(*  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 styp = Nitpick_Util.styp

  type const_table = term list Symtab.table

  type special_fun = (styp * int list * term list) * styp

  type unrolled = styp * styp

  type wf_cache = (styp * (bool * bool)) list

  type extended_context = {

    thy: theory,

    ctxt: Proof.context,

    max_bisim_depth: int,

    boxes: (typ option * bool option) list,

    stds: (typ option * bool) list,

    wfs: (styp option * bool option) list,

    user_axioms: bool option,

    debug: bool,

    binary_ints: bool option,

    destroy_constrs: bool,

    specialize: bool,

    skolemize: bool,

    star_linear_preds: bool,

    uncurry: bool,

    fast_descrs: bool,

    tac_timeout: Time.time option,

    evals: term list,

    case_names: (string * int) list,

    def_table: const_table,

    nondef_table: const_table,

    user_nondefs: term list,

    simp_table: const_table Unsynchronized.ref,

    psimp_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 * styp list) list Unsynchronized.ref}

  val name_sep : string

  val numeral_prefix : string

  val skolem_prefix : string

  val eval_prefix : string

  val original_name : string -> string

  val s_conj : term * term -> term

  val unbit_and_unbox_type : typ -> typ

  val string_for_type : Proof.context -> typ -> string

  val prefix_name : string -> string -> string

  val shortest_name : string -> string

  val short_name : string -> string

  val shorten_names_in_term : term -> term

  val type_match : theory -> typ * typ -> bool

  val const_match : theory -> styp * styp -> bool

  val term_match : theory -> term * term -> bool

  val is_TFree : typ -> bool

  val is_higher_order_type : typ -> bool

  val is_fun_type : typ -> bool

  val is_set_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_boolean_type : typ -> bool

  val is_integer_type : typ -> bool

  val is_bit_type : typ -> bool

  val is_word_type : typ -> bool

  val is_record_type : typ -> bool

  val is_number_type : theory -> typ -> bool

  val const_for_iterator_type : typ -> styp

  val nth_range_type : int -> typ -> typ

  val num_factors_in_type : typ -> int

  val num_binder_types : 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 instantiate_type : theory -> typ -> typ -> typ -> typ

  val is_real_datatype : theory -> string -> bool

  val is_quot_type : theory -> typ -> bool

  val is_codatatype : theory -> typ -> bool

  val is_pure_typedef : theory -> typ -> bool

  val is_univ_typedef : theory -> typ -> bool

  val is_datatype : theory -> typ -> bool

  val is_record_constr : styp -> bool

  val is_record_get : theory -> styp -> bool

  val is_record_update : theory -> styp -> bool

  val is_abs_fun : theory -> styp -> bool

  val is_rep_fun : theory -> styp -> bool

  val is_quot_abs_fun : Proof.context -> styp -> bool

  val is_quot_rep_fun : Proof.context -> styp -> bool

  val is_constr : theory -> styp -> bool

  val is_stale_constr : theory -> styp -> bool

  val is_sel : string -> bool

  val is_sel_like_and_no_discr : string -> bool

  val discr_for_constr : styp -> styp

  val num_sels_for_constr_type : typ -> int

  val nth_sel_name_for_constr_name : string -> int -> string

  val nth_sel_for_constr : styp -> int -> styp

  val boxed_nth_sel_for_constr : extended_context -> styp -> int -> styp

  val sel_no_from_name : string -> int

  val eta_expand : typ list -> term -> int -> term

  val extensionalize : term -> term

  val distinctness_formula : typ -> term list -> term

  val register_frac_type : string -> (string * string) list -> theory -> theory

  val unregister_frac_type : string -> theory -> theory

  val register_codatatype : typ -> string -> styp list -> theory -> theory

  val unregister_codatatype : typ -> theory -> theory

  val datatype_constrs : extended_context -> typ -> styp list

  val boxed_datatype_constrs : extended_context -> typ -> styp list

  val num_datatype_constrs : extended_context -> typ -> int

  val constr_name_for_sel_like : string -> string

  val boxed_constr_for_sel : extended_context -> styp -> styp

  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 :

    extended_context -> int -> int -> (typ * int) list -> typ -> int

  val is_finite_type : extended_context -> typ -> bool

  val all_axioms_of : theory -> term list * term list * term list

  val arity_of_built_in_const : bool -> styp -> int option

  val is_built_in_const : bool -> styp -> bool

  val case_const_names : theory -> (string * int) list

  val const_def_table : Proof.context -> term list -> const_table

  val const_nondef_table : term list -> const_table

  val const_simp_table : Proof.context -> const_table

  val const_psimp_table : Proof.context -> const_table

  val inductive_intro_table : Proof.context -> const_table -> const_table

  val ground_theorem_table : theory -> term list Inttab.table

  val ersatz_table : theory -> (string * string) list

  val def_of_const : theory -> const_table -> styp -> term option

  val is_inductive_pred : extended_context -> styp -> bool

  val is_constr_pattern_lhs : theory -> term -> bool

  val is_constr_pattern_formula : theory -> term -> bool

  val merge_type_vars_in_terms : term list -> term list

  val ground_types_in_type : extended_context -> typ -> typ list

  val ground_types_in_terms : extended_context -> term list -> typ list

  val format_type : int list -> int list -> typ -> typ

  val format_term_type :

