src/HOL/Tools/Nitpick/nitpick.ML

improved Nitpick's precision for "card" and "setsum" + fix incorrect outcome code w.r.t. "bisim_depth = -1"

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

Finite model generation for HOL formulas using Kodkod.
*)

signature NITPICK =
sig
  type styp = Nitpick_Util.styp
  type params = {
    cards_assigns: (typ option * int list) list,
    maxes_assigns: (styp option * int list) list,
    iters_assigns: (styp option * int list) list,
    bitss: int list,
    bisim_depths: int list,
    boxes: (typ option * bool option) list,
    finitizes: (typ option * bool option) list,
    monos: (typ option * bool option) list,
    stds: (typ option * bool) list,
    wfs: (styp option * bool option) list,
    sat_solver: string,
    blocking: bool,
    falsify: bool,
    debug: bool,
    verbose: bool,
    overlord: bool,
    user_axioms: bool option,
    assms: bool,
    merge_type_vars: bool,
    binary_ints: bool option,
    destroy_constrs: bool,
    specialize: bool,
    skolemize: bool,
    star_linear_preds: bool,
    uncurry: bool,
    fast_descrs: bool,
    peephole_optim: bool,
    timeout: Time.time option,
    tac_timeout: Time.time option,
    sym_break: int,
    sharing_depth: int,
    flatten_props: bool,
    max_threads: int,
    show_skolems: bool,
    show_datatypes: bool,
    show_consts: bool,
    evals: term list,
    formats: (term option * int list) list,
    max_potential: int,
    max_genuine: int,
    check_potential: bool,
    check_genuine: bool,
    batch_size: int,
    expect: string}

  val register_frac_type : string -> (string * string) list -> theory -> theory
  val unregister_frac_type : string -> theory -> theory
  val register_codatatype : typ -> string -> styp list -> theory -> theory
  val unregister_codatatype : typ -> theory -> theory
  val pick_nits_in_term :
    Proof.state -> params -> bool -> int -> int -> int -> (term * term) list
    -> term list -> term -> string * Proof.state
  val pick_nits_in_subgoal :
    Proof.state -> params -> bool -> int -> int -> string * Proof.state
end;

structure Nitpick : NITPICK =
struct

open Nitpick_Util
open Nitpick_HOL
open Nitpick_Preproc
open Nitpick_Mono
open Nitpick_Scope
open Nitpick_Peephole
open Nitpick_Rep
open Nitpick_Nut
open Nitpick_Kodkod
open Nitpick_Model

structure KK = Kodkod

type params = {
  cards_assigns: (typ option * int list) list,
  maxes_assigns: (styp option * int list) list,
  iters_assigns: (styp option * int list) list,
  bitss: int list,
  bisim_depths: int list,
  boxes: (typ option * bool option) list,
  finitizes: (typ option * bool option) list,
  monos: (typ option * bool option) list,
  stds: (typ option * bool) list,
  wfs: (styp option * bool option) list,
  sat_solver: string,
  blocking: bool,
  falsify: bool,
  debug: bool,
  verbose: bool,
  overlord: bool,
  user_axioms: bool option,
  assms: bool,
  merge_type_vars: bool,
  binary_ints: bool option,
  destroy_constrs: bool,
  specialize: bool,
  skolemize: bool,
  star_linear_preds: bool,
  uncurry: bool,
  fast_descrs: bool,
  peephole_optim: bool,
  timeout: Time.time option,
  tac_timeout: Time.time option,
  sym_break: int,
  sharing_depth: int,
  flatten_props: bool,
  max_threads: int,
  show_skolems: bool,
  show_datatypes: bool,
  show_consts: bool,
  evals: term list,
  formats: (term option * int list) list,
  max_potential: int,
  max_genuine: int,
  check_potential: bool,
  check_genuine: bool,
  batch_size: int,
  expect: string}

type problem_extension = {
  free_names: nut list,
  sel_names: nut list,
  nonsel_names: nut list,
  rel_table: nut NameTable.table,
  unsound: bool,
  scope: scope}

type rich_problem = KK.problem * problem_extension

(* Proof.context -> string -> term list -> Pretty.T list *)
fun pretties_for_formulas _ _ [] = []
  | pretties_for_formulas ctxt s ts =
    [Pretty.str (s ^ plural_s_for_list ts ^ ":"),
     Pretty.indent indent_size (Pretty.chunks
         (map2 (fn j => fn t =>
                   Pretty.block [t |> shorten_names_in_term
                                   |> Syntax.pretty_term ctxt,
                                 Pretty.str (if j = 1 then "." else ";")])
               (length ts downto 1) ts))]

(* unit -> string *)
fun install_kodkodi_message () =
  "Nitpick requires the external Java program Kodkodi. To install it, download \
  \the package from Isabelle's web page and add the \"kodkodi-x.y.z\" \
  \directory's full path to \"" ^
  Path.implode (Path.expand (Path.appends
      (Path.variable "ISABELLE_HOME_USER" ::
       map Path.basic ["etc", "components"]))) ^ "\"."

val max_unsound_delay_ms = 200
val max_unsound_delay_percent = 2

(* Time.time option -> int *)
fun unsound_delay_for_timeout NONE = max_unsound_delay_ms
  | unsound_delay_for_timeout (SOME timeout) =
    Int.max (0, Int.min (max_unsound_delay_ms,
                         Time.toMilliseconds timeout
                         * max_unsound_delay_percent div 100))

