(*  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 hol_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,

     whacks: term list,

     binary_ints: bool option,

     destroy_constrs: bool,

     specialize: bool,

     star_linear_preds: 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,

     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 * styp 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 is_higher_order_type : typ -> bool

  val s_let : 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 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 -> styp * styp -> bool

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

  val frac_from_term_pair : typ -> term -> term -> term

  val is_TFree : 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_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_record_type : typ -> bool

  val is_number_type : Proof.context -> typ -> bool

  val const_for_iterator_type : typ -> styp

  val strip_n_binders : int -> typ -> typ list * typ

  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 is_real_datatype : theory -> string -> bool

  val is_standard_datatype : theory -> (typ option * bool) list -> typ -> bool

  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_datatype : Proof.context -> (typ option * bool) list -> 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 : Proof.context -> styp -> bool

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

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

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

  val mate_of_rep_fun : Proof.context -> styp -> styp

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

  val is_constr : Proof.context -> (typ option * bool) list -> styp -> 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 : 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 binarized_and_boxed_nth_sel_for_constr :

    hol_context -> bool -> styp -> int -> styp

  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_codatatype :

    typ -> string -> styp list -> morphism -> Context.generic -> Context.generic

  val register_codatatype_global :

    typ -> string -> styp list -> theory -> theory

  val unregister_codatatype :

    typ -> morphism -> Context.generic -> Context.generic

  val unregister_codatatype_global : typ -> theory -> theory

  val datatype_constrs : hol_context -> typ -> styp list

  val binarized_and_boxed_datatype_constrs :

    hol_context -> bool -> typ -> styp list

  val num_datatype_constrs : hol_context -> typ -> int

  val constr_name_for_sel_like : string -> string

  val binarized_and_boxed_constr_for_sel : hol_context -> bool -> styp -> styp

  val discriminate_value : hol_context -> styp -> term -> term

  val select_nth_constr_arg :

    Proof.context -> (typ option * bool) list -> styp -> term -> int -> typ

    -> term

  val construct_value :

    Proof.context -> (typ option * bool) list -> styp -> term list -> term

  val coerce_term : hol_context -> typ list -> typ -> typ -> term -> term

  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 is_finite_type : hol_context -> typ -> bool

  val is_small_finite_type : hol_context -> typ -> bool

  val special_bounds : term list -> (indexname * typ) list

  val is_funky_typedef : Proof.context -> typ -> bool

  val all_axioms_of :

    Proof.context -> (term * term) list -> term list * term list * term list

  val arity_of_built_in_const :

    theory -> (typ option * bool) list -> styp -> int option

  val is_built_in_const :

    theory -> (typ option * bool) list -> styp -> bool

  val term_under_def : term -> term

  val case_const_names :

    Proof.context -> (typ option * bool) list -> (string * int) list

  val unfold_defs_in_term : hol_context -> term -> term

  val const_def_table :

    Proof.context -> (term * term) list -> term list -> 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

  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 -> styp -> term list

  val optimized_typedef_axioms : Proof.context -> string * typ list -> term list

  val optimized_quot_type_axioms :

    Proof.context -> (typ option * bool) list -> string * typ list -> term list

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

  val fixpoint_kind_of_rhs : term -> fixpoint_kind

  val fixpoint_kind_of_const :

    theory -> const_table -> string * typ -> fixpoint_kind

  val is_real_inductive_pred : hol_context -> styp -> 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 -> styp -> term list

  val is_choice_spec_fun : hol_context -> styp -> bool

  val is_choice_spec_axiom : theory -> const_table -> term -> bool

  val is_real_equational_fun : hol_context -> styp -> bool

  val is_equational_fun_but_no_plain_def : hol_context -> styp -> bool

  val codatatype_bisim_axioms : hol_context -> typ -> term list

  val is_well_founded_inductive_pred : hol_context -> styp -> bool

  val unrolled_inductive_pred_const : hol_context -> bool -> styp -> term

  val equational_fun_axioms : hol_context -> styp -> term list

  val is_equational_fun_surely_complete : hol_context -> styp -> 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 = (styp * int list * term list) * styp

