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

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2009, 2010

 Model reconstruction for Nitpick.

 *)

 signature NITPICK_MODEL =

 sig

   type styp = Nitpick_Util.styp

   type scope = Nitpick_Scope.scope

   type rep = Nitpick_Rep.rep

   type nut = Nitpick_Nut.nut

   type params =

     {show_datatypes: bool,

      show_consts: bool}

   type term_postprocessor =

     Proof.context -> string -> (typ -> term list) -> typ -> term -> term

   structure NameTable : TABLE

   val irrelevant : string

   val unknown : string

   val unrep : string

   val register_term_postprocessor :

     typ -> term_postprocessor -> theory -> theory

   val unregister_term_postprocessor : typ -> theory -> theory

   val tuple_list_for_name :

     nut NameTable.table -> Kodkod.raw_bound list -> nut -> int list list

   val dest_plain_fun : term -> bool * (term list * term list)

   val reconstruct_hol_model :

     params -> scope -> (term option * int list) list -> styp list -> nut list

     -> nut list -> nut list -> nut NameTable.table -> Kodkod.raw_bound list

     -> Pretty.T * bool

   val prove_hol_model :

     scope -> Time.time option -> nut list -> nut list -> nut NameTable.table

     -> Kodkod.raw_bound list -> term -> bool option

 end;

 structure Nitpick_Model : NITPICK_MODEL =

 struct

 open Nitpick_Util

 open Nitpick_HOL

 open Nitpick_Scope

 open Nitpick_Peephole

 open Nitpick_Rep

 open Nitpick_Nut

 structure KK = Kodkod

 type params =

   {show_datatypes: bool,

    show_consts: bool}

 type term_postprocessor =

   Proof.context -> string -> (typ -> term list) -> typ -> term -> term

 structure Data = Theory_Data(

   type T = (typ * term_postprocessor) list

   val empty = []

   val extend = I

   fun merge (x, y) = AList.merge (op =) (K true) (x, y))

 val irrelevant = "_"

 val unknown = "?"

 val unrep = "\<dots>"

 val maybe_mixfix = "_\<^sup>?"

 val base_mixfix = "_\<^bsub>base\<^esub>"

 val step_mixfix = "_\<^bsub>step\<^esub>"

 val abs_mixfix = "\<guillemotleft>_\<guillemotright>"

 val arg_var_prefix = "x"

 val cyclic_co_val_name = "\<omega>"

 val cyclic_const_prefix = "\<xi>"

 val cyclic_type_name = nitpick_prefix ^ cyclic_const_prefix

 val opt_flag = nitpick_prefix ^ "opt"

 val non_opt_flag = nitpick_prefix ^ "non_opt"

 type atom_pool = ((string * int) * int list) list

 fun add_wacky_syntax ctxt =

   let

     val name_of = fst o dest_Const

     val thy = ProofContext.theory_of ctxt |> Context.reject_draft

     val (maybe_t, thy) =

       Sign.declare_const ((@{binding nitpick_maybe}, @{typ "'a => 'a"}),

                           Mixfix (maybe_mixfix, [1000], 1000)) thy

     val (abs_t, thy) =

       Sign.declare_const ((@{binding nitpick_abs}, @{typ "'a => 'b"}),

                           Mixfix (abs_mixfix, [40], 40)) thy

     val (base_t, thy) =

       Sign.declare_const ((@{binding nitpick_base}, @{typ "'a => 'a"}),

                           Mixfix (base_mixfix, [1000], 1000)) thy

     val (step_t, thy) =

       Sign.declare_const ((@{binding nitpick_step}, @{typ "'a => 'a"}),

                           Mixfix (step_mixfix, [1000], 1000)) thy

   in

     (pairself (pairself name_of) ((maybe_t, abs_t), (base_t, step_t)),

      ProofContext.transfer_syntax thy ctxt)

   end

 (** Term reconstruction **)

 fun nth_atom_suffix pool s j k =

   (case AList.lookup (op =) (!pool) (s, k) of

      SOME js =>

      (case find_index (curry (op =) j) js of

         ~1 => (Unsynchronized.change pool (cons ((s, k), j :: js));

                length js + 1)

       | n => length js - n)

    | NONE => (Unsynchronized.change pool (cons ((s, k), [j])); 1))

   |> nat_subscript

   |> (s <> "" andalso Symbol.is_ascii_digit (List.last (explode s)))

      ? prefix "\<^isub>,"

 fun nth_atom_name pool prefix (Type (s, _)) j k =

     let val s' = shortest_name s in

       prefix ^ (if String.isPrefix "\\" s' then s' else substring (s', 0, 1)) ^

       nth_atom_suffix pool s j k

     end

   | nth_atom_name pool prefix (TFree (s, _)) j k =

     prefix ^ perhaps (try (unprefix "'")) s ^ nth_atom_suffix pool s j k

   | nth_atom_name _ _ T _ _ =

     raise TYPE ("Nitpick_Model.nth_atom_name", [T], [])

 fun nth_atom pool for_auto T j k =

   if for_auto then

     Free (nth_atom_name pool (hd (space_explode "." nitpick_prefix)) T j k, T)

   else

     Const (nth_atom_name pool "" T j k, T)

 fun extract_real_number (Const (@{const_name divide}, _) $ t1 $ t2) =

     real (snd (HOLogic.dest_number t1)) / real (snd (HOLogic.dest_number t2))

   | extract_real_number t = real (snd (HOLogic.dest_number t))

 fun nice_term_ord (Abs (_, _, t1), Abs (_, _, t2)) = nice_term_ord (t1, t2)

   | nice_term_ord tp = Real.compare (pairself extract_real_number tp)

     handle TERM ("dest_number", _) =>

            case tp of

              (t11 $ t12, t21 $ t22) =>

              (case nice_term_ord (t11, t21) of

                 EQUAL => nice_term_ord (t12, t22)

               | ord => ord)

            | _ => Term_Ord.fast_term_ord tp

 fun register_term_postprocessor T p = Data.map (cons (T, p))

 fun unregister_term_postprocessor T = Data.map (AList.delete (op =) T)

 fun tuple_list_for_name rel_table bounds name =

   the (AList.lookup (op =) bounds (the_rel rel_table name)) handle NUT _ => [[]]

 fun unarize_unbox_etc_term (Const (@{const_name FinFun}, _) $ t1) =

     unarize_unbox_etc_term t1

   | unarize_unbox_etc_term (Const (@{const_name FunBox}, _) $ t1) =

     unarize_unbox_etc_term t1

   | unarize_unbox_etc_term

         (Const (@{const_name PairBox},

                 Type (@{type_name fun}, [T1, Type (@{type_name fun}, [T2, _])]))

