src/HOL/Tools/Nitpick/nitpick_kodkod.ML

Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.

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

Kodkod problem generator part of Kodkod.
*)

signature NITPICK_KODKOD =
sig
  type hol_context = Nitpick_HOL.hol_context
  type data_type_spec = Nitpick_Scope.data_type_spec
  type kodkod_constrs = Nitpick_Peephole.kodkod_constrs
  type nut = Nitpick_Nut.nut

  structure NameTable : TABLE

  val univ_card :
    int -> int -> int -> Kodkod.bound list -> Kodkod.formula -> int
  val check_bits : int -> Kodkod.formula -> unit
  val check_arity : string -> int -> int -> unit
  val kk_tuple : bool -> int -> int list -> Kodkod.tuple
  val tuple_set_from_atom_schema : (int * int) list -> Kodkod.tuple_set
  val sequential_int_bounds : int -> Kodkod.int_bound list
  val pow_of_two_int_bounds : int -> int -> Kodkod.int_bound list
  val bounds_and_axioms_for_built_in_rels_in_formulas :
    bool -> int -> int -> int -> int -> Kodkod.formula list
    -> Kodkod.bound list * Kodkod.formula list
  val bound_for_plain_rel : Proof.context -> bool -> nut -> Kodkod.bound
  val bound_for_sel_rel :
    Proof.context -> bool -> (typ * (nut * int) list option) list
    -> data_type_spec list -> nut -> Kodkod.bound
  val merge_bounds : Kodkod.bound list -> Kodkod.bound list
  val kodkod_formula_from_nut :
    int Typtab.table -> kodkod_constrs -> nut -> Kodkod.formula
  val needed_values_for_data_type :
    nut list -> int Typtab.table -> data_type_spec -> (nut * int) list option
  val declarative_axiom_for_plain_rel : kodkod_constrs -> nut -> Kodkod.formula
  val declarative_axioms_for_data_types :
    hol_context -> bool -> nut list -> (typ * (nut * int) list option) list
    -> int -> int -> int Typtab.table -> kodkod_constrs -> nut NameTable.table
    -> data_type_spec list -> Kodkod.formula list
end;

structure Nitpick_Kodkod : NITPICK_KODKOD =
struct

open Nitpick_Util
open Nitpick_HOL
open Nitpick_Scope
open Nitpick_Peephole
open Nitpick_Rep
open Nitpick_Nut

structure KK = Kodkod

fun pull x xs = x :: filter_out (curry (op =) x) xs

fun is_data_type_acyclic ({co = false, deep = true, ...} : data_type_spec) = true
  | is_data_type_acyclic _ = false

fun flip_nums n = index_seq 1 n @ [0] |> map KK.Num

fun univ_card nat_card int_card main_j0 bounds formula =
  let
    fun rel_expr_func r k =
      Int.max (k, case r of
                    KK.Atom j => j + 1
                  | KK.AtomSeq (k', j0) => j0 + k'
                  | _ => 0)
    fun tuple_func t k =
      case t of
        KK.Tuple js => fold Integer.max (map (Integer.add 1) js) k
      | _ => k
    fun tuple_set_func ts k =
      Int.max (k, case ts of KK.TupleAtomSeq (k', j0) => j0 + k' | _ => 0)
    val expr_F = {formula_func = K I, rel_expr_func = rel_expr_func,
                  int_expr_func = K I}
    val tuple_F = {tuple_func = tuple_func, tuple_set_func = tuple_set_func}
    val card = fold (KK.fold_bound expr_F tuple_F) bounds 1
               |> KK.fold_formula expr_F formula
  in Int.max (main_j0 + fold Integer.max [2, nat_card, int_card] 0, card) end

fun check_bits bits formula =
  let
    fun int_expr_func (KK.Num k) () =
        if is_twos_complement_representable bits k then
          ()
        else
          raise TOO_SMALL ("Nitpick_Kodkod.check_bits",
                           "\"bits\" value " ^ string_of_int bits ^
                           " too small for problem")
      | int_expr_func _ () = ()
    val expr_F = {formula_func = K I, rel_expr_func = K I,
                  int_expr_func = int_expr_func}
  in KK.fold_formula expr_F formula () end

fun check_arity guilty_party univ_card n =
  if n > KK.max_arity univ_card then
    raise TOO_LARGE ("Nitpick_Kodkod.check_arity",
                     "arity " ^ string_of_int n ^
                     (if guilty_party = "" then
                        ""
                      else
                        " of Kodkod relation associated with " ^
                        quote (original_name guilty_party)) ^
                     " too large for a universe of size " ^
                     string_of_int univ_card)
  else
    ()

fun kk_tuple debug univ_card js =
  if debug then
    KK.Tuple js
  else
    KK.TupleIndex (length js,
                   fold (fn j => fn accum => accum * univ_card + j) js 0)

(* This code is not just a silly optimization: It works around a limitation in
   Kodkodi, whereby {} (i.e., KK.TupleProduct) is always given the cardinality
   of the bound in which it appears, resulting in Kodkod arity errors. *)
fun tuple_product (ts as KK.TupleSet []) _ = ts
  | tuple_product _ (ts as KK.TupleSet []) = ts
  | tuple_product ts1 ts2 = KK.TupleProduct (ts1, ts2)

val tuple_set_from_atom_schema = fold1 tuple_product o map KK.TupleAtomSeq
val upper_bound_for_rep = tuple_set_from_atom_schema o atom_schema_of_rep

val single_atom = KK.TupleSet o single o KK.Tuple o single

fun sequential_int_bounds n = [(NONE, map single_atom (index_seq 0 n))]

fun pow_of_two_int_bounds bits j0 =
  let
    fun aux 0  _ _ = []
      | aux 1 pow_of_two j = [(SOME (~ pow_of_two), [single_atom j])]
      | aux iter pow_of_two j =
        (SOME pow_of_two, [single_atom j]) ::
        aux (iter - 1) (2 * pow_of_two) (j + 1)
  in aux (bits + 1) 1 j0 end

fun built_in_rels_in_formulas formulas =
  let
    fun rel_expr_func (KK.Rel (x as (_, j))) =
        (j < 0 andalso x <> unsigned_bit_word_sel_rel andalso
         x <> signed_bit_word_sel_rel)
        ? insert (op =) x
      | rel_expr_func _ = I
    val expr_F = {formula_func = K I, rel_expr_func = rel_expr_func,
                  int_expr_func = K I}
  in fold (KK.fold_formula expr_F) formulas [] end

val max_table_size = 65536

fun check_table_size k =
  if k > max_table_size then
    raise TOO_LARGE ("Nitpick_Kodkod.check_table_size",
                     "precomputed table too large (" ^ string_of_int k ^ ")")
  else
    ()

fun tabulate_func1 debug univ_card (k, j0) f =
  (check_table_size k;
   map_filter (fn j1 => let val j2 = f j1 in
                          if j2 >= 0 then
                            SOME (kk_tuple debug univ_card [j1 + j0, j2 + j0])
                          else
                            NONE
                        end) (index_seq 0 k))

fun tabulate_op2 debug univ_card (k, j0) res_j0 f =
  (check_table_size (k * k);
   map_filter (fn j => let
                         val j1 = j div k
                         val j2 = j - j1 * k
                         val j3 = f (j1, j2)
                       in
                         if j3 >= 0 then
                           SOME (kk_tuple debug univ_card
                                          [j1 + j0, j2 + j0, j3 + res_j0])
                         else
                           NONE
                       end) (index_seq 0 (k * k)))

fun tabulate_op2_2 debug univ_card (k, j0) res_j0 f =
  (check_table_size (k * k);
   map_filter (fn j => let
                         val j1 = j div k
                         val j2 = j - j1 * k
                         val (j3, j4) = f (j1, j2)
                       in
                         if j3 >= 0 andalso j4 >= 0 then
                           SOME (kk_tuple debug univ_card
                                          [j1 + j0, j2 + j0, j3 + res_j0,
                                           j4 + res_j0])
                         else
                           NONE
                       end) (index_seq 0 (k * k)))

fun tabulate_nat_op2 debug univ_card (k, j0) f =
  tabulate_op2 debug univ_card (k, j0) j0 (atom_for_nat (k, 0) o f)

fun tabulate_int_op2 debug univ_card (k, j0) f =
  tabulate_op2 debug univ_card (k, j0) j0
               (atom_for_int (k, 0) o f o apply2 (int_for_atom (k, 0)))

fun tabulate_int_op2_2 debug univ_card (k, j0) f =
  tabulate_op2_2 debug univ_card (k, j0) j0
                 (apply2 (atom_for_int (k, 0)) o f
                  o apply2 (int_for_atom (k, 0)))

fun isa_div (m, n) = m div n handle General.Div => 0
fun isa_mod (m, n) = m mod n handle General.Div => m

fun isa_gcd (m, 0) = m
  | isa_gcd (m, n) = isa_gcd (n, isa_mod (m, n))

fun isa_lcm (m, n) = isa_div (m * n, isa_gcd (m, n))
val isa_zgcd = isa_gcd o apply2 abs

fun isa_norm_frac (m, n) =
  if n < 0 then isa_norm_frac (~m, ~n)
  else if m = 0 orelse n = 0 then (0, 1)
  else let val p = isa_zgcd (m, n) in (isa_div (m, p), isa_div (n, p)) end

fun tabulate_built_in_rel debug univ_card nat_card int_card j0
                          (x as (n, _)) =
  (check_arity "" univ_card n;
   if x = not3_rel then
     ("not3", tabulate_func1 debug univ_card (2, j0) (curry (op -) 1))
   else if x = suc_rel then
     ("suc", tabulate_func1 debug univ_card (univ_card - j0 - 1, j0)
                            (Integer.add 1))
   else if x = nat_add_rel then
     ("nat_add", tabulate_nat_op2 debug univ_card (nat_card, j0) (op +))
   else if x = int_add_rel then
     ("int_add", tabulate_int_op2 debug univ_card (int_card, j0) (op +))
   else if x = nat_subtract_rel then
     ("nat_subtract",
      tabulate_op2 debug univ_card (nat_card, j0) j0 (uncurry nat_minus))
   else if x = int_subtract_rel then
     ("int_subtract", tabulate_int_op2 debug univ_card (int_card, j0) (op -))
   else if x = nat_multiply_rel then
     ("nat_multiply", tabulate_nat_op2 debug univ_card (nat_card, j0) (op * ))
   else if x = int_multiply_rel then
     ("int_multiply", tabulate_int_op2 debug univ_card (int_card, j0) (op * ))
   else if x = nat_divide_rel then
     ("nat_divide", tabulate_nat_op2 debug univ_card (nat_card, j0) isa_div)
   else if x = int_divide_rel then
     ("int_divide", tabulate_int_op2 debug univ_card (int_card, j0) isa_div)
   else if x = nat_less_rel then
     ("nat_less", tabulate_nat_op2 debug univ_card (nat_card, j0)
                                   (int_from_bool o op <))
   else if x = int_less_rel then
     ("int_less", tabulate_int_op2 debug univ_card (int_card, j0)
                                   (int_from_bool o op <))
   else if x = gcd_rel then
     ("gcd", tabulate_nat_op2 debug univ_card (nat_card, j0) isa_gcd)
   else if x = lcm_rel then
     ("lcm", tabulate_nat_op2 debug univ_card (nat_card, j0) isa_lcm)
   else if x = norm_frac_rel then
     ("norm_frac", tabulate_int_op2_2 debug univ_card (int_card, j0)
                                      isa_norm_frac)
   else
     raise ARG ("Nitpick_Kodkod.tabulate_built_in_rel", "unknown relation"))

