src/HOL/Tools/refute.ML

declared lemmas true_eqvt and false_eqvt to be equivariant (suggested by samth at ccs.neu.edu)

(*  Title:      HOL/Tools/refute.ML
    ID:         $Id$
    Author:     Tjark Weber
    Copyright   2003-2007

Finite model generation for HOL formulas, using a SAT solver.
*)

(* ------------------------------------------------------------------------- *)
(* Declares the 'REFUTE' signature as well as a structure 'Refute'.          *)
(* Documentation is available in the Isabelle/Isar theory 'HOL/Refute.thy'.  *)
(* ------------------------------------------------------------------------- *)

signature REFUTE =
sig

  exception REFUTE of string * string

(* ------------------------------------------------------------------------- *)
(* Model/interpretation related code (translation HOL -> propositional logic *)
(* ------------------------------------------------------------------------- *)

  type params
  type interpretation
  type model
  type arguments

  exception MAXVARS_EXCEEDED

  val add_interpreter : string -> (theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option) -> theory -> theory
  val add_printer     : string -> (theory -> model -> Term.term ->
    interpretation -> (int -> bool) -> Term.term option) -> theory -> theory

  val interpret : theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments)

  val print       : theory -> model -> Term.term -> interpretation ->
    (int -> bool) -> Term.term
  val print_model : theory -> model -> (int -> bool) -> string

(* ------------------------------------------------------------------------- *)
(* Interface                                                                 *)
(* ------------------------------------------------------------------------- *)

  val set_default_param  : (string * string) -> theory -> theory
  val get_default_param  : theory -> string -> string option
  val get_default_params : theory -> (string * string) list
  val actual_params      : theory -> (string * string) list -> params

  val find_model : theory -> params -> Term.term -> bool -> unit

  (* tries to find a model for a formula: *)
  val satisfy_term   : theory -> (string * string) list -> Term.term -> unit
  (* tries to find a model that refutes a formula: *)
  val refute_term    : theory -> (string * string) list -> Term.term -> unit
  val refute_subgoal :
    theory -> (string * string) list -> Thm.thm -> int -> unit

  val setup : theory -> theory

end;  (* signature REFUTE *)

structure Refute : REFUTE =
struct

  open PropLogic;

  (* We use 'REFUTE' only for internal error conditions that should    *)
  (* never occur in the first place (i.e. errors caused by bugs in our *)
  (* code).  Otherwise (e.g. to indicate invalid input data) we use    *)
  (* 'error'.                                                          *)
  exception REFUTE of string * string;  (* ("in function", "cause") *)

  (* should be raised by an interpreter when more variables would be *)
  (* required than allowed by 'maxvars'                              *)
  exception MAXVARS_EXCEEDED;

(* ------------------------------------------------------------------------- *)
(* TREES                                                                     *)
(* ------------------------------------------------------------------------- *)

(* ------------------------------------------------------------------------- *)
(* tree: implements an arbitrarily (but finitely) branching tree as a list   *)
(*       of (lists of ...) elements                                          *)
(* ------------------------------------------------------------------------- *)

  datatype 'a tree =
      Leaf of 'a
    | Node of ('a tree) list;

  (* ('a -> 'b) -> 'a tree -> 'b tree *)

  fun tree_map f tr =
    case tr of
      Leaf x  => Leaf (f x)
    | Node xs => Node (map (tree_map f) xs);

  (* ('a * 'b -> 'a) -> 'a * ('b tree) -> 'a *)

  fun tree_foldl f =
  let
    fun itl (e, Leaf x)  = f(e,x)
      | itl (e, Node xs) = Library.foldl (tree_foldl f) (e,xs)
  in
    itl
  end;

  (* 'a tree * 'b tree -> ('a * 'b) tree *)

  fun tree_pair (t1, t2) =
    case t1 of
      Leaf x =>
      (case t2 of
          Leaf y => Leaf (x,y)
        | Node _ => raise REFUTE ("tree_pair",
            "trees are of different height (second tree is higher)"))
    | Node xs =>
      (case t2 of
          (* '~~' will raise an exception if the number of branches in   *)
          (* both trees is different at the current node                 *)
          Node ys => Node (map tree_pair (xs ~~ ys))
        | Leaf _  => raise REFUTE ("tree_pair",
            "trees are of different height (first tree is higher)"));

(* ------------------------------------------------------------------------- *)
(* params: parameters that control the translation into a propositional      *)
(*         formula/model generation                                          *)
(*                                                                           *)
(* The following parameters are supported (and required (!), except for      *)
(* "sizes"):                                                                 *)
(*                                                                           *)
(* Name          Type    Description                                         *)
(*                                                                           *)
(* "sizes"       (string * int) list                                         *)
(*                       Size of ground types (e.g. 'a=2), or depth of IDTs. *)
(* "minsize"     int     If >0, minimal size of each ground type/IDT depth.  *)
(* "maxsize"     int     If >0, maximal size of each ground type/IDT depth.  *)
(* "maxvars"     int     If >0, use at most 'maxvars' Boolean variables      *)
(*                       when transforming the term into a propositional     *)
(*                       formula.                                            *)
(* "maxtime"     int     If >0, terminate after at most 'maxtime' seconds.   *)
(* "satsolver"   string  SAT solver to be used.                              *)
(* ------------------------------------------------------------------------- *)

  type params =
    {
      sizes    : (string * int) list,
      minsize  : int,
      maxsize  : int,
      maxvars  : int,
      maxtime  : int,
      satsolver: string
    };

(* ------------------------------------------------------------------------- *)
(* interpretation: a term's interpretation is given by a variable of type    *)
(*                 'interpretation'                                          *)
(* ------------------------------------------------------------------------- *)

  type interpretation =
    prop_formula list tree;

(* ------------------------------------------------------------------------- *)
(* model: a model specifies the size of types and the interpretation of      *)
(*        terms                                                              *)
(* ------------------------------------------------------------------------- *)

  type model =
    (Term.typ * int) list * (Term.term * interpretation) list;