          $ t1 $ t2) =

     let val Ts = map uniterize_unarize_unbox_etc_type [T1, T2] in

       Const (@{const_name Pair}, Ts ---> Type (@{type_name "*"}, Ts))

       $ unarize_unbox_etc_term t1 $ unarize_unbox_etc_term t2

     end

   | unarize_unbox_etc_term (Const (s, T)) =

     Const (s, uniterize_unarize_unbox_etc_type T)

   | unarize_unbox_etc_term (t1 $ t2) =

     unarize_unbox_etc_term t1 $ unarize_unbox_etc_term t2

   | unarize_unbox_etc_term (Free (s, T)) =

     Free (s, uniterize_unarize_unbox_etc_type T)

   | unarize_unbox_etc_term (Var (x, T)) =

     Var (x, uniterize_unarize_unbox_etc_type T)

   | unarize_unbox_etc_term (Bound j) = Bound j

   | unarize_unbox_etc_term (Abs (s, T, t')) =

     Abs (s, uniterize_unarize_unbox_etc_type T, unarize_unbox_etc_term t')

 fun factor_out_types (T1 as Type (@{type_name "*"}, [T11, T12]))

                      (T2 as Type (@{type_name "*"}, [T21, T22])) =

     let val (n1, n2) = pairself num_factors_in_type (T11, T21) in

       if n1 = n2 then

         let

           val ((T11', opt_T12'), (T21', opt_T22')) = factor_out_types T12 T22

         in

           ((Type (@{type_name "*"}, [T11, T11']), opt_T12'),

            (Type (@{type_name "*"}, [T21, T21']), opt_T22'))

         end

       else if n1 < n2 then

         case factor_out_types T1 T21 of

           (p1, (T21', NONE)) => (p1, (T21', SOME T22))

         | (p1, (T21', SOME T22')) =>

           (p1, (T21', SOME (Type (@{type_name "*"}, [T22', T22]))))

       else

         swap (factor_out_types T2 T1)

     end

   | factor_out_types (Type (@{type_name "*"}, [T11, T12])) T2 =

     ((T11, SOME T12), (T2, NONE))

   | factor_out_types T1 (Type (@{type_name "*"}, [T21, T22])) =

     ((T1, NONE), (T21, SOME T22))

   | factor_out_types T1 T2 = ((T1, NONE), (T2, NONE))

 fun make_plain_fun maybe_opt T1 T2 =

   let

     fun aux T1 T2 [] =

         Const (if maybe_opt then opt_flag else non_opt_flag, T1 --> T2)

       | aux T1 T2 ((t1, t2) :: tps) =

         Const (@{const_name fun_upd}, (T1 --> T2) --> T1 --> T2 --> T1 --> T2)

         $ aux T1 T2 tps $ t1 $ t2

   in aux T1 T2 o rev end

 fun is_plain_fun (Const (s, _)) = (s = opt_flag orelse s = non_opt_flag)

   | is_plain_fun (Const (@{const_name fun_upd}, _) $ t0 $ _ $ _) =

     is_plain_fun t0

   | is_plain_fun _ = false

 val dest_plain_fun =

   let

     fun aux (Abs (_, _, Const (s, _))) = (s <> irrelevant, ([], []))

       | aux (Const (s, _)) = (s <> non_opt_flag, ([], []))

       | aux (Const (@{const_name fun_upd}, _) $ t0 $ t1 $ t2) =

         let val (maybe_opt, (ts1, ts2)) = aux t0 in

           (maybe_opt, (t1 :: ts1, t2 :: ts2))

         end

       | aux t = raise TERM ("Nitpick_Model.dest_plain_fun", [t])

   in apsnd (pairself rev) o aux end

 fun break_in_two T T1 T2 t =

   let

     val ps = HOLogic.flat_tupleT_paths T

     val cut = length (HOLogic.strip_tupleT T1)

     val (ps1, ps2) = pairself HOLogic.flat_tupleT_paths (T1, T2)

     val (ts1, ts2) = t |> HOLogic.strip_ptuple ps |> chop cut

   in (HOLogic.mk_ptuple ps1 T1 ts1, HOLogic.mk_ptuple ps2 T2 ts2) end

 fun pair_up (Type (@{type_name "*"}, [T1', T2']))

             (t1 as Const (@{const_name Pair},

                           Type (@{type_name fun},

                                 [_, Type (@{type_name fun}, [_, T1])]))

              $ t11 $ t12) t2 =

     if T1 = T1' then HOLogic.mk_prod (t1, t2)

     else HOLogic.mk_prod (t11, pair_up T2' t12 t2)

   | pair_up _ t1 t2 = HOLogic.mk_prod (t1, t2)

 fun multi_pair_up T1 t1 (ts2, ts3) = map2 (pair o pair_up T1 t1) ts2 ts3

 fun typecast_fun (Type (@{type_name fun}, [T1', T2'])) T1 T2 t =

     let

       fun do_curry T1 T1a T1b T2 t =

         let

           val (maybe_opt, tsp) = dest_plain_fun t

           val tps =

             tsp |>> map (break_in_two T1 T1a T1b)

                 |> uncurry (map2 (fn (t1a, t1b) => fn t2 => (t1a, (t1b, t2))))

                 |> AList.coalesce (op =)

                 |> map (apsnd (make_plain_fun maybe_opt T1b T2))

         in make_plain_fun maybe_opt T1a (T1b --> T2) tps end

       and do_uncurry T1 T2 t =

         let

           val (maybe_opt, tsp) = dest_plain_fun t

           val tps =

             tsp |> op ~~

                 |> maps (fn (t1, t2) =>

                             multi_pair_up T1 t1 (snd (dest_plain_fun t2)))

         in make_plain_fun maybe_opt T1 T2 tps end

       and do_arrow T1' T2' _ _ (Const (s, _)) = Const (s, T1' --> T2')

         | do_arrow T1' T2' T1 T2

                    (Const (@{const_name fun_upd}, _) $ t0 $ t1 $ t2) =

           Const (@{const_name fun_upd},

                  (T1' --> T2') --> T1' --> T2' --> T1' --> T2')

