src/HOL/Tools/Nitpick/nitpick_hol.ML

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(*  Title:      HOL/Tools/Nitpick/nitpick_hol.ML
    Author:     Jasmin Blanchette, TU Muenchen
    Copyright   2008, 2009, 2010

Auxiliary HOL-related functions used by Nitpick.
*)

signature NITPICK_HOL =
sig
  type const_table = term list Symtab.table
  type special_fun = ((string * typ) * int list * term list) * (string * typ)
  type unrolled = (string * typ) * (string * typ)
  type wf_cache = ((string * typ) * (bool * bool)) list

  type hol_context =
    {thy: theory,
     ctxt: Proof.context,
     max_bisim_depth: int,
     boxes: (typ option * bool option) list,
     wfs: ((string * typ) option * bool option) list,
     user_axioms: bool option,
     debug: bool,
     whacks: term list,
     binary_ints: bool option,
     destroy_constrs: bool,
     specialize: bool,
     star_linear_preds: bool,
     total_consts: bool option,
     needs: term list option,
     tac_timeout: Time.time,
     evals: term list,
     case_names: (string * int) list,
     def_tables: const_table * const_table,
     nondef_table: const_table,
     nondefs: term list,
     simp_table: const_table Unsynchronized.ref,
     psimp_table: const_table,
     choice_spec_table: const_table,
     intro_table: const_table,
     ground_thm_table: term list Inttab.table,
     ersatz_table: (string * string) list,
     skolems: (string * string list) list Unsynchronized.ref,
     special_funs: special_fun list Unsynchronized.ref,
     unrolled_preds: unrolled list Unsynchronized.ref,
     wf_cache: wf_cache Unsynchronized.ref,
     constr_cache: (typ * (string * typ) list) list Unsynchronized.ref}

  datatype fixpoint_kind = Lfp | Gfp | NoFp
  datatype boxability =
    InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2

  val name_sep : string
  val numeral_prefix : string
  val base_prefix : string
  val step_prefix : string
  val unrolled_prefix : string
  val ubfp_prefix : string
  val lbfp_prefix : string
  val quot_normal_prefix : string
  val skolem_prefix : string
  val special_prefix : string
  val uncurry_prefix : string
  val eval_prefix : string
  val iter_var_prefix : string
  val strip_first_name_sep : string -> string * string
  val original_name : string -> string
  val abs_var : indexname * typ -> term -> term
  val s_conj : term * term -> term
  val s_disj : term * term -> term
  val strip_any_connective : term -> term list * term
  val conjuncts_of : term -> term list
  val disjuncts_of : term -> term list
  val unarize_unbox_etc_type : typ -> typ
  val uniterize_unarize_unbox_etc_type : typ -> typ
  val string_for_type : Proof.context -> typ -> string
  val pretty_for_type : Proof.context -> typ -> Pretty.T
  val prefix_name : string -> string -> string
  val shortest_name : string -> string
  val short_name : string -> string
  val shorten_names_in_term : term -> term
  val strict_type_match : theory -> typ * typ -> bool
  val type_match : theory -> typ * typ -> bool
  val const_match : theory -> (string * typ) * (string * typ) -> bool
  val term_match : theory -> term * term -> bool
  val frac_from_term_pair : typ -> term -> term -> term
  val is_fun_type : typ -> bool
  val is_set_type : typ -> bool
  val is_fun_or_set_type : typ -> bool
  val is_set_like_type : typ -> bool
  val is_pair_type : typ -> bool
  val is_lfp_iterator_type : typ -> bool
  val is_gfp_iterator_type : typ -> bool
  val is_fp_iterator_type : typ -> bool
  val is_iterator_type : typ -> bool
  val is_boolean_type : typ -> bool
  val is_integer_type : typ -> bool
  val is_bit_type : typ -> bool
  val is_word_type : typ -> bool
  val is_integer_like_type : typ -> bool
  val is_number_type : Proof.context -> typ -> bool
  val is_higher_order_type : typ -> bool
  val elem_type : typ -> typ
  val pseudo_domain_type : typ -> typ
  val pseudo_range_type : typ -> typ
  val const_for_iterator_type : typ -> string * typ
  val strip_n_binders : int -> typ -> typ list * typ
  val nth_range_type : int -> typ -> typ
  val num_factors_in_type : typ -> int
  val curried_binder_types : typ -> typ list
  val mk_flat_tuple : typ -> term list -> term
  val dest_n_tuple : int -> term -> term list
  val is_codatatype : Proof.context -> typ -> bool
  val is_quot_type : Proof.context -> typ -> bool
  val is_pure_typedef : Proof.context -> typ -> bool
  val is_univ_typedef : Proof.context -> typ -> bool
  val is_data_type : Proof.context -> typ -> bool
  val is_record_get : theory -> string * typ -> bool
  val is_record_update : theory -> string * typ -> bool
  val is_abs_fun : Proof.context -> string * typ -> bool
  val is_rep_fun : Proof.context -> string * typ -> bool
  val is_quot_abs_fun : Proof.context -> string * typ -> bool
  val is_quot_rep_fun : Proof.context -> string * typ -> bool
  val mate_of_rep_fun : Proof.context -> string * typ -> string * typ
  val is_nonfree_constr : Proof.context -> string * typ -> bool
  val is_free_constr : Proof.context -> string * typ -> bool
  val is_constr : Proof.context -> string * typ -> bool
  val is_sel : string -> bool
  val is_sel_like_and_no_discr : string -> bool
  val box_type : hol_context -> boxability -> typ -> typ
  val binarize_nat_and_int_in_type : typ -> typ
  val binarize_nat_and_int_in_term : term -> term
  val discr_for_constr : string * typ -> string * typ
  val num_sels_for_constr_type : typ -> int
  val nth_sel_name_for_constr_name : string -> int -> string
  val nth_sel_for_constr : string * typ -> int -> string * typ
  val binarized_and_boxed_nth_sel_for_constr :
    hol_context -> bool -> string * typ -> int -> string * typ
  val sel_no_from_name : string -> int
  val close_form : term -> term
  val distinctness_formula : typ -> term list -> term
  val register_frac_type :
    string -> (string * string) list -> morphism -> Context.generic
    -> Context.generic
  val register_frac_type_global :
    string -> (string * string) list -> theory -> theory
  val unregister_frac_type :
    string -> morphism -> Context.generic -> Context.generic
  val unregister_frac_type_global : string -> theory -> theory
  val register_ersatz :
    (string * string) list -> morphism -> Context.generic -> Context.generic
  val register_ersatz_global : (string * string) list -> theory -> theory
  val register_codatatype :
    typ -> string -> (string * typ) list -> morphism -> Context.generic ->
    Context.generic
  val register_codatatype_global :
    typ -> string -> (string * typ) list -> theory -> theory
  val unregister_codatatype :
    typ -> morphism -> Context.generic -> Context.generic
  val unregister_codatatype_global : typ -> theory -> theory
  val binarized_and_boxed_data_type_constrs :
    hol_context -> bool -> typ -> (string * typ) list
  val constr_name_for_sel_like : string -> string
  val binarized_and_boxed_constr_for_sel : hol_context -> bool ->
    string * typ -> string * typ
  val card_of_type : (typ * int) list -> typ -> int
  val bounded_card_of_type : int -> int -> (typ * int) list -> typ -> int
  val bounded_exact_card_of_type :
    hol_context -> typ list -> int -> int -> (typ * int) list -> typ -> int
  val typical_card_of_type : typ -> int
  val is_finite_type : hol_context -> typ -> bool
  val is_special_eligible_arg : bool -> typ list -> term -> bool
  val s_let :
    typ list -> string -> int -> typ -> typ -> (term -> term) -> term -> term
  val s_betapply : typ list -> term * term -> term
  val s_betapplys : typ list -> term * term list -> term
  val discriminate_value : hol_context -> string * typ -> term -> term
  val select_nth_constr_arg :
    Proof.context -> string * typ -> term -> int -> typ -> term
  val construct_value : Proof.context -> string * typ -> term list -> term
  val coerce_term : hol_context -> typ list -> typ -> typ -> term -> term
  val special_bounds : term list -> (indexname * typ) list
  val is_funky_typedef : Proof.context -> typ -> bool
  val all_defs_of : theory -> (term * term) list -> term list
  val all_nondefs_of : Proof.context -> (term * term) list -> term list
  val arity_of_built_in_const : string * typ -> int option
  val is_built_in_const : string * typ -> bool
  val term_under_def : term -> term
  val case_const_names : Proof.context -> (string * int) list
  val unfold_defs_in_term : hol_context -> term -> term
  val const_def_tables :
    Proof.context -> (term * term) list -> term list
    -> const_table * const_table
  val const_nondef_table : term list -> const_table
  val const_simp_table : Proof.context -> (term * term) list -> const_table
  val const_psimp_table : Proof.context -> (term * term) list -> const_table
  val const_choice_spec_table :
    Proof.context -> (term * term) list -> const_table
  val inductive_intro_table :
    Proof.context -> (term * term) list -> const_table * const_table
    -> const_table
  val ground_theorem_table : theory -> term list Inttab.table
  val ersatz_table : Proof.context -> (string * string) list
  val add_simps : const_table Unsynchronized.ref -> string -> term list -> unit
  val inverse_axioms_for_rep_fun : Proof.context -> string * typ -> term list
  val optimized_typedef_axioms : Proof.context -> string * typ list -> term list
  val optimized_quot_type_axioms :
    Proof.context -> string * typ list -> term list
  val def_of_const : theory -> const_table * const_table -> string * typ ->
    term option
  val fixpoint_kind_of_rhs : term -> fixpoint_kind
  val fixpoint_kind_of_const :
    theory -> const_table * const_table -> string * typ -> fixpoint_kind
  val is_raw_inductive_pred : hol_context -> string * typ -> bool
  val is_constr_pattern : Proof.context -> term -> bool
  val is_constr_pattern_lhs : Proof.context -> term -> bool
  val is_constr_pattern_formula : Proof.context -> term -> bool
  val nondef_props_for_const :
    theory -> bool -> const_table -> string * typ -> term list
  val is_choice_spec_fun : hol_context -> string * typ -> bool
  val is_choice_spec_axiom : Proof.context -> const_table -> term -> bool
  val is_raw_equational_fun : hol_context -> string * typ -> bool
  val is_equational_fun : hol_context -> string * typ -> bool
  val codatatype_bisim_axioms : hol_context -> typ -> term list
  val is_well_founded_inductive_pred : hol_context -> string * typ -> bool
  val unrolled_inductive_pred_const : hol_context -> bool -> string * typ ->
    term
  val equational_fun_axioms : hol_context -> string * typ -> term list
  val is_equational_fun_surely_complete : hol_context -> string * typ -> bool
  val merged_type_var_table_for_terms :
    theory -> term list -> (sort * string) list
  val merge_type_vars_in_term :
    theory -> bool -> (sort * string) list -> term -> term
  val ground_types_in_type : hol_context -> bool -> typ -> typ list
  val ground_types_in_terms : hol_context -> bool -> term list -> typ list
end;

structure Nitpick_HOL : NITPICK_HOL =
struct

open Nitpick_Util

type const_table = term list Symtab.table
type special_fun = ((string * typ) * int list * term list) * (string * typ)
type unrolled = (string * typ) * (string * typ)
type wf_cache = ((string * typ) * (bool * bool)) list

type hol_context =
  {thy: theory,
   ctxt: Proof.context,
   max_bisim_depth: int,
   boxes: (typ option * bool option) list,
   wfs: ((string * typ) option * bool option) list,
   user_axioms: bool option,
   debug: bool,
   whacks: term list,
   binary_ints: bool option,
   destroy_constrs: bool,
   specialize: bool,
   star_linear_preds: bool,
   total_consts: bool option,
   needs: term list option,
   tac_timeout: Time.time,
   evals: term list,
   case_names: (string * int) list,
   def_tables: const_table * const_table,
   nondef_table: const_table,
   nondefs: term list,
   simp_table: const_table Unsynchronized.ref,
   psimp_table: const_table,
   choice_spec_table: const_table,
   intro_table: const_table,
   ground_thm_table: term list Inttab.table,
   ersatz_table: (string * string) list,
   skolems: (string * string list) list Unsynchronized.ref,
   special_funs: special_fun list Unsynchronized.ref,
   unrolled_preds: unrolled list Unsynchronized.ref,
   wf_cache: wf_cache Unsynchronized.ref,
   constr_cache: (typ * (string * typ) list) list Unsynchronized.ref}

datatype fixpoint_kind = Lfp | Gfp | NoFp
datatype boxability =
  InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2

(* FIXME: Get rid of 'codatatypes' and related functionality *)
structure Data = Generic_Data
(
  type T = {frac_types: (string * (string * string) list) list,
            ersatz_table: (string * string) list,
            codatatypes: (string * (string * (string * typ) list)) list}
  val empty = {frac_types = [], ersatz_table = [], codatatypes = []}
  val extend = I
  fun merge ({frac_types = fs1, ersatz_table = et1, codatatypes = cs1},
             {frac_types = fs2, ersatz_table = et2, codatatypes = cs2}) : T =
    {frac_types = AList.merge (op =) (K true) (fs1, fs2),
     ersatz_table = AList.merge (op =) (K true) (et1, et2),
     codatatypes = AList.merge (op =) (K true) (cs1, cs2)}
)

val name_sep = "$"
val numeral_prefix = nitpick_prefix ^ "num" ^ name_sep
val sel_prefix = nitpick_prefix ^ "sel"
val discr_prefix = nitpick_prefix ^ "is" ^ name_sep
val lfp_iterator_prefix = nitpick_prefix ^ "lfpit" ^ name_sep
val gfp_iterator_prefix = nitpick_prefix ^ "gfpit" ^ name_sep
val unrolled_prefix = nitpick_prefix ^ "unroll" ^ name_sep
val base_prefix = nitpick_prefix ^ "base" ^ name_sep
val step_prefix = nitpick_prefix ^ "step" ^ name_sep
val ubfp_prefix = nitpick_prefix ^ "ubfp" ^ name_sep
val lbfp_prefix = nitpick_prefix ^ "lbfp" ^ name_sep
val quot_normal_prefix = nitpick_prefix ^ "qn" ^ name_sep
val skolem_prefix = nitpick_prefix ^ "sk"
val special_prefix = nitpick_prefix ^ "sp"
val uncurry_prefix = nitpick_prefix ^ "unc"
val eval_prefix = nitpick_prefix ^ "eval"
val iter_var_prefix = "i"