fun bound_for_built_in_rel debug univ_card nat_card int_card main_j0
                           (x as (n, j)) =
  if n = 2 andalso j <= suc_rels_base then
    let val (y as (k, j0), tabulate) = atom_seq_for_suc_rel x in
      ([(x, "suc")],
       if tabulate then
         [KK.TupleSet (tabulate_func1 debug univ_card (k - 1, j0)
                       (Integer.add 1))]
       else
         [KK.TupleSet [], tuple_set_from_atom_schema [y, y]])
    end
  else
    let
      val (nick, ts) = tabulate_built_in_rel debug univ_card nat_card int_card
                                             main_j0 x
    in ([(x, nick)], [KK.TupleSet ts]) end

fun axiom_for_built_in_rel (x as (n, j)) =
  if n = 2 andalso j <= suc_rels_base then
    let val (y as (k, j0), tabulate) = atom_seq_for_suc_rel x in
      if tabulate then
        NONE
      else if k < 2 then
        SOME (KK.No (KK.Rel x))
      else
        SOME (KK.TotalOrdering (x, KK.AtomSeq y, KK.Atom j0, KK.Atom (j0 + 1)))
    end
  else
    NONE

fun bounds_and_axioms_for_built_in_rels_in_formulas debug univ_card nat_card
                                                    int_card main_j0 formulas =
  let val rels = built_in_rels_in_formulas formulas in
    (map (bound_for_built_in_rel debug univ_card nat_card int_card main_j0)
         rels,
     map_filter axiom_for_built_in_rel rels)
  end

fun bound_comment ctxt debug nick T R =
  short_name nick ^
  (if debug then " :: " ^ YXML.content_of (Syntax.string_of_typ ctxt T) else "") ^
  " : " ^ string_for_rep R

fun bound_for_plain_rel ctxt debug (u as FreeRel (x, T, R, nick)) =
    ([(x, bound_comment ctxt debug nick T R)],
     if nick = @{const_name bisim_iterator_max} then
       case R of
         Atom (k, j0) => [single_atom (k - 1 + j0)]
       | _ => raise NUT ("Nitpick_Kodkod.bound_for_plain_rel", [u])
     else
       [KK.TupleSet [], upper_bound_for_rep R])
  | bound_for_plain_rel _ _ u =
    raise NUT ("Nitpick_Kodkod.bound_for_plain_rel", [u])

fun is_data_type_nat_like ({typ, constrs, ...} : data_type_spec) =
  case constrs of
    [_, _] =>
    (case constrs |> map (snd o #const) |> List.partition is_fun_type of
       ([Type (_, Ts1)], [T2]) => forall (curry (op =) typ) (T2 :: Ts1)
     | _ => false)
  | _ => false

fun needed_values need_vals T =
  AList.lookup (op =) need_vals T |> the_default NONE |> these

fun all_values_are_needed need_vals ({typ, card, ...} : data_type_spec) =
  length (needed_values need_vals typ) = card

fun is_sel_of_constr x (Construct (sel_us, _, _, _), _) =
    exists (fn FreeRel (x', _, _, _) => x = x' | _ => false) sel_us
  | is_sel_of_constr _ _ = false

fun bound_for_sel_rel ctxt debug need_vals dtypes
        (FreeRel (x, T as Type (@{type_name fun}, [T1, T2]),
                  R as Func (Atom (_, j0), R2), nick)) =
    let
      val constr_s = original_name nick
      val {delta, epsilon, exclusive, explicit_max, ...} =
        constr_spec dtypes (constr_s, T1)
      val dtype as {card, ...} = data_type_spec dtypes T1 |> the
      val T1_need_vals = needed_values need_vals T1
    in
      ([(x, bound_comment ctxt debug nick T R)],
       let
         val discr = (R2 = Formula Neut)
         val complete_need_vals = (length T1_need_vals = card)
         val (my_need_vals, other_need_vals) =
           T1_need_vals |> List.partition (is_sel_of_constr x)
         fun atom_seq_for_self_rec j =
           if is_data_type_nat_like dtype then (1, j + j0 - 1) else (j, j0)
         fun exact_bound_for_perhaps_needy_atom j =
           case AList.find (op =) my_need_vals j of
             [constr_u] =>
             let
               val n = sel_no_from_name nick
               val arg_u =
                 case constr_u of
                   Construct (_, _, _, arg_us) => nth arg_us n
                 | _ => raise Fail "expected \"Construct\""
               val T2_need_vals = needed_values need_vals T2
             in
               case AList.lookup (op =) T2_need_vals arg_u of
                 SOME arg_j => SOME (KK.TupleAtomSeq (1, arg_j))
               | _ => NONE
             end
           | _ => NONE
         fun n_fold_tuple_union [] = KK.TupleSet []
           | n_fold_tuple_union (ts :: tss) =
             fold (curry (KK.TupleUnion o swap)) tss ts
         fun tuple_perhaps_needy_atom upper j =
           single_atom j
           |> not discr
              ? (fn ts => tuple_product ts
                              (case exact_bound_for_perhaps_needy_atom j of
                                 SOME ts => ts
                               | NONE => if upper then upper_bound_for_rep R2
                                         else KK.TupleSet []))
         fun bound_tuples upper =
           if null T1_need_vals then
             if upper then
               KK.TupleAtomSeq (epsilon - delta, delta + j0)
               |> not discr
                  ? (fn ts => tuple_product ts (upper_bound_for_rep R2))
             else
               KK.TupleSet []
           else
             (if complete_need_vals then
                my_need_vals |> map snd
              else
                index_seq (delta + j0) (epsilon - delta)
                |> filter_out (member (op = o apsnd snd) other_need_vals))
             |> map (tuple_perhaps_needy_atom upper)
             |> n_fold_tuple_union
       in
         if explicit_max = 0 orelse
            (complete_need_vals andalso null my_need_vals) then
           [KK.TupleSet []]
         else
           if discr then
             [bound_tuples true]
             |> not (exclusive orelse all_values_are_needed need_vals dtype)
                ? cons (KK.TupleSet [])
           else
             [bound_tuples false,
              if T1 = T2 andalso epsilon > delta andalso
                 is_data_type_acyclic dtype then
                index_seq delta (epsilon - delta)
                |> map (fn j => tuple_product (KK.TupleSet [KK.Tuple [j + j0]])
                                    (KK.TupleAtomSeq (atom_seq_for_self_rec j)))
                |> n_fold_tuple_union
              else
                bound_tuples true]
             |> distinct (op =)
         end)
    end
  | bound_for_sel_rel _ _ _ _ u =
    raise NUT ("Nitpick_Kodkod.bound_for_sel_rel", [u])

fun merge_bounds bs =
  let
    fun arity (zs, _) = fst (fst (hd zs))
    fun add_bound ds b [] = List.revAppend (ds, [b])
      | add_bound ds b (c :: cs) =
        if arity b = arity c andalso snd b = snd c then
          List.revAppend (ds, (fst c @ fst b, snd c) :: cs)
        else
          add_bound (c :: ds) b cs
  in fold (add_bound []) bs [] end

fun unary_var_seq j0 n = map (curry KK.Var 1) (index_seq j0 n)

val singleton_from_combination = foldl1 KK.Product o map KK.Atom

fun all_singletons_for_rep R =
  if is_lone_rep R then
    all_combinations_for_rep R |> map singleton_from_combination
  else
    raise REP ("Nitpick_Kodkod.all_singletons_for_rep", [R])

fun unpack_products (KK.Product (r1, r2)) =
    unpack_products r1 @ unpack_products r2
  | unpack_products r = [r]

fun unpack_joins (KK.Join (r1, r2)) = unpack_joins r1 @ unpack_joins r2
  | unpack_joins r = [r]

val empty_rel_for_rep = empty_n_ary_rel o arity_of_rep

fun full_rel_for_rep R =
  case atom_schema_of_rep R of
    [] => raise REP ("Nitpick_Kodkod.full_rel_for_rep", [R])
  | schema => foldl1 KK.Product (map KK.AtomSeq schema)

fun decls_for_atom_schema j0 schema =
  map2 (fn j => fn x => KK.DeclOne ((1, j), KK.AtomSeq x))
       (index_seq j0 (length schema)) schema

fun d_n_ary_function ({kk_all, kk_join, kk_lone, kk_one, ...} : kodkod_constrs)
                     R r =
  let val body_R = body_rep R in
    if is_lone_rep body_R then
      let
        val binder_schema = atom_schema_of_reps (binder_reps R)
        val body_schema = atom_schema_of_rep body_R
        val one = is_one_rep body_R
        val opt_x = case r of KK.Rel x => SOME x | _ => NONE
      in
        if opt_x <> NONE andalso length binder_schema = 1 andalso
           length body_schema = 1 then
          (if one then KK.Function else KK.Functional)
              (the opt_x, KK.AtomSeq (hd binder_schema),
               KK.AtomSeq (hd body_schema))
        else
          let
            val decls = decls_for_atom_schema ~1 binder_schema
            val vars = unary_var_seq ~1 (length binder_schema)
            val kk_xone = if one then kk_one else kk_lone
          in kk_all decls (kk_xone (fold kk_join vars r)) end
      end
    else
      KK.True
  end

fun kk_n_ary_function kk R (r as KK.Rel x) =
    if not (is_opt_rep R) then
      if x = suc_rel then
        KK.False
      else if x = nat_add_rel then
        formula_for_bool (card_of_rep (body_rep R) = 1)
      else if x = nat_multiply_rel then
        formula_for_bool (card_of_rep (body_rep R) <= 2)
      else
        d_n_ary_function kk R r
    else if x = nat_subtract_rel then
      KK.True
    else
      d_n_ary_function kk R r
  | kk_n_ary_function kk R r = d_n_ary_function kk R r

fun kk_disjoint_sets _ [] = KK.True
  | kk_disjoint_sets (kk as {kk_and, kk_no, kk_intersect, ...} : kodkod_constrs)
                     (r :: rs) =
    fold (kk_and o kk_no o kk_intersect r) rs (kk_disjoint_sets kk rs)

fun basic_rel_rel_let j ({kk_rel_let, ...} : kodkod_constrs) f r =
  if inline_rel_expr r then
    f r
  else
    let val x = (KK.arity_of_rel_expr r, j) in
      kk_rel_let [KK.AssignRelReg (x, r)] (f (KK.RelReg x))
    end

val single_rel_rel_let = basic_rel_rel_let 0

fun double_rel_rel_let kk f r1 r2 =
  single_rel_rel_let kk (fn r1 => basic_rel_rel_let 1 kk (f r1) r2) r1

fun triple_rel_rel_let kk f r1 r2 r3 =
  double_rel_rel_let kk
      (fn r1 => fn r2 => basic_rel_rel_let 2 kk (f r1 r2) r3) r1 r2

fun atom_from_formula ({kk_rel_if, ...} : kodkod_constrs) j0 f =
  kk_rel_if f (KK.Atom (j0 + 1)) (KK.Atom j0)

fun rel_expr_from_formula kk R f =
  case unopt_rep R of
    Atom (2, j0) => atom_from_formula kk j0 f
  | _ => raise REP ("Nitpick_Kodkod.rel_expr_from_formula", [R])

fun unpack_vect_in_chunks ({kk_project_seq, ...} : kodkod_constrs) chunk_arity
                          num_chunks r =
  List.tabulate (num_chunks, fn j => kk_project_seq r (j * chunk_arity)
                                                    chunk_arity)