           $ do_arrow T1' T2' T1 T2 t0 $ do_term T1' T1 t1 $ do_term T2' T2 t2

         | do_arrow _ _ _ _ t =

           raise TERM ("Nitpick_Model.typecast_fun.do_arrow", [t])

       and do_fun T1' T2' T1 T2 t =

         case factor_out_types T1' T1 of

           ((_, NONE), (_, NONE)) => t |> do_arrow T1' T2' T1 T2

         | ((_, NONE), (T1a, SOME T1b)) =>

           t |> do_curry T1 T1a T1b T2 |> do_arrow T1' T2' T1a (T1b --> T2)

         | ((T1a', SOME T1b'), (_, NONE)) =>

           t |> do_arrow T1a' (T1b' --> T2') T1 T2 |> do_uncurry T1' T2'

         | _ => raise TYPE ("Nitpick_Model.typecast_fun.do_fun", [T1, T1'], [])

       and do_term (Type (@{type_name fun}, [T1', T2']))

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

           do_fun T1' T2' T1 T2 t

         | do_term (T' as Type (@{type_name "*"}, Ts' as [T1', T2']))

                   (Type (@{type_name "*"}, [T1, T2]))

                   (Const (@{const_name Pair}, _) $ t1 $ t2) =

           Const (@{const_name Pair}, Ts' ---> T')

           $ do_term T1' T1 t1 $ do_term T2' T2 t2

         | do_term T' T t =

           if T = T' then t

           else raise TYPE ("Nitpick_Model.typecast_fun.do_term", [T, T'], [])

     in if T1' = T1 andalso T2' = T2 then t else do_fun T1' T2' T1 T2 t end

   | typecast_fun T' _ _ _ =

     raise TYPE ("Nitpick_Model.typecast_fun", [T'], [])

 fun truth_const_sort_key @{const True} = "0"

   | truth_const_sort_key @{const False} = "2"

   | truth_const_sort_key _ = "1"

 fun mk_tuple (Type (@{type_name "*"}, [T1, T2])) ts =

     HOLogic.mk_prod (mk_tuple T1 ts,

         mk_tuple T2 (List.drop (ts, length (HOLogic.flatten_tupleT T1))))

   | mk_tuple _ (t :: _) = t

   | mk_tuple T [] = raise TYPE ("Nitpick_Model.mk_tuple", [T], [])

 fun varified_type_match thy (candid_T, pat_T) =

   strict_type_match thy (candid_T, Logic.varifyT_global pat_T)

 fun all_values_of_type pool wacky_names (scope as {card_assigns, ...} : scope)

                        sel_names rel_table bounds card T =

   let

     val card = if card = 0 then card_of_type card_assigns T else card

     fun nth_value_of_type n =

       let

         fun term unfold =

           reconstruct_term unfold pool wacky_names scope sel_names rel_table

                            bounds T T (Atom (card, 0)) [[n]]

       in

         case term false of

           t as Const (s, _) =>

           if String.isPrefix cyclic_const_prefix s then

             HOLogic.mk_eq (t, term true)

           else

             t

         | t => t

       end

   in index_seq 0 card |> map nth_value_of_type |> sort nice_term_ord end

 and reconstruct_term unfold pool (wacky_names as ((maybe_name, abs_name), _))

         (scope as {hol_ctxt as {thy, ctxt, stds, ...}, binarize, card_assigns,

                    bits, datatypes, ofs, ...}) sel_names rel_table bounds =

   let

     val for_auto = (maybe_name = "")

     fun value_of_bits jss =

       let

         val j0 = offset_of_type ofs @{typ unsigned_bit}

         val js = map (Integer.add (~ j0) o the_single) jss

       in

         fold (fn j => Integer.add (reasonable_power 2 j |> j = bits ? op ~))

              js 0

       end

     val all_values =

       all_values_of_type pool wacky_names scope sel_names rel_table bounds 0

     fun postprocess_term (Type (@{type_name fun}, _)) = I

       | postprocess_term T =

         if null (Data.get thy) then

           I

         else case AList.lookup (varified_type_match thy) (Data.get thy) T of

           SOME postproc => postproc ctxt maybe_name all_values T

         | NONE => I

     fun postprocess_subterms Ts (t1 $ t2) =

         let val t = postprocess_subterms Ts t1 $ postprocess_subterms Ts t2 in

           postprocess_term (fastype_of1 (Ts, t)) t

         end

       | postprocess_subterms Ts (Abs (s, T, t')) =

         Abs (s, T, postprocess_subterms (T :: Ts) t')

       | postprocess_subterms Ts t = postprocess_term (fastype_of1 (Ts, t)) t

     fun make_set maybe_opt T1 T2 tps =

       let

         val empty_const = Const (@{const_abbrev Set.empty}, T1 --> T2)

         val insert_const = Const (@{const_name insert},

                                   T1 --> (T1 --> T2) --> T1 --> T2)

         fun aux [] =

             if maybe_opt andalso not (is_complete_type datatypes false T1) then

               insert_const $ Const (unrep, T1) $ empty_const

             else

               empty_const

           | aux ((t1, t2) :: zs) =

             aux zs

             |> t2 <> @{const False}

                ? curry (op $)

                        (insert_const

                         $ (t1 |> t2 <> @{const True}

                                  ? curry (op $)