(* Time.time option -> bool *)
fun passed_deadline NONE = false
  | passed_deadline (SOME time) = Time.compare (Time.now (), time) <> LESS

(* ('a * bool option) list -> bool *)
fun none_true assigns = forall (not_equal (SOME true) o snd) assigns

val syntactic_sorts =
  @{sort "{default,zero,one,plus,minus,uminus,times,inverse,abs,sgn,ord,eq}"} @
  @{sort number}
(* typ -> bool *)
fun has_tfree_syntactic_sort (TFree (_, S as _ :: _)) =
    subset (op =) (S, syntactic_sorts)
  | has_tfree_syntactic_sort _ = false
(* term -> bool *)
val has_syntactic_sorts = exists_type (exists_subtype has_tfree_syntactic_sort)

(* (unit -> string) -> Pretty.T *)
fun plazy f = Pretty.blk (0, pstrs (f ()))

(* Time.time -> Proof.state -> params -> bool -> int -> int -> int
   -> (term * term) list -> term list -> term -> string * Proof.state *)
fun pick_them_nits_in_term deadline state (params : params) auto i n step
                           subst orig_assm_ts orig_t =
  let
    val timer = Timer.startRealTimer ()
    val thy = Proof.theory_of state
    val ctxt = Proof.context_of state
(* FIXME: reintroduce code before new release:

    val nitpick_thy = ThyInfo.get_theory "Nitpick"
    val _ = Theory.subthy (nitpick_thy, thy) orelse
            error "You must import the theory \"Nitpick\" to use Nitpick"
*)
    val {cards_assigns, maxes_assigns, iters_assigns, bitss, bisim_depths,
         boxes, finitizes, monos, stds, wfs, sat_solver, falsify, debug,
         verbose, overlord, user_axioms, assms, merge_type_vars, binary_ints,
         destroy_constrs, specialize, skolemize, star_linear_preds, uncurry,
         fast_descrs, peephole_optim, tac_timeout, sym_break, sharing_depth,
         flatten_props, max_threads, show_skolems, show_datatypes, show_consts,
         evals, formats, max_potential, max_genuine, check_potential,
         check_genuine, batch_size, ...} =
      params
    val state_ref = Unsynchronized.ref state
    (* Pretty.T -> unit *)
    val pprint =
      if auto then
        Unsynchronized.change state_ref o Proof.goal_message o K
        o Pretty.chunks o cons (Pretty.str "") o single
        o Pretty.mark Markup.hilite
      else
        priority o Pretty.string_of
    (* (unit -> Pretty.T) -> unit *)
    fun pprint_m f = () |> not auto ? pprint o f
    fun pprint_v f = () |> verbose ? pprint o f
    fun pprint_d f = () |> debug ? pprint o f
    (* string -> unit *)
    val print = pprint o curry Pretty.blk 0 o pstrs
    (* (unit -> string) -> unit *)
    val print_m = pprint_m o K o plazy
    val print_v = pprint_v o K o plazy

    (* unit -> unit *)
    fun check_deadline () =
      if debug andalso passed_deadline deadline then raise TimeLimit.TimeOut
      else ()
    (* unit -> 'a *)
    fun do_interrupted () =
      if passed_deadline deadline then raise TimeLimit.TimeOut
      else raise Interrupt

    val orig_assm_ts = if assms orelse auto then orig_assm_ts else []
    val _ =
      if step = 0 then
        print_m (fn () => "Nitpicking formula...")
      else
        pprint_m (fn () => Pretty.chunks (
            pretties_for_formulas ctxt ("Nitpicking " ^
                (if i <> 1 orelse n <> 1 then
                   "subgoal " ^ string_of_int i ^ " of " ^ string_of_int n
                 else
                   "goal")) [Logic.list_implies (orig_assm_ts, orig_t)]))
    val neg_t = if falsify then Logic.mk_implies (orig_t, @{prop False})
                else orig_t
    val assms_t = Logic.mk_conjunction_list (neg_t :: orig_assm_ts)
    val (assms_t, evals) =
      assms_t :: evals |> merge_type_vars ? merge_type_vars_in_terms
                       |> pairf hd tl
    val original_max_potential = max_potential
    val original_max_genuine = max_genuine
(*
    val _ = priority ("*** " ^ Syntax.string_of_term ctxt orig_t)
    val _ = List.app (fn t => priority ("*** " ^ Syntax.string_of_term ctxt t))
                     orig_assm_ts
*)
    val max_bisim_depth = fold Integer.max bisim_depths ~1
    val case_names = case_const_names thy stds
    val (defs, built_in_nondefs, user_nondefs) = all_axioms_of thy subst
    val def_table = const_def_table ctxt subst defs
    val nondef_table = const_nondef_table (built_in_nondefs @ user_nondefs)
    val simp_table = Unsynchronized.ref (const_simp_table ctxt subst)
    val psimp_table = const_psimp_table ctxt subst
    val intro_table = inductive_intro_table ctxt subst def_table
    val ground_thm_table = ground_theorem_table thy
    val ersatz_table = ersatz_table thy
    val (hol_ctxt as {wf_cache, ...}) =
      {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, skolemize = skolemize,
       star_linear_preds = star_linear_preds, uncurry = uncurry,
       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,
       intro_table = intro_table, ground_thm_table = ground_thm_table,
       ersatz_table = ersatz_table, skolems = Unsynchronized.ref [],
       special_funs = Unsynchronized.ref [],
       unrolled_preds = Unsynchronized.ref [], wf_cache = Unsynchronized.ref [],
       constr_cache = Unsynchronized.ref []}
    val frees = Term.add_frees assms_t []
    val _ = null (Term.add_tvars assms_t []) orelse
            raise NOT_SUPPORTED "schematic type variables"
    val (nondef_ts, def_ts, got_all_mono_user_axioms, no_poly_user_axioms,
         binarize) = preprocess_term hol_ctxt finitizes monos assms_t
    val got_all_user_axioms =
      got_all_mono_user_axioms andalso no_poly_user_axioms