fun kk_n_fold_join
        (kk as {kk_intersect, kk_product, kk_join, kk_project_seq, ...}) one R1
        res_R r1 r2 =
  case arity_of_rep R1 of
    1 => kk_join r1 r2
  | arity1 =>
    let val unpacked_rs1 = unpack_products r1 in
      if one andalso length unpacked_rs1 = arity1 then
        fold kk_join unpacked_rs1 r2
      else if one andalso inline_rel_expr r1 then
        fold kk_join (unpack_vect_in_chunks kk 1 arity1 r1) r2
      else
        kk_project_seq
            (kk_intersect (kk_product r1 (full_rel_for_rep res_R)) r2)
            arity1 (arity_of_rep res_R)
    end

fun kk_case_switch (kk as {kk_union, kk_product, ...}) R1 R2 r rs1 rs2 =
  if rs1 = rs2 then r
  else kk_n_fold_join kk true R1 R2 r (fold1 kk_union (map2 kk_product rs1 rs2))

val lone_rep_fallback_max_card = 4096
val some_j0 = 0

fun lone_rep_fallback kk new_R old_R r =
  if old_R = new_R then
    r
  else
    let val card = card_of_rep old_R in
      if is_lone_rep old_R andalso is_lone_rep new_R andalso
         card = card_of_rep new_R then
        if card >= lone_rep_fallback_max_card then
          raise TOO_LARGE ("Nitpick_Kodkod.lone_rep_fallback",
                           "too high cardinality (" ^ string_of_int card ^ ")")
        else
          kk_case_switch kk old_R new_R r (all_singletons_for_rep old_R)
                         (all_singletons_for_rep new_R)
      else
        raise REP ("Nitpick_Kodkod.lone_rep_fallback", [old_R, new_R])
    end
and atom_from_rel_expr kk x old_R r =
  case old_R of
    Func (R1, R2) =>
    let
      val dom_card = card_of_rep R1
      val R2' = case R2 of Atom _ => R2 | _ => Atom (card_of_rep R2, some_j0)
    in
      atom_from_rel_expr kk x (Vect (dom_card, R2'))
                         (vect_from_rel_expr kk dom_card R2' old_R r)
    end
  | Opt _ => raise REP ("Nitpick_Kodkod.atom_from_rel_expr", [old_R])
  | _ => lone_rep_fallback kk (Atom x) old_R r
and struct_from_rel_expr kk Rs old_R r =
  case old_R of
    Atom _ => lone_rep_fallback kk (Struct Rs) old_R r
  | Struct Rs' =>
    if Rs' = Rs then
      r
    else if map card_of_rep Rs' = map card_of_rep Rs then
      let
        val old_arities = map arity_of_rep Rs'
        val old_offsets = offset_list old_arities
        val old_rs = map2 (#kk_project_seq kk r) old_offsets old_arities
      in
        fold1 (#kk_product kk)
              (@{map 3} (rel_expr_from_rel_expr kk) Rs Rs' old_rs)
      end
    else
      lone_rep_fallback kk (Struct Rs) old_R r
  | _ => raise REP ("Nitpick_Kodkod.struct_from_rel_expr", [old_R])
and vect_from_rel_expr kk k R old_R r =
  case old_R of
    Atom _ => lone_rep_fallback kk (Vect (k, R)) old_R r
  | Vect (k', R') =>
    if k = k' andalso R = R' then r
    else lone_rep_fallback kk (Vect (k, R)) old_R r
  | Func (R1, Formula Neut) =>
    if k = card_of_rep R1 then
      fold1 (#kk_product kk)
            (map (fn arg_r =>
                     rel_expr_from_formula kk R (#kk_subset kk arg_r r))
                 (all_singletons_for_rep R1))
    else
      raise REP ("Nitpick_Kodkod.vect_from_rel_expr", [old_R])
  | Func (R1, R2) =>
    fold1 (#kk_product kk)
          (map (fn arg_r =>
                   rel_expr_from_rel_expr kk R R2
                                         (kk_n_fold_join kk true R1 R2 arg_r r))
               (all_singletons_for_rep R1))
  | _ => raise REP ("Nitpick_Kodkod.vect_from_rel_expr", [old_R])
and func_from_no_opt_rel_expr kk R1 R2 (Atom x) r =
    let
      val dom_card = card_of_rep R1
      val R2' = case R2 of Atom _ => R2 | _ => Atom (card_of_rep R2, some_j0)
    in
      func_from_no_opt_rel_expr kk R1 R2 (Vect (dom_card, R2'))
                                (vect_from_rel_expr kk dom_card R2' (Atom x) r)
    end
  | func_from_no_opt_rel_expr kk R1 (Formula Neut) old_R r =
    (case old_R of
       Vect (k, Atom (2, j0)) =>
       let
         val args_rs = all_singletons_for_rep R1
         val vals_rs = unpack_vect_in_chunks kk 1 k r
         fun empty_or_singleton_set_for arg_r val_r =
           #kk_join kk val_r (#kk_product kk (KK.Atom (j0 + 1)) arg_r)
       in
         fold1 (#kk_union kk) (map2 empty_or_singleton_set_for args_rs vals_rs)
       end
     | Func (R1', Formula Neut) =>
       if R1 = R1' then
         r
       else
         let
           val schema = atom_schema_of_rep R1
           val r1 = fold1 (#kk_product kk) (unary_var_seq ~1 (length schema))
                    |> rel_expr_from_rel_expr kk R1' R1
           val kk_xeq = (if is_one_rep R1' then #kk_subset else #kk_rel_eq) kk
         in
           #kk_comprehension kk (decls_for_atom_schema ~1 schema) (kk_xeq r1 r)
         end
     | Func (R1', Atom (2, j0)) =>
       func_from_no_opt_rel_expr kk R1 (Formula Neut)
           (Func (R1', Formula Neut)) (#kk_join kk r (KK.Atom (j0 + 1)))
     | _ => raise REP ("Nitpick_Kodkod.func_from_no_opt_rel_expr",
                       [old_R, Func (R1, Formula Neut)]))
  | func_from_no_opt_rel_expr kk R1 R2 old_R r =
    case old_R of
      Vect (k, R) =>
      let
        val args_rs = all_singletons_for_rep R1
        val vals_rs = unpack_vect_in_chunks kk (arity_of_rep R) k r
                      |> map (rel_expr_from_rel_expr kk R2 R)
      in fold1 (#kk_union kk) (map2 (#kk_product kk) args_rs vals_rs) end
    | Func (R1', Formula Neut) =>
      (case R2 of
         Atom (x as (2, j0)) =>
         let val schema = atom_schema_of_rep R1 in
           if length schema = 1 then
             #kk_override kk (#kk_product kk (KK.AtomSeq (hd schema))
                                             (KK.Atom j0))
                             (#kk_product kk r (KK.Atom (j0 + 1)))
           else
             let
               val r1 = fold1 (#kk_product kk) (unary_var_seq ~1 (length schema))
                        |> rel_expr_from_rel_expr kk R1' R1
               val r2 = KK.Var (1, ~(length schema) - 1)
               val r3 = atom_from_formula kk j0 (#kk_subset kk r1 r)
             in
               #kk_comprehension kk (decls_for_atom_schema ~1 (schema @ [x]))
                                 (#kk_subset kk r2 r3)
             end
           end
         | _ => raise REP ("Nitpick_Kodkod.func_from_no_opt_rel_expr",
                           [old_R, Func (R1, R2)]))
    | Func (R1', R2') =>
      if R1 = R1' andalso R2 = R2' then
        r
      else
        let
          val dom_schema = atom_schema_of_rep R1
          val ran_schema = atom_schema_of_rep R2
          val dom_prod = fold1 (#kk_product kk)
                               (unary_var_seq ~1 (length dom_schema))
                         |> rel_expr_from_rel_expr kk R1' R1
          val ran_prod = fold1 (#kk_product kk)
                               (unary_var_seq (~(length dom_schema) - 1)
                                              (length ran_schema))
                         |> rel_expr_from_rel_expr kk R2' R2
          val app = kk_n_fold_join kk true R1' R2' dom_prod r
          val kk_xeq = (if is_one_rep R2' then #kk_subset else #kk_rel_eq) kk
        in
          #kk_comprehension kk (decls_for_atom_schema ~1
                                                      (dom_schema @ ran_schema))
                               (kk_xeq ran_prod app)
        end
    | _ => raise REP ("Nitpick_Kodkod.func_from_no_opt_rel_expr",
                      [old_R, Func (R1, R2)])
and rel_expr_from_rel_expr kk new_R old_R r =
  let
    val unopt_old_R = unopt_rep old_R
    val unopt_new_R = unopt_rep new_R
  in
    if unopt_old_R <> old_R andalso unopt_new_R = new_R then
      raise REP ("Nitpick_Kodkod.rel_expr_from_rel_expr", [old_R, new_R])
    else if unopt_new_R = unopt_old_R then
      r
    else
      (case unopt_new_R of
         Atom x => atom_from_rel_expr kk x
       | Struct Rs => struct_from_rel_expr kk Rs
       | Vect (k, R') => vect_from_rel_expr kk k R'
       | Func (R1, R2) => func_from_no_opt_rel_expr kk R1 R2
       | _ => raise REP ("Nitpick_Kodkod.rel_expr_from_rel_expr",
                         [old_R, new_R]))
          unopt_old_R r
  end
and rel_expr_to_func kk R1 R2 = rel_expr_from_rel_expr kk (Func (R1, R2))

fun bit_set_from_atom ({kk_join, ...} : kodkod_constrs) T r =
  kk_join r (KK.Rel (if T = @{typ "unsigned_bit word"} then
                       unsigned_bit_word_sel_rel
                     else
                       signed_bit_word_sel_rel))

val int_expr_from_atom = KK.SetSum ooo bit_set_from_atom

fun atom_from_int_expr (kk as {kk_rel_eq, kk_comprehension, ...}
                        : kodkod_constrs) T R i =
  kk_comprehension (decls_for_atom_schema ~1 (atom_schema_of_rep R))
                   (kk_rel_eq (bit_set_from_atom kk T (KK.Var (1, ~1)))
                              (KK.Bits i))