(** Constant/type information and term/type manipulation **)

fun sel_prefix_for j = sel_prefix ^ string_of_int j ^ name_sep

fun quot_normal_name_for_type ctxt T =
  quot_normal_prefix ^ YXML.content_of (Syntax.string_of_typ ctxt T)

val strip_first_name_sep =
  Substring.full #> Substring.position name_sep ##> Substring.triml 1
  #> apply2 Substring.string

fun original_name s =
  if String.isPrefix nitpick_prefix s then
    case strip_first_name_sep s of (s1, "") => s1 | (_, s2) => original_name s2
  else
    s

fun s_conj (t1, \<^const>\<open>True\<close>) = t1
  | s_conj (\<^const>\<open>True\<close>, t2) = t2
  | s_conj (t1, t2) =
    if t1 = \<^const>\<open>False\<close> orelse t2 = \<^const>\<open>False\<close> then \<^const>\<open>False\<close>
    else HOLogic.mk_conj (t1, t2)

fun s_disj (t1, \<^const>\<open>False\<close>) = t1
  | s_disj (\<^const>\<open>False\<close>, t2) = t2
  | s_disj (t1, t2) =
    if t1 = \<^const>\<open>True\<close> orelse t2 = \<^const>\<open>True\<close> then \<^const>\<open>True\<close>
    else HOLogic.mk_disj (t1, t2)

fun strip_connective conn_t (t as (t0 $ t1 $ t2)) =
    if t0 = conn_t then strip_connective t0 t2 @ strip_connective t0 t1 else [t]
  | strip_connective _ t = [t]

fun strip_any_connective (t as (t0 $ _ $ _)) =
    if t0 = \<^const>\<open>HOL.conj\<close> orelse t0 = \<^const>\<open>HOL.disj\<close> then
      (strip_connective t0 t, t0)
    else
      ([t], \<^const>\<open>Not\<close>)
  | strip_any_connective t = ([t], \<^const>\<open>Not\<close>)
val conjuncts_of = strip_connective \<^const>\<open>HOL.conj\<close>
val disjuncts_of = strip_connective \<^const>\<open>HOL.disj\<close>

(* When you add constants to these lists, make sure to handle them in
   "Nitpick_Nut.nut_from_term", and perhaps in "Nitpick_Mono.consider_term" as
   well. *)
val built_in_consts =
  [(\<^const_name>\<open>Pure.all\<close>, 1),
   (\<^const_name>\<open>Pure.eq\<close>, 2),
   (\<^const_name>\<open>Pure.imp\<close>, 2),
   (\<^const_name>\<open>Pure.conjunction\<close>, 2),
   (\<^const_name>\<open>Trueprop\<close>, 1),
   (\<^const_name>\<open>Not\<close>, 1),
   (\<^const_name>\<open>False\<close>, 0),
   (\<^const_name>\<open>True\<close>, 0),
   (\<^const_name>\<open>All\<close>, 1),
   (\<^const_name>\<open>Ex\<close>, 1),
   (\<^const_name>\<open>HOL.eq\<close>, 1),
   (\<^const_name>\<open>HOL.conj\<close>, 2),
   (\<^const_name>\<open>HOL.disj\<close>, 2),
   (\<^const_name>\<open>HOL.implies\<close>, 2),
   (\<^const_name>\<open>If\<close>, 3),
   (\<^const_name>\<open>Let\<close>, 2),
   (\<^const_name>\<open>Pair\<close>, 2),
   (\<^const_name>\<open>fst\<close>, 1),
   (\<^const_name>\<open>snd\<close>, 1),
   (\<^const_name>\<open>Set.member\<close>, 2),
   (\<^const_name>\<open>Collect\<close>, 1),
   (\<^const_name>\<open>Id\<close>, 0),
   (\<^const_name>\<open>converse\<close>, 1),
   (\<^const_name>\<open>trancl\<close>, 1),
   (\<^const_name>\<open>relcomp\<close>, 2),
   (\<^const_name>\<open>finite\<close>, 1),
   (\<^const_name>\<open>unknown\<close>, 0),
   (\<^const_name>\<open>is_unknown\<close>, 1),
   (\<^const_name>\<open>safe_The\<close>, 1),
   (\<^const_name>\<open>Frac\<close>, 0),
   (\<^const_name>\<open>norm_frac\<close>, 0),
   (\<^const_name>\<open>Suc\<close>, 0),
   (\<^const_name>\<open>nat\<close>, 0),
   (\<^const_name>\<open>nat_gcd\<close>, 0),
   (\<^const_name>\<open>nat_lcm\<close>, 0)]
val built_in_typed_consts =
  [((\<^const_name>\<open>zero_class.zero\<close>, nat_T), 0),
   ((\<^const_name>\<open>one_class.one\<close>, nat_T), 0),
   ((\<^const_name>\<open>plus_class.plus\<close>, nat_T --> nat_T --> nat_T), 0),
   ((\<^const_name>\<open>minus_class.minus\<close>, nat_T --> nat_T --> nat_T), 0),
   ((\<^const_name>\<open>times_class.times\<close>, nat_T --> nat_T --> nat_T), 0),
   ((\<^const_name>\<open>Rings.divide\<close>, nat_T --> nat_T --> nat_T), 0),
   ((\<^const_name>\<open>ord_class.less\<close>, nat_T --> nat_T --> bool_T), 2),
   ((\<^const_name>\<open>ord_class.less_eq\<close>, nat_T --> nat_T --> bool_T), 2),
   ((\<^const_name>\<open>of_nat\<close>, nat_T --> int_T), 0),
   ((\<^const_name>\<open>zero_class.zero\<close>, int_T), 0),
   ((\<^const_name>\<open>one_class.one\<close>, int_T), 0),
   ((\<^const_name>\<open>plus_class.plus\<close>, int_T --> int_T --> int_T), 0),
   ((\<^const_name>\<open>minus_class.minus\<close>, int_T --> int_T --> int_T), 0),
   ((\<^const_name>\<open>times_class.times\<close>, int_T --> int_T --> int_T), 0),
   ((\<^const_name>\<open>Rings.divide\<close>, int_T --> int_T --> int_T), 0),
   ((\<^const_name>\<open>uminus_class.uminus\<close>, int_T --> int_T), 0),
   ((\<^const_name>\<open>ord_class.less\<close>, int_T --> int_T --> bool_T), 2),
   ((\<^const_name>\<open>ord_class.less_eq\<close>, int_T --> int_T --> bool_T), 2)]

fun unarize_type \<^typ>\<open>unsigned_bit word\<close> = nat_T
  | unarize_type \<^typ>\<open>signed_bit word\<close> = int_T
  | unarize_type (Type (s, Ts as _ :: _)) = Type (s, map unarize_type Ts)
  | unarize_type T = T

fun unarize_unbox_etc_type (Type (\<^type_name>\<open>fun_box\<close>, Ts)) =
    unarize_unbox_etc_type (Type (\<^type_name>\<open>fun\<close>, Ts))
  | unarize_unbox_etc_type (Type (\<^type_name>\<open>pair_box\<close>, Ts)) =
    Type (\<^type_name>\<open>prod\<close>, map unarize_unbox_etc_type Ts)
  | unarize_unbox_etc_type \<^typ>\<open>unsigned_bit word\<close> = nat_T
  | unarize_unbox_etc_type \<^typ>\<open>signed_bit word\<close> = int_T
  | unarize_unbox_etc_type (Type (s, Ts as _ :: _)) =
    Type (s, map unarize_unbox_etc_type Ts)
  | unarize_unbox_etc_type T = T

fun uniterize_type (Type (s, Ts as _ :: _)) = Type (s, map uniterize_type Ts)
  | uniterize_type \<^typ>\<open>bisim_iterator\<close> = nat_T
  | uniterize_type T = T
val uniterize_unarize_unbox_etc_type = uniterize_type o unarize_unbox_etc_type

fun string_for_type ctxt = Syntax.string_of_typ ctxt o unarize_unbox_etc_type
fun pretty_for_type ctxt = Syntax.pretty_typ ctxt o unarize_unbox_etc_type

val prefix_name = Long_Name.qualify o Long_Name.base_name
val shortest_name = Long_Name.base_name
val prefix_abs_vars = Term.map_abs_vars o prefix_name

fun short_name s =
  case space_explode name_sep s of
    [_] => s |> String.isPrefix nitpick_prefix s ? unprefix nitpick_prefix
  | ss => map shortest_name ss |> space_implode "_"

fun shorten_names_in_type (Type (s, Ts)) =
    Type (short_name s, map shorten_names_in_type Ts)
  | shorten_names_in_type T = T

val shorten_names_in_term =
  map_aterms (fn Const (s, T) => Const (short_name s, T) | t => t)
  #> map_types shorten_names_in_type

fun strict_type_match thy (T1, T2) =
  (Sign.typ_match thy (T2, T1) Vartab.empty; true)
  handle Type.TYPE_MATCH => false

fun type_match thy = strict_type_match thy o apply2 unarize_unbox_etc_type

fun const_match thy ((s1, T1), (s2, T2)) =
  s1 = s2 andalso type_match thy (T1, T2)

fun term_match thy (Const x1, Const x2) = const_match thy (x1, x2)
  | term_match thy (Free (s1, T1), Free (s2, T2)) =
    const_match thy ((shortest_name s1, T1), (shortest_name s2, T2))
  | term_match _ (t1, t2) = t1 aconv t2

fun frac_from_term_pair T t1 t2 =
  case snd (HOLogic.dest_number t1) of
    0 => HOLogic.mk_number T 0
  | n1 => case snd (HOLogic.dest_number t2) of
            1 => HOLogic.mk_number T n1
          | n2 => Const (\<^const_name>\<open>divide\<close>, T --> T --> T)
                  $ HOLogic.mk_number T n1 $ HOLogic.mk_number T n2

fun is_fun_type (Type (\<^type_name>\<open>fun\<close>, _)) = true
  | is_fun_type _ = false

fun is_set_type (Type (\<^type_name>\<open>set\<close>, _)) = true
  | is_set_type _ = false

val is_fun_or_set_type = is_fun_type orf is_set_type

fun is_set_like_type (Type (\<^type_name>\<open>fun\<close>, [_, T'])) =
    (body_type T' = bool_T)
  | is_set_like_type (Type (\<^type_name>\<open>set\<close>, _)) = true
  | is_set_like_type _ = false

fun is_pair_type (Type (\<^type_name>\<open>prod\<close>, _)) = true
  | is_pair_type _ = false

fun is_lfp_iterator_type (Type (s, _)) = String.isPrefix lfp_iterator_prefix s
  | is_lfp_iterator_type _ = false

fun is_gfp_iterator_type (Type (s, _)) = String.isPrefix gfp_iterator_prefix s
  | is_gfp_iterator_type _ = false

val is_fp_iterator_type = is_lfp_iterator_type orf is_gfp_iterator_type

fun is_iterator_type T =
  (T = \<^typ>\<open>bisim_iterator\<close> orelse is_fp_iterator_type T)

fun is_boolean_type T = (T = prop_T orelse T = bool_T)

fun is_integer_type T = (T = nat_T orelse T = int_T)

fun is_bit_type T = (T = \<^typ>\<open>unsigned_bit\<close> orelse T = \<^typ>\<open>signed_bit\<close>)

fun is_word_type (Type (\<^type_name>\<open>word\<close>, _)) = true
  | is_word_type _ = false

val is_integer_like_type = is_iterator_type orf is_integer_type orf is_word_type

fun is_frac_type ctxt (Type (s, [])) =
    s |> AList.defined (op =) (#frac_types (Data.get (Context.Proof ctxt)))
  | is_frac_type _ _ = false

fun is_number_type ctxt = is_integer_like_type orf is_frac_type ctxt

fun is_higher_order_type (Type (\<^type_name>\<open>fun\<close>, _)) = true
  | is_higher_order_type (Type (\<^type_name>\<open>set\<close>, _)) = true
  | is_higher_order_type (Type (_, Ts)) = exists is_higher_order_type Ts
  | is_higher_order_type _ = false

fun elem_type (Type (\<^type_name>\<open>set\<close>, [T'])) = T'
  | elem_type T = raise TYPE ("Nitpick_HOL.elem_type", [T], [])

fun pseudo_domain_type (Type (\<^type_name>\<open>fun\<close>, [T1, _])) = T1
  | pseudo_domain_type T = elem_type T

fun pseudo_range_type (Type (\<^type_name>\<open>fun\<close>, [_, T2])) = T2
  | pseudo_range_type (Type (\<^type_name>\<open>set\<close>, _)) = bool_T
  | pseudo_range_type T = raise TYPE ("Nitpick_HOL.pseudo_range_type", [T], [])

fun iterator_type_for_const gfp (s, T) =
  Type ((if gfp then gfp_iterator_prefix else lfp_iterator_prefix) ^ s,
        binder_types T)

fun const_for_iterator_type (Type (s, Ts)) =
    (strip_first_name_sep s |> snd, Ts ---> bool_T)
  | const_for_iterator_type T =
    raise TYPE ("Nitpick_HOL.const_for_iterator_type", [T], [])

fun strip_n_binders 0 T = ([], T)
  | strip_n_binders n (Type (\<^type_name>\<open>fun\<close>, [T1, T2])) =
    strip_n_binders (n - 1) T2 |>> cons T1
  | strip_n_binders n (Type (\<^type_name>\<open>fun_box\<close>, Ts)) =
    strip_n_binders n (Type (\<^type_name>\<open>fun\<close>, Ts))
  | strip_n_binders _ T = raise TYPE ("Nitpick_HOL.strip_n_binders", [T], [])

val nth_range_type = snd oo strip_n_binders

fun num_factors_in_type (Type (\<^type_name>\<open>prod\<close>, [T1, T2])) =
    fold (Integer.add o num_factors_in_type) [T1, T2] 0
  | num_factors_in_type _ = 1

val curried_binder_types = maps HOLogic.flatten_tupleT o binder_types

fun maybe_curried_binder_types T =
  (if is_pair_type (body_type T) then binder_types else curried_binder_types) T

fun mk_flat_tuple _ [t] = t
  | mk_flat_tuple (Type (\<^type_name>\<open>prod\<close>, [T1, T2])) (t :: ts) =
    HOLogic.pair_const T1 T2 $ t $ (mk_flat_tuple T2 ts)
  | mk_flat_tuple T ts = raise TYPE ("Nitpick_HOL.mk_flat_tuple", [T], ts)

fun dest_n_tuple 1 t = [t]
  | dest_n_tuple n t = HOLogic.dest_prod t ||> dest_n_tuple (n - 1) |> op ::

fun typedef_info ctxt s =
  if is_frac_type ctxt (Type (s, [])) then
    SOME {abs_type = Type (s, []), rep_type = \<^typ>\<open>int * int\<close>,
          Abs_name = \<^const_name>\<open>Abs_Frac\<close>,
          Rep_name = \<^const_name>\<open>Rep_Frac\<close>,
          prop_of_Rep = \<^prop>\<open>Rep_Frac x \<in> Collect Frac\<close>
                        |> Logic.varify_global,
          Abs_inverse = NONE, Rep_inverse = NONE}
  else case Typedef.get_info ctxt s of
    (* When several entries are returned, it shouldn't matter much which one
       we take (according to Florian Haftmann). *)
    (* The "Logic.varifyT_global" calls are a temporary hack because these
       types's type variables sometimes clash with locally fixed type variables.
       Remove these calls once "Typedef" is fully localized. *)
    ({abs_type, rep_type, Abs_name, Rep_name, ...},
     {Rep, Abs_inverse, Rep_inverse, ...}) :: _ =>
    SOME {abs_type = Logic.varifyT_global abs_type,
          rep_type = Logic.varifyT_global rep_type, Abs_name = Abs_name,
          Rep_name = Rep_name, prop_of_Rep = Thm.prop_of Rep,
          Abs_inverse = SOME Abs_inverse, Rep_inverse = SOME Rep_inverse}
  | _ => NONE

val is_raw_typedef = is_some oo typedef_info
val is_raw_free_datatype = is_some oo Ctr_Sugar.ctr_sugar_of

val is_interpreted_type =
  member (op =) [\<^type_name>\<open>prod\<close>, \<^type_name>\<open>set\<close>, \<^type_name>\<open>bool\<close>,
                 \<^type_name>\<open>nat\<close>, \<^type_name>\<open>int\<close>, \<^type_name>\<open>natural\<close>,
                 \<^type_name>\<open>integer\<close>]

fun repair_constr_type (Type (_, Ts)) T =
  snd (dest_Const (Ctr_Sugar.mk_ctr Ts (Const (Name.uu, T))))

fun register_frac_type_generic frac_s ersaetze generic =
  let
    val {frac_types, ersatz_table, codatatypes} = Data.get generic
    val frac_types = AList.update (op =) (frac_s, ersaetze) frac_types
  in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
               codatatypes = codatatypes} generic end