                                          (Const (maybe_name, T1 --> T1))))

       in

         if forall (fn (_, t) => t <> @{const True} andalso t <> @{const False})

                   tps then

           Const (unknown, T1 --> T2)

         else

           aux tps

       end

     fun make_map maybe_opt T1 T2 T2' =

       let

         val update_const = Const (@{const_name fun_upd},

                                   (T1 --> T2) --> T1 --> T2 --> T1 --> T2)

         fun aux' [] = Const (@{const_abbrev Map.empty}, T1 --> T2)

           | aux' ((t1, t2) :: tps) =

             (case t2 of

                Const (@{const_name None}, _) => aux' tps

              | _ => update_const $ aux' tps $ t1 $ t2)

         fun aux tps =

           if maybe_opt andalso not (is_complete_type datatypes false T1) then

             update_const $ aux' tps $ Const (unrep, T1)

             $ (Const (@{const_name Some}, T2' --> T2) $ Const (unknown, T2'))

           else

             aux' tps

       in aux end

     fun polish_funs Ts t =

       (case fastype_of1 (Ts, t) of

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

          if is_plain_fun t then

            case T2 of

              @{typ bool} =>

              let

                val (maybe_opt, ts_pair) =

                  dest_plain_fun t ||> pairself (map (polish_funs Ts))

              in

                make_set maybe_opt T1 T2

                         (sort_wrt (truth_const_sort_key o snd) (op ~~ ts_pair))

              end

            | Type (@{type_name option}, [T2']) =>

              let

                val (maybe_opt, ts_pair) =

                  dest_plain_fun t ||> pairself (map (polish_funs Ts))

              in make_map maybe_opt T1 T2 T2' (rev (op ~~ ts_pair)) end

            | _ => raise SAME ()

          else

            raise SAME ()

        | _ => raise SAME ())

       handle SAME () =>

              case t of

                (t1 as Const (@{const_name fun_upd}, _) $ t11 $ _)

                $ (t2 as Const (s, _)) =>

                if s = unknown then polish_funs Ts t11

                else polish_funs Ts t1 $ polish_funs Ts t2

              | t1 $ t2 => polish_funs Ts t1 $ polish_funs Ts t2

              | Abs (s, T, t') => Abs (s, T, polish_funs (T :: Ts) t')

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

                if s = opt_flag orelse s = non_opt_flag then

                  Abs ("x", T1,

                       Const (if is_complete_type datatypes false T1 then

                                irrelevant

                              else

                                unknown, T2))

                else

                  t

              | t => t

     fun make_fun maybe_opt T1 T2 T' ts1 ts2 =

       ts1 ~~ ts2 |> sort (nice_term_ord o pairself fst)

                  |> make_plain_fun maybe_opt T1 T2

                  |> unarize_unbox_etc_term

                  |> typecast_fun (uniterize_unarize_unbox_etc_type T')

                                  (uniterize_unarize_unbox_etc_type T1)

                                  (uniterize_unarize_unbox_etc_type T2)

     fun term_for_atom seen (T as Type (@{type_name fun}, [T1, T2])) T' j _ =

         let

           val k1 = card_of_type card_assigns T1

           val k2 = card_of_type card_assigns T2

         in

           term_for_rep true seen T T' (Vect (k1, Atom (k2, 0)))

                        [nth_combination (replicate k1 (k2, 0)) j]

           handle General.Subscript =>

                  raise ARG ("Nitpick_Model.reconstruct_term.term_for_atom",

                             signed_string_of_int j ^ " for " ^

                             string_for_rep (Vect (k1, Atom (k2, 0))))

         end

       | term_for_atom seen (Type (@{type_name "*"}, [T1, T2])) _ j k =

         let

           val k1 = card_of_type card_assigns T1

           val k2 = k div k1

         in

           list_comb (HOLogic.pair_const T1 T2,

                      map3 (fn T => term_for_atom seen T T) [T1, T2]

                           [j div k2, j mod k2] [k1, k2]) (* ### k2 or k1? FIXME *)

         end

       | term_for_atom seen @{typ prop} _ j k =

         HOLogic.mk_Trueprop (term_for_atom seen bool_T bool_T j k)

       | term_for_atom _ @{typ bool} _ j _ =

         if j = 0 then @{const False} else @{const True}

       | term_for_atom _ @{typ unit} _ _ _ = @{const Unity}

       | term_for_atom seen T _ j k =

         if T = nat_T andalso is_standard_datatype thy stds nat_T then

           HOLogic.mk_number nat_T j

         else if T = int_T then

           HOLogic.mk_number int_T (int_for_atom (k, 0) j)

         else if is_fp_iterator_type T then

           HOLogic.mk_number nat_T (k - j - 1)

         else if T = @{typ bisim_iterator} then

           HOLogic.mk_number nat_T j

         else case datatype_spec datatypes T of

           NONE => nth_atom pool for_auto T j k

         | SOME {deep = false, ...} => nth_atom pool for_auto T j k

         | SOME {co, standard, constrs, ...} =>

           let

             fun tuples_for_const (s, T) =

               tuple_list_for_name rel_table bounds (ConstName (s, T, Any))

             fun cyclic_atom () =

               nth_atom pool for_auto (Type (cyclic_type_name, [])) j k

             fun cyclic_var () = Var ((nth_atom_name pool "" T j k, 0), T)

             val discr_jsss = map (tuples_for_const o discr_for_constr o #const)

                                  constrs

             val real_j = j + offset_of_type ofs T

             val constr_x as (constr_s, constr_T) =

               get_first (fn (jss, {const, ...}) =>

                             if member (op =) jss [real_j] then SOME const

                             else NONE)

                         (discr_jsss ~~ constrs) |> the

             val arg_Ts = curried_binder_types constr_T

             val sel_xs =

               map (binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize

                                                           constr_x)