(* ------------------------------------------------------------------------- *)
(* arguments: additional arguments required during interpretation of terms   *)
(* ------------------------------------------------------------------------- *)

  type arguments =
    {
      (* just passed unchanged from 'params': *)
      maxvars   : int,
      (* whether to use 'make_equality' or 'make_def_equality': *)
      def_eq    : bool,
      (* the following may change during the translation: *)
      next_idx  : int,
      bounds    : interpretation list,
      wellformed: prop_formula
    };


  structure RefuteDataArgs =
  struct
    val name = "HOL/refute";
    type T =
      {interpreters: (string * (theory -> model -> arguments -> Term.term ->
        (interpretation * model * arguments) option)) list,
       printers: (string * (theory -> model -> Term.term -> interpretation ->
        (int -> bool) -> Term.term option)) list,
       parameters: string Symtab.table};
    val empty = {interpreters = [], printers = [], parameters = Symtab.empty};
    val copy = I;
    val extend = I;
    fun merge _
      ({interpreters = in1, printers = pr1, parameters = pa1},
       {interpreters = in2, printers = pr2, parameters = pa2}) =
      {interpreters = AList.merge (op =) (K true) (in1, in2),
       printers = AList.merge (op =) (K true) (pr1, pr2),
       parameters = Symtab.merge (op=) (pa1, pa2)};
    fun print sg {interpreters, printers, parameters} =
      Pretty.writeln (Pretty.chunks
        [Pretty.strs ("default parameters:" :: List.concat (map
          (fn (name, value) => [name, "=", value]) (Symtab.dest parameters))),
         Pretty.strs ("interpreters:" :: map fst interpreters),
         Pretty.strs ("printers:" :: map fst printers)]);
  end;

  structure RefuteData = TheoryDataFun(RefuteDataArgs);


(* ------------------------------------------------------------------------- *)
(* interpret: interprets the term 't' using a suitable interpreter; returns  *)
(*            the interpretation and a (possibly extended) model that keeps  *)
(*            track of the interpretation of subterms                        *)
(* ------------------------------------------------------------------------- *)

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) *)

  fun interpret thy model args t =
    case get_first (fn (_, f) => f thy model args t)
      (#interpreters (RefuteData.get thy)) of
      NONE   => raise REFUTE ("interpret",
        "no interpreter for term " ^ quote (Sign.string_of_term thy t))
    | SOME x => x;

(* ------------------------------------------------------------------------- *)
(* print: converts the constant denoted by the term 't' into a term using a  *)
(*        suitable printer                                                   *)
(* ------------------------------------------------------------------------- *)

  (* theory -> model -> Term.term -> interpretation -> (int -> bool) ->
    Term.term *)

  fun print thy model t intr assignment =
    case get_first (fn (_, f) => f thy model t intr assignment)
      (#printers (RefuteData.get thy)) of
      NONE   => raise REFUTE ("print",
        "no printer for term " ^ quote (Sign.string_of_term thy t))
    | SOME x => x;

(* ------------------------------------------------------------------------- *)
(* print_model: turns the model into a string, using a fixed interpretation  *)
(*              (given by an assignment for Boolean variables) and suitable  *)
(*              printers                                                     *)
(* ------------------------------------------------------------------------- *)

  (* theory -> model -> (int -> bool) -> string *)

  fun print_model thy model assignment =
  let
    val (typs, terms) = model
    val typs_msg =
      if null typs then
        "empty universe (no type variables in term)\n"
      else
        "Size of types: " ^ commas (map (fn (T, i) =>
          Sign.string_of_typ thy T ^ ": " ^ string_of_int i) typs) ^ "\n"
    val show_consts_msg =
      if not (!show_consts) andalso Library.exists (is_Const o fst) terms then
        "set \"show_consts\" to show the interpretation of constants\n"
      else
        ""
    val terms_msg =
      if null terms then
        "empty interpretation (no free variables in term)\n"
      else
        space_implode "\n" (List.mapPartial (fn (t, intr) =>
          (* print constants only if 'show_consts' is true *)
          if (!show_consts) orelse not (is_Const t) then
            SOME (Sign.string_of_term thy t ^ ": " ^
              Sign.string_of_term thy (print thy model t intr assignment))
          else
            NONE) terms) ^ "\n"
  in
    typs_msg ^ show_consts_msg ^ terms_msg
  end;


(* ------------------------------------------------------------------------- *)
(* PARAMETER MANAGEMENT                                                      *)
(* ------------------------------------------------------------------------- *)

  (* string -> (theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option) -> theory -> theory *)

  fun add_interpreter name f thy =
  let
    val {interpreters, printers, parameters} = RefuteData.get thy
  in
    case AList.lookup (op =) interpreters name of
      NONE   => RefuteData.put {interpreters = (name, f) :: interpreters,
      printers = printers, parameters = parameters} thy
    | SOME _ => error ("Interpreter " ^ name ^ " already declared")
  end;

  (* string -> (theory -> model -> Term.term -> interpretation ->
    (int -> bool) -> Term.term option) -> theory -> theory *)

  fun add_printer name f thy =
  let
    val {interpreters, printers, parameters} = RefuteData.get thy
  in
    case AList.lookup (op =) printers name of
      NONE   => RefuteData.put {interpreters = interpreters,
      printers = (name, f) :: printers, parameters = parameters} thy
    | SOME _ => error ("Printer " ^ name ^ " already declared")
  end;

(* ------------------------------------------------------------------------- *)
(* set_default_param: stores the '(name, value)' pair in RefuteData's        *)
(*                    parameter table                                        *)
(* ------------------------------------------------------------------------- *)

  (* (string * string) -> theory -> theory *)

  fun set_default_param (name, value) thy =
  let
    val {interpreters, printers, parameters} = RefuteData.get thy
  in
    RefuteData.put (case Symtab.lookup parameters name of
      NONE   =>
      {interpreters = interpreters, printers = printers,
        parameters = Symtab.extend (parameters, [(name, value)])}
    | SOME _ =>
      {interpreters = interpreters, printers = printers,
        parameters = Symtab.update (name, value) parameters}) thy
  end;

(* ------------------------------------------------------------------------- *)
(* get_default_param: retrieves the value associated with 'name' from        *)
(*                    RefuteData's parameter table                           *)
(* ------------------------------------------------------------------------- *)

  (* theory -> string -> string option *)

  val get_default_param = Symtab.lookup o #parameters o RefuteData.get;

(* ------------------------------------------------------------------------- *)
(* get_default_params: returns a list of all '(name, value)' pairs that are  *)
(*                     stored in RefuteData's parameter table                *)
(* ------------------------------------------------------------------------- *)

  (* theory -> (string * string) list *)

  val get_default_params = Symtab.dest o #parameters o RefuteData.get;

(* ------------------------------------------------------------------------- *)
(* actual_params: takes a (possibly empty) list 'params' of parameters that  *)
(*      override the default parameters currently specified in 'thy', and    *)
(*      returns a record that can be passed to 'find_model'.                 *)
(* ------------------------------------------------------------------------- *)