(* TODO: Consider morphism. *)
fun register_frac_type frac_s ersaetze (_ : morphism) =
  register_frac_type_generic frac_s ersaetze

val register_frac_type_global = Context.theory_map oo register_frac_type_generic

fun unregister_frac_type_generic frac_s = register_frac_type_generic frac_s []
(* TODO: Consider morphism. *)

fun unregister_frac_type frac_s (_ : morphism) =
  unregister_frac_type_generic frac_s

val unregister_frac_type_global =
  Context.theory_map o unregister_frac_type_generic

fun register_ersatz_generic ersatz generic =
  let
    val {frac_types, ersatz_table, codatatypes} = Data.get generic
    val ersatz_table = AList.merge (op =) (K true) (ersatz_table, ersatz)
  in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
               codatatypes = codatatypes} generic end

(* TODO: Consider morphism. *)
fun register_ersatz ersatz (_ : morphism) =
  register_ersatz_generic ersatz

val register_ersatz_global = Context.theory_map o register_ersatz_generic

fun register_codatatype_generic coT case_name constr_xs generic =
  let
    val {frac_types, ersatz_table, codatatypes} = Data.get generic
    val constr_xs = map (apsnd (repair_constr_type coT)) constr_xs
    val (co_s, coTs) = dest_Type coT
    val _ =
      if forall is_TFree coTs andalso not (has_duplicates (op =) coTs) andalso
         co_s <> \<^type_name>\<open>fun\<close> andalso not (is_interpreted_type co_s) then
        ()
      else
        raise TYPE ("Nitpick_HOL.register_codatatype_generic", [coT], [])
    val codatatypes = AList.update (op =) (co_s, (case_name, constr_xs))
                                   codatatypes
  in Data.put {frac_types = frac_types, ersatz_table = ersatz_table,
               codatatypes = codatatypes} generic end

(* TODO: Consider morphism. *)
fun register_codatatype coT case_name constr_xs (_ : morphism) =
  register_codatatype_generic coT case_name constr_xs

val register_codatatype_global =
  Context.theory_map ooo register_codatatype_generic

fun unregister_codatatype_generic coT = register_codatatype_generic coT "" []
(* TODO: Consider morphism. *)

fun unregister_codatatype coT (_ : morphism) =
  unregister_codatatype_generic coT
val unregister_codatatype_global =
  Context.theory_map o unregister_codatatype_generic

fun is_raw_codatatype ctxt s =
  Option.map #fp (BNF_FP_Def_Sugar.fp_sugar_of ctxt s)
  = SOME BNF_Util.Greatest_FP

fun is_registered_codatatype ctxt s =
  not (null (these (Option.map snd (AList.lookup (op =)
    (#codatatypes (Data.get (Context.Proof ctxt))) s))))

fun is_codatatype ctxt (Type (s, _)) =
    is_raw_codatatype ctxt s orelse is_registered_codatatype ctxt s
  | is_codatatype _ _ = false

fun is_registered_type ctxt (T as Type (s, _)) =
    is_frac_type ctxt T orelse is_registered_codatatype ctxt s
  | is_registered_type _ _ = false

fun is_raw_quot_type ctxt (Type (s, _)) =
    is_some (Quotient_Info.lookup_quotients ctxt s)
  | is_raw_quot_type _ _ = false

fun is_quot_type ctxt T =
  is_raw_quot_type ctxt T andalso not (is_registered_type ctxt T) andalso
  T <> \<^typ>\<open>int\<close>

fun is_pure_typedef ctxt (T as Type (s, _)) =
    is_frac_type ctxt T orelse
    (is_raw_typedef ctxt s andalso
     not (is_raw_free_datatype ctxt s orelse is_raw_quot_type ctxt T orelse
       is_codatatype ctxt T orelse is_integer_like_type T))
  | is_pure_typedef _ _ = false

fun is_univ_typedef ctxt (Type (s, _)) =
    (case typedef_info ctxt s of
       SOME {prop_of_Rep, ...} =>
       let
         val t_opt =
           try (snd o HOLogic.dest_mem o HOLogic.dest_Trueprop) prop_of_Rep
       in
         case t_opt of
           SOME (Const (\<^const_name>\<open>top\<close>, _)) => true
           (* "Multiset.multiset" FIXME unchecked *)
         | SOME (Const (\<^const_name>\<open>Collect\<close>, _)
                 $ Abs (_, _, Const (\<^const_name>\<open>finite\<close>, _) $ _)) => true
           (* "FinFun.finfun" FIXME unchecked *)
         | SOME (Const (\<^const_name>\<open>Collect\<close>, _) $ Abs (_, _,
                     Const (\<^const_name>\<open>Ex\<close>, _) $ Abs (_, _,
                         Const (\<^const_name>\<open>finite\<close>, _) $ _))) => true
         | _ => false
       end
     | NONE => false)
  | is_univ_typedef _ _ = false

fun is_data_type ctxt (T as Type (s, _)) =
    (is_raw_typedef ctxt s orelse is_registered_type ctxt T orelse
     T = \<^typ>\<open>ind\<close> orelse is_raw_quot_type ctxt T) andalso
    not (is_interpreted_type s)
  | is_data_type _ _ = false

fun all_record_fields thy T =
  let val (recs, more) = Record.get_extT_fields thy T in
    recs @ more :: all_record_fields thy (snd more)
  end
  handle TYPE _ => []

val num_record_fields = Integer.add 1 o length o fst oo Record.get_extT_fields

fun no_of_record_field thy s T1 =
  find_index (curry (op =) s o fst) (Record.get_extT_fields thy T1 ||> single |> op @)

fun is_record_get thy (s, Type (\<^type_name>\<open>fun\<close>, [T1, _])) =
    exists (curry (op =) s o fst) (all_record_fields thy T1)
  | is_record_get _ _ = false

fun is_record_update thy (s, T) =
  String.isSuffix Record.updateN s andalso
  exists (curry (op =) (unsuffix Record.updateN s) o fst) (all_record_fields thy (body_type T))
  handle TYPE _ => false

fun is_abs_fun ctxt (s, Type (\<^type_name>\<open>fun\<close>, [_, Type (s', _)])) =
    (case typedef_info ctxt s' of
       SOME {Abs_name, ...} => s = Abs_name
     | NONE => false)
  | is_abs_fun _ _ = false

fun is_rep_fun ctxt (s, Type (\<^type_name>\<open>fun\<close>, [Type (s', _), _])) =
    (case typedef_info ctxt s' of
       SOME {Rep_name, ...} => s = Rep_name
     | NONE => false)
  | is_rep_fun _ _ = false

fun is_quot_abs_fun ctxt (x as (_, Type (\<^type_name>\<open>fun\<close>,
                                         [_, abs_T as Type (s', _)]))) =
    try (Quotient_Term.absrep_const_chk ctxt Quotient_Term.AbsF) s'
    = SOME (Const x) andalso not (is_registered_type ctxt abs_T)
  | is_quot_abs_fun _ _ = false

fun is_quot_rep_fun ctxt (s, Type (\<^type_name>\<open>fun\<close>,
                                   [abs_T as Type (abs_s, _), _])) =
    (case try (Quotient_Term.absrep_const_chk ctxt Quotient_Term.RepF) abs_s of
       SOME (Const (s', _)) =>
       s = s' andalso not (is_registered_type ctxt abs_T)
     | _ => false)
  | is_quot_rep_fun _ _ = false

fun mate_of_rep_fun ctxt (x as (_, Type (\<^type_name>\<open>fun\<close>,
                                         [T1 as Type (s', _), T2]))) =
    (case typedef_info ctxt s' of
       SOME {Abs_name, ...} => (Abs_name, Type (\<^type_name>\<open>fun\<close>, [T2, T1]))
     | NONE => raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x]))
  | mate_of_rep_fun _ x = raise TERM ("Nitpick_HOL.mate_of_rep_fun", [Const x])

fun rep_type_for_quot_type ctxt (T as Type (s, _)) =
    let
      val thy = Proof_Context.theory_of ctxt
      val {qtyp, rtyp, ...} = the (Quotient_Info.lookup_quotients ctxt s)
    in
      instantiate_type thy qtyp T rtyp
    end
  | rep_type_for_quot_type _ T =
    raise TYPE ("Nitpick_HOL.rep_type_for_quot_type", [T], [])

fun equiv_relation_for_quot_type thy (Type (s, Ts)) =
    let
      val {qtyp, equiv_rel, equiv_thm, ...} =
        the (Quotient_Info.lookup_quotients thy s)
      val partial =
        case Thm.prop_of equiv_thm of
          \<^const>\<open>Trueprop\<close> $ (Const (\<^const_name>\<open>equivp\<close>, _) $ _) => false
        | \<^const>\<open>Trueprop\<close> $ (Const (\<^const_name>\<open>part_equivp\<close>, _) $ _) => true
        | _ => raise NOT_SUPPORTED "Ill-formed quotient type equivalence \
                                   \relation theorem"
      val Ts' = qtyp |> dest_Type |> snd
    in (subst_atomic_types (Ts' ~~ Ts) equiv_rel, partial) end
  | equiv_relation_for_quot_type _ T =
    raise TYPE ("Nitpick_HOL.equiv_relation_for_quot_type", [T], [])

fun is_raw_free_datatype_constr ctxt (s, T) =
  case body_type T of
    dtT as Type (dt_s, _) =>
    let
      val ctrs =
        case Ctr_Sugar.ctr_sugar_of ctxt dt_s of
          SOME {ctrs, ...} => map dest_Const ctrs
        | _ => []
    in
      exists (fn (s', T') => s = s' andalso repair_constr_type dtT T' = T) ctrs
    end
  | _  => false

fun is_registered_coconstr ctxt (s, T) =
  case body_type T of
    coT as Type (co_s, _) =>
    let
      val ctrs =
        co_s
        |> AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))
        |> Option.map snd |> these
    in
      exists (fn (s', T') => s = s' andalso repair_constr_type coT T' = T) ctrs
    end
  | _ => false

fun is_nonfree_constr ctxt (s, T) =
  member (op =) [\<^const_name>\<open>FunBox\<close>, \<^const_name>\<open>PairBox\<close>,
                 \<^const_name>\<open>Quot\<close>, \<^const_name>\<open>Zero_Rep\<close>,
                 \<^const_name>\<open>Suc_Rep\<close>] s orelse
  let val (x as (_, T)) = (s, unarize_unbox_etc_type T) in
    is_raw_free_datatype_constr ctxt x orelse
    (is_abs_fun ctxt x andalso is_pure_typedef ctxt (range_type T)) orelse
    is_registered_coconstr ctxt x
  end

fun is_free_constr ctxt (s, T) =
  is_nonfree_constr ctxt (s, T) andalso
  let val (x as (_, T)) = (s, unarize_unbox_etc_type T) in
    not (is_abs_fun ctxt x) orelse is_univ_typedef ctxt (range_type T)
  end

fun is_stale_constr ctxt (x as (s, T)) =
  is_registered_type ctxt (body_type T) andalso is_nonfree_constr ctxt x andalso
  not (s = \<^const_name>\<open>Abs_Frac\<close> orelse is_registered_coconstr ctxt x)

fun is_constr ctxt (x as (_, T)) =
  is_nonfree_constr ctxt x andalso
  not (is_interpreted_type (fst (dest_Type (unarize_type (body_type T))))) andalso
  not (is_stale_constr ctxt x)

val is_sel = String.isPrefix discr_prefix orf String.isPrefix sel_prefix
val is_sel_like_and_no_discr =
  String.isPrefix sel_prefix orf
  (member (op =) [\<^const_name>\<open>fst\<close>, \<^const_name>\<open>snd\<close>])

fun in_fun_lhs_for InConstr = InSel
  | in_fun_lhs_for _ = InFunLHS

fun in_fun_rhs_for InConstr = InConstr
  | in_fun_rhs_for InSel = InSel
  | in_fun_rhs_for InFunRHS1 = InFunRHS2
  | in_fun_rhs_for _ = InFunRHS1

