(*  Title:      HOL/Tools/Nitpick/nitpick_kodkod.ML

    Author:     Jasmin Blanchette, TU Muenchen

    Copyright   2008, 2009

Kodkod problem generator part of Kodkod.

*)

signature NITPICK_KODKOD =

sig

  type extended_context = Nitpick_HOL.extended_context

  type dtype_spec = Nitpick_Scope.dtype_spec

  type kodkod_constrs = Nitpick_Peephole.kodkod_constrs

  type nut = Nitpick_Nut.nut

  type nfa_transition = Kodkod.rel_expr * typ

  type nfa_entry = typ * nfa_transition list

  type nfa_table = nfa_entry list

  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 : 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 -> int -> Kodkod.int_bound list

  val bounds_for_built_in_rels_in_formula :

    bool -> int -> int -> int -> int -> Kodkod.formula -> Kodkod.bound list

  val bound_for_plain_rel : Proof.context -> bool -> nut -> Kodkod.bound

  val bound_for_sel_rel :

    Proof.context -> bool -> dtype_spec list -> nut -> Kodkod.bound

  val merge_bounds : Kodkod.bound list -> Kodkod.bound list

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

  val declarative_axioms_for_datatypes :

    extended_context -> int -> int Typtab.table -> kodkod_constrs

    -> nut NameTable.table -> dtype_spec list -> Kodkod.formula list

  val kodkod_formula_from_nut :

    int -> int Typtab.table -> bool -> kodkod_constrs -> nut -> Kodkod.formula

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

type nfa_transition = KK.rel_expr * typ

type nfa_entry = typ * nfa_transition list

type nfa_table = nfa_entry list

structure NfaGraph = Graph(type key = typ val ord = TermOrd.typ_ord)

(* int -> KK.int_expr list *)

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

(* int -> int -> int -> KK.bound list -> KK.formula -> int *)

fun univ_card nat_card int_card main_j0 bounds formula =

  let

    (* KK.rel_expr -> int -> int *)

    fun rel_expr_func r k =

      Int.max (k, case r of

                    KK.Atom j => j + 1

                  | KK.AtomSeq (k', j0) => j0 + k'

                  | _ => 0)

    (* KK.tuple -> int -> int *)

    fun tuple_func t k =

      case t of

        KK.Tuple js => fold Integer.max (map (Integer.add 1) js) k

      | _ => k

    (* KK.tuple_set -> int -> int *)

    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

(* int -> KK.formula -> unit *)

fun check_bits bits formula =

  let

    (* KK.int_expr -> unit -> unit *)

    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

(* int -> int -> unit *)

fun check_arity univ_card n =

  if n > KK.max_arity univ_card then

    raise TOO_LARGE ("Nitpick_Kodkod.check_arity",

                     "arity " ^ string_of_int n ^ " too large for universe of \

                     \cardinality " ^ string_of_int univ_card)

  else

    ()

(* bool -> int -> int list -> KK.tuple *)

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)

(* (int * int) list -> KK.tuple_set *)

val tuple_set_from_atom_schema = foldl1 KK.TupleProduct o map KK.TupleAtomSeq

(* rep -> KK.tuple_set *)

val upper_bound_for_rep = tuple_set_from_atom_schema o atom_schema_of_rep

(* int -> KK.tuple_set *)

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

(* int -> KK.int_bound list *)

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

(* int -> int -> KK.int_bound list *)

fun pow_of_two_int_bounds bits j0 univ_card =

  let

    (* int -> int -> int -> KK.int_bound list *)

    fun aux 0  _ _ = []

      | aux 1 pow_of_two j =

        if j < univ_card then [(SOME (~ pow_of_two), [single_atom j])] else []

      | 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

(* KK.formula -> KK.n_ary_index list *)

fun built_in_rels_in_formula formula =

  let

    (* KK.rel_expr -> KK.n_ary_index list -> KK.n_ary_index list *)

    fun rel_expr_func (r as KK.Rel (x as (n, j))) =

        if x = unsigned_bit_word_sel_rel orelse x = signed_bit_word_sel_rel then

          I

        else

          (case AList.lookup (op =) (#rels initial_pool) n of

             SOME k => j < k ? insert (op =) x

           | NONE => I)

      | rel_expr_func _ = I

    val expr_F = {formula_func = K I, rel_expr_func = rel_expr_func,

                  int_expr_func = K I}

  in KK.fold_formula expr_F formula [] end

val max_table_size = 65536

(* int -> unit *)