fun kodkod_formula_from_nut ofs
        (kk as {kk_all, kk_exist, kk_formula_let, kk_formula_if, kk_or, kk_not,
                kk_iff, kk_implies, kk_and, kk_subset, kk_rel_eq, kk_no,
                kk_lone, kk_some, kk_rel_let, kk_rel_if, kk_union,
                kk_difference, kk_intersect, kk_product, kk_join, kk_closure,
                kk_comprehension, kk_project, kk_project_seq, kk_not3,
                kk_nat_less, kk_int_less, ...}) u =
  let
    val main_j0 = offset_of_type ofs bool_T
    val bool_j0 = main_j0
    val bool_atom_R = Atom (2, main_j0)
    val false_atom = KK.Atom bool_j0
    val true_atom = KK.Atom (bool_j0 + 1)
    fun formula_from_opt_atom polar j0 r =
      case polar of
        Neg => kk_not (kk_rel_eq r (KK.Atom j0))
      | _ => kk_rel_eq r (KK.Atom (j0 + 1))
    val formula_from_atom = formula_from_opt_atom Pos
    val unpack_formulas =
      map (formula_from_atom bool_j0) oo unpack_vect_in_chunks kk 1
    fun kk_vect_set_bool_op connective k r1 r2 =
      fold1 kk_and (map2 connective (unpack_formulas k r1)
                         (unpack_formulas k r2))
    fun to_f u =
      case rep_of u of
        Formula polar =>
        (case u of
           Cst (False, _, _) => KK.False
         | Cst (True, _, _) => KK.True
         | Op1 (Not, _, _, u1) =>
           kk_not (to_f_with_polarity (flip_polarity polar) u1)
         | Op1 (Finite, _, _, u1) =>
           let val opt1 = is_opt_rep (rep_of u1) in
             case polar of
               Neut =>
               if opt1 then raise NUT ("Nitpick_Kodkod.to_f (Finite)", [u])
               else KK.True (* sound? *)
             | Pos => KK.False
             | Neg => KK.True
           end
         | Op1 (IsUnknown, _, _, u1) => kk_no (to_r u1)
         | Op1 (Cast, _, _, u1) => to_f_with_polarity polar u1
         | Op2 (All, _, _, u1, u2) =>
           kk_all (untuple to_decl u1) (to_f_with_polarity polar u2)
         | Op2 (Exist, _, _, u1, u2) =>
           kk_exist (untuple to_decl u1) (to_f_with_polarity polar u2)
         | Op2 (Or, _, _, u1, u2) =>
           kk_or (to_f_with_polarity polar u1) (to_f_with_polarity polar u2)
         | Op2 (And, _, _, u1, u2) =>
           kk_and (to_f_with_polarity polar u1) (to_f_with_polarity polar u2)
         | Op2 (Less, T, Formula polar, u1, u2) =>
           formula_from_opt_atom polar bool_j0
               (to_r (Op2 (Less, T, Opt bool_atom_R, u1, u2)))
         | Op2 (DefEq, _, _, u1, u2) =>
           (case min_rep (rep_of u1) (rep_of u2) of
              Formula polar =>
              kk_iff (to_f_with_polarity polar u1) (to_f_with_polarity polar u2)
            | min_R =>
              let
                fun args (Op2 (Apply, _, _, u1, u2)) = u2 :: args u1
                  | args (Tuple (_, _, us)) = us
                  | args _ = []
                val opt_arg_us = filter (is_opt_rep o rep_of) (args u1)
              in
                if null opt_arg_us orelse not (is_Opt min_R) orelse
                   is_eval_name u1 then
                  fold (kk_or o (kk_no o to_r)) opt_arg_us
                       (kk_rel_eq (to_rep min_R u1) (to_rep min_R u2))
                else
                  kk_subset (to_rep min_R u1) (to_rep min_R u2)
              end)
         | Op2 (Eq, _, _, u1, u2) =>
           (case min_rep (rep_of u1) (rep_of u2) of
              Formula polar =>
              kk_iff (to_f_with_polarity polar u1) (to_f_with_polarity polar u2)
            | min_R =>
              if is_opt_rep min_R then
                if polar = Neut then
                  (* continuation of hackish optimization *)
                  kk_rel_eq (to_rep min_R u1) (to_rep min_R u2)
                else if is_Cst Unrep u1 then
                  to_could_be_unrep (polar = Neg) u2
                else if is_Cst Unrep u2 then
                  to_could_be_unrep (polar = Neg) u1
                else
                  let
                    val r1 = to_rep min_R u1
                    val r2 = to_rep min_R u2
                    val both_opt = forall (is_opt_rep o rep_of) [u1, u2]
                  in
                    (if polar = Pos then
                       if not both_opt then
                         kk_rel_eq r1 r2
                       else if is_lone_rep min_R andalso
                               arity_of_rep min_R = 1 then
                         kk_some (kk_intersect r1 r2)
                       else
                         raise SAME ()
                     else
                       if is_lone_rep min_R then
                         if arity_of_rep min_R = 1 then
                           kk_lone (kk_union r1 r2)
                         else if not both_opt then
                           (r1, r2) |> is_opt_rep (rep_of u2) ? swap
                                    |-> kk_subset
                         else
                           raise SAME ()
                       else
                         raise SAME ())
                    handle SAME () =>
                           formula_from_opt_atom polar bool_j0
                               (to_guard [u1, u2] bool_atom_R
                                         (rel_expr_from_formula kk bool_atom_R
                                                            (kk_rel_eq r1 r2)))
                  end
              else
                let
                  val r1 = to_rep min_R u1
                  val r2 = to_rep min_R u2
                in
                  if is_one_rep min_R then
                    let
                      val rs1 = unpack_products r1
                      val rs2 = unpack_products r2
                    in
                      if length rs1 = length rs2 andalso
                         map KK.arity_of_rel_expr rs1
                         = map KK.arity_of_rel_expr rs2 then
                        fold1 kk_and (map2 kk_subset rs1 rs2)
                      else
                        kk_subset r1 r2
                    end
                  else
                    kk_rel_eq r1 r2
                end)
         | Op2 (Apply, T, _, u1, u2) =>
           (case (polar, rep_of u1) of
              (Neg, Func (R, Formula Neut)) => kk_subset (to_opt R u2) (to_r u1)
            | _ =>
              to_f_with_polarity polar
                 (Op2 (Apply, T, Opt (Atom (2, offset_of_type ofs T)), u1, u2)))
         | Op3 (Let, _, _, u1, u2, u3) =>
           kk_formula_let [to_expr_assign u1 u2] (to_f_with_polarity polar u3)
         | Op3 (If, _, _, u1, u2, u3) =>
           kk_formula_if (to_f u1) (to_f_with_polarity polar u2)
                         (to_f_with_polarity polar u3)
         | FormulaReg (j, _, _) => KK.FormulaReg j
         | _ => raise NUT ("Nitpick_Kodkod.to_f", [u]))
      | Atom (2, j0) => formula_from_atom j0 (to_r u)
      | _ => raise NUT ("Nitpick_Kodkod.to_f", [u])
    and to_f_with_polarity polar u =
      case rep_of u of
        Formula _ => to_f u
      | Atom (2, j0) => formula_from_atom j0 (to_r u)
      | Opt (Atom (2, j0)) => formula_from_opt_atom polar j0 (to_r u)
      | _ => raise NUT ("Nitpick_Kodkod.to_f_with_polarity", [u])
    and to_r u =
      case u of
        Cst (False, _, Atom _) => false_atom
      | Cst (True, _, Atom _) => true_atom
      | Cst (Iden, _, Func (Struct [R1, R2], Formula Neut)) =>
        if R1 = R2 andalso arity_of_rep R1 = 1 then
          kk_intersect KK.Iden (kk_product (full_rel_for_rep R1) KK.Univ)
        else
          let
            val schema1 = atom_schema_of_rep R1
            val schema2 = atom_schema_of_rep R2
            val arity1 = length schema1
            val arity2 = length schema2
            val r1 = fold1 kk_product (unary_var_seq 0 arity1)
            val r2 = fold1 kk_product (unary_var_seq arity1 arity2)
            val min_R = min_rep R1 R2
          in
            kk_comprehension
                (decls_for_atom_schema 0 (schema1 @ schema2))
                (kk_rel_eq (rel_expr_from_rel_expr kk min_R R1 r1)
                           (rel_expr_from_rel_expr kk min_R R2 r2))
          end
      | Cst (Iden, _, Func (Atom (1, j0), Formula Neut)) => KK.Atom j0
      | Cst (Iden, T as Type (@{type_name set}, [T1]), R as Func (R1, _)) =>
        to_rep R (Cst (Iden, T, Func (one_rep ofs T1 R1, Formula Neut)))
      | Cst (Num j, T, R) =>
        if is_word_type T then
          atom_from_int_expr kk T R (KK.Num j)
        else if T = int_T then
          case atom_for_int (card_of_rep R, offset_of_type ofs int_T) j of
            ~1 => if is_opt_rep R then KK.None
                  else raise NUT ("Nitpick_Kodkod.to_r (Num)", [u])
          | j' => KK.Atom j'
        else
          if j < card_of_rep R then KK.Atom (j + offset_of_type ofs T)
          else if is_opt_rep R then KK.None
          else raise NUT ("Nitpick_Kodkod.to_r (Num)", [u])
      | Cst (Unknown, _, R) => empty_rel_for_rep R
      | Cst (Unrep, _, R) => empty_rel_for_rep R
      | Cst (Suc, T as @{typ "unsigned_bit word => unsigned_bit word"}, R) =>
        to_bit_word_unary_op T R (curry KK.Add (KK.Num 1))
      | Cst (Suc, @{typ "nat => nat"}, Func (Atom x, _)) =>
        kk_intersect (KK.Rel suc_rel) (kk_product KK.Univ (KK.AtomSeq x))
      | Cst (Suc, _, Func (Atom _, _)) => KK.Rel suc_rel
      | Cst (Add, Type (_, [@{typ nat}, _]), _) => KK.Rel nat_add_rel
      | Cst (Add, Type (_, [@{typ int}, _]), _) => KK.Rel int_add_rel
      | Cst (Add, T as Type (_, [@{typ "unsigned_bit word"}, _]), R) =>
        to_bit_word_binary_op T R NONE (SOME (curry KK.Add))
      | Cst (Add, T as Type (_, [@{typ "signed_bit word"}, _]), R) =>
        to_bit_word_binary_op T R
            (SOME (fn i1 => fn i2 => fn i3 =>
                 kk_implies (KK.LE (KK.Num 0, KK.BitXor (i1, i2)))
                            (KK.LE (KK.Num 0, KK.BitXor (i2, i3)))))
            (SOME (curry KK.Add))
      | Cst (Subtract, Type (_, [@{typ nat}, _]), _) =>
        KK.Rel nat_subtract_rel
      | Cst (Subtract, Type (_, [@{typ int}, _]), _) =>
        KK.Rel int_subtract_rel
      | Cst (Subtract, T as Type (_, [@{typ "unsigned_bit word"}, _]), R) =>
        to_bit_word_binary_op T R NONE
            (SOME (fn i1 => fn i2 =>
                      KK.IntIf (KK.LE (i1, i2), KK.Num 0, KK.Sub (i1, i2))))
      | Cst (Subtract, T as Type (_, [@{typ "signed_bit word"}, _]), R) =>
        to_bit_word_binary_op T R
            (SOME (fn i1 => fn i2 => fn i3 =>
                 kk_implies (KK.LT (KK.BitXor (i1, i2), KK.Num 0))
                            (KK.LT (KK.BitXor (i2, i3), KK.Num 0))))
            (SOME (curry KK.Sub))
      | Cst (Multiply, Type (_, [@{typ nat}, _]), _) =>
        KK.Rel nat_multiply_rel
      | Cst (Multiply, Type (_, [@{typ int}, _]), _) =>
        KK.Rel int_multiply_rel
      | Cst (Multiply,
             T as Type (_, [Type (@{type_name word}, [bit_T]), _]), R) =>
        to_bit_word_binary_op T R