                   (index_seq 0 (length arg_Ts))

             val sel_Rs =

               map (fn x => get_first

                                (fn ConstName (s', T', R) =>

                                    if (s', T') = x then SOME R else NONE

                                  | u => raise NUT ("Nitpick_Model.reconstruct_\

                                                    \term.term_for_atom", [u]))

                                sel_names |> the) sel_xs

             val arg_Rs = map (snd o dest_Func) sel_Rs

             val sel_jsss = map tuples_for_const sel_xs

             val arg_jsss =

               map (map_filter (fn js => if hd js = real_j then SOME (tl js)

                                         else NONE)) sel_jsss

             val uncur_arg_Ts = binder_types constr_T

             val maybe_cyclic = co orelse not standard

           in

             if maybe_cyclic andalso not (null seen) andalso

                member (op =) (seen |> unfold ? (fst o split_last)) (T, j) then

               cyclic_var ()

             else if constr_s = @{const_name Word} then

               HOLogic.mk_number

                   (if T = @{typ "unsigned_bit word"} then nat_T else int_T)

                   (value_of_bits (the_single arg_jsss))

             else

               let

                 val seen = seen |> maybe_cyclic ? cons (T, j)

                 val ts =

                   if length arg_Ts = 0 then

                     []

                   else

                     map3 (fn Ts =>

                              term_for_rep (constr_s <> @{const_name FinFun})

                                           seen Ts Ts) arg_Ts arg_Rs arg_jsss

                     |> mk_tuple (HOLogic.mk_tupleT uncur_arg_Ts)

                     |> dest_n_tuple (length uncur_arg_Ts)

                 val t =

                   if constr_s = @{const_name Abs_Frac} then

                     case ts of

                       [Const (@{const_name Pair}, _) $ t1 $ t2] =>

                       frac_from_term_pair (body_type T) t1 t2

                     | _ => raise TERM ("Nitpick_Model.reconstruct_term.\

                                        \term_for_atom (Abs_Frac)", ts)

                   else if not for_auto andalso

                           (is_abs_fun ctxt constr_x orelse

                            constr_s = @{const_name Quot}) then

                     Const (abs_name, constr_T) $ the_single ts

                   else

                     list_comb (Const constr_x, ts)

               in

                 if maybe_cyclic then

                   let val var = cyclic_var () in

                     if unfold andalso not standard andalso

                        length seen = 1 andalso

                        exists_subterm (fn Const (s, _) =>

                                           String.isPrefix cyclic_const_prefix s

                                         | t' => t' = var) t then

                       subst_atomic [(var, cyclic_atom ())] t

                     else if exists_subterm (curry (op =) var) t then

                       if co then

                         Const (@{const_name The}, (T --> bool_T) --> T)

                         $ Abs (cyclic_co_val_name, T,

                                Const (@{const_name "op ="}, T --> T --> bool_T)

                                $ Bound 0 $ abstract_over (var, t))

                       else

                         cyclic_atom ()

                     else

                       t

                   end

                 else

                   t

               end

           end

     and term_for_vect seen k R T1 T2 T' js =

       make_fun true T1 T2 T'

                (map (fn j => term_for_atom seen T1 T1 j k) (index_seq 0 k))

                (map (term_for_rep true seen T2 T2 R o single)

                     (batch_list (arity_of_rep R) js))

     and term_for_rep _ seen T T' Unit [[]] = term_for_atom seen T T' 0 1

       | term_for_rep _ seen T T' (R as Atom (k, j0)) [[j]] =

         if j >= j0 andalso j < j0 + k then term_for_atom seen T T' (j - j0) k

         else raise REP ("Nitpick_Model.reconstruct_term.term_for_rep", [R])

       | term_for_rep _ seen (Type (@{type_name "*"}, [T1, T2])) _

                      (Struct [R1, R2]) [js] =

         let

           val arity1 = arity_of_rep R1

           val (js1, js2) = chop arity1 js

         in

           list_comb (HOLogic.pair_const T1 T2,

                      map3 (fn T => term_for_rep true seen T T) [T1, T2] [R1, R2]

                           [[js1], [js2]])

         end

       | term_for_rep _ seen (Type (@{type_name fun}, [T1, T2])) T'

                      (Vect (k, R')) [js] =

         term_for_vect seen k R' T1 T2 T' js

       | term_for_rep maybe_opt seen (Type (@{type_name fun}, [T1, T2])) T'

                      (Func (R1, Formula Neut)) jss =

         let

           val jss1 = all_combinations_for_rep R1

           val ts1 = map (term_for_rep true seen T1 T1 R1 o single) jss1

           val ts2 =

             map (fn js => term_for_rep true seen T2 T2 (Atom (2, 0))

                                        [[int_from_bool (member (op =) jss js)]])

                 jss1

         in make_fun maybe_opt T1 T2 T' ts1 ts2 end

       | term_for_rep maybe_opt seen (Type (@{type_name fun}, [T1, T2])) T'

                      (Func (R1, R2)) jss =

         let

           val arity1 = arity_of_rep R1

           val jss1 = all_combinations_for_rep R1

           val ts1 = map (term_for_rep false seen T1 T1 R1 o single) jss1

           val grouped_jss2 = AList.group (op =) (map (chop arity1) jss)

           val ts2 = map (term_for_rep false seen T2 T2 R2 o the_default []

                          o AList.lookup (op =) grouped_jss2) jss1

         in make_fun maybe_opt T1 T2 T' ts1 ts2 end

       | term_for_rep _ seen T T' (Opt R) jss =

         if null jss then Const (unknown, T)

         else term_for_rep true seen T T' R jss

       | term_for_rep _ _ T _ R jss =

         raise ARG ("Nitpick_Model.reconstruct_term.term_for_rep",

                    Refute.string_of_typ T ^ " " ^ string_for_rep R ^ " " ^

                    string_of_int (length jss))

   in

     postprocess_subterms [] o polish_funs [] o unarize_unbox_etc_term

     oooo term_for_rep true []

   end

 (** Constant postprocessing **)

 fun dest_n_tuple_type 1 T = [T]