    (* styp * (bool * bool) -> unit *)
    fun print_wf (x, (gfp, wf)) =
      pprint (Pretty.blk (0,
          pstrs ("The " ^ (if gfp then "co" else "") ^ "inductive predicate \"")
          @ Syntax.pretty_term ctxt (Const x) ::
          pstrs (if wf then
                   "\" was proved well-founded. Nitpick can compute it \
                   \efficiently."
                 else
                   "\" could not be proved well-founded. Nitpick might need to \
                   \unroll it.")))
    val _ = if verbose then List.app print_wf (!wf_cache) else ()
    val _ =
      pprint_d (fn () => Pretty.chunks
          (pretties_for_formulas ctxt "Preprocessed formula" [hd nondef_ts] @
           pretties_for_formulas ctxt "Relevant definitional axiom" def_ts @
           pretties_for_formulas ctxt "Relevant nondefinitional axiom"
                                 (tl nondef_ts)))
    val _ = List.app (ignore o Term.type_of) (nondef_ts @ def_ts)
            handle TYPE (_, Ts, ts) =>
                   raise TYPE ("Nitpick.pick_them_nits_in_term", Ts, ts)

    val nondef_us = map (nut_from_term hol_ctxt Eq) nondef_ts
    val def_us = map (nut_from_term hol_ctxt DefEq) def_ts
    val (free_names, const_names) =
      fold add_free_and_const_names (nondef_us @ def_us) ([], [])
    val (sel_names, nonsel_names) =
      List.partition (is_sel o nickname_of) const_names
    val sound_finitizes =
      none_true (filter_out (fn (SOME (Type (@{type_name fun}, _)), _) => true
                          | _ => false) finitizes)
    val standard = forall snd stds
(*
    val _ = List.app (priority o string_for_nut ctxt) (nondef_us @ def_us)
*)

    val unique_scope = forall (curry (op =) 1 o length o snd) cards_assigns
    (* typ list -> string -> string *)
    fun monotonicity_message Ts extra =
      let val ss = map (quote o string_for_type ctxt) Ts in
        "The type" ^ plural_s_for_list ss ^ " " ^
        space_implode " " (serial_commas "and" ss) ^ " " ^
        (if none_true monos then
           "passed the monotonicity test"
         else
           (if length ss = 1 then "is" else "are") ^ " considered monotonic") ^
        ". " ^ extra
      end
    (* typ -> bool *)
    fun is_type_fundamentally_monotonic T =
      (is_datatype thy stds T andalso not (is_quot_type thy T) andalso
       (not (is_pure_typedef thy T) orelse is_univ_typedef thy T)) orelse
      is_number_type thy T orelse is_bit_type T
    fun is_type_actually_monotonic T =
      formulas_monotonic hol_ctxt binarize T (nondef_ts, def_ts)
    fun is_type_kind_of_monotonic T =
      case triple_lookup (type_match thy) monos T of
        SOME (SOME false) => false
      | _ => is_type_actually_monotonic T
    fun is_type_monotonic T =
      unique_scope orelse
      case triple_lookup (type_match thy) monos T of
        SOME (SOME b) => b
      | _ => is_type_fundamentally_monotonic T orelse
             is_type_actually_monotonic T
    fun is_shallow_datatype_finitizable (T as Type (@{type_name fin_fun}, _)) =
        is_type_kind_of_monotonic T
      | is_shallow_datatype_finitizable (T as Type (@{type_name fun_box}, _)) =
        is_type_kind_of_monotonic T
      | is_shallow_datatype_finitizable T =
        case triple_lookup (type_match thy) finitizes T of
          SOME (SOME b) => b
        | _ => is_type_kind_of_monotonic T
    fun is_datatype_deep T =
      not standard orelse T = nat_T orelse is_word_type T orelse
      exists (curry (op =) T o domain_type o type_of) sel_names
    val all_Ts = ground_types_in_terms hol_ctxt binarize (nondef_ts @ def_ts)
                 |> sort Term_Ord.typ_ord
    val _ = if verbose andalso binary_ints = SOME true andalso
               exists (member (op =) [nat_T, int_T]) all_Ts then
              print_v (K "The option \"binary_ints\" will be ignored because \
                         \of the presence of rationals, reals, \"Suc\", \
                         \\"gcd\", or \"lcm\" in the problem or because of the \
                         \\"non_std\" option.")
            else
              ()
    val (mono_Ts, nonmono_Ts) = List.partition is_type_monotonic all_Ts
    val _ =
      if verbose andalso not unique_scope then
        case filter_out is_type_fundamentally_monotonic mono_Ts of
          [] => ()
        | interesting_mono_Ts =>
          print_v (K (monotonicity_message interesting_mono_Ts
                         "Nitpick might be able to skip some scopes."))
      else
        ()
    val (deep_dataTs, shallow_dataTs) =
      all_Ts |> filter (is_datatype thy stds) |> List.partition is_datatype_deep
    val finitizable_dataTs =
      shallow_dataTs |> filter_out (is_finite_type hol_ctxt)
                     |> filter is_shallow_datatype_finitizable
    val _ =
      if verbose andalso not (null finitizable_dataTs) then
        print_v (K (monotonicity_message finitizable_dataTs
                        "Nitpick can use a more precise finite encoding."))
      else
        ()
    (* This detection code is an ugly hack. Fortunately, it is used only to
       provide a hint to the user. *)
    (* string * (Rule_Cases.T * bool) -> bool *)
    fun is_struct_induct_step (name, (Rule_Cases.Case {fixes, assumes, ...}, _)) =
      not (null fixes) andalso
      exists (String.isSuffix ".hyps" o fst) assumes andalso
      exists (exists (curry (op =) name o shortest_name o fst)
              o datatype_constrs hol_ctxt) deep_dataTs
    val likely_in_struct_induct_step =
      exists is_struct_induct_step (ProofContext.cases_of ctxt)
    val _ = if standard andalso likely_in_struct_induct_step then
              pprint_m (fn () => Pretty.blk (0,
                  pstrs "Hint: To check that the induction hypothesis is \
                        \general enough, try this command: " @
                  [Pretty.mark Markup.sendback (Pretty.blk (0,
                       pstrs ("nitpick [non_std, show_all]")))] @ pstrs "."))
            else
              ()
(*
    val _ = priority "Monotonic types:"
    val _ = List.app (priority o string_for_type ctxt) mono_Ts
    val _ = priority "Nonmonotonic types:"
    val _ = List.app (priority o string_for_type ctxt) nonmono_Ts
*)