  (* theory -> (string * string) list -> params *)

  fun actual_params thy override =
  let
    (* (string * string) list * string -> int *)
    fun read_int (parms, name) =
      case AList.lookup (op =) parms name of
        SOME s => (case Int.fromString s of
          SOME i => i
        | NONE   => error ("parameter " ^ quote name ^
          " (value is " ^ quote s ^ ") must be an integer value"))
      | NONE   => error ("parameter " ^ quote name ^
          " must be assigned a value")
    (* (string * string) list * string -> string *)
    fun read_string (parms, name) =
      case AList.lookup (op =) parms name of
        SOME s => s
      | NONE   => error ("parameter " ^ quote name ^
        " must be assigned a value")
    (* 'override' first, defaults last: *)
    (* (string * string) list *)
    val allparams = override @ (get_default_params thy)
    (* int *)
    val minsize   = read_int (allparams, "minsize")
    val maxsize   = read_int (allparams, "maxsize")
    val maxvars   = read_int (allparams, "maxvars")
    val maxtime   = read_int (allparams, "maxtime")
    (* string *)
    val satsolver = read_string (allparams, "satsolver")
    (* all remaining parameters of the form "string=int" are collected in *)
    (* 'sizes'                                                            *)
    (* TODO: it is currently not possible to specify a size for a type    *)
    (*       whose name is one of the other parameters (e.g. 'maxvars')   *)
    (* (string * int) list *)
    val sizes     = List.mapPartial
      (fn (name, value) => Option.map (pair name) (Int.fromString value))
      (List.filter (fn (name, _) => name<>"minsize" andalso name<>"maxsize"
        andalso name<>"maxvars" andalso name<>"maxtime"
        andalso name<>"satsolver") allparams)
  in
    {sizes=sizes, minsize=minsize, maxsize=maxsize, maxvars=maxvars,
      maxtime=maxtime, satsolver=satsolver}
  end;


(* ------------------------------------------------------------------------- *)
(* TRANSLATION HOL -> PROPOSITIONAL LOGIC, BOOLEAN ASSIGNMENT -> MODEL       *)
(* ------------------------------------------------------------------------- *)

  (* (''a * 'b) list -> ''a -> 'b *)

  fun lookup xs key =
    Option.valOf (AList.lookup (op =) xs key);

(* ------------------------------------------------------------------------- *)
(* typ_of_dtyp: converts a data type ('DatatypeAux.dtyp') into a type        *)
(*              ('Term.typ'), given type parameters for the data type's type *)
(*              arguments                                                    *)
(* ------------------------------------------------------------------------- *)

  (* DatatypeAux.descr -> (DatatypeAux.dtyp * Term.typ) list ->
    DatatypeAux.dtyp -> Term.typ *)

  fun typ_of_dtyp descr typ_assoc (DatatypeAux.DtTFree a) =
    (* replace a 'DtTFree' variable by the associated type *)
    lookup typ_assoc (DatatypeAux.DtTFree a)
    | typ_of_dtyp descr typ_assoc (DatatypeAux.DtType (s, ds)) =
    Type (s, map (typ_of_dtyp descr typ_assoc) ds)
    | typ_of_dtyp descr typ_assoc (DatatypeAux.DtRec i) =
    let
      val (s, ds, _) = lookup descr i
    in
      Type (s, map (typ_of_dtyp descr typ_assoc) ds)
    end;

(* ------------------------------------------------------------------------- *)
(* close_form: universal closure over schematic variables in 't'             *)
(* ------------------------------------------------------------------------- *)

  (* Term.term -> Term.term *)

  fun close_form t =
  let
    (* (Term.indexname * Term.typ) list *)
    val vars = sort_wrt (fst o fst) (map dest_Var (term_vars t))
  in
    Library.foldl (fn (t', ((x, i), T)) =>
      (Term.all T) $ Abs (x, T, abstract_over (Var ((x, i), T), t')))
      (t, vars)
  end;

(* ------------------------------------------------------------------------- *)
(* monomorphic_term: applies a type substitution 'typeSubs' for all type     *)
(*                   variables in a term 't'                                 *)
(* ------------------------------------------------------------------------- *)

  (* Type.tyenv -> Term.term -> Term.term *)

  fun monomorphic_term typeSubs t =
    map_types (map_type_tvar
      (fn v =>
        case Type.lookup (typeSubs, v) of
          NONE =>
          (* schematic type variable not instantiated *)
          raise REFUTE ("monomorphic_term",
            "no substitution for type variable " ^ fst (fst v) ^
            " in term " ^ Display.raw_string_of_term t)
        | SOME typ =>
          typ)) t;

(* ------------------------------------------------------------------------- *)
(* specialize_type: given a constant 's' of type 'T', which is a subterm of  *)
(*                  't', where 't' has a (possibly) more general type, the   *)
(*                  schematic type variables in 't' are instantiated to      *)
(*                  match the type 'T' (may raise Type.TYPE_MATCH)           *)
(* ------------------------------------------------------------------------- *)

  (* theory -> (string * Term.typ) -> Term.term -> Term.term *)

  fun specialize_type thy (s, T) t =
  let
    fun find_typeSubs (Const (s', T')) =
      if s=s' then
        SOME (Sign.typ_match thy (T', T) Vartab.empty)
          handle Type.TYPE_MATCH => NONE
      else
        NONE
      | find_typeSubs (Free _)           = NONE
      | find_typeSubs (Var _)            = NONE
      | find_typeSubs (Bound _)          = NONE
      | find_typeSubs (Abs (_, _, body)) = find_typeSubs body
      | find_typeSubs (t1 $ t2)          =
      (case find_typeSubs t1 of SOME x => SOME x
                              | NONE   => find_typeSubs t2)
  in
    case find_typeSubs t of
      SOME typeSubs =>
      monomorphic_term typeSubs t
    | NONE =>
      (* no match found - perhaps due to sort constraints *)
      raise Type.TYPE_MATCH
  end;

(* ------------------------------------------------------------------------- *)
(* is_const_of_class: returns 'true' iff 'Const (s, T)' is a constant that   *)
(*                    denotes membership to an axiomatic type class          *)
(* ------------------------------------------------------------------------- *)

  (* theory -> string * Term.typ -> bool *)

  fun is_const_of_class thy (s, T) =
  let
    val class_const_names = map Logic.const_of_class (Sign.all_classes thy)
  in
    (* I'm not quite sure if checking the name 's' is sufficient, *)
    (* or if we should also check the type 'T'.                   *)
    s mem_string class_const_names
  end;

(* ------------------------------------------------------------------------- *)
(* is_IDT_constructor: returns 'true' iff 'Const (s, T)' is the constructor  *)
(*                     of an inductive datatype in 'thy'                     *)
(* ------------------------------------------------------------------------- *)

  (* theory -> string * Term.typ -> bool *)

  fun is_IDT_constructor thy (s, T) =
    (case body_type T of
      Type (s', _) =>
      (case DatatypePackage.get_datatype_constrs thy s' of
        SOME constrs =>
        List.exists (fn (cname, cty) =>
          cname = s andalso Sign.typ_instance thy (T, cty)) constrs
      | NONE =>
        false)
    | _  =>
      false);

(* ------------------------------------------------------------------------- *)
(* is_IDT_recursor: returns 'true' iff 'Const (s, T)' is the recursion       *)
(*                  operator of an inductive datatype in 'thy'               *)
(* ------------------------------------------------------------------------- *)