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

     T1 :: dest_n_tuple_type (n - 1) T2

   | dest_n_tuple_type _ T =

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

 fun const_format thy def_table (x as (s, T)) =

   if String.isPrefix unrolled_prefix s then

     const_format thy def_table (original_name s, range_type T)

   else if String.isPrefix skolem_prefix s then

     let

       val k = unprefix skolem_prefix s

               |> strip_first_name_sep |> fst |> space_explode "@"

               |> hd |> Int.fromString |> the

     in [k, num_binder_types T - k] end

   else if original_name s <> s then

     [num_binder_types T]

   else case def_of_const thy def_table x of

     SOME t' => if fixpoint_kind_of_rhs t' <> NoFp then

                  let val k = length (strip_abs_vars t') in

                    [k, num_binder_types T - k]

                  end

                else

                  [num_binder_types T]

   | NONE => [num_binder_types T]

 fun intersect_formats _ [] = []

   | intersect_formats [] _ = []

   | intersect_formats ks1 ks2 =

     let val ((ks1', k1), (ks2', k2)) = pairself split_last (ks1, ks2) in

       intersect_formats (ks1' @ (if k1 > k2 then [k1 - k2] else []))

                         (ks2' @ (if k2 > k1 then [k2 - k1] else [])) @

       [Int.min (k1, k2)]

     end

 fun lookup_format thy def_table formats t =

   case AList.lookup (fn (SOME x, SOME y) =>

                         (term_match thy) (x, y) | _ => false)

                     formats (SOME t) of

     SOME format => format

   | NONE => let val format = the (AList.lookup (op =) formats NONE) in

               case t of

                 Const x => intersect_formats format

                                              (const_format thy def_table x)

               | _ => format

             end

 fun format_type default_format format T =

   let

     val T = uniterize_unarize_unbox_etc_type T

     val format = format |> filter (curry (op <) 0)

   in

     if forall (curry (op =) 1) format then

       T

     else

       let

         val (binder_Ts, body_T) = strip_type T

         val batched =

           binder_Ts

           |> map (format_type default_format default_format)

           |> rev |> chunk_list_unevenly (rev format)

           |> map (HOLogic.mk_tupleT o rev)

       in List.foldl (op -->) body_T batched end

   end

 fun format_term_type thy def_table formats t =

   format_type (the (AList.lookup (op =) formats NONE))

               (lookup_format thy def_table formats t) (fastype_of t)

 fun repair_special_format js m format =

   m - 1 downto 0 |> chunk_list_unevenly (rev format)

                  |> map (rev o filter_out (member (op =) js))

                  |> filter_out null |> map length |> rev

 fun user_friendly_const ({thy, evals, def_table, skolems, special_funs, ...}

                          : hol_context) (base_name, step_name) formats =

   let

     val default_format = the (AList.lookup (op =) formats NONE)

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

       (if String.isPrefix special_prefix s then

          let

            val do_term = map_aterms (fn Const x => fst (do_const x) | t' => t')

            val (x' as (_, T'), js, ts) =

              AList.find (op =) (!special_funs) (s, unarize_unbox_etc_type T)

              |> the_single

            val max_j = List.last js

            val Ts = List.take (binder_types T', max_j + 1)

            val missing_js = filter_out (member (op =) js) (0 upto max_j)

            val missing_Ts = filter_indices missing_js Ts

            fun nth_missing_var n =

              ((arg_var_prefix ^ nat_subscript (n + 1), 0), nth missing_Ts n)

            val missing_vars = map nth_missing_var (0 upto length missing_js - 1)

            val vars = special_bounds ts @ missing_vars

            val ts' =

              map (fn j =>

                      case AList.lookup (op =) (js ~~ ts) j of

                        SOME t => do_term t

                      | NONE =>

                        Var (nth missing_vars

                                 (find_index (curry (op =) j) missing_js)))

                  (0 upto max_j)

            val t = do_const x' |> fst

            val format =

              case AList.lookup (fn (SOME t1, SOME t2) => term_match thy (t1, t2)

                                  | _ => false) formats (SOME t) of

                SOME format =>

                repair_special_format js (num_binder_types T') format

              | NONE =>

                const_format thy def_table x'

                |> repair_special_format js (num_binder_types T')

                |> intersect_formats default_format

          in

            (list_comb (t, ts') |> fold_rev abs_var vars,

             format_type default_format format T)

          end

        else if String.isPrefix uncurry_prefix s then

          let

            val (ss, s') = unprefix uncurry_prefix s

                           |> strip_first_name_sep |>> space_explode "@"

          in

            if String.isPrefix step_prefix s' then

              do_const (s', T)

            else

              let

                val k = the (Int.fromString (hd ss))

                val j = the (Int.fromString (List.last ss))

                val (before_Ts, (tuple_T, rest_T)) =

                  strip_n_binders j T ||> (strip_n_binders 1 #>> hd)

                val T' = before_Ts ---> dest_n_tuple_type k tuple_T ---> rest_T

              in do_const (s', T') end

          end

        else if String.isPrefix unrolled_prefix s then

          let val t = Const (original_name s, range_type T) in

            (lambda (Free (iter_var_prefix, nat_T)) t,

             format_type default_format

                         (lookup_format thy def_table formats t) T)

          end

        else if String.isPrefix base_prefix s then

          (Const (base_name, T --> T) $ Const (unprefix base_prefix s, T),

           format_type default_format default_format T)

        else if String.isPrefix step_prefix s then

          (Const (step_name, T --> T) $ Const (unprefix step_prefix s, T),

           format_type default_format default_format T)

        else if String.isPrefix quot_normal_prefix s then

          let val t = Const (nitpick_prefix ^ "normalize quotient type", T) in

            (t, format_term_type thy def_table formats t)

          end

        else if String.isPrefix skolem_prefix s then

          let

            val ss = the (AList.lookup (op =) (!skolems) s)

            val (Ts, Ts') = chop (length ss) (binder_types T)

            val frees = map Free (ss ~~ Ts)

            val s' = original_name s

          in

            (fold lambda frees (Const (s', Ts' ---> T)),

             format_type default_format

                         (lookup_format thy def_table formats (Const x)) T)