    theory -> const_table -> (term option * int list) list -> term -> typ

  val user_friendly_const :

   extended_context -> string * string -> (term option * int list) list

   -> styp -> term * typ

  val assign_operator_for_const : styp -> string

  val preprocess_term :

    extended_context -> term -> ((term list * term list) * (bool * bool)) * term

end;

structure Nitpick_HOL : NITPICK_HOL =

struct

open Nitpick_Util

type const_table = term list Symtab.table

type special_fun = (styp * int list * term list) * styp

type unrolled = styp * styp

type wf_cache = (styp * (bool * bool)) list

type extended_context = {

  thy: theory,

  ctxt: Proof.context,

  max_bisim_depth: int,

  boxes: (typ option * bool option) list,

  stds: (typ option * bool) list,

  wfs: (styp option * bool option) list,

  user_axioms: bool option,

  debug: bool,

  binary_ints: bool option,

  destroy_constrs: bool,

  specialize: bool,

  skolemize: bool,

  star_linear_preds: bool,

  uncurry: bool,

  fast_descrs: bool,

  tac_timeout: Time.time option,

  evals: term list,

  case_names: (string * int) list,

  def_table: const_table,

  nondef_table: const_table,

  user_nondefs: term list,

  simp_table: const_table Unsynchronized.ref,

  psimp_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 * styp list) list Unsynchronized.ref}

structure Data = Theory_Data(

  type T = {frac_types: (string * (string * string) list) list,

            codatatypes: (string * (string * styp list)) list}

  val empty = {frac_types = [], codatatypes = []}

  val extend = I

  fun merge ({frac_types = fs1, codatatypes = cs1},

               {frac_types = fs2, codatatypes = cs2}) : T =

    {frac_types = AList.merge (op =) (K true) (fs1, fs2),

     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 set_prefix = nitpick_prefix ^ "set" ^ name_sep

val lfp_iterator_prefix = nitpick_prefix ^ "lfpit" ^ name_sep

val gfp_iterator_prefix = nitpick_prefix ^ "gfpit" ^ name_sep

val nwf_prefix = nitpick_prefix ^ "nwf" ^ 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 skolem_prefix = nitpick_prefix ^ "sk"

val special_prefix = nitpick_prefix ^ "sp"

val uncurry_prefix = nitpick_prefix ^ "unc"

val eval_prefix = nitpick_prefix ^ "eval"

val bound_var_prefix = "b"

val cong_var_prefix = "c"

val iter_var_prefix = "i"

val val_var_prefix = nitpick_prefix ^ "v"

val arg_var_prefix = "x"

(* int -> string *)

fun sel_prefix_for j = sel_prefix ^ string_of_int j ^ name_sep

fun special_prefix_for j = special_prefix ^ string_of_int j ^ name_sep

(* int -> int -> string *)

fun skolem_prefix_for k j =

  skolem_prefix ^ string_of_int k ^ "@" ^ string_of_int j ^ name_sep

fun uncurry_prefix_for k j =

  uncurry_prefix ^ string_of_int k ^ "@" ^ string_of_int j ^ name_sep

(* string -> string * string *)

val strip_first_name_sep =

  Substring.full #> Substring.position name_sep ##> Substring.triml 1

  #> pairself Substring.string

(* string -> 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

val after_name_sep = snd o strip_first_name_sep

(* term * term -> term *)

fun s_conj (t1, @{const True}) = t1

  | s_conj (@{const True}, t2) = t2

  | s_conj (t1, t2) =

    if t1 = @{const False} orelse t2 = @{const False} then @{const False}

    else HOLogic.mk_conj (t1, t2)

fun s_disj (t1, @{const False}) = t1

  | s_disj (@{const False}, t2) = t2

  | s_disj (t1, t2) =

    if t1 = @{const True} orelse t2 = @{const True} then @{const True}

    else HOLogic.mk_disj (t1, t2)

(* term -> term -> term *)

fun mk_exists v t =

  HOLogic.exists_const (fastype_of v) $ lambda v (incr_boundvars 1 t)

(* term -> term -> term list *)

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]

(* term -> term list * term *)

fun strip_any_connective (t as (t0 $ t1 $ t2)) =

    if t0 = @{const "op &"} orelse t0 = @{const "op |"} then

      (strip_connective t0 t, t0)

    else

      ([t], @{const Not})

  | strip_any_connective t = ([t], @{const Not})