  (* theory -> string * Term.typ -> bool *)

  fun is_IDT_recursor thy (s, T) =
  let
    val rec_names = Symtab.fold (append o #rec_names o snd)
      (DatatypePackage.get_datatypes thy) []
  in
    (* I'm not quite sure if checking the name 's' is sufficient, *)
    (* or if we should also check the type 'T'.                   *)
    s mem_string rec_names
  end;

(* ------------------------------------------------------------------------- *)
(* get_def: looks up the definition of a constant, as created by "constdefs" *)
(* ------------------------------------------------------------------------- *)

  (* theory -> string * Term.typ -> (string * Term.term) option *)

  fun get_def thy (s, T) =
  let
    (* maps  f ?t1 ... ?tn == rhs  to  %t1...tn. rhs *)
    fun norm_rhs eqn =
    let
      fun lambda (v as Var ((x, _), T)) t = Abs (x, T, abstract_over (v, t))
        | lambda v t                      = raise TERM ("lambda", [v, t])
      val (lhs, rhs) = Logic.dest_equals eqn
      val (_, args)  = Term.strip_comb lhs
    in
      fold lambda (rev args) rhs
    end
    (* (string * Term.term) list -> (string * Term.term) option *)
    fun get_def_ax [] = NONE
      | get_def_ax ((axname, ax) :: axioms) =
      (let
        val (lhs, _) = Logic.dest_equals ax  (* equations only *)
        val c        = Term.head_of lhs
        val (s', T') = Term.dest_Const c
      in
        if s=s' then
          let
            val typeSubs = Sign.typ_match thy (T', T) Vartab.empty
            val ax'      = monomorphic_term typeSubs ax
            val rhs      = norm_rhs ax'
          in
            SOME (axname, rhs)
          end
        else
          get_def_ax axioms
      end handle ERROR _         => get_def_ax axioms
               | TERM _          => get_def_ax axioms
               | Type.TYPE_MATCH => get_def_ax axioms)
  in
    get_def_ax (Theory.all_axioms_of thy)
  end;

(* ------------------------------------------------------------------------- *)
(* get_typedef: looks up the definition of a type, as created by "typedef"   *)
(* ------------------------------------------------------------------------- *)

  (* theory -> (string * Term.typ) -> (string * Term.term) option *)

  fun get_typedef thy T =
  let
    (* (string * Term.term) list -> (string * Term.term) option *)
    fun get_typedef_ax [] = NONE
      | get_typedef_ax ((axname, ax) :: axioms) =
      (let
        (* Term.term -> Term.typ option *)
        fun type_of_type_definition (Const (s', T')) =
          if s'="Typedef.type_definition" then
            SOME T'
          else
            NONE
          | type_of_type_definition (Free _)           = NONE
          | type_of_type_definition (Var _)            = NONE
          | type_of_type_definition (Bound _)          = NONE
          | type_of_type_definition (Abs (_, _, body)) =
          type_of_type_definition body
          | type_of_type_definition (t1 $ t2)          =
          (case type_of_type_definition t1 of
            SOME x => SOME x
          | NONE   => type_of_type_definition t2)
      in
        case type_of_type_definition ax of
          SOME T' =>
          let
            val T''      = (domain_type o domain_type) T'
            val typeSubs = Sign.typ_match thy (T'', T) Vartab.empty
          in
            SOME (axname, monomorphic_term typeSubs ax)
          end
        | NONE =>
          get_typedef_ax axioms
      end handle ERROR _         => get_typedef_ax axioms
               | MATCH           => get_typedef_ax axioms
               | Type.TYPE_MATCH => get_typedef_ax axioms)
  in
    get_typedef_ax (Theory.all_axioms_of thy)
  end;

(* ------------------------------------------------------------------------- *)
(* get_classdef: looks up the defining axiom for an axiomatic type class, as *)
(*               created by the "axclass" command                            *)
(* ------------------------------------------------------------------------- *)

  (* theory -> string -> (string * Term.term) option *)

  fun get_classdef thy class =
  let
    val axname = class ^ "_class_def"
  in
    Option.map (pair axname)
      (AList.lookup (op =) (Theory.all_axioms_of thy) axname)
  end;

(* ------------------------------------------------------------------------- *)
(* unfold_defs: unfolds all defined constants in a term 't', beta-eta        *)
(*              normalizes the result term; certain constants are not        *)
(*              unfolded (cf. 'collect_axioms' and the various interpreters  *)
(*              below): if the interpretation respects a definition anyway,  *)
(*              that definition does not need to be unfolded                 *)
(* ------------------------------------------------------------------------- *)

  (* theory -> Term.term -> Term.term *)

  (* Note: we could intertwine unfolding of constants and beta-(eta-)       *)
  (*       normalization; this would save some unfolding for terms where    *)
  (*       constants are eliminated by beta-reduction (e.g. 'K c1 c2').  On *)
  (*       the other hand, this would cause additional work for terms where *)
  (*       constants are duplicated by beta-reduction (e.g. 'S c1 c2 c3').  *)

  fun unfold_defs thy t =
  let
    (* Term.term -> Term.term *)
    fun unfold_loop t =
      case t of
      (* Pure *)
        Const ("all", _)                => t
      | Const ("==", _)                 => t
      | Const ("==>", _)                => t
      | Const ("TYPE", _)               => t  (* axiomatic type classes *)
      (* HOL *)
      | Const ("Trueprop", _)           => t
      | Const ("Not", _)                => t
      | (* redundant, since 'True' is also an IDT constructor *)
        Const ("True", _)               => t
      | (* redundant, since 'False' is also an IDT constructor *)
        Const ("False", _)              => t
      | Const ("arbitrary", _)          => t
      | Const ("The", _)                => t
      | Const ("Hilbert_Choice.Eps", _) => t
      | Const ("All", _)                => t
      | Const ("Ex", _)                 => t
      | Const ("op =", _)               => t
      | Const ("op &", _)               => t
      | Const ("op |", _)               => t
      | Const ("op -->", _)             => t
      (* sets *)
      | Const ("Collect", _)            => t
      | Const ("op :", _)               => t
      (* other optimizations *)
      | Const ("Finite_Set.card", _)    => t
      | Const ("Finite_Set.Finites", _) => t
      | Const ("Finite_Set.finite", _)  => t
      | Const ("Orderings.less", Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("bool", [])])])) => t
      | Const ("HOL.plus", Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => t
      | Const ("HOL.minus", Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => t
      | Const ("HOL.times", Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => t
      | Const ("List.op @", _)          => t
      | Const ("Lfp.lfp", _)            => t
      | Const ("Gfp.gfp", _)            => t
      | Const ("fst", _)                => t
      | Const ("snd", _)                => t
      (* simply-typed lambda calculus *)
      | Const (s, T) =>
        (if is_IDT_constructor thy (s, T)
          orelse is_IDT_recursor thy (s, T) then
          t  (* do not unfold IDT constructors/recursors *)
        (* unfold the constant if there is a defining equation *)
        else case get_def thy (s, T) of
          SOME (axname, rhs) =>
          (* Note: if the term to be unfolded (i.e. 'Const (s, T)')  *)
          (* occurs on the right-hand side of the equation, i.e. in  *)
          (* 'rhs', we must not use this equation to unfold, because *)
          (* that would loop.  Here would be the right place to      *)
          (* check this.  However, getting this really right seems   *)
          (* difficult because the user may state arbitrary axioms,  *)
          (* which could interact with overloading to create loops.  *)
          ((*Output.immediate_output (" unfolding: " ^ axname);*)unfold_loop rhs)
        | NONE => t)
      | Free _           => t
      | Var _            => t
      | Bound _          => t
      | Abs (s, T, body) => Abs (s, T, unfold_loop body)
      | t1 $ t2          => (unfold_loop t1) $ (unfold_loop t2)
    val result = Envir.beta_eta_contract (unfold_loop t)
  in
    result
  end;