          end

        else if String.isPrefix eval_prefix s then

          let

            val t = nth evals (the (Int.fromString (unprefix eval_prefix s)))

          in (t, format_term_type thy def_table formats t) end

        else if s = @{const_name undefined_fast_The} then

          (Const (nitpick_prefix ^ "The fallback", T),

           format_type default_format

                       (lookup_format thy def_table formats

                            (Const (@{const_name The}, (T --> bool_T) --> T))) T)

        else if s = @{const_name undefined_fast_Eps} then

          (Const (nitpick_prefix ^ "Eps fallback", T),

           format_type default_format

                       (lookup_format thy def_table formats

                            (Const (@{const_name Eps}, (T --> bool_T) --> T))) T)

        else

          let val t = Const (original_name s, T) in

            (t, format_term_type thy def_table formats t)

          end)

       |>> map_types uniterize_unarize_unbox_etc_type

       |>> shorten_names_in_term |>> Term.map_abs_vars shortest_name

   in do_const end

 fun assign_operator_for_const (s, T) =

   if String.isPrefix ubfp_prefix s then

     if is_fun_type T then "\<subseteq>" else "\<le>"

   else if String.isPrefix lbfp_prefix s then

     if is_fun_type T then "\<supseteq>" else "\<ge>"

   else if original_name s <> s then

     assign_operator_for_const (strip_first_name_sep s |> snd, T)

   else

     "="

 (** Model reconstruction **)

 fun term_for_name pool scope sel_names rel_table bounds name =

   let val T = type_of name in

     tuple_list_for_name rel_table bounds name

     |> reconstruct_term false pool (("", ""), ("", "")) scope sel_names

                         rel_table bounds T T (rep_of name)

   end

 fun unfold_outer_the_binders (t as Const (@{const_name The}, _)

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

                                                 $ Bound 0 $ t')) =

     betapply (Abs (s, T, t'), t) |> unfold_outer_the_binders

   | unfold_outer_the_binders t = t

 fun bisimilar_values _ 0 _ = true

   | bisimilar_values coTs max_depth (t1, t2) =

     let val T = fastype_of t1 in

       if exists_subtype (member (op =) coTs) T then

         let

           val ((head1, args1), (head2, args2)) =

             pairself (strip_comb o unfold_outer_the_binders) (t1, t2)

           val max_depth = max_depth - (if member (op =) coTs T then 1 else 0)

         in

           head1 = head2 andalso

           forall (bisimilar_values coTs max_depth) (args1 ~~ args2)

         end

       else

         t1 = t2

     end

 fun reconstruct_hol_model {show_datatypes, show_consts}

         ({hol_ctxt = {thy, ctxt, max_bisim_depth, boxes, stds, wfs, user_axioms,

                       debug, binary_ints, destroy_constrs, specialize,

                       star_linear_preds, fast_descrs, tac_timeout, evals,

                       case_names, def_table, nondef_table, user_nondefs,

                       simp_table, psimp_table, choice_spec_table, intro_table,

                       ground_thm_table, ersatz_table, skolems, special_funs,

                       unrolled_preds, wf_cache, constr_cache},

          binarize, card_assigns, bits, bisim_depth, datatypes, ofs} : scope)

         formats all_frees free_names sel_names nonsel_names rel_table bounds =

   let

     val pool = Unsynchronized.ref []

     val (wacky_names as (_, base_step_names), ctxt) =

       add_wacky_syntax ctxt

     val hol_ctxt =

       {thy = thy, ctxt = ctxt, max_bisim_depth = max_bisim_depth, boxes = boxes,

        stds = stds, wfs = wfs, user_axioms = user_axioms, debug = debug,

        binary_ints = binary_ints, destroy_constrs = destroy_constrs,

        specialize = specialize, star_linear_preds = star_linear_preds,

        fast_descrs = fast_descrs, tac_timeout = tac_timeout, evals = evals,

        case_names = case_names, def_table = def_table,

        nondef_table = nondef_table, user_nondefs = user_nondefs,

        simp_table = simp_table, psimp_table = psimp_table,

        choice_spec_table = choice_spec_table, intro_table = intro_table,

        ground_thm_table = ground_thm_table, ersatz_table = ersatz_table,

        skolems = skolems, special_funs = special_funs,

        unrolled_preds = unrolled_preds, wf_cache = wf_cache,

        constr_cache = constr_cache}

     val scope =

       {hol_ctxt = hol_ctxt, binarize = binarize, card_assigns = card_assigns,

        bits = bits, bisim_depth = bisim_depth, datatypes = datatypes, ofs = ofs}

     fun term_for_rep unfold =

       reconstruct_term unfold pool wacky_names scope sel_names rel_table bounds

     fun nth_value_of_type card T n =

       let

         fun aux unfold = term_for_rep unfold T T (Atom (card, 0)) [[n]]

       in

         case aux false of

           t as Const (s, _) =>

           if String.isPrefix cyclic_const_prefix s then

             HOLogic.mk_eq (t, aux true)

           else

             t

         | t => t

       end

     val all_values =

       all_values_of_type pool wacky_names scope sel_names rel_table bounds

     fun is_codatatype_wellformed (cos : dtype_spec list)

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

       let

         val ts = all_values card typ

         val max_depth = Integer.sum (map #card cos)

       in

         forall (not o bisimilar_values (map #typ cos) max_depth)