(* term -> term list *)

val conjuncts = strip_connective @{const "op &"}

val disjuncts = strip_connective @{const "op |"}

(* 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 all}, 1),

   (@{const_name "=="}, 2),

   (@{const_name "==>"}, 2),

   (@{const_name Pure.conjunction}, 2),

   (@{const_name Trueprop}, 1),

   (@{const_name Not}, 1),

   (@{const_name False}, 0),

   (@{const_name True}, 0),

   (@{const_name All}, 1),

   (@{const_name Ex}, 1),

   (@{const_name "op ="}, 2),

   (@{const_name "op &"}, 2),

   (@{const_name "op |"}, 2),

   (@{const_name "op -->"}, 2),

   (@{const_name If}, 3),

   (@{const_name Let}, 2),

   (@{const_name Unity}, 0),

   (@{const_name Pair}, 2),

   (@{const_name fst}, 1),

   (@{const_name snd}, 1),

   (@{const_name Id}, 0),

   (@{const_name insert}, 2),

   (@{const_name converse}, 1),

   (@{const_name trancl}, 1),

   (@{const_name rel_comp}, 2),

   (@{const_name image}, 2),

   (@{const_name Suc}, 0),

   (@{const_name finite}, 1),

   (@{const_name nat}, 0),

   (@{const_name zero_nat_inst.zero_nat}, 0),

   (@{const_name one_nat_inst.one_nat}, 0),

   (@{const_name plus_nat_inst.plus_nat}, 0),

   (@{const_name minus_nat_inst.minus_nat}, 0),

   (@{const_name times_nat_inst.times_nat}, 0),

   (@{const_name div_nat_inst.div_nat}, 0),

   (@{const_name ord_nat_inst.less_nat}, 2),

   (@{const_name ord_nat_inst.less_eq_nat}, 2),

   (@{const_name nat_gcd}, 0),

   (@{const_name nat_lcm}, 0),

   (@{const_name zero_int_inst.zero_int}, 0),

   (@{const_name one_int_inst.one_int}, 0),

   (@{const_name plus_int_inst.plus_int}, 0),

   (@{const_name minus_int_inst.minus_int}, 0),

   (@{const_name times_int_inst.times_int}, 0),

   (@{const_name div_int_inst.div_int}, 0),

   (@{const_name uminus_int_inst.uminus_int}, 0),

   (@{const_name ord_int_inst.less_int}, 2),

   (@{const_name ord_int_inst.less_eq_int}, 2),

   (@{const_name unknown}, 0),

   (@{const_name is_unknown}, 1),

   (@{const_name Tha}, 1),

   (@{const_name Frac}, 0),

   (@{const_name norm_frac}, 0)]

val built_in_descr_consts =

  [(@{const_name The}, 1),

   (@{const_name Eps}, 1)]

val built_in_typed_consts =

  [((@{const_name of_nat}, nat_T --> int_T), 0),

   ((@{const_name of_nat}, @{typ "unsigned_bit word => signed_bit word"}), 0)]

val built_in_set_consts =

  [(@{const_name lower_semilattice_fun_inst.inf_fun}, 2),

   (@{const_name upper_semilattice_fun_inst.sup_fun}, 2),

   (@{const_name minus_fun_inst.minus_fun}, 2),

   (@{const_name ord_fun_inst.less_eq_fun}, 2)]

(* typ -> typ *)

fun unbit_type @{typ "unsigned_bit word"} = nat_T

  | unbit_type @{typ "signed_bit word"} = int_T

  | unbit_type @{typ bisim_iterator} = nat_T

  | unbit_type (Type (s, Ts as _ :: _)) = Type (s, map unbit_type Ts)

  | unbit_type T = T

fun unbit_and_unbox_type (Type (@{type_name fun_box}, Ts)) =

    unbit_and_unbox_type (Type ("fun", Ts))

  | unbit_and_unbox_type (Type (@{type_name pair_box}, Ts)) =

    Type ("*", map unbit_and_unbox_type Ts)

  | unbit_and_unbox_type @{typ "unsigned_bit word"} = nat_T

  | unbit_and_unbox_type @{typ "signed_bit word"} = int_T

  | unbit_and_unbox_type @{typ bisim_iterator} = nat_T

  | unbit_and_unbox_type (Type (s, Ts as _ :: _)) =

    Type (s, map unbit_and_unbox_type Ts)

  | unbit_and_unbox_type T = T

(* Proof.context -> typ -> string *)

fun string_for_type ctxt = Syntax.string_of_typ ctxt o unbit_and_unbox_type

(* string -> string -> string *)

val prefix_name = Long_Name.qualify o Long_Name.base_name

(* string -> string *)

fun shortest_name s = List.last (space_explode "." s) handle List.Empty => ""

(* string -> term -> term *)

val prefix_abs_vars = Term.map_abs_vars o prefix_name

(* term -> term *)

val shorten_abs_vars = Term.map_abs_vars shortest_name

(* string -> string *)

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 "_"