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

    ()

(* bool -> int -> int * int -> (int -> int) -> KK.tuple list *)

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))

(* bool -> int -> int * int -> int -> (int * int -> int) -> KK.tuple list *)

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)))

(* bool -> int -> int * int -> int -> (int * int -> int * int)

   -> KK.tuple list *)

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)))

(* bool -> int -> int * int -> (int * int -> int) -> KK.tuple list *)

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)))

(* bool -> int -> int * int -> (int * int -> int * int) -> KK.tuple list *)

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)))

(* int * int -> int *)

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

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

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

(* bool -> int -> int -> int -> int -> int * int

   -> string * bool * KK.tuple list *)

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_for_bool o op <))

   else if x = int_less_rel then

     ("int_less", tabulate_int_op2 debug univ_card (int_card, j0)

                                   (int_for_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"))

(* bool -> int -> int -> int -> int -> int * int -> KK.rel_expr -> KK.bound *)

fun bound_for_built_in_rel debug univ_card nat_card int_card j0 x =

  let

    val (nick, ts) = tabulate_built_in_rel debug univ_card nat_card int_card

                                           j0 x

  in ([(x, nick)], [KK.TupleSet ts]) end

(* bool -> int -> int -> int -> int -> KK.formula -> KK.bound list *)

fun bounds_for_built_in_rels_in_formula debug univ_card nat_card int_card j0 =

  map (bound_for_built_in_rel debug univ_card nat_card int_card j0)

  o built_in_rels_in_formula

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

fun bound_comment ctxt debug nick T R =

  short_name nick ^

  (if debug then " :: " ^ plain_string_from_yxml (Syntax.string_of_typ ctxt T)

   else "") ^ " : " ^ string_for_rep R

(* Proof.context -> bool -> nut -> KK.bound *)

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])

(* Proof.context -> bool -> dtype_spec list -> nut -> KK.bound *)

fun bound_for_sel_rel ctxt debug dtypes

        (FreeRel (x, T as Type ("fun", [T1, T2]), R as Func (Atom (_, j0), R2),

                  nick)) =

    let

      val constr as {delta, epsilon, exclusive, explicit_max, ...} =

        constr_spec dtypes (original_name nick, T1)

    in

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

       if explicit_max = 0 then

         [KK.TupleSet []]

       else

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

           if R2 = Formula Neut then

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

           else

             [KK.TupleSet [], KK.TupleProduct (ts, upper_bound_for_rep R2)]

         end)

    end

  | bound_for_sel_rel _ _ _ u =

    raise NUT ("Nitpick_Kodkod.bound_for_sel_rel", [u])

(* KK.bound list -> KK.bound list *)

fun merge_bounds bs =

  let

    (* KK.bound -> int *)

    fun arity (zs, _) = fst (fst (hd zs))

    (* KK.bound list -> KK.bound -> KK.bound list -> KK.bound list *)

    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

(* int -> int -> KK.rel_expr list *)

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

(* int list -> KK.rel_expr *)

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

(* rep -> KK.rel_expr list *)

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])

(* KK.rel_expr -> KK.rel_expr list *)

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]

(* rep -> KK.rel_expr *)

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)

(* int -> int list -> KK.decl list *)

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

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

(* kodkod_constrs -> rep -> KK.rel_expr -> KK.formula *)

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

(* kodkod_constrs -> KK.rel_expr list -> KK.formula *)

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)

(* int -> kodkod_constrs -> (KK.rel_expr -> KK.rel_expr) -> KK.rel_expr

   -> KK.rel_expr *)

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

(* kodkod_constrs -> (KK.rel_expr -> KK.rel_expr) -> KK.rel_expr

   -> KK.rel_expr *)

val single_rel_rel_let = basic_rel_rel_let 0

(* kodkod_constrs -> (KK.rel_expr -> KK.rel_expr -> KK.rel_expr) -> KK.rel_expr

   -> KK.rel_expr -> KK.rel_expr *)