            (SOME (fn i1 => fn i2 => fn i3 =>
                kk_or (KK.IntEq (i2, KK.Num 0))
                      (KK.IntEq (KK.Div (i3, i2), i1)
                       |> bit_T = @{typ signed_bit}
                          ? kk_and (KK.LE (KK.Num 0,
                                           foldl1 KK.BitAnd [i1, i2, i3])))))
            (SOME (curry KK.Mult))
      | Cst (Divide, Type (_, [@{typ nat}, _]), _) => KK.Rel nat_divide_rel
      | Cst (Divide, Type (_, [@{typ int}, _]), _) => KK.Rel int_divide_rel
      | Cst (Divide, T as Type (_, [@{typ "unsigned_bit word"}, _]), R) =>
        to_bit_word_binary_op T R NONE
            (SOME (fn i1 => fn i2 =>
                      KK.IntIf (KK.IntEq (i2, KK.Num 0),
                                KK.Num 0, KK.Div (i1, i2))))
      | Cst (Divide, T as Type (_, [@{typ "signed_bit word"}, _]), R) =>
        to_bit_word_binary_op T R
            (SOME (fn i1 => fn i2 => fn i3 =>
                      KK.LE (KK.Num 0, foldl1 KK.BitAnd [i1, i2, i3])))
            (SOME (fn i1 => fn i2 =>
                 KK.IntIf (kk_and (KK.LT (i1, KK.Num 0))
                                  (KK.LT (KK.Num 0, i2)),
                     KK.Sub (KK.Div (KK.Add (i1, KK.Num 1), i2), KK.Num 1),
                     KK.IntIf (kk_and (KK.LT (KK.Num 0, i1))
                                      (KK.LT (i2, KK.Num 0)),
                         KK.Sub (KK.Div (KK.Sub (i1, KK.Num 1), i2), KK.Num 1),
                         KK.IntIf (KK.IntEq (i2, KK.Num 0),
                                   KK.Num 0, KK.Div (i1, i2))))))
      | Cst (Gcd, _, _) => KK.Rel gcd_rel
      | Cst (Lcm, _, _) => KK.Rel lcm_rel
      | Cst (Fracs, _, Func (Atom (1, _), _)) => KK.None
      | Cst (Fracs, _, Func (Struct _, _)) =>
        kk_project_seq (KK.Rel norm_frac_rel) 2 2
      | Cst (NormFrac, _, _) => KK.Rel norm_frac_rel
      | Cst (NatToInt, Type (_, [@{typ nat}, _]), Func (Atom _, Atom _)) =>
        KK.Iden
      | Cst (NatToInt, Type (_, [@{typ nat}, _]),
             Func (Atom (_, nat_j0), Opt (Atom (int_k, int_j0)))) =>
        if nat_j0 = int_j0 then
          kk_intersect KK.Iden
              (kk_product (KK.AtomSeq (max_int_for_card int_k + 1, nat_j0))
                          KK.Univ)
        else
          raise BAD ("Nitpick_Kodkod.to_r (NatToInt)", "\"nat_j0 <> int_j0\"")
      | Cst (NatToInt, T as Type (_, [@{typ "unsigned_bit word"}, _]), R) =>
        to_bit_word_unary_op T R I
      | Cst (IntToNat, Type (_, [@{typ int}, _]),
             Func (Atom (int_k, int_j0), nat_R)) =>
        let
          val abs_card = max_int_for_card int_k + 1
          val (nat_k, nat_j0) = the_single (atom_schema_of_rep nat_R)
          val overlap = Int.min (nat_k, abs_card)
        in
          if nat_j0 = int_j0 then
            kk_union (kk_product (KK.AtomSeq (int_k - abs_card,
                                              int_j0 + abs_card))
                                 (KK.Atom nat_j0))
                     (kk_intersect KK.Iden
                          (kk_product (KK.AtomSeq (overlap, int_j0)) KK.Univ))
          else
            raise BAD ("Nitpick_Kodkod.to_r (IntToNat)", "\"nat_j0 <> int_j0\"")
        end
      | Cst (IntToNat, T as Type (_, [@{typ "signed_bit word"}, _]), R) =>
        to_bit_word_unary_op T R
            (fn i => KK.IntIf (KK.LE (i, KK.Num 0), KK.Num 0, i))
      | Op1 (Not, _, R, u1) => kk_not3 (to_rep R u1)
      | Op1 (Finite, _, Opt (Atom _), _) => KK.None
      | Op1 (Converse, T, R, u1) =>
        let
          val (b_T, a_T) = HOLogic.dest_prodT (pseudo_domain_type T)
          val (b_R, a_R) =
            case R of
              Func (Struct [R1, R2], _) => (R1, R2)
            | Func (R1, _) =>
              if card_of_rep R1 <> 1 then
                raise REP ("Nitpick_Kodkod.to_r (Converse)", [R])
              else
                apply2 (Atom o pair 1 o offset_of_type ofs) (b_T, a_T)
            | _ => raise REP ("Nitpick_Kodkod.to_r (Converse)", [R])
          val body_R = body_rep R
          val a_arity = arity_of_rep a_R
          val b_arity = arity_of_rep b_R
          val ab_arity = a_arity + b_arity
          val body_arity = arity_of_rep body_R
        in
          kk_project (to_rep (Func (Struct [a_R, b_R], body_R)) u1)
                     (map KK.Num (index_seq a_arity b_arity @
                                  index_seq 0 a_arity @
                                  index_seq ab_arity body_arity))
          |> rel_expr_from_rel_expr kk R (Func (Struct [b_R, a_R], body_R))
        end
      | Op1 (Closure, _, R, u1) =>
        if is_opt_rep R then
          let
            val T1 = type_of u1
            val R' = rep_to_binary_rel_rep ofs T1 (unopt_rep (rep_of u1))
            val R'' = opt_rep ofs T1 R'
          in
            single_rel_rel_let kk
                (fn r =>
                    let
                      val true_r = kk_closure (kk_join r true_atom)
                      val full_r = full_rel_for_rep R'
                      val false_r = kk_difference full_r
                                        (kk_closure (kk_difference full_r
                                                        (kk_join r false_atom)))
                    in
                      rel_expr_from_rel_expr kk R R''
                          (kk_union (kk_product true_r true_atom)
                                    (kk_product false_r false_atom))
                    end) (to_rep R'' u1)
          end
        else
          let val R' = rep_to_binary_rel_rep ofs (type_of u1) (rep_of u1) in
            rel_expr_from_rel_expr kk R R' (kk_closure (to_rep R' u1))
          end
      | Op1 (SingletonSet, _, Func (R1, Opt _), Cst (Unrep, _, _)) =>
        kk_product (full_rel_for_rep R1) false_atom
      | Op1 (SingletonSet, _, R, u1) =>
        (case R of
           Func (R1, Formula Neut) => to_rep R1 u1
         | Func (R1, Opt _) =>
           single_rel_rel_let kk
               (fn r => kk_rel_if (kk_no r) (empty_rel_for_rep R)
                            (rel_expr_to_func kk R1 bool_atom_R
                                              (Func (R1, Formula Neut)) r))
               (to_opt R1 u1)
         | _ => raise NUT ("Nitpick_Kodkod.to_r (SingletonSet)", [u]))
      | Op1 (SafeThe, _, R, u1) =>
        if is_opt_rep R then
          kk_join (to_rep (Func (unopt_rep R, Opt bool_atom_R)) u1) true_atom
        else
          to_rep (Func (R, Formula Neut)) u1
      | Op1 (First, _, R, u1) => to_nth_pair_sel 0 R u1
      | Op1 (Second, _, R, u1) => to_nth_pair_sel 1 R u1
      | Op1 (Cast, _, R, u1) =>
        ((case rep_of u1 of
            Formula _ =>
            (case unopt_rep R of
               Atom (2, j0) => atom_from_formula kk j0 (to_f u1)
             | _ => raise SAME ())
          | _ => raise SAME ())
         handle SAME () => rel_expr_from_rel_expr kk R (rep_of u1) (to_r u1))
      | Op2 (All, T, R as Opt _, u1, u2) =>
        to_r (Op1 (Not, T, R,
                   Op2 (Exist, T, R, u1, Op1 (Not, T, rep_of u2, u2))))
      | Op2 (Exist, _, Opt _, u1, u2) =>
        let val rs1 = untuple to_decl u1 in
          if not (is_opt_rep (rep_of u2)) then
            kk_rel_if (kk_exist rs1 (to_f u2)) true_atom KK.None
          else
            let val r2 = to_r u2 in
              kk_union (kk_rel_if (kk_exist rs1 (kk_rel_eq r2 true_atom))
                                  true_atom KK.None)
                       (kk_rel_if (kk_all rs1 (kk_rel_eq r2 false_atom))
                                  false_atom KK.None)
            end
        end
      | Op2 (Or, _, _, u1, u2) =>
        if is_opt_rep (rep_of u1) then kk_rel_if (to_f u2) true_atom (to_r u1)
        else kk_rel_if (to_f u1) true_atom (to_r u2)
      | Op2 (And, _, _, u1, u2) =>
        if is_opt_rep (rep_of u1) then kk_rel_if (to_f u2) (to_r u1) false_atom
        else kk_rel_if (to_f u1) (to_r u2) false_atom
      | Op2 (Less, _, _, u1, u2) =>
        (case type_of u1 of
           @{typ nat} =>
           if is_Cst Unrep u1 then to_compare_with_unrep u2 false_atom
           else if is_Cst Unrep u2 then to_compare_with_unrep u1 true_atom
           else kk_nat_less (to_integer u1) (to_integer u2)
         | @{typ int} => kk_int_less (to_integer u1) (to_integer u2)
         | _ =>
           let
             val R1 = Opt (Atom (card_of_rep (rep_of u1),
                                 offset_of_type ofs (type_of u1)))
           in
             double_rel_rel_let kk
                 (fn r1 => fn r2 =>
                     kk_rel_if
                         (fold kk_and (map_filter (fn (u, r) =>
                              if is_opt_rep (rep_of u) then SOME (kk_some r)
                              else NONE) [(u1, r1), (u2, r2)]) KK.True)
                         (atom_from_formula kk bool_j0 (KK.LT (apply2
                             (int_expr_from_atom kk (type_of u1)) (r1, r2))))
                         KK.None)
                 (to_rep R1 u1) (to_rep R1 u2)
            end)
      | Op2 (Triad, _, Opt (Atom (2, j0)), u1, u2) =>
        let
          val f1 = to_f u1
          val f2 = to_f u2
        in
          if f1 = f2 then
            atom_from_formula kk j0 f1
          else
            kk_union (kk_rel_if f1 true_atom KK.None)
                     (kk_rel_if f2 KK.None false_atom)
        end
      | Op2 (Composition, _, R, u1, u2) =>
        let
          val (a_T, b_T) = HOLogic.dest_prodT (pseudo_domain_type (type_of u1))
          val (_, c_T) = HOLogic.dest_prodT (pseudo_domain_type (type_of u2))
          val ab_k = card_of_domain_from_rep 2 (rep_of u1)
          val bc_k = card_of_domain_from_rep 2 (rep_of u2)
          val ac_k = card_of_domain_from_rep 2 R
          val a_k = exact_root 2 (ac_k * ab_k div bc_k)
          val b_k = exact_root 2 (ab_k * bc_k div ac_k)
          val c_k = exact_root 2 (bc_k * ac_k div ab_k)
          val a_R = Atom (a_k, offset_of_type ofs a_T)
          val b_R = Atom (b_k, offset_of_type ofs b_T)
          val c_R = Atom (c_k, offset_of_type ofs c_T)
          val body_R = body_rep R
        in
          (case body_R of
             Formula Neut =>
             kk_join (to_rep (Func (Struct [a_R, b_R], Formula Neut)) u1)
                     (to_rep (Func (Struct [b_R, c_R], Formula Neut)) u2)
           | Opt (Atom (2, _)) =>
             let
               fun must R1 R2 u =
                 kk_join (to_rep (Func (Struct [R1, R2], body_R)) u) true_atom
               fun may R1 R2 u =
                 kk_difference
                     (full_rel_for_rep (Struct [R1, R2]))
                     (kk_join (to_rep (Func (Struct [R1, R2], body_R)) u)
                              false_atom)
             in
               kk_union
                   (kk_product (kk_join (must a_R b_R u1) (must b_R c_R u2))
                               true_atom)
                   (kk_product (kk_difference