(* typ -> typ *)

fun shorten_names_in_type (Type (s, Ts)) =

    Type (short_name s, map shorten_names_in_type Ts)

  | shorten_names_in_type T = T

(* term -> term *)

val shorten_names_in_term =

  map_aterms (fn Const (s, T) => Const (short_name s, T) | t => t)

  #> map_types shorten_names_in_type

(* theory -> typ * typ -> bool *)

fun type_match thy (T1, T2) =

  (Sign.typ_match thy (T2, T1) Vartab.empty; true)

  handle Type.TYPE_MATCH => false

(* theory -> styp * styp -> bool *)

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

  s1 = s2 andalso type_match thy (T1, T2)

(* theory -> term * term -> bool *)

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 thy (t1, t2) = t1 aconv t2

(* typ -> bool *)

fun is_TFree (TFree _) = true

  | is_TFree _ = false

fun is_higher_order_type (Type ("fun", _)) = true

  | is_higher_order_type (Type (_, Ts)) = exists is_higher_order_type Ts

  | is_higher_order_type _ = false

fun is_fun_type (Type ("fun", _)) = true

  | is_fun_type _ = false

fun is_set_type (Type ("fun", [_, @{typ bool}])) = true

  | is_set_type _ = false

fun is_pair_type (Type ("*", _)) = 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_boolean_type T = (T = prop_T orelse T = bool_T)

val is_integer_type =

  member (op =) [nat_T, int_T, @{typ bisim_iterator}] orf is_fp_iterator_type

fun is_bit_type T = (T = @{typ unsigned_bit} orelse T = @{typ signed_bit})

fun is_word_type (Type (@{type_name word}, _)) = true

  | is_word_type _ = false

val is_record_type = not o null o Record.dest_recTs

(* theory -> typ -> bool *)

fun is_frac_type thy (Type (s, [])) =

    not (null (these (AList.lookup (op =) (#frac_types (Data.get thy)) s)))

  | is_frac_type _ _ = false

fun is_number_type thy = is_integer_type orf is_frac_type thy

(* bool -> styp -> typ *)

fun iterator_type_for_const gfp (s, T) =

  Type ((if gfp then gfp_iterator_prefix else lfp_iterator_prefix) ^ s,

        binder_types T)

(* typ -> styp *)

fun const_for_iterator_type (Type (s, Ts)) = (after_name_sep s, Ts ---> bool_T)

  | const_for_iterator_type T =

    raise TYPE ("Nitpick_HOL.const_for_iterator_type", [T], [])

(* int -> typ -> typ * typ *)

fun strip_n_binders 0 T = ([], T)

  | strip_n_binders n (Type ("fun", [T1, T2])) =

    strip_n_binders (n - 1) T2 |>> cons T1

  | strip_n_binders n (Type (@{type_name fun_box}, Ts)) =

    strip_n_binders n (Type ("fun", Ts))

  | strip_n_binders _ T = raise TYPE ("Nitpick_HOL.strip_n_binders", [T], [])

(* typ -> typ *)

val nth_range_type = snd oo strip_n_binders

(* typ -> int *)

fun num_factors_in_type (Type ("*", [T1, T2])) =

    fold (Integer.add o num_factors_in_type) [T1, T2] 0

  | num_factors_in_type _ = 1

fun num_binder_types (Type ("fun", [_, T2])) = 1 + num_binder_types T2

  | num_binder_types _ = 0

(* typ -> typ list *)

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

(* typ -> term list -> term *)

fun mk_flat_tuple _ [t] = t

  | mk_flat_tuple (Type ("*", [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)

(* int -> term -> term list *)

fun dest_n_tuple 1 t = [t]

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

(* int -> typ -> typ list *)

fun dest_n_tuple_type 1 T = [T]

  | dest_n_tuple_type n (Type (_, [T1, T2])) =

    T1 :: dest_n_tuple_type (n - 1) T2

  | dest_n_tuple_type _ T =

    raise TYPE ("Nitpick_HOL.dest_n_tuple_type", [T], [])

(* FIXME: Use antiquotation for "code_numeral" below or detect "rep_datatype",

   e.g., by adding a field to "Datatype_Aux.info". *)

(* string -> bool *)

val is_basic_datatype =

    member (op =) [@{type_name "*"}, @{type_name bool}, @{type_name unit},

                   @{type_name nat}, @{type_name int},

                   "Code_Numeral.code_numeral"]

(* theory -> typ -> typ -> typ -> typ *)

fun instantiate_type thy T1 T1' T2 =

  Same.commit (Envir.subst_type_same

                   (Sign.typ_match thy (Logic.varifyT T1, T1') Vartab.empty))

              (Logic.varifyT T2)

  handle Type.TYPE_MATCH =>

         raise TYPE ("Nitpick_HOL.instantiate_type", [T1, T1'], [])

