(*  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 datatype_spec = Nitpick_Scope.datatype_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

    -> datatype_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_datatype :

    nut list -> int Typtab.table -> datatype_spec -> (nut * int) list option

  val declarative_axiom_for_plain_rel : kodkod_constrs -> nut -> Kodkod.formula

  val declarative_axioms_for_datatypes :

    hol_context -> bool -> nut list -> (typ * (nut * int) list option) list

    -> int -> int -> int Typtab.table -> kodkod_constrs -> nut NameTable.table

    -> datatype_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_datatype_acyclic ({co = false, standard = true, deep = true, ...}

                         : datatype_spec) = true

  | is_datatype_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 universe of cardinality " ^

                     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)

val tuple_set_from_atom_schema = foldl1 KK.TupleProduct 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 pairself (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

                 (pairself (atom_for_int (k, 0)) o f

                  o pairself (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 pairself 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 " :: " ^ unyxml (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 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, ...} = datatype_spec dtypes T1 |> the

      val need_vals =

        AList.lookup (op =) need_vals T1 |> the_default NONE |> these

    in

      ([(x, bound_comment ctxt debug nick T R)],

       let

         val ts = KK.TupleAtomSeq (epsilon - delta, delta + j0)

         val complete_need_vals = (length need_vals = card)

         val (my_need_vals, other_need_vals) =

           need_vals

           |> List.partition

                  (fn (Construct (sel_us, _, _, _), _) =>

                      exists (fn FreeRel (x', _, _, _) => x = x'

                               | _ => false) sel_us

                    | _ => false)

       in

         if explicit_max = 0 orelse

            (complete_need_vals andalso null my_need_vals) then

           [KK.TupleSet []]

         else

           if R2 = Formula Neut then

             [ts] |> not exclusive ? cons (KK.TupleSet [])

           else

             [KK.TupleSet [],

              if T1 = T2 andalso epsilon > delta andalso

                 is_datatype_acyclic dtype then

                index_seq delta (epsilon - delta)

                |> map (fn j =>

                           KK.TupleProduct (KK.TupleSet [KK.Tuple [j + j0]],

                                            KK.TupleAtomSeq (j, j0)))

                |> foldl1 KK.TupleUnion

              else

                KK.TupleProduct (ts, upper_bound_for_rep R2)]

         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)

              (map3 (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

             | Pos => formula_for_bool (not opt1)

             | 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 (Subset, _, _, u1, u2) =>

           let

             val dom_T = domain_type (type_of u1)

             val R1 = rep_of u1

             val R2 = rep_of u2

             val (dom_R, ran_R) =

               case min_rep R1 R2 of

                 Func Rp => Rp

               | R => (Atom (card_of_domain_from_rep 2 R,

                             offset_of_type ofs dom_T),

                       if is_opt_rep R then Opt bool_atom_R else Formula Neut)

             val set_R = Func (dom_R, ran_R)

           in

             if not (is_opt_rep ran_R) then

               to_set_bool_op kk_implies kk_subset u1 u2

             else if polar = Neut then

               raise NUT ("Nitpick_Kodkod.to_f (Subset)", [u])

             else

               let

                 (* FIXME: merge with similar code below *)

                 fun set_to_r widen u =

                   if widen then

                     kk_difference (full_rel_for_rep dom_R)

                                   (kk_join (to_rep set_R u) false_atom)

                   else

                     kk_join (to_rep set_R u) true_atom

                 val widen1 = (polar = Pos andalso is_opt_rep R1)

                 val widen2 = (polar = Neg andalso is_opt_rep R2)

               in kk_subset (set_to_r widen1 u1) (set_to_r widen2 u2) end

           end

         | 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 fun}, [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)