432 fun unfold_mutually_inductive_preds thy table =
  1433   map_aterms (fn t as Const x =>
  1434                  (case def_of_const thy table x of
  1435                     SOME t' =>
  1436                     let val t' = Envir.eta_contract t' in
  1437                       if is_mutually_inductive_pred_def thy table t' then t'
  1438                       else t
  1439                     end
  1440                  | NONE => t)
  1441                | t => t)
  1442 
  1443 fun case_const_names ctxt stds =
  1444   let val thy = ProofContext.theory_of ctxt in
  1445     Symtab.fold (fn (dtype_s, {index, descr, case_name, ...}) =>
  1446                     if is_basic_datatype thy stds dtype_s then
  1447                       I
  1448                     else
  1449                       cons (case_name, AList.lookup (op =) descr index
  1450                                        |> the |> #3 |> length))
  1451                 (Datatype.get_all thy) [] @
  1452     map (apsnd length o snd) (#codatatypes (Data.get (Context.Proof ctxt)))
  1453   end
  1454 
  1455 fun fixpoint_kind_of_const thy table x =
  1456   if is_built_in_const thy [(NONE, false)] x then NoFp
  1457   else fixpoint_kind_of_rhs (the (def_of_const thy table x))
  1458   handle Option.Option => NoFp
  1459 
  1460 fun is_real_inductive_pred ({thy, stds, def_tables, intro_table, ...}
  1461                             : hol_context) x =
  1462   fixpoint_kind_of_const thy def_tables x <> NoFp andalso
  1463   not (null (def_props_for_const thy stds intro_table x))
  1464 fun is_inductive_pred hol_ctxt (x as (s, _)) =
  1465   is_real_inductive_pred hol_ctxt x orelse String.isPrefix ubfp_prefix s orelse
  1466   String.isPrefix lbfp_prefix s
  1467 
  1468 fun lhs_of_equation t =
  1469   case t of
  1470     Const (@{const_name all}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1471   | Const (@{const_name "=="}, _) $ t1 $ _ => SOME t1
  1472   | @{const "==>"} $ _ $ t2 => lhs_of_equation t2
  1473   | @{const Trueprop} $ t1 => lhs_of_equation t1
  1474   | Const (@{const_name All}, _) $ Abs (_, _, t1) => lhs_of_equation t1
  1475   | Const (@{const_name HOL.eq}, _) $ t1 $ _ => SOME t1
  1476   | @{const HOL.implies} $ _ $ t2 => lhs_of_equation t2
  1477   | _ => NONE
  1478 fun is_constr_pattern _ (Bound _) = true
  1479   | is_constr_pattern _ (Var _) = true
  1480   | is_constr_pattern ctxt t =
  1481     case strip_comb t of
  1482       (Const x, args) =>
  1483       is_constr_like ctxt x andalso forall (is_constr_pattern ctxt) args
  1484     | _ => false
  1485 fun is_constr_pattern_lhs ctxt t =
  1486   forall (is_constr_pattern ctxt) (snd (strip_comb t))
  1487 fun is_constr_pattern_formula ctxt t =
  1488   case lhs_of_equation t of
  1489     SOME t' => is_constr_pattern_lhs ctxt t'
  1490   | NONE => false
  1491 
  1492 (* Similar to "specialize_type" but returns all matches rather than only the
  1493    first (preorder) match. *)
  1494 fun multi_specialize_type thy slack (s, T) t =
  1495   let
  1496     fun aux (Const (s', T')) ys =
  1497         if s = s' then
  1498           ys |> (if AList.defined (op =) ys T' then
  1499                    I
  1500                  else
  1501                    cons (T', monomorphic_term (Sign.typ_match thy (T', T)
  1502                                                               Vartab.empty) t)
  1503                    handle Type.TYPE_MATCH => I
  1504                         | TERM _ =>
  1505                           if slack then
  1506                             I
  1507                           else
  1508                             raise NOT_SUPPORTED
  1509                                       ("too much polymorphism in axiom \"" ^
  1510                                        Syntax.string_of_term_global thy t ^
  1511                                        "\" involving " ^ quote s))
  1512         else
  1513           ys
  1514       | aux _ ys = ys
  1515   in map snd (fold_aterms aux t []) end
  1516 fun nondef_props_for_const thy slack table (x as (s, _)) =
  1517   these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)
  1518 
  1519 fun unvarify_term (t1 $ t2) = unvarify_term t1 $ unvarify_term t2
  1520   | unvarify_term (Var ((s, 0), T)) = Free (s, T)
  1521   | unvarify_term (Abs (s, T, t')) = Abs (s, T, unvarify_term t')
  1522   | unvarify_term t = t
  1523 fun axiom_for_choice_spec thy =
  1524   unvarify_term
  1525   #> Object_Logic.atomize_term thy
  1526   #> Choice_Specification.close_form
  1527   #> HOLogic.mk_Trueprop
  1528 fun is_choice_spec_fun ({thy, def_tables, nondef_table, choice_spec_table, ...}
  1529                         : hol_context) x =
  1530   case nondef_props_for_const thy true choice_spec_table x of
  1531     [] => false
  1532   | ts => case def_of_const thy def_tables x of
  1533             SOME (Const (@{const_name Eps}, _) $ _) => true
  1534           | SOME _ => false
  1535           | NONE =>
  1536             let val ts' = nondef_props_for_const thy true nondef_table x in
  1537               length ts' = length ts andalso
  1538               forall (fn t =>
  1539                          exists (curry (op aconv) (axiom_for_choice_spec thy t))
  1540                                 ts') ts
  1541             end
  1542 
  1543 fun is_choice_spec_axiom thy choice_spec_table t =
  1544   Symtab.exists (fn (_, ts) =>
  1545                     exists (curry (op aconv) t o axiom_for_choice_spec thy) ts)
  1546                 choice_spec_table
  1547 
  1548 fun is_real_equational_fun ({thy, stds, simp_table, psimp_table, ...}
  1549                             : hol_context) x =
  1550   exists (fn table => not (null (def_props_for_const thy stds table x)))