(* theory -> typ -> typ -> styp *)

fun repair_constr_type thy body_T' T =

  instantiate_type thy (body_type T) body_T' T

(* string -> (string * string) list -> theory -> theory *)

fun register_frac_type frac_s ersaetze thy =

  let

    val {frac_types, codatatypes} = Data.get thy

    val frac_types = AList.update (op =) (frac_s, ersaetze) frac_types

  in Data.put {frac_types = frac_types, codatatypes = codatatypes} thy end

(* string -> theory -> theory *)

fun unregister_frac_type frac_s = register_frac_type frac_s []

(* typ -> string -> styp list -> theory -> theory *)

fun register_codatatype co_T case_name constr_xs thy =

  let

    val {frac_types, codatatypes} = Data.get thy

    val constr_xs = map (apsnd (repair_constr_type thy co_T)) constr_xs

    val (co_s, co_Ts) = dest_Type co_T

    val _ =

      if forall is_TFree co_Ts andalso not (has_duplicates (op =) co_Ts) andalso

         co_s <> "fun" andalso not (is_basic_datatype co_s) then

        ()

      else

        raise TYPE ("Nitpick_HOL.register_codatatype", [co_T], [])

    val codatatypes = AList.update (op =) (co_s, (case_name, constr_xs))

                                   codatatypes

  in Data.put {frac_types = frac_types, codatatypes = codatatypes} thy end

(* typ -> theory -> theory *)

fun unregister_codatatype co_T = register_codatatype co_T "" []

type typedef_info =

  {rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string,

   set_def: thm option, prop_of_Rep: thm, set_name: string,

   Abs_inverse: thm option, Rep_inverse: thm option}

(* theory -> string -> typedef_info *)

fun typedef_info thy s =

  if is_frac_type thy (Type (s, [])) then

    SOME {abs_type = Type (s, []), rep_type = @{typ "int * int"},

          Abs_name = @{const_name Abs_Frac}, Rep_name = @{const_name Rep_Frac},

          set_def = NONE, prop_of_Rep = @{prop "Rep_Frac x \<in> Frac"}

                          |> Logic.varify,

          set_name = @{const_name Frac}, Abs_inverse = NONE, Rep_inverse = NONE}

  else case Typedef.get_info thy s of

    SOME {abs_type, rep_type, Abs_name, Rep_name, set_def, Rep, Abs_inverse,

          Rep_inverse, ...} =>

    SOME {abs_type = abs_type, rep_type = rep_type, Abs_name = Abs_name,

          Rep_name = Rep_name, set_def = set_def, prop_of_Rep = prop_of Rep,

          set_name = set_prefix ^ s, Abs_inverse = SOME Abs_inverse,

          Rep_inverse = SOME Rep_inverse}

  | NONE => NONE

(* theory -> string -> bool *)

val is_typedef = is_some oo typedef_info

val is_real_datatype = is_some oo Datatype.get_info

(* theory -> typ -> bool *)

fun is_quot_type _ (Type ("IntEx.my_int", _)) = true (* FIXME *)

  | is_quot_type _ (Type ("FSet.fset", _)) = true (* FIXME *)

  | is_quot_type _ _ = false

fun is_codatatype thy (T as Type (s, _)) =

    not (null (AList.lookup (op =) (#codatatypes (Data.get thy)) s

               |> Option.map snd |> these))

  | is_codatatype _ _ = false

fun is_pure_typedef thy (T as Type (s, _)) =

    is_typedef thy s andalso

    not (is_real_datatype thy s orelse is_quot_type thy T orelse

         is_codatatype thy T orelse is_record_type T orelse is_integer_type T)

  | is_pure_typedef _ _ = false

fun is_univ_typedef thy (Type (s, _)) =

    (case typedef_info thy s of

       SOME {set_def, prop_of_Rep, ...} =>

       (case set_def of

          SOME thm =>

          try (fst o dest_Const o snd o Logic.dest_equals o prop_of) thm

        | NONE =>

          try (fst o dest_Const o snd o HOLogic.dest_mem

               o HOLogic.dest_Trueprop) prop_of_Rep) = SOME @{const_name top}

     | NONE => false)

  | is_univ_typedef _ _ = false

fun is_datatype thy (T as Type (s, _)) =

    (is_typedef thy s orelse is_codatatype thy T orelse T = @{typ ind} orelse

     is_quot_type thy T) andalso

    not (is_basic_datatype s)

  | is_datatype _ _ = false

(* theory -> typ -> (string * typ) list * (string * typ) *)

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 _ => []

(* styp -> bool *)

fun is_record_constr (x as (s, T)) =

  String.isSuffix Record.extN s andalso

  let val dataT = body_type T in

    is_record_type dataT andalso

    s = unsuffix Record.ext_typeN (fst (dest_Type dataT)) ^ Record.extN

  end

(* theory -> typ -> int *)

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

(* theory -> string -> typ -> int *)

fun no_of_record_field thy s T1 =

  find_index (curry (op =) s o fst)

             (Record.get_extT_fields thy T1 ||> single |> op @)

(* theory -> styp -> bool *)

fun is_record_get thy (s, Type ("fun", [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 thy (s, Type ("fun", [_, Type (s', _)])) =