(* ------------------------------------------------------------------------- *)
(* collect_axioms: collects (monomorphic, universally quantified, unfolded   *)
(*                 versions of) all HOL axioms that are relevant w.r.t 't'   *)
(* ------------------------------------------------------------------------- *)

  (* Note: to make the collection of axioms more easily extensible, this    *)
  (*       function could be based on user-supplied "axiom collectors",     *)
  (*       similar to 'interpret'/interpreters or 'print'/printers          *)

  (* Note: currently we use "inverse" functions to the definitional         *)
  (*       mechanisms provided by Isabelle/HOL, e.g. for "axclass",         *)
  (*       "typedef", "constdefs".  A more general approach could consider  *)
  (*       *every* axiom of the theory and collect it if it has a constant/ *)
  (*       type/typeclass in common with the term 't'.                      *)

  (* theory -> Term.term -> Term.term list *)

  (* Which axioms are "relevant" for a particular term/type goes hand in    *)
  (* hand with the interpretation of that term/type by its interpreter (see *)
  (* way below): if the interpretation respects an axiom anyway, the axiom  *)
  (* does not need to be added as a constraint here.                        *)

  (* To avoid collecting the same axiom multiple times, we use an           *)
  (* accumulator 'axs' which contains all axioms collected so far.          *)

  fun collect_axioms thy t =
  let
    val _ = Output.immediate_output "Adding axioms..."
    (* (string * Term.term) list *)
    val axioms = Theory.all_axioms_of thy
    (* string * Term.term -> Term.term list -> Term.term list *)
    fun collect_this_axiom (axname, ax) axs =
    let
      val ax' = unfold_defs thy ax
    in
      if member (op aconv) axs ax' then
        axs
      else (
        Output.immediate_output (" " ^ axname);
        collect_term_axioms (ax' :: axs, ax')
      )
    end
    (* Term.term list * Term.typ -> Term.term list *)
    and collect_sort_axioms (axs, T) =
    let
      (* string list *)
      val sort = (case T of
          TFree (_, sort) => sort
        | TVar (_, sort)  => sort
        | _               => raise REFUTE ("collect_axioms", "type " ^
          Sign.string_of_typ thy T ^ " is not a variable"))
      (* obtain axioms for all superclasses *)
      val superclasses = sort @ (maps (Sign.super_classes thy) sort)
      (* merely an optimization, because 'collect_this_axiom' disallows *)
      (* duplicate axioms anyway:                                       *)
      val superclasses = distinct (op =) superclasses
      val class_axioms = maps (fn class => map (fn ax =>
        ("<" ^ class ^ ">", Thm.prop_of ax))
        (#axioms (AxClass.get_definition thy class) handle ERROR _ => []))
        superclasses
      (* replace the (at most one) schematic type variable in each axiom *)
      (* by the actual type 'T'                                          *)
      val monomorphic_class_axioms = map (fn (axname, ax) =>
        (case Term.term_tvars ax of
          [] =>
          (axname, ax)
        | [(idx, S)] =>
          (axname, monomorphic_term (Vartab.make [(idx, (S, T))]) ax)
        | _ =>
          raise REFUTE ("collect_axioms", "class axiom " ^ axname ^ " (" ^
            Sign.string_of_term thy ax ^
            ") contains more than one type variable")))
        class_axioms
    in
      fold collect_this_axiom monomorphic_class_axioms axs
    end
    (* Term.term list * Term.typ -> Term.term list *)
    and collect_type_axioms (axs, T) =
      case T of
      (* simple types *)
        Type ("prop", [])      => axs
      | Type ("fun", [T1, T2]) => collect_type_axioms
        (collect_type_axioms (axs, T1), T2)
      | Type ("set", [T1])     => collect_type_axioms (axs, T1)
      (* axiomatic type classes *)
      | Type ("itself", [T1])  => collect_type_axioms (axs, T1)
      | Type (s, Ts)           =>
        (case DatatypePackage.get_datatype thy s of
          SOME info =>  (* inductive datatype *)
            (* only collect relevant type axioms for the argument types *)
            Library.foldl collect_type_axioms (axs, Ts)
        | NONE =>
          (case get_typedef thy T of
            SOME (axname, ax) =>
            collect_this_axiom (axname, ax) axs
          | NONE =>
            (* unspecified type, perhaps introduced with "typedecl" *)
            (* at least collect relevant type axioms for the argument types *)
            Library.foldl collect_type_axioms (axs, Ts)))
      (* axiomatic type classes *)
      | TFree _                => collect_sort_axioms (axs, T)
      (* axiomatic type classes *)
      | TVar _                 => collect_sort_axioms (axs, T)
    (* Term.term list * Term.term -> Term.term list *)
    and collect_term_axioms (axs, t) =
      case t of
      (* Pure *)
        Const ("all", _)                => axs
      | Const ("==", _)                 => axs
      | Const ("==>", _)                => axs
      (* axiomatic type classes *)
      | Const ("TYPE", T)               => collect_type_axioms (axs, T)
      (* HOL *)
      | Const ("Trueprop", _)           => axs
      | Const ("Not", _)                => axs
      (* redundant, since 'True' is also an IDT constructor *)
      | Const ("True", _)               => axs
      (* redundant, since 'False' is also an IDT constructor *)
      | Const ("False", _)              => axs
      | Const ("arbitrary", T)          => collect_type_axioms (axs, T)
      | Const ("The", T)                =>
        let
          val ax = specialize_type thy ("The", T)
            (lookup axioms "HOL.the_eq_trivial")
        in
          collect_this_axiom ("HOL.the_eq_trivial", ax) axs
        end
      | Const ("Hilbert_Choice.Eps", T) =>
        let
          val ax = specialize_type thy ("Hilbert_Choice.Eps", T)
            (lookup axioms "Hilbert_Choice.someI")
        in
          collect_this_axiom ("Hilbert_Choice.someI", ax) axs
        end
      | Const ("All", T)                => collect_type_axioms (axs, T)
      | Const ("Ex", T)                 => collect_type_axioms (axs, T)
      | Const ("op =", T)               => collect_type_axioms (axs, T)
      | Const ("op &", _)               => axs
      | Const ("op |", _)               => axs
      | Const ("op -->", _)             => axs
      (* sets *)
      | Const ("Collect", T)            => collect_type_axioms (axs, T)
      | Const ("op :", T)               => collect_type_axioms (axs, T)
      (* other optimizations *)
      | Const ("Finite_Set.card", T)    => collect_type_axioms (axs, T)
      | Const ("Finite_Set.Finites", T) => collect_type_axioms (axs, T)
      | Const ("Finite_Set.finite", T)  => collect_type_axioms (axs, T)
      | Const ("Orderings.less", T as Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("bool", [])])])) =>
          collect_type_axioms (axs, T)
      | Const ("HOL.plus", T as Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
          collect_type_axioms (axs, T)
      | Const ("HOL.minus", T as Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
          collect_type_axioms (axs, T)
      | Const ("HOL.times", T as Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
          collect_type_axioms (axs, T)
      | Const ("List.op @", T)          => collect_type_axioms (axs, T)
      | Const ("Lfp.lfp", T)            => collect_type_axioms (axs, T)
      | Const ("Gfp.gfp", T)            => collect_type_axioms (axs, T)
      | Const ("fst", T)                => collect_type_axioms (axs, T)
      | Const ("snd", T)                => collect_type_axioms (axs, T)
      (* simply-typed lambda calculus *)
      | Const (s, T)                    =>
          if is_const_of_class thy (s, T) then
            (* axiomatic type classes: add "OFCLASS(?'a::c, c_class)" *)
            (* and the class definition                               *)
            let
              val class   = Logic.class_of_const s
              val inclass = Logic.mk_inclass (TVar (("'a", 0), [class]), class)
              val ax_in   = SOME (specialize_type thy (s, T) inclass)
                (* type match may fail due to sort constraints *)
                handle Type.TYPE_MATCH => NONE
              val ax_1 = Option.map (fn ax => (Sign.string_of_term thy ax, ax))
                ax_in
              val ax_2 = Option.map (apsnd (specialize_type thy (s, T)))
                (get_classdef thy class)
            in
              collect_type_axioms (fold collect_this_axiom
                (map_filter I [ax_1, ax_2]) axs, T)
            end
          else if is_IDT_constructor thy (s, T)
            orelse is_IDT_recursor thy (s, T) then
            (* only collect relevant type axioms *)
            collect_type_axioms (axs, T)
          else
            (* other constants should have been unfolded, with some *)
            (* exceptions: e.g. Abs_xxx/Rep_xxx functions for       *)
            (* typedefs, or type-class related constants            *)
            (* only collect relevant type axioms *)
            collect_type_axioms (axs, T)
      | Free (_, T)      => collect_type_axioms (axs, T)
      | Var (_, T)       => collect_type_axioms (axs, T)
      | Bound i          => axs
      | Abs (_, T, body) => collect_term_axioms
        (collect_type_axioms (axs, T), body)
      | t1 $ t2          => collect_term_axioms
        (collect_term_axioms (axs, t1), t2)
    (* Term.term list *)
    val result = map close_form (collect_term_axioms ([], t))
    val _ = writeln " ...done."
  in
    result
  end;