fun is_boxing_worth_it (hol_ctxt : hol_context) boxy T =
  case T of
    Type (\<^type_name>\<open>fun\<close>, _) =>
    (boxy = InPair orelse boxy = InFunLHS) andalso
    not (is_boolean_type (body_type T))
  | Type (\<^type_name>\<open>prod\<close>, Ts) =>
    boxy = InPair orelse boxy = InFunRHS1 orelse boxy = InFunRHS2 orelse
    ((boxy = InExpr orelse boxy = InFunLHS) andalso
     exists (is_boxing_worth_it hol_ctxt InPair)
            (map (box_type hol_ctxt InPair) Ts))
  | _ => false
and should_box_type (hol_ctxt as {thy, boxes, ...}) boxy z =
  case triple_lookup (type_match thy) boxes (Type z) of
    SOME (SOME box_me) => box_me
  | _ => is_boxing_worth_it hol_ctxt boxy (Type z)
and box_type hol_ctxt boxy T =
  case T of
    Type (z as (\<^type_name>\<open>fun\<close>, [T1, T2])) =>
    if boxy <> InConstr andalso boxy <> InSel andalso
       should_box_type hol_ctxt boxy z then
      Type (\<^type_name>\<open>fun_box\<close>,
            [box_type hol_ctxt InFunLHS T1, box_type hol_ctxt InFunRHS1 T2])
    else
      box_type hol_ctxt (in_fun_lhs_for boxy) T1
      --> box_type hol_ctxt (in_fun_rhs_for boxy) T2
  | Type (z as (\<^type_name>\<open>prod\<close>, Ts)) =>
    if boxy <> InConstr andalso boxy <> InSel
       andalso should_box_type hol_ctxt boxy z then
      Type (\<^type_name>\<open>pair_box\<close>, map (box_type hol_ctxt InSel) Ts)
    else
      Type (\<^type_name>\<open>prod\<close>,
            map (box_type hol_ctxt
                          (if boxy = InConstr orelse boxy = InSel then boxy
                           else InPair)) Ts)
  | _ => T

fun binarize_nat_and_int_in_type \<^typ>\<open>nat\<close> = \<^typ>\<open>unsigned_bit word\<close>
  | binarize_nat_and_int_in_type \<^typ>\<open>int\<close> = \<^typ>\<open>signed_bit word\<close>
  | binarize_nat_and_int_in_type (Type (s, Ts)) =
    Type (s, map binarize_nat_and_int_in_type Ts)
  | binarize_nat_and_int_in_type T = T
val binarize_nat_and_int_in_term = map_types binarize_nat_and_int_in_type

fun discr_for_constr (s, T) = (discr_prefix ^ s, body_type T --> bool_T)

fun num_sels_for_constr_type T = length (maybe_curried_binder_types T)

fun nth_sel_name_for_constr_name s n =
  if s = \<^const_name>\<open>Pair\<close> then
    if n = 0 then \<^const_name>\<open>fst\<close> else \<^const_name>\<open>snd\<close>
  else
    sel_prefix_for n ^ s

fun nth_sel_for_constr x ~1 = discr_for_constr x
  | nth_sel_for_constr (s, T) n =
    (nth_sel_name_for_constr_name s n,
     body_type T --> nth (maybe_curried_binder_types T) n)

fun binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize =
  apsnd ((binarize ? binarize_nat_and_int_in_type) o box_type hol_ctxt InSel)
  oo nth_sel_for_constr

fun sel_no_from_name s =
  if String.isPrefix discr_prefix s then
    ~1
  else if String.isPrefix sel_prefix s then
    s |> unprefix sel_prefix |> Int.fromString |> the
  else if s = \<^const_name>\<open>snd\<close> then
    1
  else
    0

val close_form =
  let
    fun close_up zs zs' =
      fold (fn (z as ((s, _), T)) => fn t' =>
               Logic.all_const T $ Abs (s, T, abstract_over (Var z, t')))
           (take (length zs' - length zs) zs')
    fun aux zs (\<^const>\<open>Pure.imp\<close> $ t1 $ t2) =
        let val zs' = Term.add_vars t1 zs in
          close_up zs zs' (Logic.mk_implies (t1, aux zs' t2))
        end
      | aux zs t = close_up zs (Term.add_vars t zs) t
  in aux [] end

fun distinctness_formula T =
  all_distinct_unordered_pairs_of
  #> map (fn (t1, t2) => \<^const>\<open>Not\<close> $ (HOLogic.eq_const T $ t1 $ t2))
  #> List.foldr (s_conj o swap) \<^const>\<open>True\<close>

fun zero_const T = Const (\<^const_name>\<open>zero_class.zero\<close>, T)
fun suc_const T = Const (\<^const_name>\<open>Suc\<close>, T --> T)

fun uncached_data_type_constrs ({ctxt, ...} : hol_context) (T as Type (s, _)) =
    if is_interpreted_type s then
      []
    else
      (case AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt))) s of
         SOME (_, xs' as (_ :: _)) => map (apsnd (repair_constr_type T)) xs'
       | _ =>
         if is_frac_type ctxt T then
           case typedef_info ctxt s of
             SOME {abs_type, rep_type, Abs_name, ...} =>
             [(Abs_name, varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
           | NONE => [] (* impossible *)
         else
           case Ctr_Sugar.ctr_sugar_of ctxt s of
             SOME {ctrs, ...} =>
             map (apsnd (repair_constr_type T) o dest_Const) ctrs
           | NONE =>
             if is_raw_quot_type ctxt T then
               [(\<^const_name>\<open>Quot\<close>, rep_type_for_quot_type ctxt T --> T)]
             else case typedef_info ctxt s of
               SOME {abs_type, rep_type, Abs_name, ...} =>
               [(Abs_name, varify_and_instantiate_type ctxt abs_type T rep_type --> T)]
             | NONE =>
               if T = \<^typ>\<open>ind\<close> then [dest_Const \<^const>\<open>Zero_Rep\<close>, dest_Const \<^const>\<open>Suc_Rep\<close>]
               else [])
  | uncached_data_type_constrs _ _ = []

fun data_type_constrs (hol_ctxt as {constr_cache, ...}) T =
  case AList.lookup (op =) (!constr_cache) T of
    SOME xs => xs
  | NONE =>
    let val xs = uncached_data_type_constrs hol_ctxt T in
      (Unsynchronized.change constr_cache (cons (T, xs)); xs)
    end

fun binarized_and_boxed_data_type_constrs hol_ctxt binarize =
  map (apsnd ((binarize ? binarize_nat_and_int_in_type)
              o box_type hol_ctxt InConstr)) o data_type_constrs hol_ctxt

fun constr_name_for_sel_like \<^const_name>\<open>fst\<close> = \<^const_name>\<open>Pair\<close>
  | constr_name_for_sel_like \<^const_name>\<open>snd\<close> = \<^const_name>\<open>Pair\<close>
  | constr_name_for_sel_like s' = original_name s'

fun binarized_and_boxed_constr_for_sel hol_ctxt binarize (s', T') =
  let val s = constr_name_for_sel_like s' in
    AList.lookup (op =)
        (binarized_and_boxed_data_type_constrs hol_ctxt binarize (domain_type T'))
        s
    |> the |> pair s
  end

fun card_of_type assigns (Type (\<^type_name>\<open>fun\<close>, [T1, T2])) =
    reasonable_power (card_of_type assigns T2) (card_of_type assigns T1)
  | card_of_type assigns (Type (\<^type_name>\<open>prod\<close>, [T1, T2])) =
    card_of_type assigns T1 * card_of_type assigns T2
  | card_of_type assigns (Type (\<^type_name>\<open>set\<close>, [T'])) =
    reasonable_power 2 (card_of_type assigns T')
  | card_of_type _ (Type (\<^type_name>\<open>itself\<close>, _)) = 1
  | card_of_type _ \<^typ>\<open>prop\<close> = 2
  | card_of_type _ \<^typ>\<open>bool\<close> = 2
  | card_of_type assigns T =
    case AList.lookup (op =) assigns T of
      SOME k => k
    | NONE => if T = \<^typ>\<open>bisim_iterator\<close> then 0
              else raise TYPE ("Nitpick_HOL.card_of_type", [T], [])

fun bounded_card_of_type max default_card assigns
                         (Type (\<^type_name>\<open>fun\<close>, [T1, T2])) =
    let
      val k1 = bounded_card_of_type max default_card assigns T1
      val k2 = bounded_card_of_type max default_card assigns T2
    in
      if k1 = max orelse k2 = max then max
      else Int.min (max, reasonable_power k2 k1)
      handle TOO_LARGE _ => max
    end
  | bounded_card_of_type max default_card assigns
                         (Type (\<^type_name>\<open>prod\<close>, [T1, T2])) =
    let
      val k1 = bounded_card_of_type max default_card assigns T1
      val k2 = bounded_card_of_type max default_card assigns T2
    in if k1 = max orelse k2 = max then max else Int.min (max, k1 * k2) end
  | bounded_card_of_type max default_card assigns
                         (Type (\<^type_name>\<open>set\<close>, [T'])) =
    bounded_card_of_type max default_card assigns (T' --> bool_T)
  | bounded_card_of_type max default_card assigns T =
    Int.min (max, if default_card = ~1 then
                    card_of_type assigns T
                  else
                    card_of_type assigns T
                    handle TYPE ("Nitpick_HOL.card_of_type", _, _) =>
                           default_card)

(* Similar to "ATP_Util.tiny_card_of_type". *)
fun bounded_exact_card_of_type hol_ctxt finitizable_dataTs max default_card
                               assigns T =
  let
    fun aux avoid T =
      (if member (op =) avoid T then
         0
       else if member (op =) finitizable_dataTs T then
         raise SAME ()
       else case T of
         Type (\<^type_name>\<open>fun\<close>, [T1, T2]) =>
         (case (aux avoid T1, aux avoid T2) of
            (_, 1) => 1
          | (0, _) => 0
          | (_, 0) => 0
          | (k1, k2) =>
            if k1 >= max orelse k2 >= max then max
            else Int.min (max, reasonable_power k2 k1))
       | Type (\<^type_name>\<open>prod\<close>, [T1, T2]) =>
         (case (aux avoid T1, aux avoid T2) of
            (0, _) => 0
          | (_, 0) => 0
          | (k1, k2) =>
            if k1 >= max orelse k2 >= max then max
            else Int.min (max, k1 * k2))
       | Type (\<^type_name>\<open>set\<close>, [T']) => aux avoid (T' --> bool_T)
       | Type (\<^type_name>\<open>itself\<close>, _) => 1
       | \<^typ>\<open>prop\<close> => 2
       | \<^typ>\<open>bool\<close> => 2
       | Type _ =>
         (case data_type_constrs hol_ctxt T of
            [] => if is_integer_type T orelse is_bit_type T then 0
                  else raise SAME ()
          | constrs =>
            let
              val constr_cards =
                map (Integer.prod o map (aux (T :: avoid)) o binder_types o snd)
                    constrs
            in
              if exists (curry (op =) 0) constr_cards then 0
              else Int.min (max, Integer.sum constr_cards)
            end)
       | _ => raise SAME ())
      handle SAME () =>
             AList.lookup (op =) assigns T |> the_default default_card
  in Int.min (max, aux [] T) end

val typical_atomic_card = 4
val typical_card_of_type = bounded_card_of_type 16777217 typical_atomic_card []

fun is_finite_type hol_ctxt T =
  bounded_exact_card_of_type hol_ctxt [] 1 2 [] T > 0

fun is_special_eligible_arg strict Ts t =
  case map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t) of
    [] => true
  | bad_Ts =>
    let
      val bad_Ts_cost =
        if strict then fold (curry (op *) o typical_card_of_type) bad_Ts 1
        else fold (Integer.max o typical_card_of_type) bad_Ts 0
      val T_cost = typical_card_of_type (fastype_of1 (Ts, t))
    in (bad_Ts_cost, T_cost) |> (if strict then op < else op <=) end

fun abs_var ((s, j), T) body = Abs (s, T, abstract_over (Var ((s, j), T), body))

fun let_var s = (nitpick_prefix ^ s, 999)
val let_inline_threshold = 20

fun s_let Ts s n abs_T body_T f t =
  if (n - 1) * (size_of_term t - 1) <= let_inline_threshold orelse
     is_special_eligible_arg false Ts t then
    f t
  else
    let val z = (let_var s, abs_T) in
      Const (\<^const_name>\<open>Let\<close>, abs_T --> (abs_T --> body_T) --> body_T)
      $ t $ abs_var z (incr_boundvars 1 (f (Var z)))
    end

fun loose_bvar1_count (Bound i, k) = if i = k then 1 else 0
  | loose_bvar1_count (t1 $ t2, k) =
    loose_bvar1_count (t1, k) + loose_bvar1_count (t2, k)
  | loose_bvar1_count (Abs (_, _, t), k) = loose_bvar1_count (t, k + 1)
  | loose_bvar1_count _ = 0

fun s_betapply _ (t1 as Const (\<^const_name>\<open>Pure.eq\<close>, _) $ t1', t2) =
    if t1' aconv t2 then \<^prop>\<open>True\<close> else t1 $ t2
  | s_betapply _ (t1 as Const (\<^const_name>\<open>HOL.eq\<close>, _) $ t1', t2) =
    if t1' aconv t2 then \<^term>\<open>True\<close> else t1 $ t2
  | s_betapply _ (Const (\<^const_name>\<open>If\<close>, _) $ \<^const>\<open>True\<close> $ t1', _) = t1'
  | s_betapply _ (Const (\<^const_name>\<open>If\<close>, _) $ \<^const>\<open>False\<close> $ _, t2) = t2
  | s_betapply Ts (Const (\<^const_name>\<open>Let\<close>,
                          Type (_, [bound_T, Type (_, [_, body_T])]))
                   $ t12 $ Abs (s, T, t13'), t2) =
    let val body_T' = range_type body_T in
      Const (\<^const_name>\<open>Let\<close>, bound_T --> (bound_T --> body_T') --> body_T')
      $ t12 $ Abs (s, T, s_betapply (T :: Ts) (t13', incr_boundvars 1 t2))
    end
  | s_betapply Ts (t1 as Abs (s1, T1, t1'), t2) =
    (s_let Ts s1 (loose_bvar1_count (t1', 0)) T1 (fastype_of1 (T1 :: Ts, t1'))
           (curry betapply t1) t2
     (* FIXME: fix all "s_betapply []" calls *)
     handle TERM _ => betapply (t1, t2)
          | General.Subscript => betapply (t1, t2))
  | s_betapply _ (t1, t2) = t1 $ t2

fun s_betapplys Ts = Library.foldl (s_betapply Ts)

fun s_beta_norm Ts t =
  let
    fun aux _ (Var _) = raise Same.SAME
      | aux Ts (Abs (s, T, t')) = Abs (s, T, aux (T :: Ts) t')
      | aux Ts ((t1 as Abs _) $ t2) =
        Same.commit (aux Ts) (s_betapply Ts (t1, t2))
      | aux Ts (t1 $ t2) =
        ((case aux Ts t1 of
           t1 as Abs _ => Same.commit (aux Ts) (s_betapply Ts (t1, t2))
         | t1 => t1 $ Same.commit (aux Ts) t2)
        handle Same.SAME => t1 $ aux Ts t2)
      | aux _ _ = raise Same.SAME
  in aux Ts t handle Same.SAME => t end