    val need_incremental = Int.max (max_potential, max_genuine) >= 2
    val effective_sat_solver =
      if sat_solver <> "smart" then
        if need_incremental andalso
           not (member (op =) (Kodkod_SAT.configured_sat_solvers true)
                       sat_solver) then
          (print_m (K ("An incremental SAT solver is required: \"SAT4J\" will \
                       \be used instead of " ^ quote sat_solver ^ "."));
           "SAT4J")
        else
          sat_solver
      else
        Kodkod_SAT.smart_sat_solver_name need_incremental
    val _ =
      if sat_solver = "smart" then
        print_v (fn () => "Using SAT solver " ^ quote effective_sat_solver ^
                          ". The following" ^
                          (if need_incremental then " incremental " else " ") ^
                          "solvers are configured: " ^
                          commas (map quote (Kodkod_SAT.configured_sat_solvers
                                                       need_incremental)) ^ ".")
      else
        ()

    val too_big_scopes = Unsynchronized.ref []

    (* bool -> scope -> rich_problem option *)
    fun problem_for_scope unsound
            (scope as {card_assigns, bits, bisim_depth, datatypes, ofs, ...}) =
      let
        val _ = not (exists (fn other => scope_less_eq other scope)
                            (!too_big_scopes)) orelse
                raise TOO_LARGE ("Nitpick.pick_them_nits_in_term.\
                                 \problem_for_scope", "too large scope")
(*
        val _ = priority "Offsets:"
        val _ = List.app (fn (T, j0) =>
                             priority (string_for_type ctxt T ^ " = " ^
                                       string_of_int j0))
                         (Typtab.dest ofs)
*)
        val all_exact = forall (is_exact_type datatypes true) all_Ts
        (* nut list -> rep NameTable.table -> nut list * rep NameTable.table *)
        val repify_consts = choose_reps_for_consts scope all_exact
        val main_j0 = offset_of_type ofs bool_T
        val (nat_card, nat_j0) = spec_of_type scope nat_T
        val (int_card, int_j0) = spec_of_type scope int_T
        val _ = (nat_j0 = main_j0 andalso int_j0 = main_j0) orelse
                raise BAD ("Nitpick.pick_them_nits_in_term.problem_for_scope",
                           "bad offsets")
        val kk = kodkod_constrs peephole_optim nat_card int_card main_j0
        val (free_names, rep_table) =
          choose_reps_for_free_vars scope free_names NameTable.empty
        val (sel_names, rep_table) = choose_reps_for_all_sels scope rep_table
        val (nonsel_names, rep_table) = repify_consts nonsel_names rep_table
        val min_highest_arity =
          NameTable.fold (curry Int.max o arity_of_rep o snd) rep_table 1
        val min_univ_card =
          NameTable.fold (curry Int.max o min_univ_card_of_rep o snd) rep_table
                         (univ_card nat_card int_card main_j0 [] KK.True)
        val _ = check_arity min_univ_card min_highest_arity