(* ------------------------------------------------------------------------- *)
(* ground_types: collects all ground types in a term (including argument     *)
(*               types of other types), suppressing duplicates.  Does not    *)
(*               return function types, set types, non-recursive IDTs, or    *)
(*               'propT'.  For IDTs, also the argument types of constructors *)
(*               are considered.                                             *)
(* ------------------------------------------------------------------------- *)

  (* theory -> Term.term -> Term.typ list *)

  fun ground_types thy t =
  let
    (* Term.typ * Term.typ list -> Term.typ list *)
    fun collect_types (T, acc) =
      if T mem acc then
        acc  (* prevent infinite recursion (for IDTs) *)
      else
        (case T of
          Type ("fun", [T1, T2]) => collect_types (T1, collect_types (T2, acc))
        | Type ("prop", [])      => acc
        | Type ("set", [T1])     => collect_types (T1, acc)
        | Type (s, Ts)           =>
          (case DatatypePackage.get_datatype thy s of
            SOME info =>  (* inductive datatype *)
            let
              val index               = #index info
              val descr               = #descr info
              val (_, dtyps, constrs) = lookup descr index
              val typ_assoc           = dtyps ~~ Ts
              (* sanity check: every element in 'dtyps' must be a 'DtTFree' *)
              val _ = (if Library.exists (fn d =>
                  case d of DatatypeAux.DtTFree _ => false | _ => true) dtyps
                then
                  raise REFUTE ("ground_types", "datatype argument (for type "
                    ^ Sign.string_of_typ thy (Type (s, Ts))
                    ^ ") is not a variable")
                else
                  ())
              (* if the current type is a recursive IDT (i.e. a depth is *)
              (* required), add it to 'acc'                              *)
              val acc' = (if Library.exists (fn (_, ds) => Library.exists
                DatatypeAux.is_rec_type ds) constrs then
                  insert (op =) T acc
                else
                  acc)
              (* collect argument types *)
              val acc_args = foldr collect_types acc' Ts
              (* collect constructor types *)
              val acc_constrs = foldr collect_types acc_args (List.concat
                (map (fn (_, ds) => map (typ_of_dtyp descr typ_assoc) ds)
                  constrs))
            in
              acc_constrs
            end
          | NONE =>
            (* not an inductive datatype, e.g. defined via "typedef" or *)
            (* "typedecl"                                               *)
            insert (op =) T (foldr collect_types acc Ts))
        | TFree _                => insert (op =) T acc
        | TVar _                 => insert (op =) T acc)
  in
    it_term_types collect_types (t, [])
  end;

(* ------------------------------------------------------------------------- *)
(* string_of_typ: (rather naive) conversion from types to strings, used to   *)
(*                look up the size of a type in 'sizes'.  Parameterized      *)
(*                types with different parameters (e.g. "'a list" vs. "bool  *)
(*                list") are identified.                                     *)
(* ------------------------------------------------------------------------- *)

  (* Term.typ -> string *)

  fun string_of_typ (Type (s, _))     = s
    | string_of_typ (TFree (s, _))    = s
    | string_of_typ (TVar ((s,_), _)) = s;

(* ------------------------------------------------------------------------- *)
(* first_universe: returns the "first" (i.e. smallest) universe by assigning *)
(*                 'minsize' to every type for which no size is specified in *)
(*                 'sizes'                                                   *)
(* ------------------------------------------------------------------------- *)