type unrolled = styp * styp

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

type hol_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,

   whacks: term list,

   binary_ints: bool option,

   destroy_constrs: bool,

   specialize: bool,

   star_linear_preds: 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,

   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 * styp list) list Unsynchronized.ref}

datatype fixpoint_kind = Lfp | Gfp | NoFp

datatype boxability =

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

structure Data = Generic_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 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 ^ unyxml (Syntax.string_of_typ ctxt T)

val strip_first_name_sep =

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

  #> pairself 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 is_higher_order_type (Type (@{type_name fun}, _)) = true

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

  | is_higher_order_type _ = false

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 s n abs_T body_T f t =

  if (n - 1) * (size_of_term t - 1) <= let_inline_threshold orelse

     is_higher_order_type abs_T then

    f t

  else

    let val z = (let_var s, abs_T) in

      Const (@{const_name Let}, 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 _ (Const (@{const_name If}, _) $ @{const True} $ t1', _) = t1'

  | s_betapply _ (Const (@{const_name If}, _) $ @{const False} $ _, t2) = t2

  | s_betapply Ts (Const (@{const_name Let},

                          Type (_, [bound_T, Type (_, [_, body_T])]))

                   $ t12 $ Abs (s, T, t13'), t2) =

    let val body_T' = range_type body_T in

      Const (@{const_name Let}, 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 s1 (loose_bvar1_count (t1', 0)) T1 (fastype_of1 (T1 :: Ts, t1'))

              (curry betapply t1) t2

     handle TERM _ => betapply (t1, t2)) (* FIXME: fix all uses *)

  | 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 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)

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 HOL.conj} orelse t0 = @{const HOL.disj} then

      (strip_connective t0 t, t0)

    else

      ([t], @{const Not})

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

val conjuncts_of = strip_connective @{const HOL.conj}

val disjuncts_of = strip_connective @{const HOL.disj}

(* 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 HOL.eq}, 1),

   (@{const_name HOL.conj}, 2),

   (@{const_name HOL.disj}, 2),

   (@{const_name HOL.implies}, 2),

   (@{const_name If}, 3),

   (@{const_name Let}, 2),

   (@{const_name Pair}, 2),

   (@{const_name fst}, 1),

   (@{const_name snd}, 1),

   (@{const_name Id}, 0),

   (@{const_name converse}, 1),

   (@{const_name trancl}, 1),

   (@{const_name rel_comp}, 2),

   (@{const_name image}, 2),

   (@{const_name finite}, 1),

   (@{const_name unknown}, 0),

   (@{const_name is_unknown}, 1),

   (@{const_name safe_The}, 1),

   (@{const_name Frac}, 0),

   (@{const_name norm_frac}, 0)]

val built_in_nat_consts =

  [(@{const_name Suc}, 0),

   (@{const_name nat}, 0),

   (@{const_name nat_gcd}, 0),

   (@{const_name nat_lcm}, 0)]

val built_in_typed_consts =

  [((@{const_name zero_class.zero}, int_T), 0),

   ((@{const_name one_class.one}, int_T), 0),

   ((@{const_name plus_class.plus}, int_T --> int_T --> int_T), 0),

   ((@{const_name minus_class.minus}, int_T --> int_T --> int_T), 0),

   ((@{const_name times_class.times}, int_T --> int_T --> int_T), 0),

   ((@{const_name div_class.div}, int_T --> int_T --> int_T), 0),

   ((@{const_name uminus_class.uminus}, int_T --> int_T), 0),

   ((@{const_name ord_class.less}, int_T --> int_T --> bool_T), 2),

   ((@{const_name ord_class.less_eq}, int_T --> int_T --> bool_T), 2)]