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

(* kodkod_constrs -> (KK.rel_expr -> KK.rel_expr -> KK.rel_expr -> KK.rel_expr)

   -> KK.rel_expr -> KK.rel_expr -> KK.rel_expr -> KK.rel_expr *)

fun tripl_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

(* kodkod_constrs -> int -> KK.formula -> KK.rel_expr *)

fun atom_from_formula ({kk_rel_if, ...} : kodkod_constrs) j0 f =

  kk_rel_if f (KK.Atom (j0 + 1)) (KK.Atom j0)

(* kodkod_constrs -> rep -> KK.formula -> KK.rel_expr *)

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])

(* kodkod_cotrs -> int -> int -> KK.rel_expr -> KK.rel_expr list *)

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)

(* kodkod_constrs -> bool -> rep -> rep -> KK.rel_expr -> KK.rel_expr

   -> KK.rel_expr *)

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 =

        if inline_rel_expr r1 then unpack_vect_in_chunks kk 1 arity1 r1

        else unpack_products r1

    in

      if one andalso length unpacked_rs1 = arity1 then

        fold kk_join unpacked_rs1 r2

      else

        kk_project_seq

            (kk_intersect (kk_product r1 (full_rel_for_rep res_R)) r2)

            arity1 (arity_of_rep res_R)

    end

(* kodkod_constrs -> rep -> rep -> KK.rel_expr -> KK.rel_expr list

   -> KK.rel_expr list -> KK.rel_expr *)

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

(* kodkod_constrs -> rep -> rep -> KK.rel_expr -> KK.rel_expr *)

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

(* kodkod_constrs -> int * int -> rep -> KK.rel_expr -> KK.rel_expr *)

and atom_from_rel_expr kk (x as (k, j0)) 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

(* kodkod_constrs -> rep list -> rep -> KK.rel_expr -> KK.rel_expr *)

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' =>

    let

      val Rs = filter (not_equal Unit) Rs

      val Rs' = filter (not_equal Unit) Rs'

    in

      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

    end

  | _ => raise REP ("Nitpick_Kodkod.struct_from_rel_expr", [old_R])

(* kodkod_constrs -> int -> rep -> rep -> KK.rel_expr -> KK.rel_expr *)

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 (Unit, R2) => rel_expr_from_rel_expr kk R R2 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])

(* kodkod_constrs -> rep -> rep -> rep -> KK.rel_expr -> KK.rel_expr *)

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 Unit R2 old_R r =

    (case old_R of

       Vect (k, R') => rel_expr_from_rel_expr kk R2 R' r

     | Func (Unit, R2') => rel_expr_from_rel_expr kk R2 R2' r

     | Func (Atom (1, _), Formula Neut) =>

       (case unopt_rep R2 of

          Atom (2, j0) => atom_from_formula kk j0 (#kk_some kk r)

        | _ => raise REP ("Nitpick_Kodkod.func_from_no_opt_rel_expr",

                          [old_R, Func (Unit, R2)]))

     | Func (R1', R2') =>

       rel_expr_from_rel_expr kk R2 R2' (#kk_project_seq kk r (arity_of_rep R1')

                              (arity_of_rep R2'))

     | _ => raise REP ("Nitpick_Kodkod.func_from_no_opt_rel_expr",

                       [old_R, Func (Unit, R2)]))

  | 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

         (* KK.rel_expr -> KK.rel_expr -> KK.rel_expr *)

         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 (Unit, (Atom (2, j0))) =>

       #kk_rel_if kk (#kk_rel_eq kk r (KK.Atom (j0 + 1)))

                  (full_rel_for_rep R1) (empty_rel_for_rep R1)

     | 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 (Unit, R2') =>

      let val j0 = some_j0 in

        func_from_no_opt_rel_expr kk R1 R2 (Func (Atom (1, j0), R2'))

                                  (#kk_product kk (KK.Atom j0) r)

      end

    | 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)])

(* kodkod_constrs -> rep -> rep -> KK.rel_expr -> KK.rel_expr *)