  (* Term.typ list -> (string * int) list -> int -> (Term.typ * int) list *)

  fun first_universe xs sizes minsize =
  let
    fun size_of_typ T =
      case AList.lookup (op =) sizes (string_of_typ T) of
        SOME n => n
      | NONE   => minsize
  in
    map (fn T => (T, size_of_typ T)) xs
  end;

(* ------------------------------------------------------------------------- *)
(* next_universe: enumerates all universes (i.e. assignments of sizes to     *)
(*                types), where the minimal size of a type is given by       *)
(*                'minsize', the maximal size is given by 'maxsize', and a   *)
(*                type may have a fixed size given in 'sizes'                *)
(* ------------------------------------------------------------------------- *)

  (* (Term.typ * int) list -> (string * int) list -> int -> int ->
    (Term.typ * int) list option *)

  fun next_universe xs sizes minsize maxsize =
  let
    (* creates the "first" list of length 'len', where the sum of all list *)
    (* elements is 'sum', and the length of the list is 'len'              *)
    (* int -> int -> int -> int list option *)
    fun make_first _ 0 sum =
      if sum=0 then
        SOME []
      else
        NONE
      | make_first max len sum =
      if sum<=max orelse max<0 then
        Option.map (fn xs' => sum :: xs') (make_first max (len-1) 0)
      else
        Option.map (fn xs' => max :: xs') (make_first max (len-1) (sum-max))
    (* enumerates all int lists with a fixed length, where 0<=x<='max' for *)
    (* all list elements x (unless 'max'<0)                                *)
    (* int -> int -> int -> int list -> int list option *)
    fun next max len sum [] =
      NONE
      | next max len sum [x] =
      (* we've reached the last list element, so there's no shift possible *)
      make_first max (len+1) (sum+x+1)  (* increment 'sum' by 1 *)
      | next max len sum (x1::x2::xs) =
      if x1>0 andalso (x2<max orelse max<0) then
        (* we can shift *)
        SOME (valOf (make_first max (len+1) (sum+x1-1)) @ (x2+1) :: xs)
      else
        (* continue search *)
        next max (len+1) (sum+x1) (x2::xs)
    (* only consider those types for which the size is not fixed *)
    val mutables = List.filter
      (not o (AList.defined (op =) sizes) o string_of_typ o fst) xs
    (* subtract 'minsize' from every size (will be added again at the end) *)
    val diffs = map (fn (_, n) => n-minsize) mutables
  in
    case next (maxsize-minsize) 0 0 diffs of
      SOME diffs' =>
      (* merge with those types for which the size is fixed *)
      SOME (snd (foldl_map (fn (ds, (T, _)) =>
        case AList.lookup (op =) sizes (string_of_typ T) of
        (* return the fixed size *)
          SOME n => (ds, (T, n))
        (* consume the head of 'ds', add 'minsize' *)
        | NONE   => (tl ds, (T, minsize + hd ds)))
        (diffs', xs)))
    | NONE =>
      NONE
  end;

(* ------------------------------------------------------------------------- *)
(* toTrue: converts the interpretation of a Boolean value to a propositional *)
(*         formula that is true iff the interpretation denotes "true"        *)
(* ------------------------------------------------------------------------- *)

  (* interpretation -> prop_formula *)

  fun toTrue (Leaf [fm, _]) =
    fm
    | toTrue _              =
    raise REFUTE ("toTrue", "interpretation does not denote a Boolean value");

(* ------------------------------------------------------------------------- *)
(* toFalse: converts the interpretation of a Boolean value to a              *)
(*          propositional formula that is true iff the interpretation        *)
(*          denotes "false"                                                  *)
(* ------------------------------------------------------------------------- *)

  (* interpretation -> prop_formula *)

  fun toFalse (Leaf [_, fm]) =
    fm
    | toFalse _              =
    raise REFUTE ("toFalse", "interpretation does not denote a Boolean value");

(* ------------------------------------------------------------------------- *)
(* find_model: repeatedly calls 'interpret' with appropriate parameters,     *)
(*             applies a SAT solver, and (in case a model is found) displays *)
(*             the model to the user by calling 'print_model'                *)
(* thy       : the current theory                                            *)
(* {...}     : parameters that control the translation/model generation      *)
(* t         : term to be translated into a propositional formula            *)
(* negate    : if true, find a model that makes 't' false (rather than true) *)
(* ------------------------------------------------------------------------- *)

  (* theory -> params -> Term.term -> bool -> unit *)