  1551          [!simp_table, psimp_table]
  1552 fun is_equational_fun_but_no_plain_def hol_ctxt =
  1553   is_real_equational_fun hol_ctxt orf is_inductive_pred hol_ctxt
  1554 
  1555 (** Constant unfolding **)
  1556 
  1557 fun constr_case_body ctxt stds (func_t, (x as (_, T))) =
  1558   let val arg_Ts = binder_types T in
  1559     s_betapplys [] (func_t, map2 (select_nth_constr_arg ctxt stds x (Bound 0))
  1560                                  (index_seq 0 (length arg_Ts)) arg_Ts)
  1561   end
  1562 fun add_constr_case res_T (body_t, guard_t) res_t =
  1563   if res_T = bool_T then
  1564     s_conj (HOLogic.mk_imp (guard_t, body_t), res_t)
  1565   else
  1566     Const (@{const_name If}, bool_T --> res_T --> res_T --> res_T)
  1567     $ guard_t $ body_t $ res_t
  1568 fun optimized_case_def (hol_ctxt as {ctxt, stds, ...}) dataT res_T func_ts =
  1569   let
  1570     val xs = datatype_constrs hol_ctxt dataT
  1571     val cases =
  1572       func_ts ~~ xs
  1573       |> map (fn (func_t, x) =>
  1574                  (constr_case_body ctxt stds (incr_boundvars 1 func_t, x),
  1575                   discriminate_value hol_ctxt x (Bound 0)))
  1576       |> AList.group (op aconv)
  1577       |> map (apsnd (List.foldl s_disj @{const False}))
  1578       |> sort (int_ord o pairself (size_of_term o snd))
  1579       |> rev
  1580   in
  1581     if res_T = bool_T then
  1582       if forall (member (op =) [@{const False}, @{const True}] o fst) cases then
  1583         case cases of
  1584           [(body_t, _)] => body_t
  1585         | [_, (@{const True}, head_t2)] => head_t2
  1586         | [_, (@{const False}, head_t2)] => @{const Not} $ head_t2
  1587         | _ => raise BAD ("Nitpick_HOL.optimized_case_def", "impossible cases")
  1588       else
  1589         @{const True} |> fold_rev (add_constr_case res_T) cases
  1590     else
  1591       fst (hd cases) |> fold_rev (add_constr_case res_T) (tl cases)
  1592   end
  1593   |> curry absdummy dataT
  1594 
  1595 fun optimized_record_get (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T res_T t =
  1596   let val constr_x = hd (datatype_constrs hol_ctxt rec_T) in
  1597     case no_of_record_field thy s rec_T of
  1598       ~1 => (case rec_T of
  1599                Type (_, Ts as _ :: _) =>
  1600                let
  1601                  val rec_T' = List.last Ts
  1602                  val j = num_record_fields thy rec_T - 1
  1603                in
  1604                  select_nth_constr_arg ctxt stds constr_x t j res_T
  1605                  |> optimized_record_get hol_ctxt s rec_T' res_T
  1606                end
  1607              | _ => raise TYPE ("Nitpick_HOL.optimized_record_get", [rec_T],
  1608                                 []))
  1609     | j => select_nth_constr_arg ctxt stds constr_x t j res_T
  1610   end
  1611 fun optimized_record_update (hol_ctxt as {thy, ctxt, stds, ...}) s rec_T fun_t
  1612                             rec_t =
  1613   let
  1614     val constr_x as (_, constr_T) = hd (datatype_constrs hol_ctxt rec_T)
  1615     val Ts = binder_types constr_T
  1616     val n = length Ts
  1617     val special_j = no_of_record_field thy s rec_T
  1618     val ts =
  1619       map2 (fn j => fn T =>
  1620                let val t = select_nth_constr_arg ctxt stds constr_x rec_t j T in
  1621                  if j = special_j then
  1622                    s_betapply [] (fun_t, t)
  1623                  else if j = n - 1 andalso special_j = ~1 then
  1624                    optimized_record_update hol_ctxt s
  1625                        (rec_T |> dest_Type |> snd |> List.last) fun_t t
  1626                  else
  1627                    t
  1628                end) (index_seq 0 n) Ts
  1629   in list_comb (Const constr_x, ts) end
  1630 
  1631 (* Prevents divergence in case of cyclic or infinite definition dependencies. *)
  1632 val unfold_max_depth = 255
  1633 
  1634 (* Inline definitions or define as an equational constant? Booleans tend to
  1635    benefit more from inlining, due to the polarity analysis. (However, if
  1636    "total_consts" is set, the polarity analysis is likely not to be so
  1637    crucial.) *)
  1638 val def_inline_threshold_for_booleans = 60
  1639 val def_inline_threshold_for_non_booleans = 20
  1640 
  1641 fun unfold_defs_in_term
  1642         (hol_ctxt as {thy, ctxt, stds, whacks, total_consts, case_names,
  1643                       def_tables, ground_thm_table, ersatz_table, ...}) =
  1644   let
  1645     fun do_term depth Ts t =
  1646       case t of
  1647         (t0 as Const (@{const_name Int.number_class.number_of},
  1648                       Type (@{type_name fun}, [_, ran_T]))) $ t1 =>
  1649         ((if is_number_type ctxt ran_T then
  1650             let
  1651               val j = t1 |> HOLogic.dest_numeral
  1652                          |> ran_T = nat_T ? Integer.max 0
  1653               val s = numeral_prefix ^ signed_string_of_int j
  1654             in
  1655               if is_integer_like_type ran_T then
  1656                 if is_standard_datatype thy stds ran_T then
  1657                   Const (s, ran_T)
  1658                 else
  1659                   funpow j (curry (op $) (suc_const ran_T)) (zero_const ran_T)
  1660               else
  1661                 do_term depth Ts (Const (@{const_name of_int}, int_T --> ran_T)
  1662                                   $ Const (s, int_T))
  1663             end
  1664             handle TERM _ => raise SAME ()
  1665           else
  1666             raise SAME ())
  1667          handle SAME () =>
  1668                 s_betapply [] (do_term depth Ts t0, do_term depth Ts t1))
  1669       | Const (@{const_name refl_on}, T) $ Const (@{const_name top}, _) $ t2 =>