fun discr_term_for_constr hol_ctxt (x as (s, T)) =
  let val dataT = body_type T in
    if s = \<^const_name>\<open>Suc\<close> then
      Abs (Name.uu, dataT, \<^const>\<open>Not\<close> $ HOLogic.mk_eq (zero_const dataT, Bound 0))
    else if length (data_type_constrs hol_ctxt dataT) >= 2 then
      Const (discr_for_constr x)
    else
      Abs (Name.uu, dataT, \<^const>\<open>True\<close>)
  end

fun discriminate_value (hol_ctxt as {ctxt, ...}) x t =
  case head_of t of
    Const x' =>
    if x = x' then \<^const>\<open>True\<close>
    else if is_nonfree_constr ctxt x' then \<^const>\<open>False\<close>
    else s_betapply [] (discr_term_for_constr hol_ctxt x, t)
  | _ => s_betapply [] (discr_term_for_constr hol_ctxt x, t)

fun nth_arg_sel_term_for_constr (x as (s, T)) n =
  let val (arg_Ts, dataT) = strip_type T in
    if dataT = nat_T then
      \<^term>\<open>%n::nat. n - 1\<close>
    else if is_pair_type dataT then
      Const (nth_sel_for_constr x n)
    else
      let
        fun aux m (Type (\<^type_name>\<open>prod\<close>, [T1, T2])) =
            let
              val (m, t1) = aux m T1
              val (m, t2) = aux m T2
            in (m, HOLogic.mk_prod (t1, t2)) end
          | aux m T =
            (m + 1, Const (nth_sel_name_for_constr_name s m, dataT --> T)
                    $ Bound 0)
        val m = fold (Integer.add o num_factors_in_type)
                     (List.take (arg_Ts, n)) 0
      in Abs ("x", dataT, aux m (nth arg_Ts n) |> snd) end
  end

fun select_nth_constr_arg ctxt x t n res_T =
  (case strip_comb t of
     (Const x', args) =>
     if x = x' then
        if is_free_constr ctxt x then nth args n else raise SAME ()
     else if is_nonfree_constr ctxt x' then
       Const (\<^const_name>\<open>unknown\<close>, res_T)
     else
       raise SAME ()
   | _ => raise SAME())
  handle SAME () => s_betapply [] (nth_arg_sel_term_for_constr x n, t)

fun construct_value _ x [] = Const x
  | construct_value ctxt (x as (s, _)) args =
    let val args = map Envir.eta_contract args in
      case hd args of
        Const (s', _) $ t =>
        if is_sel_like_and_no_discr s' andalso
           constr_name_for_sel_like s' = s andalso
           forall (fn (n, t') => select_nth_constr_arg ctxt x t n dummyT = t')
                  (index_seq 0 (length args) ~~ args) then
          t
        else
          list_comb (Const x, args)
      | _ => list_comb (Const x, args)
    end

fun constr_expand (hol_ctxt as {ctxt, ...}) T t =
  (case head_of t of
     Const x => if is_nonfree_constr ctxt x then t else raise SAME ()
   | _ => raise SAME ())
  handle SAME () =>
         let
           val x' as (_, T') =
             if is_pair_type T then
               let val (T1, T2) = HOLogic.dest_prodT T in
                 (\<^const_name>\<open>Pair\<close>, T1 --> T2 --> T)
               end
             else
               data_type_constrs hol_ctxt T |> hd
           val arg_Ts = binder_types T'
         in
           list_comb (Const x', map2 (select_nth_constr_arg ctxt x' t)
                                     (index_seq 0 (length arg_Ts)) arg_Ts)
         end

fun coerce_bound_no f j t =
  case t of
    t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2
  | Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')
  | Bound j' => if j' = j then f t else t
  | _ => t

fun coerce_bound_0_in_term hol_ctxt new_T old_T =
  old_T <> new_T ? coerce_bound_no (coerce_term hol_ctxt [new_T] old_T new_T) 0
and coerce_term (hol_ctxt as {ctxt, ...}) Ts new_T old_T t =
  if old_T = new_T then
    t
  else
    case (new_T, old_T) of
      (Type (new_s, new_Ts as [new_T1, new_T2]),
       Type (\<^type_name>\<open>fun\<close>, [old_T1, old_T2])) =>
      (case eta_expand Ts t 1 of
         Abs (s, _, t') =>
         Abs (s, new_T1,
              t' |> coerce_bound_0_in_term hol_ctxt new_T1 old_T1
                 |> coerce_term hol_ctxt (new_T1 :: Ts) new_T2 old_T2)
         |> Envir.eta_contract
         |> new_s <> \<^type_name>\<open>fun\<close>
            ? construct_value ctxt
                  (\<^const_name>\<open>FunBox\<close>,
                   Type (\<^type_name>\<open>fun\<close>, new_Ts) --> new_T)
              o single
       | t' => raise TERM ("Nitpick_HOL.coerce_term", [t']))
    | (Type (new_s, new_Ts as [new_T1, new_T2]),
       Type (old_s, old_Ts as [old_T1, old_T2])) =>
      if old_s = \<^type_name>\<open>fun_box\<close> orelse
         old_s = \<^type_name>\<open>pair_box\<close> orelse old_s = \<^type_name>\<open>prod\<close> then
        case constr_expand hol_ctxt old_T t of
          Const (old_s, _) $ t1 =>
          if new_s = \<^type_name>\<open>fun\<close> then
            coerce_term hol_ctxt Ts new_T (Type (\<^type_name>\<open>fun\<close>, old_Ts)) t1
          else
            construct_value ctxt
                (old_s, Type (\<^type_name>\<open>fun\<close>, new_Ts) --> new_T)
                [coerce_term hol_ctxt Ts (Type (\<^type_name>\<open>fun\<close>, new_Ts))
                             (Type (\<^type_name>\<open>fun\<close>, old_Ts)) t1]
        | Const _ $ t1 $ t2 =>
          construct_value ctxt
              (if new_s = \<^type_name>\<open>prod\<close> then \<^const_name>\<open>Pair\<close>
               else \<^const_name>\<open>PairBox\<close>, new_Ts ---> new_T)
              (@{map 3} (coerce_term hol_ctxt Ts) [new_T1, new_T2] [old_T1, old_T2]
                    [t1, t2])
        | t' => raise TERM ("Nitpick_HOL.coerce_term", [t'])
      else
        raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])
    | _ => raise TYPE ("Nitpick_HOL.coerce_term", [new_T, old_T], [t])

fun is_ground_term (t1 $ t2) = is_ground_term t1 andalso is_ground_term t2
  | is_ground_term (Const _) = true
  | is_ground_term _ = false

fun special_bounds ts =
  fold Term.add_vars ts [] |> sort (Term_Ord.fast_indexname_ord o apply2 fst)

fun is_funky_typedef_name ctxt s =
  member (op =) [\<^type_name>\<open>unit\<close>, \<^type_name>\<open>prod\<close>, \<^type_name>\<open>set\<close>,
                 \<^type_name>\<open>Sum_Type.sum\<close>, \<^type_name>\<open>int\<close>] s orelse
  is_frac_type ctxt (Type (s, []))

fun is_funky_typedef ctxt (Type (s, _)) = is_funky_typedef_name ctxt s
  | is_funky_typedef _ _ = false

fun all_defs_of thy subst =
  let
    val def_names =
      thy |> Theory.defs_of
          |> Defs.all_specifications_of
          |> maps snd |> map_filter #def
          |> Ord_List.make fast_string_ord
  in
    Theory.nodes_of thy
    |> maps Thm.axioms_of
    |> map (apsnd (subst_atomic subst o Thm.prop_of))
    |> sort (fast_string_ord o apply2 fst)
    |> Ord_List.inter (fast_string_ord o apsnd fst) def_names
    |> map snd
  end

(* Ideally we would check against "Complex_Main", not "Hilbert_Choice", but any
   theory will do as long as it contains all the "axioms" and "axiomatization"
   commands. *)
fun is_built_in_theory thy_id =
  Context.subthy_id (thy_id, Context.theory_id \<^theory>\<open>Hilbert_Choice\<close>)

fun all_nondefs_of ctxt subst =
  ctxt |> Spec_Rules.get
       |> filter (Spec_Rules.is_unknown o fst)
       |> maps (snd o snd)
       |> filter_out (is_built_in_theory o Thm.theory_id)
       |> map (subst_atomic subst o Thm.prop_of)

fun arity_of_built_in_const (s, T) =
  if s = \<^const_name>\<open>If\<close> then
    if nth_range_type 3 T = \<^typ>\<open>bool\<close> then NONE else SOME 3
  else
    case AList.lookup (op =) built_in_consts s of
      SOME n => SOME n
    | NONE =>
      case AList.lookup (op =) built_in_typed_consts (s, unarize_type T) of
        SOME n => SOME n
      | NONE =>
        case s of
          \<^const_name>\<open>zero_class.zero\<close> => if is_iterator_type T then SOME 0 else NONE
        | \<^const_name>\<open>Suc\<close> => if is_iterator_type (domain_type T) then SOME 0 else NONE
        | _ => NONE

val is_built_in_const = is_some o arity_of_built_in_const

(* This function is designed to work for both real definition axioms and
   simplification rules (equational specifications). *)
fun term_under_def t =
  case t of
    \<^const>\<open>Pure.imp\<close> $ _ $ t2 => term_under_def t2
  | Const (\<^const_name>\<open>Pure.eq\<close>, _) $ t1 $ _ => term_under_def t1
  | \<^const>\<open>Trueprop\<close> $ t1 => term_under_def t1
  | Const (\<^const_name>\<open>HOL.eq\<close>, _) $ t1 $ _ => term_under_def t1
  | Abs (_, _, t') => term_under_def t'
  | t1 $ _ => term_under_def t1
  | _ => t

(* Here we crucially rely on "specialize_type" performing a preorder traversal
   of the term, without which the wrong occurrence of a constant could be
   matched in the face of overloading. *)
fun def_props_for_const thy table (x as (s, _)) =
  if is_built_in_const x then
    []
  else
    these (Symtab.lookup table s)
    |> map_filter (try (specialize_type thy x))
    |> filter (curry (op =) (Const x) o term_under_def)

fun normalized_rhs_of t =
  let
    fun aux (v as Var _) (SOME t) = SOME (lambda v t)
      | aux (c as Const (\<^const_name>\<open>Pure.type\<close>, _)) (SOME t) = SOME (lambda c t)
      | aux _ _ = NONE
    val (lhs, rhs) =
      case t of
        Const (\<^const_name>\<open>Pure.eq\<close>, _) $ t1 $ t2 => (t1, t2)
      | \<^const>\<open>Trueprop\<close> $ (Const (\<^const_name>\<open>HOL.eq\<close>, _) $ t1 $ t2) =>
        (t1, t2)
      | _ => raise TERM ("Nitpick_HOL.normalized_rhs_of", [t])
    val args = strip_comb lhs |> snd
  in fold_rev aux args (SOME rhs) end

fun get_def_of_const thy table (x as (s, _)) =
  x |> def_props_for_const thy table |> List.last
    |> normalized_rhs_of |> Option.map (prefix_abs_vars s)
  handle List.Empty => NONE
       | TERM _ => NONE

fun def_of_const_ext thy (unfold_table, fallback_table) (x as (s, _)) =
  if is_built_in_const x orelse original_name s <> s then
    NONE
  else case get_def_of_const thy unfold_table x of
    SOME def => SOME (true, def)
  | NONE => get_def_of_const thy fallback_table x |> Option.map (pair false)

val def_of_const = Option.map snd ooo def_of_const_ext

fun fixpoint_kind_of_rhs (Abs (_, _, t)) = fixpoint_kind_of_rhs t
  | fixpoint_kind_of_rhs (Const (\<^const_name>\<open>lfp\<close>, _) $ Abs _) = Lfp
  | fixpoint_kind_of_rhs (Const (\<^const_name>\<open>gfp\<close>, _) $ Abs _) = Gfp
  | fixpoint_kind_of_rhs _ = NoFp

fun is_mutually_inductive_pred_def thy table t =
  let
    fun is_good_arg (Bound _) = true
      | is_good_arg (Const (s, _)) =
        s = \<^const_name>\<open>True\<close> orelse s = \<^const_name>\<open>False\<close> orelse
        s = \<^const_name>\<open>undefined\<close>
      | is_good_arg _ = false
  in
    case t |> strip_abs_body |> strip_comb of
      (Const x, ts as (_ :: _)) =>
      (case def_of_const thy table x of
         SOME t' => fixpoint_kind_of_rhs t' <> NoFp andalso
                    forall is_good_arg ts
       | NONE => false)
    | _ => false
  end

fun unfold_mutually_inductive_preds thy table =
  map_aterms (fn t as Const x =>
      (case def_of_const thy table x of
         SOME t' =>
         let val t' = Envir.eta_contract t' in
           if is_mutually_inductive_pred_def thy table t' then t' else t
         end
      | NONE => t)
    | t => t)

fun case_const_names ctxt =
  map_filter (fn {casex = Const (s, T), ...} =>
      (case rev (binder_types T) of
        [] => NONE
      | T :: Ts => if is_data_type ctxt T then SOME (s, length Ts) else NONE))
    (Ctr_Sugar.ctr_sugars_of ctxt) @
  map (apsnd length o snd) (#codatatypes (Data.get (Context.Proof ctxt)))

fun fixpoint_kind_of_const thy table x =
  if is_built_in_const x then NoFp
  else fixpoint_kind_of_rhs (the (def_of_const thy table x))
  handle Option.Option => NoFp

fun is_raw_inductive_pred ({thy, def_tables, intro_table, ...} : hol_context) x =
  fixpoint_kind_of_const thy def_tables x <> NoFp andalso
  not (null (def_props_for_const thy intro_table x))

fun is_inductive_pred hol_ctxt (x as (s, _)) =
  String.isPrefix ubfp_prefix s orelse String.isPrefix lbfp_prefix s orelse
  is_raw_inductive_pred hol_ctxt x

fun lhs_of_equation t =
  case t of
    Const (\<^const_name>\<open>Pure.all\<close>, _) $ Abs (_, _, t1) => lhs_of_equation t1
  | Const (\<^const_name>\<open>Pure.eq\<close>, _) $ t1 $ _ => SOME t1
  | \<^const>\<open>Pure.imp\<close> $ _ $ t2 => lhs_of_equation t2
  | \<^const>\<open>Trueprop\<close> $ t1 => lhs_of_equation t1
  | Const (\<^const_name>\<open>All\<close>, _) $ Abs (_, _, t1) => lhs_of_equation t1
  | Const (\<^const_name>\<open>HOL.eq\<close>, _) $ t1 $ _ => SOME t1
  | \<^const>\<open>HOL.implies\<close> $ _ $ t2 => lhs_of_equation t2
  | _ => NONE

fun is_constr_pattern _ (Bound _) = true
  | is_constr_pattern _ (Var _) = true
  | is_constr_pattern ctxt t =
    case strip_comb t of
      (Const x, args) =>
      is_nonfree_constr ctxt x andalso forall (is_constr_pattern ctxt) args
    | _ => false

fun is_constr_pattern_lhs ctxt t =
  forall (is_constr_pattern ctxt) (snd (strip_comb t))

fun is_constr_pattern_formula ctxt t =
  case lhs_of_equation t of
    SOME t' => is_constr_pattern_lhs ctxt t'
  | NONE => false