    (case typedef_info thy s' of

       SOME {Abs_name, ...} => s = Abs_name

     | NONE => false)

  | is_abs_fun _ _ = false

fun is_rep_fun thy (s, Type ("fun", [Type (s', _), _])) =

    (case typedef_info thy s' of

       SOME {Rep_name, ...} => s = Rep_name

     | NONE => false)

  | is_rep_fun _ _ = false

(* Proof.context -> styp -> bool *)

fun is_quot_abs_fun _ ("IntEx.abs_my_int", _) = true (* FIXME *)

  | is_quot_abs_fun _ ("FSet.abs_fset", _) = true (* FIXME *)

  | is_quot_abs_fun _ _ = false

fun is_quot_rep_fun _ ("IntEx.rep_my_int", _) = true (* FIXME *)

  | is_quot_rep_fun _ ("FSet.rep_fset", _) = true (* FIXME *)

  | is_quot_rep_fun _ _ = false

(* theory -> styp -> styp *)

fun mate_of_rep_fun thy (x as (_, Type ("fun", [T1 as Type (s', _), T2]))) =

    (case typedef_info thy s' of

       SOME {Abs_name, ...} => (Abs_name, Type ("fun", [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])

(* theory -> typ -> typ *)

fun rep_type_for_quot_type _ (Type ("IntEx.my_int", [])) = @{typ "nat * nat"}

  | rep_type_for_quot_type _ (Type ("FSet.fset", [T])) =

    Type (@{type_name list}, [T])

  | rep_type_for_quot_type _ T =

    raise TYPE ("Nitpick_HOL.rep_type_for_quot_type", [T], [])

(* theory -> typ -> term *)

fun equiv_relation_for_quot_type _ (Type ("IntEx.my_int", [])) =

    Const ("IntEx.intrel", @{typ "(nat * nat) => (nat * nat) => bool"})

  | equiv_relation_for_quot_type _ (Type ("FSet.fset", [T])) =

    Const ("FSet.list_eq",

           Type (@{type_name list}, [T]) --> Type (@{type_name list}, [T])

           --> bool_T)

  | equiv_relation_for_quot_type _ T =

    raise TYPE ("Nitpick_HOL.equiv_relation_for_quot_type", [T], [])

(* theory -> styp -> bool *)

fun is_coconstr thy (s, T) =

  let

    val {codatatypes, ...} = Data.get thy

    val co_T = body_type T

    val co_s = dest_Type co_T |> fst

  in

    exists (fn (s', T') => s = s' andalso repair_constr_type thy co_T T' = T)

           (AList.lookup (op =) codatatypes co_s |> Option.map snd |> these)

  end

  handle TYPE ("dest_Type", _, _) => false

fun is_constr_like thy (s, T) =

  member (op =) [@{const_name FunBox}, @{const_name PairBox},

                 @{const_name Quot}, @{const_name Zero_Rep},

                 @{const_name Suc_Rep}] s orelse

  let val (x as (s, T)) = (s, unbit_and_unbox_type T) in

    Refute.is_IDT_constructor thy x orelse is_record_constr x orelse

    (is_abs_fun thy x andalso is_pure_typedef thy (range_type T)) orelse

    x = (@{const_name zero_nat_inst.zero_nat}, nat_T) orelse

    is_coconstr thy x

  end

fun is_stale_constr thy (x as (_, T)) =

  is_codatatype thy (body_type T) andalso is_constr_like thy x andalso

  not (is_coconstr thy x)

fun is_constr thy (x as (_, T)) =

  is_constr_like thy x andalso

  not (is_basic_datatype (fst (dest_Type (unbit_type (body_type T))))) andalso

  not (is_stale_constr thy x)

(* string -> bool *)

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 fst}, @{const_name snd}])

datatype boxability =

  InConstr | InSel | InExpr | InPair | InFunLHS | InFunRHS1 | InFunRHS2

(* boxability -> boxability *)

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

(* extended_context -> boxability -> typ -> bool *)

fun is_boxing_worth_it (ext_ctxt : extended_context) boxy T =

  case T of

    Type ("fun", _) =>

    (boxy = InPair orelse boxy = InFunLHS) andalso

    not (is_boolean_type (body_type T))

  | Type ("*", Ts) =>

    boxy = InPair orelse boxy = InFunRHS1 orelse boxy = InFunRHS2 orelse

    ((boxy = InExpr orelse boxy = InFunLHS) andalso

     exists (is_boxing_worth_it ext_ctxt InPair)

            (map (box_type ext_ctxt InPair) Ts))

  | _ => false

(* extended_context -> boxability -> string * typ list -> string *)

and should_box_type (ext_ctxt as {thy, boxes, ...}) boxy (z as (s, Ts)) =

  case triple_lookup (type_match thy) boxes (Type z) of

    SOME (SOME box_me) => box_me

  | _ => is_boxing_worth_it ext_ctxt boxy (Type z)