  1670         do_const depth Ts t (@{const_name refl'}, range_type T) [t2]
  1671       | (t0 as Const (@{const_name Sigma}, Type (_, [T1, Type (_, [T2, T3])])))
  1672         $ t1 $ (t2 as Abs (_, _, t2')) =>
  1673         if loose_bvar1 (t2', 0) then
  1674           s_betapplys Ts (do_term depth Ts t0, map (do_term depth Ts) [t1, t2])
  1675         else
  1676           do_term depth Ts
  1677                   (Const (@{const_name prod}, T1 --> range_type T2 --> T3)
  1678                    $ t1 $ incr_boundvars ~1 t2')
  1679       | Const (x as (@{const_name distinct},
  1680                Type (@{type_name fun}, [Type (@{type_name list}, [T']), _])))
  1681         $ (t1 as _ $ _) =>
  1682         (t1 |> HOLogic.dest_list |> distinctness_formula T'
  1683          handle TERM _ => do_const depth Ts t x [t1])
  1684       | Const (x as (@{const_name If}, _)) $ t1 $ t2 $ t3 =>
  1685         if is_ground_term t1 andalso
  1686            exists (Pattern.matches thy o rpair t1)
  1687                   (Inttab.lookup_list ground_thm_table (hash_term t1)) then
  1688           do_term depth Ts t2
  1689         else
  1690           do_const depth Ts t x [t1, t2, t3]
  1691       | Const (@{const_name Let}, _) $ t1 $ t2 =>
  1692         s_betapply Ts (pairself (do_term depth Ts) (t2, t1))
  1693       | Const x => do_const depth Ts t x []
  1694       | t1 $ t2 =>
  1695         (case strip_comb t of
  1696            (Const x, ts) => do_const depth Ts t x ts
  1697          | _ => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2))
  1698       | Bound _ => t
  1699       | Abs (s, T, body) => Abs (s, T, do_term depth (T :: Ts) body)
  1700       | _ => if member (term_match thy) whacks t then
  1701                Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1702              else
  1703                t
  1704     and select_nth_constr_arg_with_args _ _ (x as (_, T)) [] n res_T =
  1705         (Abs (Name.uu, body_type T,
  1706               select_nth_constr_arg ctxt stds x (Bound 0) n res_T), [])
  1707       | select_nth_constr_arg_with_args depth Ts x (t :: ts) n res_T =
  1708         (select_nth_constr_arg ctxt stds x (do_term depth Ts t) n res_T, ts)
  1709     and quot_rep_of depth Ts abs_T rep_T ts =
  1710       select_nth_constr_arg_with_args depth Ts
  1711           (@{const_name Quot}, rep_T --> abs_T) ts 0 rep_T
  1712     and do_const depth Ts t (x as (s, T)) ts =
  1713       if member (term_match thy) whacks (Const x) then
  1714         Const (@{const_name unknown}, fastype_of1 (Ts, t))
  1715       else case AList.lookup (op =) ersatz_table s of
  1716         SOME s' =>
  1717         do_const (depth + 1) Ts (list_comb (Const (s', T), ts)) (s', T) ts
  1718       | NONE =>
  1719         let
  1720           fun def_inline_threshold () =
  1721             if is_boolean_type (nth_range_type (length ts) T) andalso
  1722                total_consts <> SOME true then
  1723               def_inline_threshold_for_booleans
  1724             else
  1725               def_inline_threshold_for_non_booleans
  1726           val (const, ts) =
  1727             if is_built_in_const thy stds x then
  1728               (Const x, ts)
  1729             else case AList.lookup (op =) case_names s of
  1730               SOME n =>
  1731               if length ts < n then
  1732                 (do_term depth Ts (eta_expand Ts t (n - length ts)), [])
  1733               else
  1734                 let
  1735                   val (dataT, res_T) = nth_range_type n T
  1736                                        |> pairf domain_type range_type
  1737                 in
  1738                   (optimized_case_def hol_ctxt dataT res_T
  1739                                       (map (do_term depth Ts) (take n ts)),
  1740                    drop n ts)
  1741                 end
  1742             | _ =>
  1743               if is_constr ctxt stds x then
  1744                 (Const x, ts)
  1745               else if is_stale_constr ctxt x then
  1746                 raise NOT_SUPPORTED ("(non-co)constructors of codatatypes \
  1747                                      \(\"" ^ s ^ "\")")
  1748               else if is_quot_abs_fun ctxt x then
  1749                 let
  1750                   val rep_T = domain_type T
  1751                   val abs_T = range_type T
  1752                 in
  1753                   (Abs (Name.uu, rep_T,
  1754                         Const (@{const_name Quot}, rep_T --> abs_T)
  1755                                $ (Const (quot_normal_name_for_type ctxt abs_T,
  1756                                          rep_T --> rep_T) $ Bound 0)), ts)
  1757                 end
  1758               else if is_quot_rep_fun ctxt x then
  1759                 quot_rep_of depth Ts (domain_type T) (range_type T) ts
  1760               else if is_record_get thy x then
  1761                 case length ts of
  1762                   0 => (do_term depth Ts (eta_expand Ts t 1), [])
  1763                 | _ => (optimized_record_get hol_ctxt s (domain_type T)
  1764                             (range_type T) (do_term depth Ts (hd ts)), tl ts)
  1765               else if is_record_update thy x then
  1766                 case length ts of
  1767                   2 => (optimized_record_update hol_ctxt
  1768                             (unsuffix Record.updateN s) (nth_range_type 2 T)
  1769                             (do_term depth Ts (hd ts))
  1770                             (do_term depth Ts (nth ts 1)), [])
  1771                 | n => (do_term depth Ts (eta_expand Ts t (2 - n)), [])
  1772               else if is_abs_fun ctxt x andalso
  1773                       is_quot_type ctxt (range_type T) then