(* Similar to "specialize_type" but returns all matches rather than only the
   first (preorder) match. *)
fun multi_specialize_type thy slack (s, T) t =
  let
    fun aux (Const (s', T')) ys =
        if s = s' then
          ys |> (if AList.defined (op =) ys T' then
                   I
                 else
                   cons (T', Envir.subst_term_types (Sign.typ_match thy (T', T)
                     Vartab.empty) t)
                   handle Type.TYPE_MATCH => I
                        | TERM _ =>
                          if slack then
                            I
                          else
                            raise NOT_SUPPORTED
                                      ("too much polymorphism in axiom \"" ^
                                       Syntax.string_of_term_global thy t ^
                                       "\" involving " ^ quote s))
        else
          ys
      | aux _ ys = ys
  in map snd (fold_aterms aux t []) end

fun nondef_props_for_const thy slack table (x as (s, _)) =
  these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)

fun unvarify_term (t1 $ t2) = unvarify_term t1 $ unvarify_term t2
  | unvarify_term (Var ((s, 0), T)) = Free (s, T)
  | unvarify_term (Abs (s, T, t')) = Abs (s, T, unvarify_term t')
  | unvarify_term t = t

fun axiom_for_choice_spec ctxt =
  unvarify_term
  #> Object_Logic.atomize_term ctxt
  #> Choice_Specification.close_form
  #> HOLogic.mk_Trueprop

fun is_choice_spec_fun ({thy, ctxt, def_tables, nondef_table, choice_spec_table, ...}
                        : hol_context) x =
  case nondef_props_for_const thy true choice_spec_table x of
    [] => false
  | ts => case def_of_const thy def_tables x of
            SOME (Const (\<^const_name>\<open>Eps\<close>, _) $ _) => true
          | SOME _ => false
          | NONE =>
            let val ts' = nondef_props_for_const thy true nondef_table x in
              length ts' = length ts andalso
              forall (fn t =>
                         exists (curry (op aconv) (axiom_for_choice_spec ctxt t))
                                ts') ts
            end

fun is_choice_spec_axiom thy choice_spec_table t =
  Symtab.exists (exists (curry (op aconv) t o axiom_for_choice_spec thy) o snd)
                choice_spec_table

fun is_raw_equational_fun ({thy, simp_table, psimp_table, ...} : hol_context)
                          x =
  exists (fn table => not (null (def_props_for_const thy table x)))
         [!simp_table, psimp_table]

fun is_equational_fun hol_ctxt =
  is_raw_equational_fun hol_ctxt orf is_inductive_pred hol_ctxt

(** Constant unfolding **)

fun constr_case_body ctxt Ts (func_t, (x as (_, T))) =
  let val arg_Ts = binder_types T in
    s_betapplys Ts (func_t, map2 (select_nth_constr_arg ctxt x (Bound 0))
                                 (index_seq 0 (length arg_Ts)) arg_Ts)
  end

fun add_constr_case res_T (body_t, guard_t) res_t =
  if res_T = bool_T then
    s_conj (HOLogic.mk_imp (guard_t, body_t), res_t)
  else
    Const (\<^const_name>\<open>If\<close>, bool_T --> res_T --> res_T --> res_T)
    $ guard_t $ body_t $ res_t

fun optimized_case_def (hol_ctxt as {ctxt, ...}) Ts dataT res_T func_ts =
  let
    val xs = data_type_constrs hol_ctxt dataT
    val cases =
      func_ts ~~ xs
      |> map (fn (func_t, x) =>
                 (constr_case_body ctxt (dataT :: Ts)
                                   (incr_boundvars 1 func_t, x),
                  discriminate_value hol_ctxt x (Bound 0)))
      |> AList.group (op aconv)
      |> map (apsnd (List.foldl s_disj \<^const>\<open>False\<close>))
      |> sort (int_ord o apply2 (size_of_term o snd))
      |> rev
  in
    if res_T = bool_T then
      if forall (member (op =) [\<^const>\<open>False\<close>, \<^const>\<open>True\<close>] o fst) cases then
        case cases of
          [(body_t, _)] => body_t
        | [_, (\<^const>\<open>True\<close>, head_t2)] => head_t2
        | [_, (\<^const>\<open>False\<close>, head_t2)] => \<^const>\<open>Not\<close> $ head_t2
        | _ => raise BAD ("Nitpick_HOL.optimized_case_def", "impossible cases")
      else
        \<^const>\<open>True\<close> |> fold_rev (add_constr_case res_T) cases
    else
      fst (hd cases) |> fold_rev (add_constr_case res_T) (tl cases)
  end
  |> absdummy dataT

fun optimized_record_get (hol_ctxt as {thy, ctxt, ...}) s rec_T res_T t =
  let val constr_x = hd (data_type_constrs hol_ctxt rec_T) in
    case no_of_record_field thy s rec_T of
      ~1 => (case rec_T of
               Type (_, Ts as _ :: _) =>
               let
                 val rec_T' = List.last Ts
                 val j = num_record_fields thy rec_T - 1
               in
                 select_nth_constr_arg ctxt constr_x t j res_T
                 |> optimized_record_get hol_ctxt s rec_T' res_T
               end
             | _ => raise TYPE ("Nitpick_HOL.optimized_record_get", [rec_T], []))
    | j => select_nth_constr_arg ctxt constr_x t j res_T
  end

fun optimized_record_update (hol_ctxt as {thy, ctxt, ...}) s rec_T fun_t rec_t =
  let
    val constr_x as (_, constr_T) = hd (data_type_constrs hol_ctxt rec_T)
    val Ts = binder_types constr_T
    val n = length Ts
    val special_j = no_of_record_field thy s rec_T
    val ts =
      map2 (fn j => fn T =>
               let val t = select_nth_constr_arg ctxt constr_x rec_t j T in
                 if j = special_j then
                   s_betapply [] (fun_t, t)
                 else if j = n - 1 andalso special_j = ~1 then
                   optimized_record_update hol_ctxt s
                       (rec_T |> dest_Type |> snd |> List.last) fun_t t
                 else
                   t
               end) (index_seq 0 n) Ts
  in list_comb (Const constr_x, ts) end

(* Prevents divergence in case of cyclic or infinite definition dependencies. *)
val unfold_max_depth = 255

(* Inline definitions or define as an equational constant? Booleans tend to
   benefit more from inlining, due to the polarity analysis. (However, if
   "total_consts" is set, the polarity analysis is likely not to be so
   crucial.) *)
val def_inline_threshold_for_booleans = 60
val def_inline_threshold_for_non_booleans = 20

fun unfold_defs_in_term
        (hol_ctxt as {thy, ctxt, whacks, total_consts, case_names,
                      def_tables, ground_thm_table, ersatz_table, ...}) =
  let
    fun do_numeral depth Ts mult T some_t0 t1 t2 =
      (if is_number_type ctxt T then
         let
           val j = mult * HOLogic.dest_numeral t2
         in
           if j = 1 then
             raise SAME ()
           else
             let
               val s = numeral_prefix ^ signed_string_of_int j
             in
               if is_integer_like_type T then
                 Const (s, T)
               else
                 do_term depth Ts (Const (\<^const_name>\<open>of_int\<close>, int_T --> T)
                                   $ Const (s, int_T))
             end
         end
         handle TERM _ => raise SAME ()
       else
         raise SAME ())
      handle SAME () => (case some_t0 of NONE => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2)
         | SOME t0 => s_betapply [] (do_term depth Ts t0, s_betapply [] (do_term depth Ts t1, do_term depth Ts t2)))
    and do_term depth Ts t =
      case t of
        (t0 as Const (\<^const_name>\<open>uminus\<close>, _) $ ((t1 as Const (\<^const_name>\<open>numeral\<close>,
                      Type (\<^type_name>\<open>fun\<close>, [_, ran_T]))) $ t2)) =>
        do_numeral depth Ts ~1 ran_T (SOME t0) t1 t2
      | (t1 as Const (\<^const_name>\<open>numeral\<close>,
                      Type (\<^type_name>\<open>fun\<close>, [_, ran_T]))) $ t2 =>
        do_numeral depth Ts 1 ran_T NONE t1 t2
      | Const (\<^const_name>\<open>refl_on\<close>, T) $ Const (\<^const_name>\<open>top\<close>, _) $ t2 =>
        do_const depth Ts t (\<^const_name>\<open>refl'\<close>, range_type T) [t2]
      | (t0 as Const (\<^const_name>\<open>Sigma\<close>, Type (_, [T1, Type (_, [T2, T3])])))
        $ t1 $ (t2 as Abs (_, _, t2')) =>
        if loose_bvar1 (t2', 0) then
          s_betapplys Ts (do_term depth Ts t0, map (do_term depth Ts) [t1, t2])
        else
          do_term depth Ts
                  (Const (\<^const_name>\<open>prod\<close>, T1 --> range_type T2 --> T3)
                   $ t1 $ incr_boundvars ~1 t2')
      | Const (x as (\<^const_name>\<open>distinct\<close>,
               Type (\<^type_name>\<open>fun\<close>, [Type (\<^type_name>\<open>list\<close>, [T']), _])))
        $ (t1 as _ $ _) =>
        (t1 |> HOLogic.dest_list |> distinctness_formula T'
         handle TERM _ => do_const depth Ts t x [t1])
      | Const (x as (\<^const_name>\<open>If\<close>, _)) $ t1 $ t2 $ t3 =>
        if is_ground_term t1 andalso
           exists (Pattern.matches thy o rpair t1)
                  (Inttab.lookup_list ground_thm_table (hash_term t1)) then
          do_term depth Ts t2
        else
          do_const depth Ts t x [t1, t2, t3]
      | Const (\<^const_name>\<open>Let\<close>, _) $ t1 $ t2 =>
        s_betapply Ts (apply2 (do_term depth Ts) (t2, t1))
      | Const x => do_const depth Ts t x []
      | t1 $ t2 =>
        (case strip_comb t of
           (Const x, ts) => do_const depth Ts t x ts
         | _ => s_betapply [] (do_term depth Ts t1, do_term depth Ts t2))
      | Bound _ => t
      | Abs (s, T, body) => Abs (s, T, do_term depth (T :: Ts) body)
      | _ => if member (term_match thy) whacks t then
               Const (\<^const_name>\<open>unknown\<close>, fastype_of1 (Ts, t))
             else
               t
    and select_nth_constr_arg_with_args _ _ (x as (_, T)) [] n res_T =
        (Abs (Name.uu, body_type T,
              select_nth_constr_arg ctxt x (Bound 0) n res_T), [])
      | select_nth_constr_arg_with_args depth Ts x (t :: ts) n res_T =
        (select_nth_constr_arg ctxt x (do_term depth Ts t) n res_T, ts)
    and quot_rep_of depth Ts abs_T rep_T ts =
      select_nth_constr_arg_with_args depth Ts
          (\<^const_name>\<open>Quot\<close>, rep_T --> abs_T) ts 0 rep_T
    and do_const depth Ts t (x as (s, T)) ts =
      if member (term_match thy) whacks (Const x) then
        Const (\<^const_name>\<open>unknown\<close>, fastype_of1 (Ts, t))
      else case AList.lookup (op =) ersatz_table s of
        SOME s' =>
        do_const (depth + 1) Ts (list_comb (Const (s', T), ts)) (s', T) ts
      | NONE =>
        let
          fun def_inline_threshold () =
            if is_boolean_type (body_type T) andalso
               total_consts <> SOME true then
              def_inline_threshold_for_booleans
            else
              def_inline_threshold_for_non_booleans
          val (const, ts) =
            if is_built_in_const x then
              (Const x, ts)
            else case AList.lookup (op =) case_names s of
              SOME n =>
              if length ts < n then
                (do_term depth Ts (eta_expand Ts t (n - length ts)), [])
              else
                let
                  val (dataT, res_T) = nth_range_type n T
                                       |> pairf domain_type range_type
                in
                  (optimized_case_def hol_ctxt Ts dataT res_T
                                      (map (do_term depth Ts) (take n ts)),
                   drop n ts)
                end
            | _ =>
              if is_constr ctxt x then
                (Const x, ts)
              else if is_stale_constr ctxt x then
                raise NOT_SUPPORTED ("(non-co)constructors of codatatypes \
                                     \(\"" ^ s ^ "\")")
              else if is_quot_abs_fun ctxt x then
                case T of
                  Type (\<^type_name>\<open>fun\<close>, [rep_T, abs_T as Type (abs_s, _)]) =>
                  if is_interpreted_type abs_s then
                    raise NOT_SUPPORTED ("abstraction function on " ^
                                         quote abs_s)
                  else
                    (Abs (Name.uu, rep_T,
                          Const (\<^const_name>\<open>Quot\<close>, rep_T --> abs_T)
                                 $ (Const (quot_normal_name_for_type ctxt abs_T,
                                           rep_T --> rep_T) $ Bound 0)), ts)
              else if is_quot_rep_fun ctxt x then
                case T of
                  Type (\<^type_name>\<open>fun\<close>, [abs_T as Type (abs_s, _), rep_T]) =>
                  if is_interpreted_type abs_s then
                    raise NOT_SUPPORTED ("representation function on " ^
                                         quote abs_s)
                  else
                    quot_rep_of depth Ts abs_T rep_T ts
              else if is_record_get thy x then
                case length ts of
                  0 => (do_term depth Ts (eta_expand Ts t 1), [])
                | _ => (optimized_record_get hol_ctxt s (domain_type T)
                            (range_type T) (do_term depth Ts (hd ts)), tl ts)
              else if is_record_update thy x then
                case length ts of
                  2 => (optimized_record_update hol_ctxt
                            (unsuffix Record.updateN s) (nth_range_type 2 T)
                            (do_term depth Ts (hd ts))
                            (do_term depth Ts (nth ts 1)), [])
                | n => (do_term depth Ts (eta_expand Ts t (2 - n)), [])
              else if is_abs_fun ctxt x andalso
                      is_quot_type ctxt (range_type T) then
                let
                  val abs_T = range_type T
                  val rep_T = elem_type (domain_type T)
                  val eps_fun = Const (\<^const_name>\<open>Eps\<close>,
                                       (rep_T --> bool_T) --> rep_T)
                  val normal_fun =
                    Const (quot_normal_name_for_type ctxt abs_T,
                           rep_T --> rep_T)
                  val abs_fun = Const (\<^const_name>\<open>Quot\<close>, rep_T --> abs_T)
                  val pred =
                    Abs (Name.uu, rep_T,
                         Const (\<^const_name>\<open>Set.member\<close>,
                                rep_T --> domain_type T --> bool_T)
                         $ Bound 0 $ Bound 1)
                in
                  (Abs (Name.uu, HOLogic.mk_setT rep_T,
                        abs_fun $ (normal_fun $ (eps_fun $ pred)))
                   |> do_term (depth + 1) Ts, ts)
                end
              else if is_rep_fun ctxt x then
                let val x' = mate_of_rep_fun ctxt x in
                  if is_constr ctxt x' then
                    select_nth_constr_arg_with_args depth Ts x' ts 0
                                                    (range_type T)
                  else if is_quot_type ctxt (domain_type T) then
                    let
                      val abs_T = domain_type T
                      val rep_T = elem_type (range_type T)
                      val (rep_fun, _) = quot_rep_of depth Ts abs_T rep_T []
                      val (equiv_rel, _) =
                        equiv_relation_for_quot_type ctxt abs_T
                    in
                      (Abs (Name.uu, abs_T,
                            HOLogic.Collect_const rep_T
                            $ (equiv_rel $ (rep_fun $ Bound 0))),
                       ts)
                    end
                  else
                    (Const x, ts)
                end
              else if is_equational_fun hol_ctxt x orelse
                      is_choice_spec_fun hol_ctxt x then
                (Const x, ts)
              else case def_of_const_ext thy def_tables x of
                SOME (unfold, def) =>
                if depth > unfold_max_depth then
                  raise TOO_LARGE ("Nitpick_HOL.unfold_defs_in_term",
                                   "too many nested definitions (" ^
                                   string_of_int depth ^ ") while expanding " ^
                                   quote s)
                else if s = \<^const_name>\<open>wfrec'\<close> then
                  (do_term (depth + 1) Ts (s_betapplys Ts (def, ts)), [])
                else if not unfold andalso
                     size_of_term def > def_inline_threshold () then
                  (Const x, ts)
                else
                  (do_term (depth + 1) Ts def, ts)
              | NONE => (Const x, ts)
        in
          s_betapplys Ts (const, map (do_term depth Ts) ts)
          |> s_beta_norm Ts
        end
  in do_term 0 [] end