        val def_us = map (choose_reps_in_nut scope unsound rep_table true)
                         def_us
        val nondef_us = map (choose_reps_in_nut scope unsound rep_table false)
                            nondef_us
(*
        val _ = List.app (priority o string_for_nut ctxt)
                         (free_names @ sel_names @ nonsel_names @
                          nondef_us @ def_us)
*)
        val (free_rels, pool, rel_table) =
          rename_free_vars free_names initial_pool NameTable.empty
        val (sel_rels, pool, rel_table) =
          rename_free_vars sel_names pool rel_table
        val (other_rels, pool, rel_table) =
          rename_free_vars nonsel_names pool rel_table
        val nondef_us = map (rename_vars_in_nut pool rel_table) nondef_us
        val def_us = map (rename_vars_in_nut pool rel_table) def_us
        val nondef_fs = map (kodkod_formula_from_nut ofs kk) nondef_us
        val def_fs = map (kodkod_formula_from_nut ofs kk) def_us
        val formula = fold (fold s_and) [def_fs, nondef_fs] KK.True
        val comment = (if unsound then "unsound" else "sound") ^ "\n" ^
                      PrintMode.setmp [] multiline_string_for_scope scope
        val kodkod_sat_solver =
          Kodkod_SAT.sat_solver_spec effective_sat_solver |> snd
        val bit_width = if bits = 0 then 16 else bits + 1
        val delay = if unsound then
                      Option.map (fn time => Time.- (time, Time.now ()))
                                 deadline
                      |> unsound_delay_for_timeout
                    else
                      0
        val settings = [("solver", commas (map quote kodkod_sat_solver)),
                        ("skolem_depth", "-1"),
                        ("bit_width", string_of_int bit_width),
                        ("symmetry_breaking", signed_string_of_int sym_break),
                        ("sharing", signed_string_of_int sharing_depth),
                        ("flatten", Bool.toString flatten_props),
                        ("delay", signed_string_of_int delay)]
        val plain_rels = free_rels @ other_rels
        val plain_bounds = map (bound_for_plain_rel ctxt debug) plain_rels
        val plain_axioms = map (declarative_axiom_for_plain_rel kk) plain_rels
        val sel_bounds = map (bound_for_sel_rel ctxt debug datatypes) sel_rels
        val dtype_axioms =
          declarative_axioms_for_datatypes hol_ctxt binarize bits ofs kk
                                           rel_table datatypes
        val declarative_axioms = plain_axioms @ dtype_axioms
        val univ_card = Int.max (univ_card nat_card int_card main_j0
                                     (plain_bounds @ sel_bounds) formula,
                                 main_j0 |> bits > 0 ? Integer.add (bits + 1))
        val built_in_bounds = bounds_for_built_in_rels_in_formula debug
                                  univ_card nat_card int_card main_j0 formula
        val bounds = built_in_bounds @ plain_bounds @ sel_bounds
                     |> not debug ? merge_bounds
        val highest_arity =
          fold Integer.max (map (fst o fst) (maps fst bounds)) 0
        val formula = fold_rev s_and declarative_axioms formula
        val _ = if bits = 0 then () else check_bits bits formula
        val _ = if formula = KK.False then ()
                else check_arity univ_card highest_arity
      in
        SOME ({comment = comment, settings = settings, univ_card = univ_card,
               tuple_assigns = [], bounds = bounds,
               int_bounds = if bits = 0 then sequential_int_bounds univ_card
                            else pow_of_two_int_bounds bits main_j0,
               expr_assigns = [], formula = formula},
              {free_names = free_names, sel_names = sel_names,
               nonsel_names = nonsel_names, rel_table = rel_table,
               unsound = unsound, scope = scope})
      end
      handle TOO_LARGE (loc, msg) =>
             if loc = "Nitpick_Kodkod.check_arity" andalso
                not (Typtab.is_empty ofs) then
               problem_for_scope unsound
                   {hol_ctxt = hol_ctxt, binarize = binarize,
                    card_assigns = card_assigns, bits = bits,
                    bisim_depth = bisim_depth, datatypes = datatypes,
                    ofs = Typtab.empty}
             else if loc = "Nitpick.pick_them_nits_in_term.\
                           \problem_for_scope" then
               NONE
             else
               (Unsynchronized.change too_big_scopes (cons scope);
                print_v (fn () => ("Limit reached: " ^ msg ^
                                   ". Skipping " ^ (if unsound then "potential"
                                                    else "genuine") ^
                                   " component of scope."));
                NONE)
           | TOO_SMALL (_, msg) =>
             (print_v (fn () => ("Limit reached: " ^ msg ^
                                 ". Skipping " ^ (if unsound then "potential"
                                                  else "genuine") ^
                                 " component of scope."));
              NONE)

    val das_wort_model =
      (if falsify then "counterexample" else "model")
      |> not standard ? prefix "nonstandard "

    val scopes = Unsynchronized.ref []
    val generated_scopes = Unsynchronized.ref []
    val generated_problems = Unsynchronized.ref ([] : rich_problem list)
    val checked_problems = Unsynchronized.ref (SOME [])
    val met_potential = Unsynchronized.ref 0

    (* rich_problem list -> int list -> unit *)
    fun update_checked_problems problems =
      List.app (Unsynchronized.change checked_problems o Option.map o cons
                o nth problems)
    (* string -> unit *)
    fun show_kodkod_warning "" = ()
      | show_kodkod_warning s = print_m (fn () => "Kodkod warning: " ^ s ^ ".")