  1774                 let
  1775                   val abs_T = range_type T
  1776                   val rep_T = domain_type (domain_type T)
  1777                   val eps_fun = Const (@{const_name Eps},
  1778                                        (rep_T --> bool_T) --> rep_T)
  1779                   val normal_fun =
  1780                     Const (quot_normal_name_for_type ctxt abs_T,
  1781                            rep_T --> rep_T)
  1782                   val abs_fun = Const (@{const_name Quot}, rep_T --> abs_T)
  1783                 in
  1784                   (Abs (Name.uu, rep_T --> bool_T,
  1785                         abs_fun $ (normal_fun $ (eps_fun $ Bound 0)))
  1786                    |> do_term (depth + 1) Ts, ts)
  1787                 end
  1788               else if is_rep_fun ctxt x then
  1789                 let val x' = mate_of_rep_fun ctxt x in
  1790                   if is_constr ctxt stds x' then
  1791                     select_nth_constr_arg_with_args depth Ts x' ts 0
  1792                                                     (range_type T)
  1793                   else if is_quot_type ctxt (domain_type T) then
  1794                     let
  1795                       val abs_T = domain_type T
  1796                       val rep_T = domain_type (range_type T)
  1797                       val (rep_fun, _) = quot_rep_of depth Ts abs_T rep_T []
  1798                       val (equiv_rel, _) =
  1799                         equiv_relation_for_quot_type thy abs_T
  1800                     in
  1801                       (Abs (Name.uu, abs_T, equiv_rel $ (rep_fun $ Bound 0)),
  1802                        ts)
  1803                     end
  1804                   else
  1805                     (Const x, ts)
  1806                 end
  1807               else if is_equational_fun_but_no_plain_def hol_ctxt x orelse
  1808                       is_choice_spec_fun hol_ctxt x then
  1809                 (Const x, ts)
  1810               else case def_of_const_ext thy def_tables x of
  1811                 SOME (unfold, def) =>
  1812                 if depth > unfold_max_depth then
  1813                   raise TOO_LARGE ("Nitpick_HOL.unfold_defs_in_term",
  1814                                    "too many nested definitions (" ^
  1815                                    string_of_int depth ^ ") while expanding " ^
  1816                                    quote s)
  1817                 else if s = @{const_name wfrec'} then
  1818                   (do_term (depth + 1) Ts (s_betapplys Ts (def, ts)), [])
  1819                 else if not unfold andalso
  1820                      size_of_term def > def_inline_threshold () then
  1821                   (Const x, ts)
  1822                 else
  1823                   (do_term (depth + 1) Ts def, ts)
  1824               | NONE => (Const x, ts)
  1825         in
  1826           s_betapplys Ts (const, map (do_term depth Ts) ts)
  1827           |> s_beta_norm Ts
  1828         end
  1829   in do_term 0 [] end
  1830 
  1831 (** Axiom extraction/generation **)
  1832 
  1833 fun extensional_equal j (Type (@{type_name fun}, [dom_T, ran_T])) t1 t2 =
  1834     let val var_t = Var (("x", j), dom_T) in
  1835       extensional_equal (j + 1) ran_T (betapply (t1, var_t))
  1836                         (betapply (t2, var_t))
  1837     end
  1838   | extensional_equal _ T t1 t2 =
  1839     Const (@{const_name HOL.eq}, T --> T --> bool_T) $ t1 $ t2
  1840 
  1841 fun equationalize_term ctxt tag t =
  1842   let
  1843     val j = maxidx_of_term t + 1
  1844     val (prems, concl) = Logic.strip_horn t
  1845   in
  1846     Logic.list_implies (prems,
  1847         case concl of
  1848           @{const Trueprop} $ (Const (@{const_name HOL.eq}, Type (_, [T, _]))
  1849                                $ t1 $ t2) =>
  1850           @{const Trueprop} $ extensional_equal j T t1 t2
  1851         | @{const Trueprop} $ t' =>
  1852           @{const Trueprop} $ HOLogic.mk_eq (t', @{const True})
  1853         | Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
  1854           @{const Trueprop} $ extensional_equal j T t1 t2
  1855         | _ => (warning ("Ignoring " ^ quote tag ^ " for non-equation" ^
  1856                          quote (Syntax.string_of_term ctxt t) ^ ".");
  1857                 raise SAME ()))
  1858     |> SOME
  1859   end
  1860   handle SAME () => NONE
  1861 
  1862 fun pair_for_prop t =
  1863   case term_under_def t of
  1864     Const (s, _) => (s, t)
  1865   | t' => raise TERM ("Nitpick_HOL.pair_for_prop", [t, t'])
  1866 
  1867 fun def_table_for get ctxt subst =
  1868   ctxt |> get |> map (pair_for_prop o subst_atomic subst)
  1869        |> AList.group (op =) |> Symtab.make
  1870 
  1871 fun const_def_tables ctxt subst ts =
  1872   (def_table_for (map prop_of o Nitpick_Unfolds.get) ctxt subst,
  1873    fold (fn (s, t) => Symtab.map_default (s, []) (cons t))
  1874         (map pair_for_prop ts) Symtab.empty)
  1875 
  1876 fun paired_with_consts t = map (rpair t) (Term.add_const_names t [])
  1877 fun const_nondef_table ts =
  1878   fold (append o paired_with_consts) ts [] |> AList.group (op =) |> Symtab.make
  1879 
  1880 fun const_simp_table ctxt =
  1881   def_table_for (map_filter (equationalize_term ctxt "nitpick_simp" o prop_of)
  1882                  o Nitpick_Simps.get) ctxt
  1883 fun const_psimp_table ctxt =
  1884   def_table_for (map_filter (equationalize_term ctxt "nitpick_psimp" o prop_of)
  1885                  o Nitpick_Psimps.get) ctxt
  1886 
  1887 fun const_choice_spec_table ctxt subst =
  1888   map (subst_atomic subst o prop_of) (Nitpick_Choice_Specs.get ctxt)
  1889   |> const_nondef_table
  1890 
  1891 fun inductive_intro_table ctxt subst def_tables =