(* extended_context -> boxability -> typ -> typ *)

and box_type ext_ctxt boxy T =

  case T of

    Type (z as ("fun", [T1, T2])) =>

    if boxy <> InConstr andalso boxy <> InSel andalso

       should_box_type ext_ctxt boxy z then

      Type (@{type_name fun_box},

            [box_type ext_ctxt InFunLHS T1, box_type ext_ctxt InFunRHS1 T2])

    else

      box_type ext_ctxt (in_fun_lhs_for boxy) T1

      --> box_type ext_ctxt (in_fun_rhs_for boxy) T2

  | Type (z as ("*", Ts)) =>

    if boxy <> InConstr andalso boxy <> InSel

       andalso should_box_type ext_ctxt boxy z then

      Type (@{type_name pair_box}, map (box_type ext_ctxt InSel) Ts)

    else

      Type ("*", map (box_type ext_ctxt

                          (if boxy = InConstr orelse boxy = InSel then boxy

                           else InPair)) Ts)

  | _ => T

(* styp -> styp *)

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

(* typ -> int *)

fun num_sels_for_constr_type T = length (maybe_curried_binder_types T)

(* string -> int -> string *)

fun nth_sel_name_for_constr_name s n =

  if s = @{const_name Pair} then

    if n = 0 then @{const_name fst} else @{const_name snd}

  else

    sel_prefix_for n ^ s

(* styp -> int -> styp *)

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)

(* extended_context -> styp -> int -> styp *)

fun boxed_nth_sel_for_constr ext_ctxt =

  apsnd (box_type ext_ctxt InSel) oo nth_sel_for_constr

(* string -> int *)

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 snd} then

    1

  else

    0

(* typ list -> term -> int -> term *)

fun eta_expand _ t 0 = t

  | eta_expand Ts (Abs (s, T, t')) n =

    Abs (s, T, eta_expand (T :: Ts) t' (n - 1))

  | eta_expand Ts t n =

    fold_rev (curry3 Abs ("x\<^isub>\<eta>" ^ nat_subscript n))

             (List.take (binder_types (fastype_of1 (Ts, t)), n))

             (list_comb (incr_boundvars n t, map Bound (n - 1 downto 0)))

(* term -> term *)

fun extensionalize t =

  case t of

    (t0 as @{const Trueprop}) $ t1 => t0 $ extensionalize t1

  | Const (@{const_name "op ="}, _) $ t1 $ Abs (s, T, t2) =>

    let val v = Var ((s, maxidx_of_term t + 1), T) in

      extensionalize (HOLogic.mk_eq (t1 $ v, subst_bound (v, t2)))

    end

  | _ => t

(* typ -> term list -> term *)

fun distinctness_formula T =

  all_distinct_unordered_pairs_of

  #> map (fn (t1, t2) => @{const Not} $ (HOLogic.eq_const T $ t1 $ t2))

  #> List.foldr (s_conj o swap) @{const True}

(* typ -> term *)

fun zero_const T = Const (@{const_name zero_nat_inst.zero_nat}, T)

fun suc_const T = Const (@{const_name Suc}, T --> T)

(* extended_context -> typ -> styp list *)

fun uncached_datatype_constrs ({thy, stds, ...} : extended_context)

                              (T as Type (s, Ts)) =

    (case AList.lookup (op =) (#codatatypes (Data.get thy)) s of

       SOME (_, xs' as (_ :: _)) => map (apsnd (repair_constr_type thy T)) xs'

     | _ =>

       if is_datatype thy T then

         case Datatype.get_info thy s of

           SOME {index, descr, ...} =>

           let

             val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the

           in

             map (fn (s', Us) =>

                     (s', map (Refute.typ_of_dtyp descr (dtyps ~~ Ts)) Us

                          ---> T)) constrs

             |> (triple_lookup (type_match thy) stds T |> the |> not) ?

                cons (@{const_name NonStd}, @{typ \<xi>} --> T)

           end

         | NONE =>

           if is_record_type T then

             let

               val s' = unsuffix Record.ext_typeN s ^ Record.extN

               val T' = (Record.get_extT_fields thy T

                        |> apsnd single |> uncurry append |> map snd) ---> T

             in [(s', T')] end

           else if is_quot_type thy T then

             [(@{const_name Quot}, rep_type_for_quot_type thy T --> T)]

           else case typedef_info thy s of

             SOME {abs_type, rep_type, Abs_name, ...} =>

             [(Abs_name, instantiate_type thy abs_type T rep_type --> T)]

           | NONE =>

             if T = @{typ ind} then

               [dest_Const @{const Zero_Rep}, dest_Const @{const Suc_Rep}]

             else

               []

       else

         [])

  | uncached_datatype_constrs _ _ = []