    (* bool -> KK.raw_bound list -> problem_extension -> bool * bool option *)
    fun print_and_check_model genuine bounds
            ({free_names, sel_names, nonsel_names, rel_table, scope, ...}
             : problem_extension) =
      let
        val (reconstructed_model, codatatypes_ok) =
          reconstruct_hol_model {show_skolems = show_skolems,
                                 show_datatypes = show_datatypes,
                                 show_consts = show_consts}
              scope formats frees free_names sel_names nonsel_names rel_table
              bounds
        val genuine_means_genuine =
          got_all_user_axioms andalso none_true wfs andalso
          sound_finitizes andalso codatatypes_ok
      in
        (pprint (Pretty.chunks
             [Pretty.blk (0,
                  (pstrs ("Nitpick found a" ^
                          (if not genuine then " potential "
                           else if genuine_means_genuine then " "
                           else " quasi genuine ") ^ das_wort_model) @
                   (case pretties_for_scope scope verbose of
                      [] => []
                    | pretties => pstrs " for " @ pretties) @
                   [Pretty.str ":\n"])),
              Pretty.indent indent_size reconstructed_model]);
         if genuine then
           (if check_genuine andalso standard then
              case prove_hol_model scope tac_timeout free_names sel_names
                                   rel_table bounds assms_t of
                SOME true =>
                print ("Confirmation by \"auto\": The above " ^
                       das_wort_model ^ " is really genuine.")
              | SOME false =>
                if genuine_means_genuine then
                  error ("A supposedly genuine " ^ das_wort_model ^ " was \
                         \shown to be spurious by \"auto\".\nThis should never \
                         \happen.\nPlease send a bug report to blanchet\
                         \[e-mail].")
                 else
                   print ("Refutation by \"auto\": The above " ^
                          das_wort_model ^ " is spurious.")
               | NONE => print "No confirmation by \"auto\"."
            else
              ();
            if not standard andalso likely_in_struct_induct_step then
              print "The existence of a nonstandard model suggests that the \
                    \induction hypothesis is not general enough or perhaps \
                    \even wrong. See the \"Inductive Properties\" section of \
                    \the Nitpick manual for details (\"isabelle doc nitpick\")."
            else
              ();
            if has_syntactic_sorts orig_t then
              print "Hint: Maybe you forgot a type constraint?"
            else
              ();
            if not genuine_means_genuine then
              if no_poly_user_axioms then
                let
                  val options =
                    [] |> not got_all_mono_user_axioms
                          ? cons ("user_axioms", "\"true\"")
                       |> not (none_true wfs)
                          ? cons ("wf", "\"smart\" or \"false\"")
                       |> not sound_finitizes
                          ? cons ("finitize", "\"smart\" or \"false\"")
                       |> not codatatypes_ok
                          ? cons ("bisim_depth", "a nonnegative value")
                  val ss =
                    map (fn (name, value) => quote name ^ " set to " ^ value)
                        options
                in
                  print ("Try again with " ^
                         space_implode " " (serial_commas "and" ss) ^
                         " to confirm that the " ^ das_wort_model ^
                         " is genuine.")
                end
              else
                print ("Nitpick is unable to guarantee the authenticity of \
                       \the " ^ das_wort_model ^ " in the presence of \
                       \polymorphic axioms.")
            else
              ();
            NONE)
         else
           if not genuine then
             (Unsynchronized.inc met_potential;
              if check_potential then
                let
                  val status = prove_hol_model scope tac_timeout free_names
                                              sel_names rel_table bounds assms_t
                in
                  (case status of
                     SOME true => print ("Confirmation by \"auto\": The \
                                         \above " ^ das_wort_model ^
                                         " is genuine.")
                   | SOME false => print ("Refutation by \"auto\": The above " ^
                                          das_wort_model ^ " is spurious.")
                   | NONE => print "No confirmation by \"auto\".");
                  status
                end
              else
                NONE)
           else
             NONE)
        |> pair genuine_means_genuine
      end
    (* bool * int * int * int -> bool -> rich_problem list
       -> bool * int * int * int *)
    fun solve_any_problem (found_really_genuine, max_potential, max_genuine,
                           donno) first_time problems =
      let
        val max_potential = Int.max (0, max_potential)
        val max_genuine = Int.max (0, max_genuine)
        (* bool -> int * KK.raw_bound list -> bool * bool option *)
        fun print_and_check genuine (j, bounds) =
          print_and_check_model genuine bounds (snd (nth problems j))
        val max_solutions = max_potential + max_genuine
                            |> not need_incremental ? curry Int.min 1
      in
        if max_solutions <= 0 then
          (found_really_genuine, 0, 0, donno)
        else
          case KK.solve_any_problem overlord deadline max_threads max_solutions
                                    (map fst problems) of
            KK.NotInstalled =>
            (print_m install_kodkodi_message;
             (found_really_genuine, max_potential, max_genuine, donno + 1))
          | KK.Normal ([], unsat_js, s) =>
            (update_checked_problems problems unsat_js; show_kodkod_warning s;
             (found_really_genuine, max_potential, max_genuine, donno))
          | KK.Normal (sat_ps, unsat_js, s) =>
            let
              val (lib_ps, con_ps) =
                List.partition (#unsound o snd o nth problems o fst) sat_ps
            in
              update_checked_problems problems (unsat_js @ map fst lib_ps);
              show_kodkod_warning s;
              if null con_ps then
                let
                  val genuine_codes =
                    lib_ps |> take max_potential
                           |> map (print_and_check false)
                           |> filter (curry (op =) (SOME true) o snd)
                  val found_really_genuine =
                    found_really_genuine orelse exists fst genuine_codes
                  val num_genuine = length genuine_codes
                  val max_genuine = max_genuine - num_genuine
                  val max_potential = max_potential
                                      - (length lib_ps - num_genuine)
                in
                  if max_genuine <= 0 then
                    (found_really_genuine, 0, 0, donno)
                  else
                    let
                      (* "co_js" is the list of sound problems whose unsound
                         pendants couldn't be satisfied and hence that most
                         probably can't be satisfied themselves. *)
                      val co_js =
                        map (fn j => j - 1) unsat_js
                        |> filter (fn j =>
                                      j >= 0 andalso
                                      scopes_equivalent
                                          (#scope (snd (nth problems j)))
                                          (#scope (snd (nth problems (j + 1)))))
                      val bye_js = sort_distinct int_ord (map fst sat_ps @
                                                          unsat_js @ co_js)
                      val problems =
                        problems |> filter_out_indices bye_js
                                 |> max_potential <= 0
                                    ? filter_out (#unsound o snd)
                    in
                      solve_any_problem (found_really_genuine, max_potential,
                                         max_genuine, donno) false problems
                    end
                end
              else
                let
                  val genuine_codes =
                    con_ps |> take max_genuine
                           |> map (print_and_check true)
                  val max_genuine = max_genuine - length genuine_codes
                  val found_really_genuine =
                    found_really_genuine orelse exists fst genuine_codes
                in
                  if max_genuine <= 0 orelse not first_time then
                    (found_really_genuine, 0, max_genuine, donno)
                  else
                    let
                      val bye_js = sort_distinct int_ord
                                                 (map fst sat_ps @ unsat_js)
                      val problems =
                        problems |> filter_out_indices bye_js
                                 |> filter_out (#unsound o snd)
                    in
                      solve_any_problem (found_really_genuine, 0, max_genuine,
                                         donno) false problems
                    end
                end
            end
          | KK.TimedOut unsat_js =>
            (update_checked_problems problems unsat_js; raise TimeLimit.TimeOut)
          | KK.Interrupted NONE => (checked_problems := NONE; do_interrupted ())
          | KK.Interrupted (SOME unsat_js) =>
            (update_checked_problems problems unsat_js; do_interrupted ())
          | KK.Error (s, unsat_js) =>
            (update_checked_problems problems unsat_js;
             print_v (K ("Kodkod error: " ^ s ^ "."));
             (found_really_genuine, max_potential, max_genuine, donno + 1))
      end