  1892   let val thy = ProofContext.theory_of ctxt in
  1893     def_table_for
  1894         (maps (map (unfold_mutually_inductive_preds thy def_tables o prop_of)
  1895                o snd o snd)
  1896          o filter (fn (cat, _) => cat = Spec_Rules.Inductive orelse
  1897                                   cat = Spec_Rules.Co_Inductive)
  1898          o Spec_Rules.get) ctxt subst
  1899   end
  1900 
  1901 fun ground_theorem_table thy =
  1902   fold ((fn @{const Trueprop} $ t1 =>
  1903             is_ground_term t1 ? Inttab.map_default (hash_term t1, []) (cons t1)
  1904           | _ => I) o prop_of o snd) (Global_Theory.all_thms_of thy) Inttab.empty
  1905 
  1906 (* TODO: Move to "Nitpick.thy" *)
  1907 val basic_ersatz_table =
  1908   [(@{const_name card}, @{const_name card'}),
  1909    (@{const_name setsum}, @{const_name setsum'}),
  1910    (@{const_name fold_graph}, @{const_name fold_graph'}),
  1911    (@{const_name wf}, @{const_name wf'}),
  1912    (@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
  1913    (@{const_name wfrec}, @{const_name wfrec'})]
  1914 
  1915 fun ersatz_table ctxt =
  1916  basic_ersatz_table
  1917  |> fold (append o snd) (#frac_types (Data.get (Context.Proof ctxt)))
  1918 
  1919 fun add_simps simp_table s eqs =
  1920   Unsynchronized.change simp_table
  1921       (Symtab.update (s, eqs @ these (Symtab.lookup (!simp_table) s)))
  1922 
  1923 fun inverse_axioms_for_rep_fun ctxt (x as (_, T)) =
  1924   let
  1925     val thy = ProofContext.theory_of ctxt
  1926     val abs_T = domain_type T
  1927   in
  1928     typedef_info ctxt (fst (dest_Type abs_T)) |> the
  1929     |> pairf #Abs_inverse #Rep_inverse
  1930     |> pairself (specialize_type thy x o prop_of o the)
  1931     ||> single |> op ::
  1932   end
  1933 fun optimized_typedef_axioms ctxt (abs_z as (abs_s, _)) =
  1934   let
  1935     val thy = ProofContext.theory_of ctxt
  1936     val abs_T = Type abs_z
  1937   in
  1938     if is_univ_typedef ctxt abs_T then
  1939       []
  1940     else case typedef_info ctxt abs_s of
  1941       SOME {abs_type, rep_type, Rep_name, prop_of_Rep, set_name, ...} =>
  1942       let
  1943         val rep_T = varify_and_instantiate_type ctxt abs_type abs_T rep_type
  1944         val rep_t = Const (Rep_name, abs_T --> rep_T)
  1945         val set_t = Const (set_name, rep_T --> bool_T)
  1946         val set_t' =
  1947           prop_of_Rep |> HOLogic.dest_Trueprop
  1948                       |> specialize_type thy (dest_Const rep_t)
  1949                       |> HOLogic.dest_mem |> snd
  1950       in
  1951         [HOLogic.all_const abs_T
  1952          $ Abs (Name.uu, abs_T, set_t $ (rep_t $ Bound 0))]
  1953         |> set_t <> set_t' ? cons (HOLogic.mk_eq (set_t, set_t'))
  1954         |> map HOLogic.mk_Trueprop
  1955       end
  1956     | NONE => []
  1957   end
  1958 fun optimized_quot_type_axioms ctxt stds abs_z =
  1959   let
  1960     val thy = ProofContext.theory_of ctxt
  1961     val abs_T = Type abs_z
  1962     val rep_T = rep_type_for_quot_type thy abs_T
  1963     val (equiv_rel, partial) = equiv_relation_for_quot_type thy abs_T
  1964     val a_var = Var (("a", 0), abs_T)
  1965     val x_var = Var (("x", 0), rep_T)
  1966     val y_var = Var (("y", 0), rep_T)
  1967     val x = (@{const_name Quot}, rep_T --> abs_T)
  1968     val sel_a_t = select_nth_constr_arg ctxt stds x a_var 0 rep_T
  1969     val normal_fun =
  1970       Const (quot_normal_name_for_type ctxt abs_T, rep_T --> rep_T)
  1971     val normal_x = normal_fun $ x_var
  1972     val normal_y = normal_fun $ y_var
  1973     val is_unknown_t = Const (@{const_name is_unknown}, rep_T --> bool_T)
  1974   in
  1975     [Logic.mk_equals (normal_fun $ sel_a_t, sel_a_t),
  1976      Logic.list_implies
  1977          ([@{const Not} $ (is_unknown_t $ normal_x),
  1978            @{const Not} $ (is_unknown_t $ normal_y),
  1979            equiv_rel $ x_var $ y_var] |> map HOLogic.mk_Trueprop,
  1980            Logic.mk_equals (normal_x, normal_y)),
  1981      Logic.list_implies
  1982          ([HOLogic.mk_Trueprop (@{const Not} $ (is_unknown_t $ normal_x)),
  1983            HOLogic.mk_Trueprop (@{const Not} $ HOLogic.mk_eq (normal_x, x_var))],
  1984           HOLogic.mk_Trueprop (equiv_rel $ x_var $ normal_x))]
  1985     |> partial ? cons (HOLogic.mk_Trueprop (equiv_rel $ sel_a_t $ sel_a_t))
  1986   end
  1987 
  1988 fun codatatype_bisim_axioms (hol_ctxt as {ctxt, stds, ...}) T =
  1989   let
  1990     val xs = datatype_constrs hol_ctxt T
  1991     val set_T = T --> bool_T
  1992     val iter_T = @{typ bisim_iterator}
  1993     val bisim_max = @{const bisim_iterator_max}
  1994     val n_var = Var (("n", 0), iter_T)
  1995     val n_var_minus_1 =
  1996       Const (@{const_name safe_The}, (iter_T --> bool_T) --> iter_T)
  1997       $ Abs ("m", iter_T, HOLogic.eq_const iter_T
  1998                           $ (suc_const iter_T $ Bound 0) $ n_var)
  1999     val x_var = Var (("x", 0), T)
  2000     val y_var = Var (("y", 0), T)
  2001     fun bisim_const T =
  2002       Const (@{const_name bisim}, iter_T --> T --> T --> bool_T)
  2003     fun nth_sub_bisim x n nth_T =
  2004       (if is_codatatype ctxt nth_T then bisim_const nth_T $ n_var_minus_1
  2005        else HOLogic.eq_const nth_T)
  2006       $ select_nth_constr_arg ctxt stds x x_var n nth_T
  2007       $ select_nth_constr_arg ctxt stds x y_var n nth_T
  2008     fun case_func (x as (_, T)) =
  2009       let
  2010         val arg_Ts = binder_types T
  2011         val core_t =
  2012           discriminate_value hol_ctxt x y_var ::
  2013           map2 (nth_sub_bisim x) (index_seq 0 (length arg_Ts)) arg_Ts
  2014           |> foldr1 s_conj
  2015       in List.foldr absdummy core_t arg_Ts end