                                   (full_rel_for_rep (Struct [a_R, c_R]))
                                   (kk_join (may a_R b_R u1) (may b_R c_R u2)))
                               false_atom)
             end
           | _ => raise NUT ("Nitpick_Kodkod.to_r (Composition)", [u]))
          |> rel_expr_from_rel_expr kk R (Func (Struct [a_R, c_R], body_R))
        end
      | Op2 (Apply, @{typ nat}, _,
             Op2 (Apply, _, _, Cst (Subtract, _, _), u1), u2) =>
        if is_Cst Unrep u2 andalso not (is_opt_rep (rep_of u1)) then
          KK.Atom (offset_of_type ofs nat_T)
        else
          fold kk_join (map to_integer [u1, u2]) (KK.Rel nat_subtract_rel)
      | Op2 (Apply, _, R, u1, u2) => to_apply R u1 u2
      | Op2 (Lambda, _, R as Opt (Atom (1, j0)), u1, u2) =>
        to_guard [u1, u2] R (KK.Atom j0)
      | Op2 (Lambda, _, Func (_, Formula Neut), u1, u2) =>
        kk_comprehension (untuple to_decl u1) (to_f u2)
      | Op2 (Lambda, _, Func (_, R2), u1, u2) =>
        let
          val dom_decls = untuple to_decl u1
          val ran_schema = atom_schema_of_rep R2
          val ran_decls = decls_for_atom_schema ~1 ran_schema
          val ran_vars = unary_var_seq ~1 (length ran_decls)
        in
          kk_comprehension (dom_decls @ ran_decls)
                           (kk_subset (fold1 kk_product ran_vars)
                                      (to_rep R2 u2))
        end
      | Op3 (Let, _, R, u1, u2, u3) =>
        kk_rel_let [to_expr_assign u1 u2] (to_rep R u3)
      | Op3 (If, _, R, u1, u2, u3) =>
        if is_opt_rep (rep_of u1) then
          triple_rel_rel_let kk
              (fn r1 => fn r2 => fn r3 =>
                  let val empty_r = empty_rel_for_rep R in
                    fold1 kk_union
                          [kk_rel_if (kk_rel_eq r1 true_atom) r2 empty_r,
                           kk_rel_if (kk_rel_eq r1 false_atom) r3 empty_r,
                           kk_rel_if (kk_rel_eq r2 r3)
                                (if inline_rel_expr r2 then r2 else r3) empty_r]
                  end)
              (to_r u1) (to_rep R u2) (to_rep R u3)
        else
          kk_rel_if (to_f u1) (to_rep R u2) (to_rep R u3)
      | Tuple (_, R, us) =>
        (case unopt_rep R of
           Struct Rs => to_product Rs us
         | Vect (k, R) => to_product (replicate k R) us
         | Atom (1, j0) =>
           kk_rel_if (kk_some (fold1 kk_product (map to_r us)))
                     (KK.Atom j0) KK.None
         | _ => raise NUT ("Nitpick_Kodkod.to_r (Tuple)", [u]))
      | Construct ([u'], _, _, []) => to_r u'
      | Construct (discr_u :: sel_us, _, _, arg_us) =>
        let
          val set_rs =
            map2 (fn sel_u => fn arg_u =>
                     let
                       val (R1, R2) = dest_Func (rep_of sel_u)
                       val sel_r = to_r sel_u
                       val arg_r = to_opt R2 arg_u
                     in
                       if is_one_rep R2 then
                         kk_n_fold_join kk true R2 R1 arg_r
                              (kk_project sel_r (flip_nums (arity_of_rep R2)))
                       else
                         kk_comprehension [KK.DeclOne ((1, ~1), to_r discr_u)]
                             (kk_rel_eq (kk_join (KK.Var (1, ~1)) sel_r) arg_r)
                         |> is_opt_rep (rep_of arg_u) ? to_guard [arg_u] R1
                     end) sel_us arg_us
        in fold1 kk_intersect set_rs end
      | BoundRel (x, _, _, _) => KK.Var x
      | FreeRel (x, _, _, _) => KK.Rel x
      | RelReg (j, _, R) => KK.RelReg (arity_of_rep R, j)
      | u => raise NUT ("Nitpick_Kodkod.to_r", [u])
    and to_decl (BoundRel (x, _, R, _)) =
        KK.DeclOne (x, KK.AtomSeq (the_single (atom_schema_of_rep R)))
      | to_decl u = raise NUT ("Nitpick_Kodkod.to_decl", [u])
    and to_expr_assign (FormulaReg (j, _, _)) u =
        KK.AssignFormulaReg (j, to_f u)
      | to_expr_assign (RelReg (j, _, R)) u =
        KK.AssignRelReg ((arity_of_rep R, j), to_r u)
      | to_expr_assign u1 _ = raise NUT ("Nitpick_Kodkod.to_expr_assign", [u1])
    and to_atom (x as (_, j0)) u =
      case rep_of u of
        Formula _ => atom_from_formula kk j0 (to_f u)
      | R => atom_from_rel_expr kk x R (to_r u)
    and to_struct Rs u = struct_from_rel_expr kk Rs (rep_of u) (to_r u)
    and to_vect k R u = vect_from_rel_expr kk k R (rep_of u) (to_r u)
    and to_func R1 R2 u = rel_expr_to_func kk R1 R2 (rep_of u) (to_r u)
    and to_opt R u =
      let val old_R = rep_of u in
        if is_opt_rep old_R then
          rel_expr_from_rel_expr kk (Opt R) old_R (to_r u)
        else
          to_rep R u
      end
    and to_rep (Atom x) u = to_atom x u
      | to_rep (Struct Rs) u = to_struct Rs u
      | to_rep (Vect (k, R)) u = to_vect k R u
      | to_rep (Func (R1, R2)) u = to_func R1 R2 u
      | to_rep (Opt R) u = to_opt R u
      | to_rep R _ = raise REP ("Nitpick_Kodkod.to_rep", [R])
    and to_integer u = to_opt (one_rep ofs (type_of u) (rep_of u)) u
    and to_guard guard_us R r =
      let
        val unpacked_rs = unpack_joins r
        val plain_guard_rs =
          map to_r (filter (is_Opt o rep_of) guard_us)
          |> filter_out (member (op =) unpacked_rs)
        val func_guard_us =
          filter ((is_Func andf is_opt_rep) o rep_of) guard_us
        val func_guard_rs = map to_r func_guard_us
        val guard_fs =
          map kk_no plain_guard_rs @
          map2 (kk_not oo kk_n_ary_function kk)
               (map (unopt_rep o rep_of) func_guard_us) func_guard_rs
      in
        if null guard_fs then r
        else kk_rel_if (fold1 kk_or guard_fs) (empty_rel_for_rep R) r
      end
    and to_project new_R old_R r j0 =
      rel_expr_from_rel_expr kk new_R old_R
                             (kk_project_seq r j0 (arity_of_rep old_R))
    and to_product Rs us = fold1 kk_product (map (uncurry to_opt) (Rs ~~ us))
    and to_nth_pair_sel n res_R u =
      case u of
        Tuple (_, _, us) => to_rep res_R (nth us n)
      | _ => let
               val R = rep_of u
               val (a_T, b_T) = HOLogic.dest_prodT (type_of u)
               val Rs =
                 case unopt_rep R of
                   Struct (Rs as [_, _]) => Rs
                 | _ =>
                   let
                     val res_card = card_of_rep res_R
                     val other_card = card_of_rep R div res_card
                     val (a_card, b_card) = (res_card, other_card)
                                            |> n = 1 ? swap
                   in
                     [Atom (a_card, offset_of_type ofs a_T),
                      Atom (b_card, offset_of_type ofs b_T)]
                   end
               val nth_R = nth Rs n
               val j0 = if n = 0 then 0 else arity_of_rep (hd Rs)
             in to_project res_R nth_R (to_rep (Opt (Struct Rs)) u) j0 end
    and to_set_bool_op connective set_oper u1 u2 =
      let
        val min_R = min_rep (rep_of u1) (rep_of u2)
        val r1 = to_rep min_R u1
        val r2 = to_rep min_R u2
      in
        case min_R of
          Vect (k, Atom _) => kk_vect_set_bool_op connective k r1 r2
        | Func (_, R') =>
          (case body_rep R' of
             Formula Neut => set_oper r1 r2
           | Atom _ => set_oper (kk_join r1 true_atom) (kk_join r2 true_atom)
           | _ => raise REP ("Nitpick_Kodkod.to_set_bool_op", [min_R]))
        | _ => raise REP ("Nitpick_Kodkod.to_set_bool_op", [min_R])
      end
    and to_bit_word_unary_op T R oper =
      let
        val Ts = strip_type T ||> single |> op @
        fun int_arg j = int_expr_from_atom kk (nth Ts j) (KK.Var (1, j))
      in
        kk_comprehension (decls_for_atom_schema 0 (atom_schema_of_rep R))
            (KK.FormulaLet
                 (map (fn j => KK.AssignIntReg (j, int_arg j)) (0 upto 1),
                  KK.IntEq (KK.IntReg 1, oper (KK.IntReg 0))))
      end
    and to_bit_word_binary_op T R opt_guard opt_oper =
      let
        val Ts = strip_type T ||> single |> op @
        fun int_arg j = int_expr_from_atom kk (nth Ts j) (KK.Var (1, j))
      in
        kk_comprehension (decls_for_atom_schema 0 (atom_schema_of_rep R))
            (KK.FormulaLet
                 (map (fn j => KK.AssignIntReg (j, int_arg j)) (0 upto 2),
                  fold1 kk_and
                        ((case opt_guard of
                            NONE => []
                          | SOME guard =>
                            [guard (KK.IntReg 0) (KK.IntReg 1) (KK.IntReg 2)]) @
                         (case opt_oper of
                            NONE => []
                          | SOME oper =>
                            [KK.IntEq (KK.IntReg 2,
                                       oper (KK.IntReg 0) (KK.IntReg 1))]))))
      end
    and to_apply (R as Formula _) _ _ =
        raise REP ("Nitpick_Kodkod.to_apply", [R])
      | to_apply res_R func_u arg_u =
        case unopt_rep (rep_of func_u) of
          Atom (1, j0) =>
          to_guard [arg_u] res_R
                   (rel_expr_from_rel_expr kk res_R (Atom (1, j0)) (to_r func_u))
        | Atom (k, _) =>
          let
            val dom_card = card_of_rep (rep_of arg_u)
            val ran_R =
              Atom (exact_root dom_card k,
                    offset_of_type ofs (pseudo_range_type (type_of func_u)))
          in
            to_apply_vect dom_card ran_R res_R (to_vect dom_card ran_R func_u)
                          arg_u
          end
        | Vect (1, R') =>
          to_guard [arg_u] res_R
                   (rel_expr_from_rel_expr kk res_R R' (to_r func_u))
        | Vect (k, R') => to_apply_vect k R' res_R (to_r func_u) arg_u
        | Func (R, Formula Neut) =>
          to_guard [arg_u] res_R (rel_expr_from_formula kk res_R
                                      (kk_subset (to_opt R arg_u) (to_r func_u)))
        | Func (R1, R2) =>
          rel_expr_from_rel_expr kk res_R R2
              (kk_n_fold_join kk true R1 R2 (to_opt R1 arg_u) (to_r func_u))
          |> body_rep R2 = Formula Neut ? to_guard [arg_u] res_R
        | _ => raise NUT ("Nitpick_Kodkod.to_apply", [func_u])
    and to_apply_vect k R' res_R func_r arg_u =
      let
        val arg_R = one_rep ofs (type_of arg_u) (unopt_rep (rep_of arg_u))
        val vect_r = vect_from_rel_expr kk k res_R (Vect (k, R')) func_r
        val vect_rs = unpack_vect_in_chunks kk (arity_of_rep res_R) k vect_r
      in
        kk_case_switch kk arg_R res_R (to_opt arg_R arg_u)
                       (all_singletons_for_rep arg_R) vect_rs
      end
    and to_could_be_unrep neg u =
      if neg andalso is_opt_rep (rep_of u) then kk_no (to_r u) else KK.False
    and to_compare_with_unrep u r =
      if is_opt_rep (rep_of u) then
        kk_rel_if (kk_some (to_r u)) r (empty_rel_for_rep (rep_of u))
      else
        r
  in to_f_with_polarity Pos u end