(* extended_context -> typ -> styp list *)

fun datatype_constrs (ext_ctxt as {constr_cache, ...}) T =

  case AList.lookup (op =) (!constr_cache) T of

    SOME xs => xs

  | NONE =>

    let val xs = uncached_datatype_constrs ext_ctxt T in

      (Unsynchronized.change constr_cache (cons (T, xs)); xs)

    end

fun boxed_datatype_constrs ext_ctxt =

  map (apsnd (box_type ext_ctxt InConstr)) o datatype_constrs ext_ctxt

(* extended_context -> typ -> int *)

val num_datatype_constrs = length oo datatype_constrs

(* string -> string *)

fun constr_name_for_sel_like @{const_name fst} = @{const_name Pair}

  | constr_name_for_sel_like @{const_name snd} = @{const_name Pair}

  | constr_name_for_sel_like s' = original_name s'

(* extended_context -> styp -> styp *)

fun boxed_constr_for_sel ext_ctxt (s', T') =

  let val s = constr_name_for_sel_like s' in

    AList.lookup (op =) (boxed_datatype_constrs ext_ctxt (domain_type T')) s

    |> the |> pair s

  end

(* extended_context -> styp -> term *)

fun discr_term_for_constr ext_ctxt (x as (s, T)) =

  let val dataT = body_type T in

    if s = @{const_name Suc} then

      Abs (Name.uu, dataT,

           @{const Not} $ HOLogic.mk_eq (zero_const dataT, Bound 0))

    else if num_datatype_constrs ext_ctxt dataT >= 2 then

      Const (discr_for_constr x)

    else

      Abs (Name.uu, dataT, @{const True})

  end

(* extended_context -> styp -> term -> term *)

fun discriminate_value (ext_ctxt as {thy, ...}) (x as (_, T)) t =

  case strip_comb t of

    (Const x', args) =>

    if x = x' then @{const True}

    else if is_constr_like thy x' then @{const False}

    else betapply (discr_term_for_constr ext_ctxt x, t)

  | _ => betapply (discr_term_for_constr ext_ctxt x, t)

(* styp -> term -> term *)

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 "%n::nat. minus_nat_inst.minus_nat n one_nat_inst.one_nat"}

    else if is_pair_type dataT then

      Const (nth_sel_for_constr x n)

    else

      let

        (* int -> typ -> int * term *)

        fun aux m (Type ("*", [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

(* theory -> styp -> term -> int -> typ -> term *)

fun select_nth_constr_arg thy x t n res_T =

  case strip_comb t of

    (Const x', args) =>

    if x = x' then nth args n

    else if is_constr_like thy x' then Const (@{const_name unknown}, res_T)

    else betapply (nth_arg_sel_term_for_constr x n, t)

  | _ => betapply (nth_arg_sel_term_for_constr x n, t)

(* theory -> styp -> term list -> term *)

fun construct_value _ x [] = Const x

  | construct_value thy (x as (s, _)) args =

    let val args = map Envir.eta_contract args in

      case hd args of

        Const (x' as (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 thy 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

(* extended_context -> typ -> term -> term *)

fun constr_expand (ext_ctxt as {thy, ...}) T t =

  (case head_of t of

     Const x => if is_constr_like thy 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 Pair}, T1 --> T2 --> T)

               end

             else

               datatype_constrs ext_ctxt T |> hd

           val arg_Ts = binder_types T'

         in

           list_comb (Const x', map2 (select_nth_constr_arg thy x' t)

                                     (index_seq 0 (length arg_Ts)) arg_Ts)

         end

(* (typ * int) list -> typ -> int *)

fun card_of_type assigns (Type ("fun", [T1, T2])) =

    reasonable_power (card_of_type assigns T2) (card_of_type assigns T1)

  | card_of_type assigns (Type ("*", [T1, T2])) =

    card_of_type assigns T1 * card_of_type assigns T2

  | card_of_type _ (Type (@{type_name itself}, _)) = 1

  | card_of_type _ @{typ prop} = 2

  | card_of_type _ @{typ bool} = 2

  | card_of_type _ @{typ unit} = 1

  | card_of_type assigns T =

    case AList.lookup (op =) assigns T of

      SOME k => k

    | NONE => if T = @{typ bisim_iterator} then 0

              else raise TYPE ("Nitpick_HOL.card_of_type", [T], [])

(* int -> (typ * int) list -> typ -> int *)

fun bounded_card_of_type max default_card assigns (Type ("fun", [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)

    end

  | bounded_card_of_type max default_card assigns (Type ("*", [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 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)

(* extended_context -> int -> (typ * int) list -> typ -> int *)

fun bounded_exact_card_of_type ext_ctxt max default_card assigns T =

  let

    (* typ list -> typ -> int *)

    fun aux avoid T =

      (if member (op =) avoid T then

         0

       else case T of

         Type ("fun", [T1, T2]) =>

         let

           val k1 = aux avoid T1

           val k2 = aux avoid T2

         in

           if k1 = 0 orelse k2 = 0 then 0

           else if k1 >= max orelse k2 >= max then max