  2016   in
  2017     [HOLogic.mk_imp
  2018        (HOLogic.mk_disj (HOLogic.eq_const iter_T $ n_var $ zero_const iter_T,
  2019             s_betapply [] (optimized_case_def hol_ctxt T bool_T
  2020                                               (map case_func xs), x_var)),
  2021         bisim_const T $ n_var $ x_var $ y_var),
  2022      HOLogic.eq_const set_T $ (bisim_const T $ bisim_max $ x_var)
  2023      $ (Const (@{const_name insert}, T --> set_T --> set_T)
  2024         $ x_var $ Const (@{const_name bot_class.bot}, set_T))]
  2025     |> map HOLogic.mk_Trueprop
  2026   end
  2027 
  2028 exception NO_TRIPLE of unit
  2029 
  2030 fun triple_for_intro_rule thy x t =
  2031   let
  2032     val prems = Logic.strip_imp_prems t |> map (Object_Logic.atomize_term thy)
  2033     val concl = Logic.strip_imp_concl t |> Object_Logic.atomize_term thy
  2034     val (main, side) = List.partition (exists_Const (curry (op =) x)) prems
  2035     val is_good_head = curry (op =) (Const x) o head_of
  2036   in
  2037     if forall is_good_head main then (side, main, concl) else raise NO_TRIPLE ()
  2038   end
  2039 
  2040 val tuple_for_args = HOLogic.mk_tuple o snd o strip_comb
  2041 fun wf_constraint_for rel side concl main =
  2042   let
  2043     val core = HOLogic.mk_mem (HOLogic.mk_prod
  2044                                (pairself tuple_for_args (main, concl)), Var rel)
  2045     val t = List.foldl HOLogic.mk_imp core side
  2046     val vars = filter_out (curry (op =) rel) (Term.add_vars t [])
  2047   in
  2048     Library.foldl (fn (t', ((x, j), T)) =>
  2049                       HOLogic.all_const T
  2050                       $ Abs (x, T, abstract_over (Var ((x, j), T), t')))
  2051                   (t, vars)
  2052   end
  2053 fun wf_constraint_for_triple rel (side, main, concl) =
  2054   map (wf_constraint_for rel side concl) main |> foldr1 s_conj
  2055 
  2056 fun terminates_by ctxt timeout goal tac =
  2057   can (SINGLE (Classical.safe_tac (claset_of ctxt)) #> the
  2058        #> SINGLE (DETERM_TIMEOUT timeout
  2059                                  (tac ctxt (auto_tac (clasimpset_of ctxt))))
  2060        #> the #> Goal.finish ctxt) goal
  2061 
  2062 val max_cached_wfs = 50
  2063 val cached_timeout =
  2064   Synchronized.var "Nitpick_HOL.cached_timeout" (SOME Time.zeroTime)
  2065 val cached_wf_props =
  2066   Synchronized.var "Nitpick_HOL.cached_wf_props" ([] : (term * bool) list)
  2067 
  2068 val termination_tacs = [Lexicographic_Order.lex_order_tac true,
  2069                         ScnpReconstruct.sizechange_tac]
  2070 
  2071 fun uncached_is_well_founded_inductive_pred
  2072         ({thy, ctxt, stds, debug, tac_timeout, intro_table, ...} : hol_context)
  2073         (x as (_, T)) =
  2074   case def_props_for_const thy stds intro_table x of
  2075     [] => raise TERM ("Nitpick_HOL.uncached_is_well_founded_inductive",
  2076                       [Const x])
  2077   | intro_ts =>
  2078     (case map (triple_for_intro_rule thy x) intro_ts
  2079           |> filter_out (null o #2) of
  2080        [] => true
  2081      | triples =>
  2082        let
  2083          val binders_T = HOLogic.mk_tupleT (binder_types T)
  2084          val rel_T = HOLogic.mk_prodT (binders_T, binders_T) --> bool_T
  2085          val j = fold Integer.max (map maxidx_of_term intro_ts) 0 + 1
  2086          val rel = (("R", j), rel_T)
  2087          val prop = Const (@{const_name wf}, rel_T --> bool_T) $ Var rel ::
  2088                     map (wf_constraint_for_triple rel) triples
  2089                     |> foldr1 s_conj |> HOLogic.mk_Trueprop
  2090          val _ = if debug then
  2091                    Output.urgent_message ("Wellfoundedness goal: " ^
  2092                              Syntax.string_of_term ctxt prop ^ ".")
  2093                  else
  2094                    ()
  2095        in
  2096          if tac_timeout = Synchronized.value cached_timeout andalso
  2097             length (Synchronized.value cached_wf_props) < max_cached_wfs then
  2098            ()
  2099          else
  2100            (Synchronized.change cached_wf_props (K []);
  2101             Synchronized.change cached_timeout (K tac_timeout));
  2102          case AList.lookup (op =) (Synchronized.value cached_wf_props) prop of
  2103            SOME wf => wf
  2104          | NONE =>
  2105            let
  2106              val goal = prop |> cterm_of thy |> Goal.init
  2107              val wf = exists (terminates_by ctxt tac_timeout goal)
  2108                              termination_tacs
  2109            in Synchronized.change cached_wf_props (cons (prop, wf)); wf end
  2110        end)
  2111     handle List.Empty => false | NO_TRIPLE () => false
  2112 
  2113 (* The type constraint below is a workaround for a Poly/ML crash. *)
  2114 
  2115 fun is_well_founded_inductive_pred
  2116         (hol_ctxt as {thy, wfs, def_tables, wf_cache, ...} : hol_context)
  2117         (x as (s, _)) =
  2118   case triple_lookup (const_match thy) wfs x of
  2119     SOME (SOME b) => b
  2120   | _ => s = @{const_name Nats} orelse s = @{const_name fold_graph'} orelse
  2121          case AList.lookup (op =) (!wf_cache) x of
  2122            SOME (_, wf) => wf
  2123          | NONE =>
  2124            let
  2125              val gfp = (fixpoint_kind_of_const thy def_tables x = Gfp)
  2126              val wf = uncached_is_well_founded_inductive_pred hol_ctxt x
  2127            in
  2128              Unsynchronized.change wf_cache (cons (x, (gfp, wf))); wf
  2129            end
  2130 
  2131 fun ap_curry [_] _ t = t
  2132   | ap_curry arg_Ts tuple_T t =
  2133     let val n = length arg_Ts in
  2134       list_abs (map (pair "c") arg_Ts,