    (* int -> int -> scope list -> bool * int * int * int
       -> bool * int * int * int *)
    fun run_batch j n scopes (found_really_genuine, max_potential, max_genuine,
                              donno) =
      let
        val _ =
          if null scopes then
            print_m (K "The scope specification is inconsistent.")
          else if verbose then
            pprint (Pretty.chunks
                [Pretty.blk (0,
                     pstrs ((if n > 1 then
                               "Batch " ^ string_of_int (j + 1) ^ " of " ^
                               signed_string_of_int n ^ ": "
                             else
                               "") ^
                            "Trying " ^ string_of_int (length scopes) ^
                            " scope" ^ plural_s_for_list scopes ^ ":")),
                 Pretty.indent indent_size
                     (Pretty.chunks (map2
                          (fn j => fn scope =>
                              Pretty.block (
                                  (case pretties_for_scope scope true of
                                     [] => [Pretty.str "Empty"]
                                   | pretties => pretties) @
                                  [Pretty.str (if j = 1 then "." else ";")]))
                          (length scopes downto 1) scopes))])
          else
            ()
        (* scope * bool -> rich_problem list * bool
           -> rich_problem list * bool *)
        fun add_problem_for_scope (scope, unsound) (problems, donno) =
          (check_deadline ();
           case problem_for_scope unsound scope of
             SOME problem =>
             (problems
              |> (null problems orelse
                  not (KK.problems_equivalent (fst problem)
                                              (fst (hd problems))))
                  ? cons problem, donno)
           | NONE => (problems, donno + 1))
        val (problems, donno) =
          fold add_problem_for_scope
               (map_product pair scopes
                    ((if max_genuine > 0 then [false] else []) @
                     (if max_potential > 0 then [true] else [])))
               ([], donno)
        val _ = Unsynchronized.change generated_problems (append problems)
        val _ = Unsynchronized.change generated_scopes (append scopes)
        val _ =
          if j + 1 = n then
            let
              val (unsound_problems, sound_problems) =
                List.partition (#unsound o snd) (!generated_problems)
            in
              if not (null sound_problems) andalso
                 forall (KK.is_problem_trivially_false o fst)
                        sound_problems then
                print_m (fn () =>
                    "Warning: The conjecture either trivially holds for the \
                    \given scopes or lies outside Nitpick's supported \
                    \fragment." ^
                    (if exists (not o KK.is_problem_trivially_false o fst)
                               unsound_problems then
                       " Only potential counterexamples may be found."
                     else
                       ""))
              else
                ()
            end
          else
            ()
      in
        solve_any_problem (found_really_genuine, max_potential, max_genuine,
                           donno) true (rev problems)
      end

    (* rich_problem list -> scope -> int *)
    fun scope_count (problems : rich_problem list) scope =
      length (filter (scopes_equivalent scope o #scope o snd) problems)
    (* string -> string *)
    fun excipit did_so_and_so =
      let
        (* rich_problem list -> rich_problem list *)
        val do_filter =
          if !met_potential = max_potential then filter_out (#unsound o snd)
          else I
        val total = length (!scopes)
        val unsat =
          fold (fn scope =>
                   case scope_count (do_filter (!generated_problems)) scope of
                     0 => I
                   | n =>
                     scope_count (do_filter (these (!checked_problems)))
                                 scope = n
                     ? Integer.add 1) (!generated_scopes) 0
      in
        "Nitpick " ^ did_so_and_so ^
        (if is_some (!checked_problems) andalso total > 0 then
           " after checking " ^
           string_of_int (Int.min (total - 1, unsat)) ^ " of " ^
           string_of_int total ^ " scope" ^ plural_s total
         else
           "") ^ "."
      end