(** Axiom extraction/generation **)

fun extensional_equal j T t1 t2 =
  if is_fun_type T then
    let
      val dom_T = pseudo_domain_type T
      val ran_T = pseudo_range_type T
      val var_t = Var (("x", j), dom_T)
    in
      extensional_equal (j + 1) ran_T (betapply (t1, var_t))
                        (betapply (t2, var_t))
    end
  else
    Const (\<^const_name>\<open>HOL.eq\<close>, T --> T --> bool_T) $ t1 $ t2

(* FIXME: needed? *)
fun equationalize_term ctxt tag t =
  let
    val j = maxidx_of_term t + 1
    val (prems, concl) = Logic.strip_horn t
  in
    Logic.list_implies (prems,
        case concl of
          \<^const>\<open>Trueprop\<close> $ (Const (\<^const_name>\<open>HOL.eq\<close>, Type (_, [T, _]))
                               $ t1 $ t2) =>
          \<^const>\<open>Trueprop\<close> $ extensional_equal j T t1 t2
        | \<^const>\<open>Trueprop\<close> $ t' =>
          \<^const>\<open>Trueprop\<close> $ HOLogic.mk_eq (t', \<^const>\<open>True\<close>)
        | Const (\<^const_name>\<open>Pure.eq\<close>, Type (_, [T, _])) $ t1 $ t2 =>
          \<^const>\<open>Trueprop\<close> $ extensional_equal j T t1 t2
        | _ => (warning ("Ignoring " ^ quote tag ^ " for non-equation " ^
                         quote (Syntax.string_of_term ctxt t));
                raise SAME ()))
    |> SOME
  end
  handle SAME () => NONE

fun pair_for_prop t =
  case term_under_def t of
    Const (s, _) => (s, t)
  | t' => raise TERM ("Nitpick_HOL.pair_for_prop", [t, t'])

fun def_table_for ts subst =
  ts |> map (pair_for_prop o subst_atomic subst)
       |> AList.group (op =) |> Symtab.make

fun const_def_tables ctxt subst ts =
  (def_table_for
    (map Thm.prop_of (rev (Named_Theorems.get ctxt \<^named_theorems>\<open>nitpick_unfold\<close>))) subst,
   fold (fn (s, t) => Symtab.map_default (s, []) (cons t))
        (map pair_for_prop ts) Symtab.empty)

fun paired_with_consts t = map (rpair t) (Term.add_const_names t [])

fun const_nondef_table ts =
  fold (append o paired_with_consts) ts [] |> AList.group (op =) |> Symtab.make

fun const_simp_table ctxt =
  def_table_for (map_filter (equationalize_term ctxt "nitpick_simp" o Thm.prop_of)
    (rev (Named_Theorems.get ctxt \<^named_theorems>\<open>nitpick_simp\<close>)))

fun const_psimp_table ctxt =
  def_table_for (map_filter (equationalize_term ctxt "nitpick_psimp" o Thm.prop_of)
    (rev (Named_Theorems.get ctxt \<^named_theorems>\<open>nitpick_psimp\<close>)))

fun const_choice_spec_table ctxt subst =
  map (subst_atomic subst o Thm.prop_of)
    (rev (Named_Theorems.get ctxt \<^named_theorems>\<open>nitpick_choice_spec\<close>))
  |> const_nondef_table

fun inductive_intro_table ctxt subst def_tables =
  let val thy = Proof_Context.theory_of ctxt in
    def_table_for
        (maps (map (unfold_mutually_inductive_preds thy def_tables o Thm.prop_of)
               o snd o snd)
         (filter ((Spec_Rules.is_inductive orf Spec_Rules.is_co_inductive) o #1)
           (Spec_Rules.get ctxt))) subst
  end

fun ground_theorem_table thy =
  fold ((fn \<^const>\<open>Trueprop\<close> $ t1 =>
            is_ground_term t1 ? Inttab.map_default (hash_term t1, []) (cons t1)
          | _ => I) o Thm.prop_of o snd) (Global_Theory.all_thms_of thy true) Inttab.empty

fun ersatz_table ctxt =
 #ersatz_table (Data.get (Context.Proof ctxt))
 |> fold (append o snd) (#frac_types (Data.get (Context.Proof ctxt)))

fun add_simps simp_table s eqs =
  Unsynchronized.change simp_table
      (Symtab.update (s, eqs @ these (Symtab.lookup (!simp_table) s)))

fun inverse_axioms_for_rep_fun ctxt (x as (_, T)) =
  let
    val thy = Proof_Context.theory_of ctxt
    val abs_T = domain_type T
  in
    typedef_info ctxt (fst (dest_Type abs_T)) |> the
    |> pairf #Abs_inverse #Rep_inverse
    |> apply2 (specialize_type thy x o Thm.prop_of o the)
    ||> single |> op ::
  end

fun optimized_typedef_axioms ctxt (abs_z as (abs_s, _)) =
  let
    val thy = Proof_Context.theory_of ctxt
    val abs_T = Type abs_z
  in
    if is_univ_typedef ctxt abs_T then
      []
    else case typedef_info ctxt abs_s of
      SOME {abs_type, rep_type, Rep_name, prop_of_Rep, ...} =>
      let
        val rep_T = varify_and_instantiate_type ctxt abs_type abs_T rep_type
        val rep_t = Const (Rep_name, abs_T --> rep_T)
        val set_t =
          prop_of_Rep |> HOLogic.dest_Trueprop
                      |> specialize_type thy (dest_Const rep_t)
                      |> HOLogic.dest_mem |> snd
      in
        [HOLogic.all_const abs_T
             $ Abs (Name.uu, abs_T, HOLogic.mk_mem (rep_t $ Bound 0, set_t))
         |> HOLogic.mk_Trueprop]
      end
    | NONE => []
  end

fun optimized_quot_type_axioms ctxt abs_z =
  let
    val abs_T = Type abs_z
    val rep_T = rep_type_for_quot_type ctxt abs_T
    val (equiv_rel, partial) = equiv_relation_for_quot_type ctxt abs_T
    val a_var = Var (("a", 0), abs_T)
    val x_var = Var (("x", 0), rep_T)
    val y_var = Var (("y", 0), rep_T)
    val x = (\<^const_name>\<open>Quot\<close>, rep_T --> abs_T)
    val sel_a_t = select_nth_constr_arg ctxt x a_var 0 rep_T
    val normal_fun =
      Const (quot_normal_name_for_type ctxt abs_T, rep_T --> rep_T)
    val normal_x = normal_fun $ x_var
    val normal_y = normal_fun $ y_var
    val is_unknown_t = Const (\<^const_name>\<open>is_unknown\<close>, rep_T --> bool_T)
  in
    [Logic.mk_equals (normal_fun $ sel_a_t, sel_a_t),
     Logic.list_implies
         ([\<^const>\<open>Not\<close> $ (is_unknown_t $ normal_x),
           \<^const>\<open>Not\<close> $ (is_unknown_t $ normal_y),
           equiv_rel $ x_var $ y_var] |> map HOLogic.mk_Trueprop,
           Logic.mk_equals (normal_x, normal_y)),
     Logic.list_implies
         ([HOLogic.mk_Trueprop (\<^const>\<open>Not\<close> $ (is_unknown_t $ normal_x)),
           HOLogic.mk_Trueprop (\<^const>\<open>Not\<close> $ HOLogic.mk_eq (normal_x, x_var))],
          HOLogic.mk_Trueprop (equiv_rel $ x_var $ normal_x))]
    |> partial ? cons (HOLogic.mk_Trueprop (equiv_rel $ sel_a_t $ sel_a_t))
  end

fun codatatype_bisim_axioms (hol_ctxt as {ctxt, ...}) T =
  let
    val xs = data_type_constrs hol_ctxt T
    val pred_T = T --> bool_T
    val iter_T = \<^typ>\<open>bisim_iterator\<close>
    val bisim_max = \<^const>\<open>bisim_iterator_max\<close>
    val n_var = Var (("n", 0), iter_T)
    val n_var_minus_1 =
      Const (\<^const_name>\<open>safe_The\<close>, (iter_T --> bool_T) --> iter_T)
      $ Abs ("m", iter_T, HOLogic.eq_const iter_T $ (suc_const iter_T $ Bound 0) $ n_var)
    val x_var = Var (("x", 0), T)
    val y_var = Var (("y", 0), T)
    fun bisim_const T = Const (\<^const_name>\<open>bisim\<close>, [iter_T, T, T] ---> bool_T)
    fun nth_sub_bisim x n nth_T =
      (if is_codatatype ctxt nth_T then bisim_const nth_T $ n_var_minus_1
       else HOLogic.eq_const nth_T)
      $ select_nth_constr_arg ctxt x x_var n nth_T
      $ select_nth_constr_arg ctxt x y_var n nth_T
    fun case_func (x as (_, T)) =
      let
        val arg_Ts = binder_types T
        val core_t =
          discriminate_value hol_ctxt x y_var ::
          map2 (nth_sub_bisim x) (index_seq 0 (length arg_Ts)) arg_Ts
          |> foldr1 s_conj
      in fold_rev absdummy arg_Ts core_t end
  in
    [HOLogic.mk_imp
       (HOLogic.mk_disj (HOLogic.eq_const iter_T $ n_var $ zero_const iter_T,
            s_betapply [] (optimized_case_def hol_ctxt [] T bool_T (map case_func xs), x_var)),
        bisim_const T $ n_var $ x_var $ y_var),
     HOLogic.eq_const pred_T $ (bisim_const T $ bisim_max $ x_var)
     $ Abs (Name.uu, T, HOLogic.mk_eq (x_var, Bound 0))]
    |> map HOLogic.mk_Trueprop
  end

exception NO_TRIPLE of unit

fun triple_for_intro_rule ctxt x t =
  let
    val prems = Logic.strip_imp_prems t |> map (Object_Logic.atomize_term ctxt)
    val concl = Logic.strip_imp_concl t |> Object_Logic.atomize_term ctxt
    val (main, side) = List.partition (exists_Const (curry (op =) x)) prems
    val is_good_head = curry (op =) (Const x) o head_of
  in
    if forall is_good_head main then (side, main, concl) else raise NO_TRIPLE ()
  end

val tuple_for_args = HOLogic.mk_tuple o snd o strip_comb

fun wf_constraint_for rel side concl main =
  let
    val core = HOLogic.mk_mem (HOLogic.mk_prod
                               (apply2 tuple_for_args (main, concl)), Var rel)
    val t = List.foldl HOLogic.mk_imp core side
    val vars = filter_out (curry (op =) rel) (Term.add_vars t [])
  in
    Library.foldl (fn (t', ((x, j), T)) =>
                      HOLogic.all_const T
                      $ Abs (x, T, abstract_over (Var ((x, j), T), t')))
                  (t, vars)
  end

fun wf_constraint_for_triple rel (side, main, concl) =
  map (wf_constraint_for rel side concl) main |> foldr1 s_conj

fun terminates_by ctxt timeout goal tac =
  can (SINGLE (Classical.safe_tac ctxt) #> the
       #> SINGLE (DETERM_TIMEOUT timeout (tac ctxt (auto_tac ctxt)))
       #> the #> Goal.finish ctxt) goal

val max_cached_wfs = 50
val cached_timeout = Synchronized.var "Nitpick_HOL.cached_timeout" Time.zeroTime
val cached_wf_props =
  Synchronized.var "Nitpick_HOL.cached_wf_props" ([] : (term * bool) list)

val termination_tacs = [Lexicographic_Order.lex_order_tac true,
                        ScnpReconstruct.sizechange_tac]

fun uncached_is_well_founded_inductive_pred
        ({thy, ctxt, debug, tac_timeout, intro_table, ...} : hol_context)
        (x as (_, T)) =
  case def_props_for_const thy intro_table x of
    [] => raise TERM ("Nitpick_HOL.uncached_is_well_founded_inductive",
                      [Const x])
  | intro_ts =>
    (case map (triple_for_intro_rule ctxt x) intro_ts
          |> filter_out (null o #2) of
       [] => true
     | triples =>
       let
         val binders_T = HOLogic.mk_tupleT (binder_types T)
         val rel_T = HOLogic.mk_setT (HOLogic.mk_prodT (binders_T, binders_T))
         val j = fold Integer.max (map maxidx_of_term intro_ts) 0 + 1
         val rel = (("R", j), rel_T)
         val prop = Const (\<^const_name>\<open>wf\<close>, rel_T --> bool_T) $ Var rel ::
                    map (wf_constraint_for_triple rel) triples
                    |> foldr1 s_conj |> HOLogic.mk_Trueprop
         val _ = if debug then
                   writeln ("Wellfoundedness goal: " ^ Syntax.string_of_term ctxt prop)
                 else
                   ()
       in
         if tac_timeout = Synchronized.value cached_timeout andalso
            length (Synchronized.value cached_wf_props) < max_cached_wfs then
           ()
         else
           (Synchronized.change cached_wf_props (K []);
            Synchronized.change cached_timeout (K tac_timeout));
         case AList.lookup (op =) (Synchronized.value cached_wf_props) prop of
           SOME wf => wf
         | NONE =>
           let
             val goal = prop |> Thm.cterm_of ctxt |> Goal.init
             val wf = exists (terminates_by ctxt tac_timeout goal)
                             termination_tacs
           in Synchronized.change cached_wf_props (cons (prop, wf)); wf end
       end)
    handle List.Empty => false | NO_TRIPLE () => false