                (all_distinct_unordered_pairs_of ts)

       end

     fun pretty_for_assign name =

       let

         val (oper, (t1, T'), T) =

           case name of

             FreeName (s, T, _) =>

             let val t = Free (s, uniterize_unarize_unbox_etc_type T) in

               ("=", (t, format_term_type thy def_table formats t), T)

             end

           | ConstName (s, T, _) =>

             (assign_operator_for_const (s, T),

              user_friendly_const hol_ctxt base_step_names formats (s, T), T)

           | _ => raise NUT ("Nitpick_Model.reconstruct_hol_model.\

                             \pretty_for_assign", [name])

         val t2 = if rep_of name = Any then

                    Const (@{const_name undefined}, T')

                  else

                    tuple_list_for_name rel_table bounds name

                    |> term_for_rep false T T' (rep_of name)

       in

         Pretty.block (Pretty.breaks

             [setmp_show_all_types (Syntax.pretty_term ctxt) t1,

              Pretty.str oper, Syntax.pretty_term ctxt t2])

       end

     fun pretty_for_datatype ({typ, card, complete, ...} : dtype_spec) =

       Pretty.block (Pretty.breaks

           (Syntax.pretty_typ ctxt (uniterize_unarize_unbox_etc_type typ) ::

            (case typ of

               Type (@{type_name fin_fun}, _) => [Pretty.str "[finite]"]

             | Type (@{type_name fun_box}, _) => [Pretty.str "[boxed]"]

             | Type (@{type_name pair_box}, _) => [Pretty.str "[boxed]"]

             | _ => []) @

            [Pretty.str "=",

             Pretty.enum "," "{" "}"

                 (map (Syntax.pretty_term ctxt) (all_values card typ) @

                  (if fun_from_pair complete false then []

                   else [Pretty.str unrep]))]))

     fun integer_datatype T =

       [{typ = T, card = card_of_type card_assigns T, co = false,

         standard = true, complete = (false, false), concrete = (true, true),

         deep = true, constrs = []}]

       handle TYPE ("Nitpick_HOL.card_of_type", _, _) => []

     val (codatatypes, datatypes) =

       datatypes |> filter #deep |> List.partition #co

                 ||> append (integer_datatype int_T

                             |> is_standard_datatype thy stds nat_T

                                ? append (integer_datatype nat_T))

     val block_of_datatypes =

       if show_datatypes andalso not (null datatypes) then

         [Pretty.big_list ("Datatype" ^ plural_s_for_list datatypes ^ ":")

                          (map pretty_for_datatype datatypes)]

       else

         []

     val block_of_codatatypes =

       if show_datatypes andalso not (null codatatypes) then

         [Pretty.big_list ("Codatatype" ^ plural_s_for_list codatatypes ^ ":")

                          (map pretty_for_datatype codatatypes)]

       else

         []

     fun block_of_names show title names =

       if show andalso not (null names) then

         Pretty.str (title ^ plural_s_for_list names ^ ":")

         :: map (Pretty.indent indent_size o pretty_for_assign)

                (sort_wrt (original_name o nickname_of) names)

       else

         []

     val (skolem_names, nonskolem_nonsel_names) =

       List.partition is_skolem_name nonsel_names

     val (eval_names, noneval_nonskolem_nonsel_names) =

       List.partition (String.isPrefix eval_prefix o nickname_of)

                      nonskolem_nonsel_names

       ||> filter_out (member (op =) [@{const_name bisim},

                                      @{const_name bisim_iterator_max}]

                       o nickname_of)

     val free_names =

       map (fn x as (s, T) =>

               case filter (curry (op =) x

                        o pairf nickname_of

                                (uniterize_unarize_unbox_etc_type o type_of))

                        free_names of

                 [name] => name

               | [] => FreeName (s, T, Any)

               | _ => raise TERM ("Nitpick_Model.reconstruct_hol_model",

                                  [Const x])) all_frees

     val chunks = block_of_names true "Free variable" free_names @

                  block_of_names true "Skolem constant" skolem_names @

                  block_of_names true "Evaluated term" eval_names @

                  block_of_datatypes @ block_of_codatatypes @

                  block_of_names show_consts "Constant"

                                 noneval_nonskolem_nonsel_names

   in

     (Pretty.chunks (if null chunks then [Pretty.str "Empty assignment"]

                     else chunks),

      bisim_depth >= 0 orelse

      forall (is_codatatype_wellformed codatatypes) codatatypes)

   end

 fun prove_hol_model (scope as {hol_ctxt = {thy, ctxt, debug, ...},

                                card_assigns, ...})

                     auto_timeout free_names sel_names rel_table bounds prop =

   let

     val pool = Unsynchronized.ref []

     fun free_type_assm (T, k) =

       let

         fun atom j = nth_atom pool true T j k

         fun equation_for_atom j = HOLogic.eq_const T $ Bound 0 $ atom j

         val eqs = map equation_for_atom (index_seq 0 k)

         val compreh_assm =

           Const (@{const_name All}, (T --> bool_T) --> bool_T)

               $ Abs ("x", T, foldl1 HOLogic.mk_disj eqs)

         val distinct_assm = distinctness_formula T (map atom (index_seq 0 k))

       in s_conj (compreh_assm, distinct_assm) end

     fun free_name_assm name =

       HOLogic.mk_eq (Free (nickname_of name, type_of name),

                      term_for_name pool scope sel_names rel_table bounds name)

     val freeT_assms = map free_type_assm (filter (is_TFree o fst) card_assigns)

     val model_assms = map free_name_assm free_names

     val assm = foldr1 s_conj (freeT_assms @ model_assms)

     fun try_out negate =

       let

         val concl = (negate ? curry (op $) @{const Not})

                     (Object_Logic.atomize_term thy prop)

         val prop = HOLogic.mk_Trueprop (HOLogic.mk_imp (assm, concl))

                    |> map_types (map_type_tfree

                                      (fn (s, []) => TFree (s, HOLogic.typeS)

                                        | x => TFree x))

        val _ = if debug then

                  priority ((if negate then "Genuineness" else "Spuriousness") ^

                            " goal: " ^ Syntax.string_of_term ctxt prop ^ ".")

                else

                  ()

         val goal = prop |> cterm_of thy |> Goal.init

       in

         (goal |> SINGLE (DETERM_TIMEOUT auto_timeout

                                         (auto_tac (clasimpset_of ctxt)))

               |> the |> Goal.finish ctxt; true)

         handle THM _ => false

              | TimeLimit.TimeOut => false

       end

   in

     if try_out false then SOME true

     else if try_out true then SOME false

     else NONE

   end

 end;