val built_in_typed_nat_consts =

  [((@{const_name zero_class.zero}, nat_T), 0),

   ((@{const_name one_class.one}, nat_T), 0),

   ((@{const_name plus_class.plus}, nat_T --> nat_T --> nat_T), 0),

   ((@{const_name minus_class.minus}, nat_T --> nat_T --> nat_T), 0),

   ((@{const_name times_class.times}, nat_T --> nat_T --> nat_T), 0),

   ((@{const_name div_class.div}, nat_T --> nat_T --> nat_T), 0),

   ((@{const_name ord_class.less}, nat_T --> nat_T --> bool_T), 2),

   ((@{const_name ord_class.less_eq}, nat_T --> nat_T --> bool_T), 2),

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

val built_in_set_consts =

  [(@{const_name ord_class.less_eq}, 2)]

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

  | unarize_type @{typ "signed_bit word"} = 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 fin_fun}, Ts)) =

    unarize_unbox_etc_type (Type (@{type_name fun}, Ts))

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

    unarize_unbox_etc_type (Type (@{type_name fun}, Ts))

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

    Type (@{type_name prod}, map unarize_unbox_etc_type Ts)

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

  | unarize_unbox_etc_type @{typ "signed_bit word"} = 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 bisim_iterator} = 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

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

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 pairself 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 divide}, T --> T --> T)

                  $ HOLogic.mk_number T n1 $ HOLogic.mk_number T n2

fun is_TFree (TFree _) = true

  | is_TFree _ = false

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

  | is_fun_type _ = false

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

  | is_set_type _ = false

fun is_pair_type (Type (@{type_name prod}, _)) = 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 bisim_iterator} 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 unsigned_bit} orelse T = @{typ signed_bit})

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

  | is_word_type _ = false

val is_integer_like_type = is_iterator_type orf is_integer_type orf is_word_type

val is_record_type = not o null o Record.dest_recTs

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

    s |> AList.lookup (op =) (#frac_types (Data.get (Context.Proof ctxt)))

      |> these |> null |> not

  | is_frac_type _ _ = false

fun is_number_type ctxt = is_integer_like_type orf is_frac_type ctxt

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 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 (@{type_name fun}, 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 prod}, [T1, T2])) =

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

  | num_factors_in_type _ = 1

fun num_binder_types (Type (@{type_name fun}, [_, T2])) =

    1 + num_binder_types T2

  | num_binder_types _ = 0

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 prod}, [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 ::

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}

fun typedef_info ctxt s =

  if is_frac_type ctxt (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_global,

          set_name = @{const_name Frac}, 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, ...},

     {set_def, 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, 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

val is_typedef = is_some oo typedef_info

val is_real_datatype = is_some oo Datatype.get_info

fun is_standard_datatype thy = the oo triple_lookup (type_match thy)

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

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

fun is_basic_datatype thy stds s =

  member (op =) [@{type_name prod}, @{type_name bool}, @{type_name int},

                 "Code_Numeral.code_numeral"] s orelse

  (s = @{type_name nat} andalso is_standard_datatype thy stds nat_T)

(* TODO: use "Term_Subst.instantiateT" instead? *)

fun instantiate_type thy T1 T1' T2 =

  Same.commit (Envir.subst_type_same

                   (Sign.typ_match thy (T1, T1') Vartab.empty)) T2

  handle Type.TYPE_MATCH =>

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

fun varify_and_instantiate_type ctxt T1 T1' T2 =

  let val thy = ProofContext.theory_of ctxt in

    instantiate_type thy (varify_type ctxt T1) T1' (varify_type ctxt T2)

  end

fun repair_constr_type ctxt body_T' T =

  varify_and_instantiate_type ctxt (body_type T) body_T' T

fun register_frac_type_generic frac_s ersaetze generic =

  let

    val {frac_types, codatatypes} = Data.get generic

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

  in Data.put {frac_types = frac_types, 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_codatatype_generic co_T case_name constr_xs generic =

  let

    val ctxt = Context.proof_of generic

    val thy = Context.theory_of generic

    val {frac_types, codatatypes} = Data.get generic

    val constr_xs = map (apsnd (repair_constr_type ctxt 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 <> @{type_name fun} andalso

         not (is_basic_datatype thy [(NONE, true)] co_s) then

        ()

      else

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

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

                                   codatatypes

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

(* TODO: Consider morphism. *)

fun register_codatatype co_T case_name constr_xs (_ : morphism) =

  register_codatatype_generic co_T case_name constr_xs

val register_codatatype_global =

  Context.theory_map ooo register_codatatype_generic

fun unregister_codatatype_generic co_T = register_codatatype_generic co_T "" []