    (* int -> int -> scope list -> bool * int * int * int -> KK.outcome *)
    fun run_batches _ _ []
                    (found_really_genuine, max_potential, max_genuine, donno) =
        if donno > 0 andalso max_genuine > 0 then
          (print_m (fn () => excipit "ran out of some resource"); "unknown")
        else if max_genuine = original_max_genuine then
          if max_potential = original_max_potential then
            (print_m (fn () =>
                 "Nitpick found no " ^ das_wort_model ^ "." ^
                 (if not standard andalso likely_in_struct_induct_step then
                    " This suggests that the induction hypothesis might be \
                    \general enough to prove this subgoal."
                  else
                    "")); "none")
          else
            (print_m (fn () =>
                 "Nitpick could not find a" ^
                 (if max_genuine = 1 then " better " ^ das_wort_model ^ "."
                  else "ny better " ^ das_wort_model ^ "s.")); "potential")
        else if found_really_genuine then
          "genuine"
        else
          "quasi_genuine"
      | run_batches j n (batch :: batches) z =
        let val (z as (_, _, max_genuine, _)) = run_batch j n batch z in
          run_batches (j + 1) n (if max_genuine > 0 then batches else []) z
        end

    val (skipped, the_scopes) =
      all_scopes hol_ctxt binarize sym_break cards_assigns maxes_assigns
                 iters_assigns bitss bisim_depths mono_Ts nonmono_Ts deep_dataTs
                 finitizable_dataTs
    val _ = if skipped > 0 then
              print_m (fn () => "Too many scopes. Skipping " ^
                                string_of_int skipped ^ " scope" ^
                                plural_s skipped ^
                                ". (Consider using \"mono\" or \
                                \\"merge_type_vars\" to prevent this.)")
            else
              ()
    val _ = scopes := the_scopes

    val batches = batch_list batch_size (!scopes)
    val outcome_code =
      (run_batches 0 (length batches) batches
                   (false, max_potential, max_genuine, 0)
       handle Exn.Interrupt => do_interrupted ())
      handle TimeLimit.TimeOut =>
             (print_m (fn () => excipit "ran out of time");
              if !met_potential > 0 then "potential" else "unknown")
           | Exn.Interrupt => if auto orelse debug then raise Interrupt
                              else error (excipit "was interrupted")
    val _ = print_v (fn () => "Total time: " ^
                              signed_string_of_int (Time.toMilliseconds
                                    (Timer.checkRealTimer timer)) ^ " ms.")
  in (outcome_code, !state_ref) end
  handle Exn.Interrupt =>
         if auto orelse #debug params then
           raise Interrupt
         else
           if passed_deadline deadline then
             (priority "Nitpick ran out of time."; ("unknown", state))
           else
             error "Nitpick was interrupted."

(* Proof.state -> params -> bool -> int -> int -> int -> (term * term) list
   -> term list -> term -> string * Proof.state *)
fun pick_nits_in_term state (params as {debug, timeout, expect, ...}) auto i n
                      step subst orig_assm_ts orig_t =
  if getenv "KODKODI" = "" then
    (if auto then ()
     else warning (Pretty.string_of (plazy install_kodkodi_message));
     ("unknown", state))
  else
    let
      val deadline = Option.map (curry Time.+ (Time.now ())) timeout
      val outcome as (outcome_code, _) =
        time_limit (if debug then NONE else timeout)
            (pick_them_nits_in_term deadline state params auto i n step subst
                                    orig_assm_ts) orig_t
    in
      if expect = "" orelse outcome_code = expect then outcome
      else error ("Unexpected outcome: " ^ quote outcome_code ^ ".")
    end

(* string list -> term -> bool *)
fun is_fixed_equation fixes
                      (Const (@{const_name "=="}, _) $ Free (s, _) $ Const _) =
    member (op =) fixes s
  | is_fixed_equation _ _ = false
(* Proof.context -> term list * term -> (term * term) list * term list * term *)
fun extract_fixed_frees ctxt (assms, t) =
  let
    val fixes = Variable.fixes_of ctxt |> map snd
    val (subst, other_assms) =
      List.partition (is_fixed_equation fixes) assms
      |>> map Logic.dest_equals
  in (subst, other_assms, subst_atomic subst t) end

(* Proof.state -> params -> bool -> int -> int -> string * Proof.state *)
fun pick_nits_in_subgoal state params auto i step =
  let
    val ctxt = Proof.context_of state
    val t = state |> Proof.raw_goal |> #goal |> prop_of
  in
    case Logic.count_prems t of
      0 => (priority "No subgoal!"; ("none", state))
    | n =>
      let
        val (t, frees) = Logic.goal_params t i
        val t = subst_bounds (frees, t)
        val assms = map term_of (Assumption.all_assms_of ctxt)
        val (subst, assms, t) = extract_fixed_frees ctxt (assms, t)
      in pick_nits_in_term state params auto i n step subst assms t end
  end

end;