  2135                 incr_boundvars n t
  2136                 $ mk_flat_tuple tuple_T (map Bound (n - 1 downto 0)))
  2137     end
  2138 
  2139 fun num_occs_of_bound_in_term j (t1 $ t2) =
  2140     op + (pairself (num_occs_of_bound_in_term j) (t1, t2))
  2141   | num_occs_of_bound_in_term j (Abs (_, _, t')) =
  2142     num_occs_of_bound_in_term (j + 1) t'
  2143   | num_occs_of_bound_in_term j (Bound j') = if j' = j then 1 else 0
  2144   | num_occs_of_bound_in_term _ _ = 0
  2145 
  2146 val is_linear_inductive_pred_def =
  2147   let
  2148     fun do_disjunct j (Const (@{const_name Ex}, _) $ Abs (_, _, t2)) =
  2149         do_disjunct (j + 1) t2
  2150       | do_disjunct j t =
  2151         case num_occs_of_bound_in_term j t of
  2152           0 => true
  2153         | 1 => exists (curry (op =) (Bound j) o head_of) (conjuncts_of t)
  2154         | _ => false
  2155     fun do_lfp_def (Const (@{const_name lfp}, _) $ t2) =
  2156         let val (xs, body) = strip_abs t2 in
  2157           case length xs of
  2158             1 => false
  2159           | n => forall (do_disjunct (n - 1)) (disjuncts_of body)
  2160         end
  2161       | do_lfp_def _ = false
  2162   in do_lfp_def o strip_abs_body end
  2163 
  2164 fun n_ptuple_paths 0 = []
  2165   | n_ptuple_paths 1 = []
  2166   | n_ptuple_paths n = [] :: map (cons 2) (n_ptuple_paths (n - 1))
  2167 val ap_n_split = HOLogic.mk_psplits o n_ptuple_paths
  2168 
  2169 val linear_pred_base_and_step_rhss =
  2170   let
  2171     fun aux (Const (@{const_name lfp}, _) $ t2) =
  2172         let
  2173           val (xs, body) = strip_abs t2
  2174           val arg_Ts = map snd (tl xs)
  2175           val tuple_T = HOLogic.mk_tupleT arg_Ts
  2176           val j = length arg_Ts
  2177           fun repair_rec j (Const (@{const_name Ex}, T1) $ Abs (s2, T2, t2')) =
  2178               Const (@{const_name Ex}, T1)
  2179               $ Abs (s2, T2, repair_rec (j + 1) t2')
  2180             | repair_rec j (@{const HOL.conj} $ t1 $ t2) =
  2181               @{const HOL.conj} $ repair_rec j t1 $ repair_rec j t2
  2182             | repair_rec j t =
  2183               let val (head, args) = strip_comb t in
  2184                 if head = Bound j then
  2185                   HOLogic.eq_const tuple_T $ Bound j
  2186                   $ mk_flat_tuple tuple_T args
  2187                 else
  2188                   t
  2189               end
  2190           val (nonrecs, recs) =
  2191             List.partition (curry (op =) 0 o num_occs_of_bound_in_term j)
  2192                            (disjuncts_of body)
  2193           val base_body = nonrecs |> List.foldl s_disj @{const False}
  2194           val step_body = recs |> map (repair_rec j)
  2195                                |> List.foldl s_disj @{const False}
  2196         in
  2197           (list_abs (tl xs, incr_bv (~1, j, base_body))
  2198            |> ap_n_split (length arg_Ts) tuple_T bool_T,
  2199            Abs ("y", tuple_T, list_abs (tl xs, step_body)
  2200                               |> ap_n_split (length arg_Ts) tuple_T bool_T))
  2201         end
  2202       | aux t =
  2203         raise TERM ("Nitpick_HOL.linear_pred_base_and_step_rhss.aux", [t])
  2204   in aux end
  2205 
  2206 fun starred_linear_pred_const (hol_ctxt as {simp_table, ...}) (s, T) def =
  2207   let
  2208     val j = maxidx_of_term def + 1
  2209     val (outer, fp_app) = strip_abs def
  2210     val outer_bounds = map Bound (length outer - 1 downto 0)
  2211     val outer_vars = map (fn (s, T) => Var ((s, j), T)) outer
  2212     val fp_app = subst_bounds (rev outer_vars, fp_app)
  2213     val (outer_Ts, rest_T) = strip_n_binders (length outer) T
  2214     val tuple_arg_Ts = strip_type rest_T |> fst
  2215     val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
  2216     val set_T = tuple_T --> bool_T
  2217     val curried_T = tuple_T --> set_T
  2218     val uncurried_T = Type (@{type_name prod}, [tuple_T, tuple_T]) --> bool_T
  2219     val (base_rhs, step_rhs) = linear_pred_base_and_step_rhss fp_app
  2220     val base_x as (base_s, _) = (base_prefix ^ s, outer_Ts ---> set_T)
  2221     val base_eq = HOLogic.mk_eq (list_comb (Const base_x, outer_vars), base_rhs)
  2222                   |> HOLogic.mk_Trueprop
  2223     val _ = add_simps simp_table base_s [base_eq]
  2224     val step_x as (step_s, _) = (step_prefix ^ s, outer_Ts ---> curried_T)
  2225     val step_eq = HOLogic.mk_eq (list_comb (Const step_x, outer_vars), step_rhs)
  2226                   |> HOLogic.mk_Trueprop
  2227     val _ = add_simps simp_table step_s [step_eq]
  2228   in
  2229     list_abs (outer,
  2230               Const (@{const_name Image}, uncurried_T --> set_T --> set_T)
  2231               $ (Const (@{const_name rtrancl}, uncurried_T --> uncurried_T)
  2232                  $ (Const (@{const_name prod_case}, curried_T --> uncurried_T)
  2233                     $ list_comb (Const step_x, outer_bounds)))
  2234               $ list_comb (Const base_x, outer_bounds)
  2235               |> ap_curry tuple_arg_Ts tuple_T)
  2236     |> unfold_defs_in_term hol_ctxt
  2237   end
  2238 
  2239 fun is_good_starred_linear_pred_type (Type (@{type_name fun}, Ts)) =
  2240     forall (not o (is_fun_type orf is_pair_type)) Ts
  2241   | is_good_starred_linear_pred_type _ = false
  2242 
  2243 fun unrolled_inductive_pred_const (hol_ctxt as {thy, star_linear_preds,
  2244                                                 def_tables, simp_table, ...})
  2245                                   gfp (x as (s, T)) =
  2246   let
  2247     val iter_T = iterator_type_for_const gfp x
  2248     val x' as (s', _) = (unrolled_prefix ^ s, iter_T --> T)
  2249     val unrolled_const = Const x' $ zero_const iter_T