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

(* kodkod_constrs -> rep -> rep -> rep -> KK.rel_expr -> KK.rel_expr *)

and rel_expr_to_func kk R1 R2 = rel_expr_from_rel_expr kk (Func (R1, R2))

(* kodkod_constrs -> typ -> KK.rel_expr -> KK.rel_expr *)

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))

(* kodkod_constrs -> typ -> KK.rel_expr -> KK.int_expr *)

val int_expr_from_atom = KK.SetSum ooo bit_set_from_atom

(* kodkod_constrs -> typ -> rep -> KK.int_expr -> KK.rel_expr *)

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))

(* kodkod_constrs -> nut -> KK.formula *)

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])

(* nut NameTable.table -> styp -> KK.rel_expr * rep * int *)

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])

(* nut NameTable.table -> styp -> KK.rel_expr *)

fun discr_rel_expr rel_table = #1 o const_triple rel_table o discr_for_constr

(* extended_context -> kodkod_constrs -> nut NameTable.table -> dtype_spec list

   -> styp -> int -> nfa_transition list *)

fun nfa_transitions_for_sel ext_ctxt ({kk_project, ...} : kodkod_constrs)

                            rel_table (dtypes : dtype_spec list) constr_x n =

  let

    val x as (_, T) = boxed_nth_sel_for_constr ext_ctxt 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 (kk_project r (map KK.Num [0, j]), T))

               (index_seq 1 (arity - 1) ~~ tl type_schema)

  end

(* extended_context -> kodkod_constrs -> nut NameTable.table -> dtype_spec list

   -> styp -> nfa_transition list *)

fun nfa_transitions_for_constr ext_ctxt kk rel_table dtypes (x as (_, T)) =

  maps (nfa_transitions_for_sel ext_ctxt kk rel_table dtypes x)

       (index_seq 0 (num_sels_for_constr_type T))

(* extended_context -> kodkod_constrs -> nut NameTable.table -> dtype_spec list

   -> dtype_spec -> nfa_entry option *)

fun nfa_entry_for_datatype _ _ _ _ ({co = true, ...} : dtype_spec) = NONE

  | nfa_entry_for_datatype _ _ _ _ {shallow = true, ...} = NONE

  | nfa_entry_for_datatype ext_ctxt kk rel_table dtypes {typ, constrs, ...} =

    SOME (typ, maps (nfa_transitions_for_constr ext_ctxt kk rel_table dtypes

                     o #const) constrs)

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

(* nfa_table -> typ -> typ -> KK.rel_expr list *)

fun direct_path_rel_exprs nfa start final =

  case AList.lookup (op =) nfa final of

    SOME trans => map fst (filter (curry (op =) start o snd) trans)

  | NONE => []

(* kodkod_constrs -> nfa_table -> typ list -> typ -> typ -> KK.rel_expr *)

and any_path_rel_expr ({kk_union, ...} : kodkod_constrs) nfa [] start final =

    fold kk_union (direct_path_rel_exprs nfa start final)

         (if start = final then KK.Iden else empty_rel)

  | any_path_rel_expr (kk as {kk_union, ...}) nfa (q :: qs) start final =

    kk_union (any_path_rel_expr kk nfa qs start final)

             (knot_path_rel_expr kk nfa qs start q final)

(* kodkod_constrs -> nfa_table -> typ list -> typ -> typ -> typ

   -> KK.rel_expr *)

and knot_path_rel_expr (kk as {kk_join, kk_reflexive_closure, ...}) nfa qs start

                       knot final =

  kk_join (kk_join (any_path_rel_expr kk nfa qs knot final)

                   (kk_reflexive_closure (loop_path_rel_expr kk nfa qs knot)))

          (any_path_rel_expr kk nfa qs start knot)

(* kodkod_constrs -> nfa_table -> typ list -> typ -> KK.rel_expr *)

and loop_path_rel_expr ({kk_union, ...} : kodkod_constrs) nfa [] start =

    fold kk_union (direct_path_rel_exprs nfa start start) empty_rel

  | loop_path_rel_expr (kk as {kk_union, kk_closure, ...}) nfa (q :: qs) start =

    if start = q then

      kk_closure (loop_path_rel_expr kk nfa qs start)

    else

      kk_union (loop_path_rel_expr kk nfa qs start)