fun declarative_axiom_for_plain_rel kk (FreeRel (x, _, R as Func _, nick)) =
    kk_n_ary_function kk (R |> nick = @{const_name List.set} ? unopt_rep)
                      (KK.Rel x)
  | declarative_axiom_for_plain_rel ({kk_lone, kk_one, ...} : kodkod_constrs)
                                    (FreeRel (x, _, R, _)) =
    if is_one_rep R then kk_one (KK.Rel x)
    else if is_lone_rep R andalso card_of_rep R > 1 then kk_lone (KK.Rel x)
    else KK.True
  | declarative_axiom_for_plain_rel _ u =
    raise NUT ("Nitpick_Kodkod.declarative_axiom_for_plain_rel", [u])

fun const_triple rel_table (x as (s, T)) =
  case the_name rel_table (ConstName (s, T, Any)) of
    FreeRel ((n, j), _, R, _) => (KK.Rel (n, j), R, n)
  | _ => raise TERM ("Nitpick_Kodkod.const_triple", [Const x])

fun discr_rel_expr rel_table = #1 o const_triple rel_table o discr_for_constr

fun nfa_transitions_for_sel hol_ctxt binarize
                            ({kk_project, ...} : kodkod_constrs) rel_table
                            (dtypes : data_type_spec list) constr_x n =
  let
    val x as (_, T) =
      binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize constr_x n
    val (r, R, arity) = const_triple rel_table x
    val type_schema = type_schema_of_rep T R
  in
    map_filter (fn (j, T) =>
                   if forall (not_equal T o #typ) dtypes then NONE
                   else SOME ((x, kk_project r (map KK.Num [0, j])), T))
               (index_seq 1 (arity - 1) ~~ tl type_schema)
  end

fun nfa_transitions_for_constr hol_ctxt binarize kk rel_table dtypes
                               (x as (_, T)) =
  maps (nfa_transitions_for_sel hol_ctxt binarize kk rel_table dtypes x)
       (index_seq 0 (num_sels_for_constr_type T))

fun nfa_entry_for_data_type _ _ _ _ _ ({co = true, ...} : data_type_spec) = NONE
  | nfa_entry_for_data_type _ _ _ _ _ {deep = false, ...} = NONE
  | nfa_entry_for_data_type hol_ctxt binarize kk rel_table dtypes
                            {typ, constrs, ...} =
    SOME (typ, maps (nfa_transitions_for_constr hol_ctxt binarize kk rel_table
                                                dtypes o #const) constrs)

val empty_rel = KK.Product (KK.None, KK.None)

fun direct_path_rel_exprs nfa start_T final_T =
  case AList.lookup (op =) nfa final_T of
    SOME trans => map (snd o fst) (filter (curry (op =) start_T o snd) trans)
  | NONE => []
and any_path_rel_expr ({kk_union, ...} : kodkod_constrs) nfa [] start_T
                      final_T =
    fold kk_union (direct_path_rel_exprs nfa start_T final_T)
         (if start_T = final_T then KK.Iden else empty_rel)
  | any_path_rel_expr (kk as {kk_union, ...}) nfa (T :: Ts) start_T final_T =
    kk_union (any_path_rel_expr kk nfa Ts start_T final_T)
             (knot_path_rel_expr kk nfa Ts start_T T final_T)
and knot_path_rel_expr (kk as {kk_join, kk_reflexive_closure, ...}) nfa Ts
                       start_T knot_T final_T =
  kk_join (kk_join (any_path_rel_expr kk nfa Ts knot_T final_T)
                   (kk_reflexive_closure (loop_path_rel_expr kk nfa Ts knot_T)))
          (any_path_rel_expr kk nfa Ts start_T knot_T)
and loop_path_rel_expr ({kk_union, ...} : kodkod_constrs) nfa [] start_T =
    fold kk_union (direct_path_rel_exprs nfa start_T start_T) empty_rel
  | loop_path_rel_expr (kk as {kk_union, kk_closure, ...}) nfa (T :: Ts)
                       start_T =
    if start_T = T then
      kk_closure (loop_path_rel_expr kk nfa Ts start_T)
    else
      kk_union (loop_path_rel_expr kk nfa Ts start_T)
               (knot_path_rel_expr kk nfa Ts start_T T start_T)

fun add_nfa_to_graph [] = I
  | add_nfa_to_graph ((_, []) :: nfa) = add_nfa_to_graph nfa
  | add_nfa_to_graph ((T, ((_, T') :: transitions)) :: nfa) =
    add_nfa_to_graph ((T, transitions) :: nfa) o Typ_Graph.add_edge (T, T') o
    Typ_Graph.default_node (T', ()) o Typ_Graph.default_node (T, ())

fun strongly_connected_sub_nfas nfa =
  add_nfa_to_graph nfa Typ_Graph.empty
  |> Typ_Graph.strong_conn
  |> map (fn keys => filter (member (op =) keys o fst) nfa)

(* Cycle breaking in the bounds takes care of singly recursive data types, hence
   the first equation. *)
fun acyclicity_axioms_for_data_type _ [_] _ = []
  | acyclicity_axioms_for_data_type (kk as {kk_no, kk_intersect, ...}) nfa
                                    start_T =
    [kk_no (kk_intersect
                (loop_path_rel_expr kk nfa (pull start_T (map fst nfa)) start_T)
                KK.Iden)]

fun acyclicity_axioms_for_data_types kk =
  maps (fn nfa => maps (acyclicity_axioms_for_data_type kk nfa o fst) nfa)

fun atom_equation_for_nut ofs kk (u, j) =
  let val dummy_u = RelReg (0, type_of u, rep_of u) in
    case Op2 (DefEq, bool_T, Formula Pos, dummy_u, u)
         |> kodkod_formula_from_nut ofs kk of
      KK.RelEq (KK.RelReg _, r) => SOME (KK.RelEq (KK.Atom j, r))
    | _ => raise BAD ("Nitpick_Kodkod.atom_equation_for_nut",
                      "malformed Kodkod formula")
  end

fun needed_values_for_data_type [] _ _ = SOME []
  | needed_values_for_data_type need_us ofs
                                ({typ, card, constrs, ...} : data_type_spec) =
    let
      fun aux (u as Construct (FreeRel (_, _, _, s) :: _, T, _, us)) =
          fold aux us
          #> (fn NONE => NONE
               | accum as SOME (loose, fixed) =>
                 if T = typ then
                   case AList.lookup (op =) fixed u of
                     SOME _ => accum
                   | NONE =>
                     let
                       val constr_s = constr_name_for_sel_like s
                       val {delta, epsilon, ...} =
                         constrs
                         |> List.find (fn {const, ...} => fst const = constr_s)
                         |> the
                       val j0 = offset_of_type ofs T
                     in
                       case find_first (fn j => j >= delta andalso
                                        j < delta + epsilon) loose of
                         SOME j =>
                         SOME (remove (op =) j loose, (u, j0 + j) :: fixed)
                       | NONE => NONE
                     end
                 else
                   accum)
        | aux _ = I
    in
      SOME (index_seq 0 card, []) |> fold aux need_us |> Option.map (rev o snd)
    end

fun needed_value_axioms_for_data_type _ _ _ (_, NONE) = [KK.False]
  | needed_value_axioms_for_data_type ofs kk dtypes (T, SOME fixed) =
    if is_data_type_nat_like (the (data_type_spec dtypes T)) then []
    else fixed |> map_filter (atom_equation_for_nut ofs kk)

fun all_ge ({kk_join, kk_reflexive_closure, ...} : kodkod_constrs) z r =
  kk_join r (kk_reflexive_closure (KK.Rel (suc_rel_for_atom_seq z)))

fun gt ({kk_subset, kk_join, kk_closure, ...} : kodkod_constrs) z r1 r2 =
  kk_subset r1 (kk_join r2 (kk_closure (KK.Rel (suc_rel_for_atom_seq z))))

fun constr_quadruple ({const = (s, T), delta, epsilon, ...} : constr_spec) =
  (delta, (epsilon, (num_binder_types T, s)))
val constr_ord =
  prod_ord int_ord (prod_ord int_ord (prod_ord int_ord string_ord))
  o apply2 constr_quadruple

fun data_type_ord (({card = card1, self_rec = self_rec1, constrs = constr1, ...},
                    {card = card2, self_rec = self_rec2, constrs = constr2, ...})
                   : data_type_spec * data_type_spec) =
  prod_ord int_ord (prod_ord bool_ord int_ord)
           ((card1, (self_rec1, length constr1)),
            (card2, (self_rec2, length constr2)))