  2250     val def = the (def_of_const thy def_tables x)
  2251   in
  2252     if is_equational_fun_but_no_plain_def hol_ctxt x' then
  2253       unrolled_const (* already done *)
  2254     else if not gfp andalso star_linear_preds andalso
  2255          is_linear_inductive_pred_def def andalso
  2256          is_good_starred_linear_pred_type T then
  2257       starred_linear_pred_const hol_ctxt x def
  2258     else
  2259       let
  2260         val j = maxidx_of_term def + 1
  2261         val (outer, fp_app) = strip_abs def
  2262         val outer_bounds = map Bound (length outer - 1 downto 0)
  2263         val cur = Var ((iter_var_prefix, j + 1), iter_T)
  2264         val next = suc_const iter_T $ cur
  2265         val rhs =
  2266           case fp_app of
  2267             Const _ $ t =>
  2268             s_betapply [] (t, list_comb (Const x', next :: outer_bounds))
  2269           | _ => raise TERM ("Nitpick_HOL.unrolled_inductive_pred_const",
  2270                              [fp_app])
  2271         val (inner, naked_rhs) = strip_abs rhs
  2272         val all = outer @ inner
  2273         val bounds = map Bound (length all - 1 downto 0)
  2274         val vars = map (fn (s, T) => Var ((s, j), T)) all
  2275         val eq = HOLogic.mk_eq (list_comb (Const x', cur :: bounds), naked_rhs)
  2276                  |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2277         val _ = add_simps simp_table s' [eq]
  2278       in unrolled_const end
  2279   end
  2280 
  2281 fun raw_inductive_pred_axiom ({thy, def_tables, ...} : hol_context) x =
  2282   let
  2283     val def = the (def_of_const thy def_tables x)
  2284     val (outer, fp_app) = strip_abs def
  2285     val outer_bounds = map Bound (length outer - 1 downto 0)
  2286     val rhs =
  2287       case fp_app of
  2288         Const _ $ t => s_betapply [] (t, list_comb (Const x, outer_bounds))
  2289       | _ => raise TERM ("Nitpick_HOL.raw_inductive_pred_axiom", [fp_app])
  2290     val (inner, naked_rhs) = strip_abs rhs
  2291     val all = outer @ inner
  2292     val bounds = map Bound (length all - 1 downto 0)
  2293     val j = maxidx_of_term def + 1
  2294     val vars = map (fn (s, T) => Var ((s, j), T)) all
  2295   in
  2296     HOLogic.mk_eq (list_comb (Const x, bounds), naked_rhs)
  2297     |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  2298   end
  2299 fun inductive_pred_axiom hol_ctxt (x as (s, T)) =
  2300   if String.isPrefix ubfp_prefix s orelse String.isPrefix lbfp_prefix s then
  2301     let val x' = (strip_first_name_sep s |> snd, T) in
  2302       raw_inductive_pred_axiom hol_ctxt x' |> subst_atomic [(Const x', Const x)]
  2303     end
  2304   else
  2305     raw_inductive_pred_axiom hol_ctxt x
  2306 
  2307 fun equational_fun_axioms (hol_ctxt as {thy, ctxt, stds, def_tables, simp_table,
  2308                                         psimp_table, ...}) x =
  2309   case def_props_for_const thy stds (!simp_table) x of
  2310     [] => (case def_props_for_const thy stds psimp_table x of
  2311              [] => (if is_inductive_pred hol_ctxt x then
  2312                       [inductive_pred_axiom hol_ctxt x]
  2313                     else case def_of_const thy def_tables x of
  2314                       SOME def =>
  2315                       @{const Trueprop} $ HOLogic.mk_eq (Const x, def)
  2316                       |> equationalize_term ctxt "" |> the |> single
  2317                     | NONE => [])
  2318            | psimps => psimps)
  2319   | simps => simps
  2320 fun is_equational_fun_surely_complete hol_ctxt x =
  2321   case equational_fun_axioms hol_ctxt x of
  2322     [@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t1 $ _)] =>
  2323     strip_comb t1 |> snd |> forall is_Var
  2324   | _ => false
  2325 
  2326 (** Type preprocessing **)
  2327 
  2328 fun merged_type_var_table_for_terms thy ts =
  2329   let
  2330     fun add (s, S) table =
  2331       table
  2332       |> (case AList.lookup (Sign.subsort thy o swap) table S of
  2333             SOME _ => I
  2334           | NONE =>
  2335             filter_out (fn (S', _) => Sign.subsort thy (S, S'))
  2336             #> cons (S, s))
  2337     val tfrees = [] |> fold Term.add_tfrees ts
  2338                     |> sort (string_ord o pairself fst)
  2339   in [] |> fold add tfrees |> rev end
  2340 
  2341 fun merge_type_vars_in_term thy merge_type_vars table =
  2342   merge_type_vars
  2343   ? map_types (map_atyps
  2344         (fn TFree (_, S) =>
  2345             TFree (table |> find_first (fn (S', _) => Sign.subsort thy (S', S))
  2346                          |> the |> swap)
  2347           | T => T))
  2348 
  2349 fun add_ground_types hol_ctxt binarize =
  2350   let
  2351     fun aux T accum =
  2352       case T of
  2353         Type (@{type_name fun}, Ts) => fold aux Ts accum
  2354       | Type (@{type_name prod}, Ts) => fold aux Ts accum
  2355       | Type (@{type_name itself}, [T1]) => aux T1 accum
  2356       | Type (_, Ts) =>
  2357         if member (op =) (@{typ prop} :: @{typ bool} :: accum) T then
  2358           accum
  2359         else
  2360           T :: accum
  2361           |> fold aux (case binarized_and_boxed_datatype_constrs hol_ctxt
  2362                                                                  binarize T of
  2363                          [] => Ts
  2364                        | xs => map snd xs)
  2365       | _ => insert (op =) T accum
  2366   in aux end
  2367 
  2368 fun ground_types_in_type hol_ctxt binarize T =
  2369   add_ground_types hol_ctxt binarize T []
  2370 fun ground_types_in_terms hol_ctxt binarize ts =
  2371   fold (fold_types (add_ground_types hol_ctxt binarize)) ts []
  2372 
  2373 end;