               (knot_path_rel_expr kk nfa qs start q start)

(* nfa_table -> unit NfaGraph.T *)

fun graph_for_nfa nfa =

  let

    (* typ -> unit NfaGraph.T -> unit NfaGraph.T *)

    fun new_node q = perhaps (try (NfaGraph.new_node (q, ())))

    (* nfa_table -> unit NfaGraph.T -> unit NfaGraph.T *)

    fun add_nfa [] = I

      | add_nfa ((_, []) :: nfa) = add_nfa nfa

      | add_nfa ((q, ((_, q') :: transitions)) :: nfa) =

        add_nfa ((q, transitions) :: nfa) o NfaGraph.add_edge (q, q') o

        new_node q' o new_node q

  in add_nfa nfa NfaGraph.empty end

(* nfa_table -> nfa_table list *)

fun strongly_connected_sub_nfas nfa =

  nfa |> graph_for_nfa |> NfaGraph.strong_conn

      |> map (fn keys => filter (member (op =) keys o fst) nfa)

(* dtype_spec list -> kodkod_constrs -> nfa_table -> typ -> KK.formula *)

fun acyclicity_axiom_for_datatype dtypes kk nfa start =

  #kk_no kk (#kk_intersect kk

                 (loop_path_rel_expr kk nfa (map fst nfa) start) KK.Iden)

(* extended_context -> kodkod_constrs -> nut NameTable.table -> dtype_spec list

   -> KK.formula list *)

fun acyclicity_axioms_for_datatypes ext_ctxt kk rel_table dtypes =

  map_filter (nfa_entry_for_datatype ext_ctxt kk rel_table dtypes) dtypes

  |> strongly_connected_sub_nfas

  |> maps (fn nfa => map (acyclicity_axiom_for_datatype dtypes kk nfa o fst)

                         nfa)

(* extended_context -> int -> kodkod_constrs -> nut NameTable.table

   -> KK.rel_expr -> constr_spec -> int -> KK.formula *)

fun sel_axiom_for_sel ext_ctxt j0

        (kk as {kk_all, kk_implies, kk_formula_if, kk_subset, kk_rel_eq, kk_no,

                kk_join, ...}) rel_table dom_r

        ({const, delta, epsilon, exclusive, explicit_max, ...} : constr_spec)

        n =

  let

    val x as (_, T) = boxed_nth_sel_for_constr ext_ctxt const n

    val (r, R, arity) = const_triple rel_table x

    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

      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

(* extended_context -> int -> int -> kodkod_constrs -> nut NameTable.table

   -> constr_spec -> KK.formula list *)

fun sel_axioms_for_constr ext_ctxt bits j0 kk rel_table

        (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 ran_r = discr_rel_expr rel_table const

        val max_axiom =

          if honors_explicit_max then

            KK.True

          else if is_twos_complement_representable bits (epsilon - delta) then

            KK.LE (KK.Cardinality ran_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 ::

        map (sel_axiom_for_sel ext_ctxt j0 kk rel_table ran_r constr)

            (index_seq 0 (num_sels_for_constr_type (snd const)))

      end

  end

(* extended_context -> int -> int -> kodkod_constrs -> nut NameTable.table

   -> dtype_spec -> KK.formula list *)

fun sel_axioms_for_datatype ext_ctxt bits j0 kk rel_table

                            ({constrs, ...} : dtype_spec) =

  maps (sel_axioms_for_constr ext_ctxt bits j0 kk rel_table) constrs

(* extended_context -> kodkod_constrs -> nut NameTable.table -> constr_spec

   -> KK.formula list *)

fun uniqueness_axiom_for_constr ext_ctxt

        ({kk_all, kk_implies, kk_and, kk_rel_eq, kk_lone, kk_join, ...}

         : kodkod_constrs) rel_table ({const, ...} : constr_spec) =

  let

    (* KK.rel_expr -> KK.formula *)

    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 boxed_nth_sel_for_constr ext_ctxt const)