(* The type constraint below is a workaround for a Poly/ML crash. *)

fun is_well_founded_inductive_pred
        (hol_ctxt as {thy, wfs, def_tables, wf_cache, ...} : hol_context)
        (x as (s, _)) =
  case triple_lookup (const_match thy) wfs x of
    SOME (SOME b) => b
  | _ => s = \<^const_name>\<open>Nats\<close> orelse s = \<^const_name>\<open>fold_graph'\<close> orelse
         case AList.lookup (op =) (!wf_cache) x of
           SOME (_, wf) => wf
         | NONE =>
           let
             val gfp = (fixpoint_kind_of_const thy def_tables x = Gfp)
             val wf = uncached_is_well_founded_inductive_pred hol_ctxt x
           in
             Unsynchronized.change wf_cache (cons (x, (gfp, wf))); wf
           end

fun ap_curry [_] _ t = t
  | ap_curry arg_Ts tuple_T t =
    let val n = length arg_Ts in
      fold_rev (Term.abs o pair "c") arg_Ts
                (incr_boundvars n t $ mk_flat_tuple tuple_T (map Bound (n - 1 downto 0)))
    end

fun num_occs_of_bound_in_term j (t1 $ t2) =
    op + (apply2 (num_occs_of_bound_in_term j) (t1, t2))
  | num_occs_of_bound_in_term j (Abs (_, _, t')) =
    num_occs_of_bound_in_term (j + 1) t'
  | num_occs_of_bound_in_term j (Bound j') = if j' = j then 1 else 0
  | num_occs_of_bound_in_term _ _ = 0

val is_linear_inductive_pred_def =
  let
    fun do_disjunct j (Const (\<^const_name>\<open>Ex\<close>, _) $ Abs (_, _, t2)) =
        do_disjunct (j + 1) t2
      | do_disjunct j t =
        case num_occs_of_bound_in_term j t of
          0 => true
        | 1 => exists (curry (op =) (Bound j) o head_of) (conjuncts_of t)
        | _ => false
    fun do_lfp_def (Const (\<^const_name>\<open>lfp\<close>, _) $ t2) =
        let val (xs, body) = strip_abs t2 in
          case length xs of
            1 => false
          | n => forall (do_disjunct (n - 1)) (disjuncts_of body)
        end
      | do_lfp_def _ = false
  in do_lfp_def o strip_abs_body end

fun n_ptuple_paths 0 = []
  | n_ptuple_paths 1 = []
  | n_ptuple_paths n = [] :: map (cons 2) (n_ptuple_paths (n - 1))
val ap_n_split = HOLogic.mk_ptupleabs o n_ptuple_paths

val linear_pred_base_and_step_rhss =
  let
    fun aux (Const (\<^const_name>\<open>lfp\<close>, _) $ t2) =
        let
          val (xs, body) = strip_abs t2
          val arg_Ts = map snd (tl xs)
          val tuple_T = HOLogic.mk_tupleT arg_Ts
          val j = length arg_Ts
          fun repair_rec j (Const (\<^const_name>\<open>Ex\<close>, T1) $ Abs (s2, T2, t2')) =
              Const (\<^const_name>\<open>Ex\<close>, T1)
              $ Abs (s2, T2, repair_rec (j + 1) t2')
            | repair_rec j (\<^const>\<open>HOL.conj\<close> $ t1 $ t2) =
              \<^const>\<open>HOL.conj\<close> $ repair_rec j t1 $ repair_rec j t2
            | repair_rec j t =
              let val (head, args) = strip_comb t in
                if head = Bound j then
                  HOLogic.eq_const tuple_T $ Bound j
                  $ mk_flat_tuple tuple_T args
                else
                  t
              end
          val (nonrecs, recs) =
            List.partition (curry (op =) 0 o num_occs_of_bound_in_term j)
                           (disjuncts_of body)
          val base_body = nonrecs |> List.foldl s_disj \<^const>\<open>False\<close>
          val step_body = recs |> map (repair_rec j)
                               |> List.foldl s_disj \<^const>\<open>False\<close>
        in
          (fold_rev Term.abs (tl xs) (incr_bv (~1, j, base_body))
           |> ap_n_split (length arg_Ts) tuple_T bool_T,
           Abs ("y", tuple_T, fold_rev Term.abs (tl xs) step_body
                              |> ap_n_split (length arg_Ts) tuple_T bool_T))
        end
      | aux t =
        raise TERM ("Nitpick_HOL.linear_pred_base_and_step_rhss.aux", [t])
  in aux end

fun predicatify T t =
  let val set_T = HOLogic.mk_setT T in
    Abs (Name.uu, T,
         Const (\<^const_name>\<open>Set.member\<close>, T --> set_T --> bool_T)
         $ Bound 0 $ incr_boundvars 1 t)
  end

fun starred_linear_pred_const (hol_ctxt as {simp_table, ...}) (s, T) def =
  let
    val j = maxidx_of_term def + 1
    val (outer, fp_app) = strip_abs def
    val outer_bounds = map Bound (length outer - 1 downto 0)
    val outer_vars = map (fn (s, T) => Var ((s, j), T)) outer
    val fp_app = subst_bounds (rev outer_vars, fp_app)
    val (outer_Ts, rest_T) = strip_n_binders (length outer) T
    val tuple_arg_Ts = strip_type rest_T |> fst
    val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
    val prod_T = HOLogic.mk_prodT (tuple_T, tuple_T)
    val set_T = HOLogic.mk_setT tuple_T
    val rel_T = HOLogic.mk_setT prod_T
    val pred_T = tuple_T --> bool_T
    val curried_T = tuple_T --> pred_T
    val uncurried_T = prod_T --> bool_T
    val (base_rhs, step_rhs) = linear_pred_base_and_step_rhss fp_app
    val base_x as (base_s, _) = (base_prefix ^ s, outer_Ts ---> pred_T)
    val base_eq = HOLogic.mk_eq (list_comb (Const base_x, outer_vars), base_rhs)
                  |> HOLogic.mk_Trueprop
    val _ = add_simps simp_table base_s [base_eq]
    val step_x as (step_s, _) = (step_prefix ^ s, outer_Ts ---> curried_T)
    val step_eq = HOLogic.mk_eq (list_comb (Const step_x, outer_vars), step_rhs)
                  |> HOLogic.mk_Trueprop
    val _ = add_simps simp_table step_s [step_eq]
    val image_const = Const (\<^const_name>\<open>Image\<close>, rel_T --> set_T --> set_T)
    val rtrancl_const = Const (\<^const_name>\<open>rtrancl\<close>, rel_T --> rel_T)
    val base_set =
      HOLogic.Collect_const tuple_T $ list_comb (Const base_x, outer_bounds)
    val step_set =
      HOLogic.Collect_const prod_T
      $ (Const (\<^const_name>\<open>case_prod\<close>, curried_T --> uncurried_T)
                $ list_comb (Const step_x, outer_bounds))
    val image_set =
      image_const $ (rtrancl_const $ step_set) $ base_set
      |> predicatify tuple_T
  in
    fold_rev Term.abs outer (image_set |> ap_curry tuple_arg_Ts tuple_T)
    |> unfold_defs_in_term hol_ctxt
  end

fun is_good_starred_linear_pred_type (Type (\<^type_name>\<open>fun\<close>, Ts)) =
    forall (not o (is_fun_or_set_type orf is_pair_type)) Ts
  | is_good_starred_linear_pred_type _ = false

fun unrolled_inductive_pred_const (hol_ctxt as {thy, star_linear_preds,
                                                def_tables, simp_table, ...})
                                  gfp (x as (s, T)) =
  let
    val iter_T = iterator_type_for_const gfp x
    val x' as (s', _) = (unrolled_prefix ^ s, iter_T --> T)
    val unrolled_const = Const x' $ zero_const iter_T
    val def = the (def_of_const thy def_tables x)
  in
    if is_equational_fun hol_ctxt x' then
      unrolled_const (* already done *)
    else if not gfp andalso star_linear_preds andalso
         is_linear_inductive_pred_def def andalso
         is_good_starred_linear_pred_type T then
      starred_linear_pred_const hol_ctxt x def
    else
      let
        val j = maxidx_of_term def + 1
        val (outer, fp_app) = strip_abs def
        val outer_bounds = map Bound (length outer - 1 downto 0)
        val cur = Var ((iter_var_prefix, j + 1), iter_T)
        val next = suc_const iter_T $ cur
        val rhs =
          case fp_app of
            Const _ $ t =>
            s_betapply [] (t, list_comb (Const x', next :: outer_bounds))
          | _ => raise TERM ("Nitpick_HOL.unrolled_inductive_pred_const",
                             [fp_app])
        val (inner, naked_rhs) = strip_abs rhs
        val all = outer @ inner
        val bounds = map Bound (length all - 1 downto 0)
        val vars = map (fn (s, T) => Var ((s, j), T)) all
        val eq = HOLogic.mk_eq (list_comb (Const x', cur :: bounds), naked_rhs)
                 |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
        val _ = add_simps simp_table s' [eq]
      in unrolled_const end
  end

fun raw_inductive_pred_axiom ({thy, def_tables, ...} : hol_context) x =
  let
    val def = the (def_of_const thy def_tables x)
    val (outer, fp_app) = strip_abs def
    val outer_bounds = map Bound (length outer - 1 downto 0)
    val rhs =
      case fp_app of
        Const _ $ t => s_betapply [] (t, list_comb (Const x, outer_bounds))
      | _ => raise TERM ("Nitpick_HOL.raw_inductive_pred_axiom", [fp_app])
    val (inner, naked_rhs) = strip_abs rhs
    val all = outer @ inner
    val bounds = map Bound (length all - 1 downto 0)
    val j = maxidx_of_term def + 1
    val vars = map (fn (s, T) => Var ((s, j), T)) all
  in
    HOLogic.mk_eq (list_comb (Const x, bounds), naked_rhs)
    |> HOLogic.mk_Trueprop |> curry subst_bounds (rev vars)
  end

fun inductive_pred_axiom hol_ctxt (x as (s, T)) =
  if String.isPrefix ubfp_prefix s orelse String.isPrefix lbfp_prefix s then
    let val x' = (strip_first_name_sep s |> snd, T) in
      raw_inductive_pred_axiom hol_ctxt x' |> subst_atomic [(Const x', Const x)]
    end
  else
    raw_inductive_pred_axiom hol_ctxt x

fun equational_fun_axioms (hol_ctxt as {thy, ctxt, def_tables, simp_table,
                                        psimp_table, ...}) x =
  case def_props_for_const thy (!simp_table) x of
    [] => (case def_props_for_const thy psimp_table x of
             [] => (if is_inductive_pred hol_ctxt x then
                      [inductive_pred_axiom hol_ctxt x]
                    else case def_of_const thy def_tables x of
                      SOME def =>
                      \<^const>\<open>Trueprop\<close> $ HOLogic.mk_eq (Const x, def)
                      |> equationalize_term ctxt "" |> the |> single
                    | NONE => [])
           | psimps => psimps)
  | simps => simps

fun is_equational_fun_surely_complete hol_ctxt x =
  case equational_fun_axioms hol_ctxt x of
    [\<^const>\<open>Trueprop\<close> $ (Const (\<^const_name>\<open>HOL.eq\<close>, _) $ t1 $ _)] =>
    strip_comb t1 |> snd |> forall is_Var
  | _ => false

(** Type preprocessing **)

fun merged_type_var_table_for_terms thy ts =
  let
    fun add (s, S) table =
      table
      |> (case AList.lookup (Sign.subsort thy o swap) table S of
            SOME _ => I
          | NONE =>
            filter_out (fn (S', _) => Sign.subsort thy (S, S'))
            #> cons (S, s))
    val tfrees = [] |> fold Term.add_tfrees ts
                    |> sort (string_ord o apply2 fst)
  in [] |> fold add tfrees |> rev end

fun merge_type_vars_in_term thy merge_type_vars table =
  merge_type_vars
  ? map_types (map_atyps
        (fn TFree (_, S) =>
            TFree (table |> find_first (fn (S', _) => Sign.subsort thy (S', S))
                         |> the |> swap)
          | T => T))

fun add_ground_types hol_ctxt binarize =
  let
    fun aux T accum =
      case T of
        Type (\<^type_name>\<open>fun\<close>, Ts) => fold aux Ts accum
      | Type (\<^type_name>\<open>prod\<close>, Ts) => fold aux Ts accum
      | Type (\<^type_name>\<open>set\<close>, Ts) => fold aux Ts accum
      | Type (\<^type_name>\<open>itself\<close>, [T1]) => aux T1 accum
      | Type (_, Ts) =>
        if member (op =) (\<^typ>\<open>prop\<close> :: \<^typ>\<open>bool\<close> :: accum) T then
          accum
        else
          T :: accum
          |> fold aux (case binarized_and_boxed_data_type_constrs hol_ctxt
                                                                  binarize T of
                         [] => Ts
                       | xs => map snd xs)
      | _ => insert (op =) T accum
  in aux end

fun ground_types_in_type hol_ctxt binarize T =
  add_ground_types hol_ctxt binarize T []

fun ground_types_in_terms hol_ctxt binarize ts =
  fold (fold_types (add_ground_types hol_ctxt binarize)) ts []

end;