(* We must absolutely tabulate "suc" for all data types whose selector bounds
   break cycles; otherwise, we may end up with two incompatible symmetry
   breaking orders, leading to spurious models. *)
fun should_tabulate_suc_for_type dtypes T =
  is_asymmetric_non_data_type T orelse
  case data_type_spec dtypes T of
    SOME {self_rec, ...} => self_rec
  | NONE => false

fun lex_order_rel_expr (kk as {kk_implies, kk_and, kk_subset, kk_join, ...})
                       dtypes sel_quadruples =
  case sel_quadruples of
    [] => KK.True
  | ((r, Func (Atom _, Atom x), 2), (_, Type (_, [_, T]))) :: sel_quadruples' =>
    let val z = (x, should_tabulate_suc_for_type dtypes T) in
      if null sel_quadruples' then
        gt kk z (kk_join (KK.Var (1, 1)) r) (kk_join (KK.Var (1, 0)) r)
      else
        kk_and (kk_subset (kk_join (KK.Var (1, 1)) r)
                          (all_ge kk z (kk_join (KK.Var (1, 0)) r)))
               (kk_implies (kk_subset (kk_join (KK.Var (1, 1)) r)
                                      (kk_join (KK.Var (1, 0)) r))
                           (lex_order_rel_expr kk dtypes sel_quadruples'))
    end
    (* Skip constructors components that aren't atoms, since we cannot compare
       these easily. *)
  | _ :: sel_quadruples' => lex_order_rel_expr kk dtypes sel_quadruples'

fun is_nil_like_constr_type dtypes T =
  case data_type_spec dtypes T of
    SOME {constrs, ...} =>
    (case filter_out (is_self_recursive_constr_type o snd o #const) constrs of
       [{const = (_, T'), ...}] => T = T'
     | _ => false)
  | NONE => false

fun sym_break_axioms_for_constr_pair hol_ctxt binarize
       (kk as {kk_all, kk_or, kk_implies, kk_and, kk_some, kk_intersect,
               kk_join, ...}) rel_table nfas dtypes
       (constr_ord,
        ({const = const1 as (_, T1), delta = delta1, epsilon = epsilon1, ...},
         {const = const2 as (_, _), delta = delta2, epsilon = epsilon2, ...})
        : constr_spec * constr_spec) =
  let
    val dataT = body_type T1
    val nfa = nfas |> find_first (exists (curry (op =) dataT o fst)) |> these
    val rec_Ts = nfa |> map fst
    fun rec_and_nonrec_sels (x as (_, T)) =
      index_seq 0 (num_sels_for_constr_type T)
      |> map (binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize x)
      |> List.partition (member (op =) rec_Ts o range_type o snd)
    val sel_xs1 = rec_and_nonrec_sels const1 |> op @
  in
    if constr_ord = EQUAL andalso null sel_xs1 then
      []
    else
      let
        val z =
          (case #2 (const_triple rel_table (discr_for_constr const1)) of
             Func (Atom x, Formula _) => x
           | R => raise REP ("Nitpick_Kodkod.sym_break_axioms_for_constr_pair",
                             [R]), should_tabulate_suc_for_type dtypes dataT)
        val (rec_sel_xs2, nonrec_sel_xs2) = rec_and_nonrec_sels const2
        val sel_xs2 = rec_sel_xs2 @ nonrec_sel_xs2
        fun sel_quadruples2 () = sel_xs2 |> map (`(const_triple rel_table))
        (* If the two constructors are the same, we drop the first selector
           because that one is always checked by the lexicographic order.
           We sometimes also filter out direct subterms, because those are
           already handled by the acyclicity breaking in the bound
           declarations. *)
        fun filter_out_sels no_direct sel_xs =
          apsnd (filter_out
                     (fn ((x, _), T) =>
                         (constr_ord = EQUAL andalso x = hd sel_xs) orelse
                         (T = dataT andalso
                          (no_direct orelse not (member (op =) sel_xs x)))))
        fun subterms_r no_direct sel_xs j =
          loop_path_rel_expr kk (map (filter_out_sels no_direct sel_xs) nfa)
                           (filter_out (curry (op =) dataT) (map fst nfa)) dataT
          |> kk_join (KK.Var (1, j))
      in
        [kk_all [KK.DeclOne ((1, 0), discr_rel_expr rel_table const1),
                 KK.DeclOne ((1, 1), discr_rel_expr rel_table const2)]
             (kk_implies
                 (if delta2 >= epsilon1 then KK.True
                  else if delta1 >= epsilon2 - 1 then KK.False
                  else gt kk z (KK.Var (1, 1)) (KK.Var (1, 0)))
                 (kk_or
                      (if is_nil_like_constr_type dtypes T1 then
                         KK.True
                       else
                         kk_some (kk_intersect (subterms_r false sel_xs2 1)
                                               (all_ge kk z (KK.Var (1, 0)))))
                      (case constr_ord of
                         EQUAL =>
                         kk_and
                             (lex_order_rel_expr kk dtypes (sel_quadruples2 ()))
                             (kk_all [KK.DeclOne ((1, 2),
                                                  subterms_r true sel_xs1 0)]
                                     (gt kk z (KK.Var (1, 1)) (KK.Var (1, 2))))
                       | LESS =>
                         kk_all [KK.DeclOne ((1, 2),
                                 subterms_r false sel_xs1 0)]
                                (gt kk z (KK.Var (1, 1)) (KK.Var (1, 2)))
                       | GREATER => KK.False)))]
      end
  end

fun sym_break_axioms_for_data_type hol_ctxt binarize kk rel_table nfas dtypes
                                   ({constrs, ...} : data_type_spec) =
  let
    val constrs = sort constr_ord constrs
    val constr_pairs = all_distinct_unordered_pairs_of constrs
  in
    map (pair EQUAL) (constrs ~~ constrs) @
    map (pair LESS) constr_pairs @
    map (pair GREATER) (map swap constr_pairs)
    |> maps (sym_break_axioms_for_constr_pair hol_ctxt binarize kk rel_table
                                              nfas dtypes)
  end

fun is_data_type_in_needed_value T (Construct (_, T', _, us)) =
    T = T' orelse exists (is_data_type_in_needed_value T) us
  | is_data_type_in_needed_value _ _ = false

val min_sym_break_card = 7

fun sym_break_axioms_for_data_types hol_ctxt binarize need_us
      datatype_sym_break kk rel_table nfas dtypes =
  if datatype_sym_break = 0 then
    []
  else
    dtypes |> filter is_data_type_acyclic
           |> filter (fn {constrs = [_], ...} => false
                       | {card, constrs, ...} =>
                         card >= min_sym_break_card andalso
                         forall (forall (not o is_higher_order_type)
                                 o binder_types o snd o #const) constrs)
           |> filter_out
                  (fn {typ, ...} =>
                      exists (is_data_type_in_needed_value typ) need_us)
           |> (fn dtypes' =>
                  dtypes' |> length dtypes' > datatype_sym_break
                             ? (sort (data_type_ord o swap)
                                #> take datatype_sym_break))
           |> maps (sym_break_axioms_for_data_type hol_ctxt binarize kk rel_table
                                                   nfas dtypes)

fun sel_axioms_for_sel hol_ctxt binarize j0
        (kk as {kk_all, kk_formula_if, kk_subset, kk_no, kk_join, ...})
        need_vals rel_table dom_r (dtype as {typ, ...})
        ({const, delta, epsilon, exclusive, ...} : constr_spec) n =
  let
    val x = binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize const n
    val (r, R, _) = const_triple rel_table x
    val rel_x =
      case r of
        KK.Rel x => x
      | _ => raise BAD ("Nitpick_Kodkod.sel_axioms_for_sel", "non-Rel")
    val R2 = dest_Func R |> snd
    val z = (epsilon - delta, delta + j0)
  in
    if exclusive then
      [kk_n_ary_function kk (Func (Atom z, R2)) r]
    else if all_values_are_needed need_vals dtype then
      typ |> needed_values need_vals
          |> filter (is_sel_of_constr rel_x)
          |> map (fn (_, j) => kk_n_ary_function kk R2 (kk_join (KK.Atom j) r))
    else
      let val r' = kk_join (KK.Var (1, 0)) r in
        [kk_all [KK.DeclOne ((1, 0), KK.AtomSeq z)]
                (kk_formula_if (kk_subset (KK.Var (1, 0)) dom_r)
                               (kk_n_ary_function kk R2 r') (kk_no r'))]
      end
  end

fun sel_axioms_for_constr hol_ctxt binarize bits j0 kk need_vals rel_table
        dtype (constr as {const, delta, epsilon, explicit_max, ...}) =
  let
    val honors_explicit_max =
      explicit_max < 0 orelse epsilon - delta <= explicit_max
  in
    if explicit_max = 0 then
      [formula_for_bool honors_explicit_max]
    else
      let
        val dom_r = discr_rel_expr rel_table const
        val max_axiom =
          if honors_explicit_max then
            KK.True
          else if bits = 0 orelse
               is_twos_complement_representable bits (epsilon - delta) then
            KK.LE (KK.Cardinality dom_r, KK.Num explicit_max)
          else
            raise TOO_SMALL ("Nitpick_Kodkod.sel_axioms_for_constr",
                             "\"bits\" value " ^ string_of_int bits ^
                             " too small for \"max\"")
      in
        max_axiom ::
        maps (sel_axioms_for_sel hol_ctxt binarize j0 kk need_vals rel_table
                                 dom_r dtype constr)
             (index_seq 0 (num_sels_for_constr_type (snd const)))
      end
  end

fun sel_axioms_for_data_type hol_ctxt binarize bits j0 kk rel_table need_vals
                             (dtype as {constrs, ...}) =
  maps (sel_axioms_for_constr hol_ctxt binarize bits j0 kk rel_table need_vals
                              dtype) constrs

fun uniqueness_axioms_for_constr hol_ctxt binarize
        ({kk_all, kk_implies, kk_and, kk_rel_eq, kk_lone, kk_join, ...}
         : kodkod_constrs) need_vals rel_table dtype
        ({const, ...} : constr_spec) =
  let
    fun conjunct_for_sel r =
      kk_rel_eq (kk_join (KK.Var (1, 0)) r) (kk_join (KK.Var (1, 1)) r)
    val num_sels = num_sels_for_constr_type (snd const)
    val triples =
      map (const_triple rel_table
           o binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize const)
          (~1 upto num_sels - 1)
    val set_r = triples |> hd |> #1
  in
    if num_sels = 0 then
      [kk_lone set_r]
    else if all_values_are_needed need_vals dtype then
      []
    else
      [kk_all (map (KK.DeclOne o rpair set_r o pair 1) [0, 1])
              (kk_implies
                   (fold1 kk_and (map (conjunct_for_sel o #1) (tl triples)))
                   (kk_rel_eq (KK.Var (1, 0)) (KK.Var (1, 1))))]
  end

fun uniqueness_axioms_for_data_type hol_ctxt binarize kk need_vals rel_table
                                    (dtype as {constrs, ...}) =
  maps (uniqueness_axioms_for_constr hol_ctxt binarize kk need_vals rel_table
                                     dtype) constrs

fun effective_constr_max ({delta, epsilon, ...} : constr_spec) = epsilon - delta

fun partition_axioms_for_data_type j0 (kk as {kk_rel_eq, kk_union, ...})
        need_vals rel_table (dtype as {card, constrs, ...}) =
  if forall #exclusive constrs then
    [Integer.sum (map effective_constr_max constrs) = card |> formula_for_bool]
  else if all_values_are_needed need_vals dtype then
    []
  else
    let val rs = map (discr_rel_expr rel_table o #const) constrs in
      [kk_rel_eq (fold1 kk_union rs) (KK.AtomSeq (card, j0)),
       kk_disjoint_sets kk rs]
    end

fun other_axioms_for_data_type _ _ _ _ _ _ _ {deep = false, ...} = []
  | other_axioms_for_data_type hol_ctxt binarize need_vals bits ofs kk rel_table
                               (dtype as {typ, ...}) =
    let val j0 = offset_of_type ofs typ in
      sel_axioms_for_data_type hol_ctxt binarize bits j0 kk need_vals rel_table
                               dtype @
      uniqueness_axioms_for_data_type hol_ctxt binarize kk need_vals rel_table
                                      dtype @
      partition_axioms_for_data_type j0 kk need_vals rel_table dtype
    end

fun declarative_axioms_for_data_types hol_ctxt binarize need_us need_vals
        datatype_sym_break bits ofs kk rel_table dtypes =
  let
    val nfas =
      dtypes |> map_filter (nfa_entry_for_data_type hol_ctxt binarize kk
                                                    rel_table dtypes)
             |> strongly_connected_sub_nfas
  in
    acyclicity_axioms_for_data_types kk nfas @
    maps (needed_value_axioms_for_data_type ofs kk dtypes) need_vals @
    sym_break_axioms_for_data_types hol_ctxt binarize need_us datatype_sym_break
                                    kk rel_table nfas dtypes @
    maps (other_axioms_for_data_type hol_ctxt binarize need_vals bits ofs kk
                                     rel_table) dtypes
  end

end;