                      (~1 upto num_sels - 1)

    val j0 = case triples |> hd |> #2 of

               Func (Atom (_, j0), _) => j0

             | R => raise REP ("Nitpick_Kodkod.uniqueness_axiom_for_constr",

                               [R])

    val set_r = triples |> hd |> #1

  in

    if num_sels = 0 then

      kk_lone set_r

    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

(* extended_context -> kodkod_constrs -> nut NameTable.table -> dtype_spec

   -> KK.formula list *)

fun uniqueness_axioms_for_datatype ext_ctxt kk rel_table

                                   ({constrs, ...} : dtype_spec) =

  map (uniqueness_axiom_for_constr ext_ctxt kk rel_table) constrs

(* constr_spec -> int *)

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

(* int -> kodkod_constrs -> nut NameTable.table -> dtype_spec

   -> KK.formula list *)

fun partition_axioms_for_datatype j0 (kk as {kk_rel_eq, kk_union, ...})

                                  rel_table

                                  ({card, constrs, ...} : dtype_spec) =

  if forall #exclusive constrs then

    [Integer.sum (map effective_constr_max constrs) = card |> formula_for_bool]

  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

(* extended_context -> int -> int Typtab.table -> kodkod_constrs

   -> nut NameTable.table -> dtype_spec -> KK.formula list *)

fun other_axioms_for_datatype _ _ _ _ _ {shallow = true, ...} = []

  | other_axioms_for_datatype ext_ctxt bits ofs kk rel_table

                              (dtype as {typ, ...}) =

    let val j0 = offset_of_type ofs typ in

      sel_axioms_for_datatype ext_ctxt bits j0 kk rel_table dtype @

      uniqueness_axioms_for_datatype ext_ctxt kk rel_table dtype @

      partition_axioms_for_datatype j0 kk rel_table dtype

    end

(* extended_context -> int -> int Typtab.table -> kodkod_constrs

   -> nut NameTable.table -> dtype_spec list -> KK.formula list *)

fun declarative_axioms_for_datatypes ext_ctxt bits ofs kk rel_table dtypes =

  acyclicity_axioms_for_datatypes ext_ctxt kk rel_table dtypes @

  maps (other_axioms_for_datatype ext_ctxt bits ofs kk rel_table) dtypes

(* int -> int Typtab.table -> bool -> kodkod_constrs -> nut -> KK.formula *)

fun kodkod_formula_from_nut bits ofs liberal

        (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_one,

                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)

    (* polarity -> int -> KK.rel_expr -> KK.formula *)

    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))

    (* int -> KK.rel_expr -> KK.formula *)

    val formula_from_atom = formula_from_opt_atom Pos

    (* KK.formula -> KK.formula -> KK.formula *)

    fun kk_notimplies f1 f2 = kk_and f1 (kk_not f2)

    (* KK.rel_expr -> KK.rel_expr -> KK.rel_expr *)

    val kk_or3 =

      double_rel_rel_let kk

          (fn r1 => fn r2 =>

              kk_rel_if (kk_subset true_atom (kk_union r1 r2)) true_atom

                        (kk_intersect r1 r2))

    val kk_and3 =

      double_rel_rel_let kk

          (fn r1 => fn r2 =>

              kk_rel_if (kk_subset false_atom (kk_union r1 r2)) false_atom

                        (kk_intersect r1 r2))

    fun kk_notimplies3 r1 r2 = kk_and3 r1 (kk_not3 r2)

    (* int -> KK.rel_expr -> KK.formula list *)

    val unpack_formulas =

      map (formula_from_atom bool_j0) oo unpack_vect_in_chunks kk 1

    (* (KK.formula -> KK.formula -> KK.formula) -> int -> KK.rel_expr

       -> KK.rel_expr -> KK.rel_expr *)

    fun kk_vect_set_op connective k r1 r2 =

      fold1 kk_product (map2 (atom_from_formula kk bool_j0 oo connective)

                             (unpack_formulas k r1) (unpack_formulas k r2))

    (* (KK.formula -> KK.formula -> KK.formula) -> int -> KK.rel_expr

       -> KK.rel_expr -> KK.formula *)

    fun kk_vect_set_bool_op connective k r1 r2 =

      fold1 kk_and (map2 connective (unpack_formulas k r1)