(* TODO: Consider morphism. *)

fun unregister_codatatype co_T (_ : morphism) =

  unregister_codatatype_generic co_T

val unregister_codatatype_global =

  Context.theory_map o unregister_codatatype_generic

fun is_codatatype ctxt (Type (s, _)) =

    s |> AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))

      |> Option.map snd |> these |> null |> not

  | is_codatatype _ _ = false

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

    is_some (Quotient_Info.quotdata_lookup_raw thy s)

  | is_real_quot_type _ _ = false

fun is_quot_type ctxt T =

  let val thy = ProofContext.theory_of ctxt in

    is_real_quot_type thy T andalso not (is_codatatype ctxt T)

  end

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

    let val thy = ProofContext.theory_of ctxt in

      is_typedef ctxt s andalso

      not (is_real_datatype thy s orelse is_real_quot_type thy T orelse

           is_codatatype ctxt T orelse is_record_type T orelse

           is_integer_like_type T)

    end

  | is_pure_typedef _ _ = false

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

    (case typedef_info ctxt s of

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

       let

         val t_opt =

           case set_def of

             SOME thm => try (snd o Logic.dest_equals o prop_of) thm

           | NONE => try (snd o HOLogic.dest_mem o HOLogic.dest_Trueprop)

                         prop_of_Rep

       in

         case t_opt of

           SOME (Const (@{const_name top}, _)) => true

           (* "Multiset.multiset" *)

         | SOME (Const (@{const_name Collect}, _)

                 $ Abs (_, _, Const (@{const_name finite}, _) $ _)) => true

           (* "FinFun.finfun" *)

         | SOME (Const (@{const_name Collect}, _) $ Abs (_, _,

                     Const (@{const_name Ex}, _) $ Abs (_, _,

                         Const (@{const_name finite}, _) $ _))) => true

         | _ => false

       end

     | NONE => false)

  | is_univ_typedef _ _ = false

fun is_datatype ctxt stds (T as Type (s, _)) =

    let val thy = ProofContext.theory_of ctxt in

      (is_typedef ctxt s orelse is_codatatype ctxt T orelse

       T = @{typ ind} orelse is_real_quot_type thy T) andalso

      not (is_basic_datatype thy stds s)

    end

  | is_datatype _ _ _ = 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 _ => []

fun is_record_constr (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

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 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 ctxt (s, Type (@{type_name fun}, [_, 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 fun}, [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 fun},

                                         [_, abs_T as Type (s', _)]))) =

    try (Quotient_Term.absrep_const_chk Quotient_Term.AbsF ctxt) s'

    = SOME (Const x) andalso not (is_codatatype ctxt abs_T)

  | is_quot_abs_fun _ _ = false

fun is_quot_rep_fun ctxt (x as (_, Type (@{type_name fun},

                                         [abs_T as Type (s', _), _]))) =

    try (Quotient_Term.absrep_const_chk Quotient_Term.RepF ctxt) s'

    = SOME (Const x) andalso not (is_codatatype ctxt abs_T)

  | is_quot_rep_fun _ _ = false

fun mate_of_rep_fun ctxt (x as (_, Type (@{type_name fun},

                                         [T1 as Type (s', _), T2]))) =

    (case typedef_info ctxt s' of

       SOME {Abs_name, ...} => (Abs_name, Type (@{type_name 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])

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

    let val {qtyp, rtyp, ...} = Quotient_Info.quotdata_lookup thy 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, ...} =

        Quotient_Info.quotdata_lookup thy s

      val partial =

        case prop_of equiv_thm of

          @{const Trueprop} $ (Const (@{const_name equivp}, _) $ _) => false

        | @{const Trueprop} $ (Const (@{const_name part_equivp}, _) $ _) => 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_coconstr ctxt (s, T) =

  case body_type T of

    co_T as Type (co_s, _) =>

    let val {codatatypes, ...} = Data.get (Context.Proof ctxt) in

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

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

    end

  | _ => false

fun is_constr_like ctxt (s, T) =

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

                 @{const_name PairBox}, @{const_name Quot},

                 @{const_name Zero_Rep}, @{const_name Suc_Rep}] s orelse

  let

    val thy = ProofContext.theory_of ctxt

    val (x as (_, T)) = (s, unarize_unbox_etc_type T)

  in

    is_real_constr thy x orelse is_record_constr x orelse

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

    is_coconstr ctxt x

  end

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

  is_codatatype ctxt (body_type T) andalso is_constr_like ctxt x andalso

  not (is_coconstr ctxt x)

fun is_constr ctxt stds (x as (_, T)) =

  let val thy = ProofContext.theory_of ctxt in

    is_constr_like ctxt x andalso

    not (is_basic_datatype thy stds

                         (fst (dest_Type (unarize_type (body_type T))))) andalso

    not (is_stale_constr ctxt x)

  end

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}])

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 fun}, _) =>

    (boxy = InPair orelse boxy = InFunLHS) andalso

    not (is_boolean_type (body_type T))

  | Type (@{type_name prod}, 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 fun}, [T1, T2])) =>

    if boxy <> InConstr andalso boxy <> InSel andalso

       should_box_type hol_ctxt boxy z then

      Type (@{type_name fun_box},

            [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 prod}, Ts)) =>

    if boxy <> InConstr andalso boxy <> InSel

       andalso should_box_type hol_ctxt boxy z then

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

    else

      Type (@{type_name prod},

            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 nat} = @{typ "unsigned_bit word"}

  | binarize_nat_and_int_in_type @{typ int} = @{typ "signed_bit word"}

  | 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 Pair} then

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

  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 snd} then

    1

  else

    0

val close_form =

  let

    fun close_up zs zs' =

      fold (fn (z as ((s, _), T)) => fn t' =>

               Term.all T $ Abs (s, T, abstract_over (Var z, t')))

           (take (length zs' - length zs) zs')

    fun aux zs (@{const "==>"} $ 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 Not} $ (HOLogic.eq_const T $ t1 $ t2))

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

fun zero_const T = Const (@{const_name zero_class.zero}, T)

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

fun uncached_datatype_constrs ({thy, ctxt, stds, ...} : hol_context)

                              (T as Type (s, Ts)) =

    (case AList.lookup (op =) (#codatatypes (Data.get (Context.Proof ctxt)))

                       s of

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

     | _ =>

       if is_datatype ctxt stds T then

         case Datatype.get_info thy s of

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

           let

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

           in

             map (apsnd (fn Us =>

                            map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))

                 constrs

           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_real_quot_type thy T then

             [(@{const_name Quot}, rep_type_for_quot_type thy 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 ind} then

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

             else

               []

       else

         [])

  | uncached_datatype_constrs _ _ = []

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

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

    SOME xs => xs

  | NONE =>

    let val xs = uncached_datatype_constrs hol_ctxt T in

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

    end

fun binarized_and_boxed_datatype_constrs hol_ctxt binarize =

  map (apsnd ((binarize ? binarize_nat_and_int_in_type)

              o box_type hol_ctxt InConstr)) o datatype_constrs hol_ctxt

val num_datatype_constrs = length oo datatype_constrs

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'

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_datatype_constrs hol_ctxt binarize (domain_type T'))

        s

    |> the |> pair s

  end

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

  let val dataT = body_type T in

    if s = @{const_name Suc} then

      Abs (Name.uu, dataT,