  fun find_model thy {sizes, minsize, maxsize, maxvars, maxtime, satsolver} t
    negate =
  let
    (* unit -> unit *)
    fun wrapper () =
    let
      val u      = unfold_defs thy t
      val _      = writeln ("Unfolded term: " ^ Sign.string_of_term thy u)
      val axioms = collect_axioms thy u
      (* Term.typ list *)
      val types = Library.foldl (fn (acc, t') =>
        acc union (ground_types thy t')) ([], u :: axioms)
      val _     = writeln ("Ground types: "
        ^ (if null types then "none."
           else commas (map (Sign.string_of_typ thy) types)))
      (* we can only consider fragments of recursive IDTs, so we issue a  *)
      (* warning if the formula contains a recursive IDT                  *)
      (* TODO: no warning needed for /positive/ occurrences of IDTs       *)
      val _ = if Library.exists (fn
          Type (s, _) =>
          (case DatatypePackage.get_datatype thy s of
            SOME info =>  (* inductive datatype *)
            let
              val index           = #index info
              val descr           = #descr info
              val (_, _, constrs) = lookup descr index
            in
              (* recursive datatype? *)
              Library.exists (fn (_, ds) =>
                Library.exists DatatypeAux.is_rec_type ds) constrs
            end
          | NONE => false)
        | _ => false) types then
          warning ("Term contains a recursive datatype; "
            ^ "countermodel(s) may be spurious!")
        else
          ()
      (* (Term.typ * int) list -> unit *)
      fun find_model_loop universe =
      let
        val init_model = (universe, [])
        val init_args  = {maxvars = maxvars, def_eq = false, next_idx = 1,
          bounds = [], wellformed = True}
        val _          = Output.immediate_output ("Translating term (sizes: "
          ^ commas (map (fn (_, n) => string_of_int n) universe) ^ ") ...")
        (* translate 'u' and all axioms *)
        val ((model, args), intrs) = foldl_map (fn ((m, a), t') =>
          let
            val (i, m', a') = interpret thy m a t'
          in
            (* set 'def_eq' to 'true' *)
            ((m', {maxvars = #maxvars a', def_eq = true,
              next_idx = #next_idx a', bounds = #bounds a',
              wellformed = #wellformed a'}), i)
          end) ((init_model, init_args), u :: axioms)
        (* make 'u' either true or false, and make all axioms true, and *)
        (* add the well-formedness side condition                       *)
        val fm_u  = (if negate then toFalse else toTrue) (hd intrs)
        val fm_ax = PropLogic.all (map toTrue (tl intrs))
        val fm    = PropLogic.all [#wellformed args, fm_ax, fm_u]
      in
        Output.immediate_output " invoking SAT solver...";
        (case SatSolver.invoke_solver satsolver fm of
          SatSolver.SATISFIABLE assignment =>
          (writeln " model found!";
          writeln ("*** Model found: ***\n" ^ print_model thy model
            (fn i => case assignment i of SOME b => b | NONE => true)))
        | SatSolver.UNSATISFIABLE _ =>
          (Output.immediate_output " no model exists.\n";
          case next_universe universe sizes minsize maxsize of
            SOME universe' => find_model_loop universe'
          | NONE           => writeln
            "Search terminated, no larger universe within the given limits.")
        | SatSolver.UNKNOWN =>
          (Output.immediate_output " no model found.\n";
          case next_universe universe sizes minsize maxsize of
            SOME universe' => find_model_loop universe'
          | NONE           => writeln
            "Search terminated, no larger universe within the given limits.")
        ) handle SatSolver.NOT_CONFIGURED =>
          error ("SAT solver " ^ quote satsolver ^ " is not configured.")
      end handle MAXVARS_EXCEEDED =>
        writeln ("\nSearch terminated, number of Boolean variables ("
          ^ string_of_int maxvars ^ " allowed) exceeded.")
      in
        find_model_loop (first_universe types sizes minsize)
      end
    in
      (* some parameter sanity checks *)
      minsize>=1 orelse
        error ("\"minsize\" is " ^ string_of_int minsize ^ ", must be at least 1");
      maxsize>=1 orelse
        error ("\"maxsize\" is " ^ string_of_int maxsize ^ ", must be at least 1");
      maxsize>=minsize orelse
        error ("\"maxsize\" (=" ^ string_of_int maxsize ^
        ") is less than \"minsize\" (=" ^ string_of_int minsize ^ ").");
      maxvars>=0 orelse
        error ("\"maxvars\" is " ^ string_of_int maxvars ^ ", must be at least 0");
      maxtime>=0 orelse
        error ("\"maxtime\" is " ^ string_of_int maxtime ^ ", must be at least 0");
      (* enter loop with or without time limit *)
      writeln ("Trying to find a model that "
        ^ (if negate then "refutes" else "satisfies") ^ ": "
        ^ Sign.string_of_term thy t);
      if maxtime>0 then (
        interrupt_timeout (Time.fromSeconds (Int.toLarge maxtime))
          wrapper ()
        handle Interrupt =>
          writeln ("\nSearch terminated, time limit (" ^ string_of_int maxtime
            ^ (if maxtime=1 then " second" else " seconds") ^ ") exceeded.")
      ) else
        wrapper ()
    end;


(* ------------------------------------------------------------------------- *)
(* INTERFACE, PART 2: FINDING A MODEL                                        *)
(* ------------------------------------------------------------------------- *)

(* ------------------------------------------------------------------------- *)
(* satisfy_term: calls 'find_model' to find a model that satisfies 't'       *)
(* params      : list of '(name, value)' pairs used to override default      *)
(*               parameters                                                  *)
(* ------------------------------------------------------------------------- *)

  (* theory -> (string * string) list -> Term.term -> unit *)

  fun satisfy_term thy params t =
    find_model thy (actual_params thy params) t false;

(* ------------------------------------------------------------------------- *)
(* refute_term: calls 'find_model' to find a model that refutes 't'          *)
(* params     : list of '(name, value)' pairs used to override default       *)
(*              parameters                                                   *)
(* ------------------------------------------------------------------------- *)

  (* theory -> (string * string) list -> Term.term -> unit *)

  fun refute_term thy params t =
  let
    (* disallow schematic type variables, since we cannot properly negate  *)
    (* terms containing them (their logical meaning is that there EXISTS a *)
    (* type s.t. ...; to refute such a formula, we would have to show that *)
    (* for ALL types, not ...)                                             *)
    val _ = null (term_tvars t) orelse
      error "Term to be refuted contains schematic type variables"

    (* existential closure over schematic variables *)
    (* (Term.indexname * Term.typ) list *)
    val vars = sort_wrt (fst o fst) (map dest_Var (term_vars t))
    (* Term.term *)
    val ex_closure = Library.foldl (fn (t', ((x, i), T)) =>
      (HOLogic.exists_const T) $
        Abs (x, T, abstract_over (Var ((x, i), T), t')))
      (t, vars)
    (* Note: If 't' is of type 'propT' (rather than 'boolT'), applying   *)
    (* 'HOLogic.exists_const' is not type-correct.  However, this is not *)
    (* really a problem as long as 'find_model' still interprets the     *)
    (* resulting term correctly, without checking its type.              *)

    (* replace outermost universally quantified variables by Free's:     *)
    (* refuting a term with Free's is generally faster than refuting a   *)
    (* term with (nested) quantifiers, because quantifiers are expanded, *)
    (* while the SAT solver searches for an interpretation for Free's.   *)
    (* Also we get more information back that way, namely an             *)
    (* interpretation which includes values for the (formerly)           *)
    (* quantified variables.                                             *)
    (* maps  !!x1...xn. !xk...xm. t   to   t  *)
    fun strip_all_body (Const ("all", _) $ Abs (_, _, t)) = strip_all_body t
      | strip_all_body (Const ("Trueprop", _) $ t)        = strip_all_body t
      | strip_all_body (Const ("All", _) $ Abs (_, _, t)) = strip_all_body t
      | strip_all_body t                                  = t
    (* maps  !!x1...xn. !xk...xm. t   to   [x1, ..., xn, xk, ..., xm]  *)
    fun strip_all_vars (Const ("all", _) $ Abs (a, T, t)) =
      (a, T) :: strip_all_vars t
      | strip_all_vars (Const ("Trueprop", _) $ t)        =
      strip_all_vars t
      | strip_all_vars (Const ("All", _) $ Abs (a, T, t)) =
      (a, T) :: strip_all_vars t
      | strip_all_vars t                                  =
      [] : (string * typ) list
    val strip_t = strip_all_body ex_closure
    val frees   = Term.rename_wrt_term strip_t (strip_all_vars ex_closure)
    val subst_t = Term.subst_bounds (map Free frees, strip_t)
  in
    find_model thy (actual_params thy params) subst_t true
  end;

(* ------------------------------------------------------------------------- *)
(* refute_subgoal: calls 'refute_term' on a specific subgoal                 *)
(* params        : list of '(name, value)' pairs used to override default    *)
(*                 parameters                                                *)
(* subgoal       : 0-based index specifying the subgoal number               *)
(* ------------------------------------------------------------------------- *)

  (* theory -> (string * string) list -> Thm.thm -> int -> unit *)

  fun refute_subgoal thy params thm subgoal =
    refute_term thy params (List.nth (Thm.prems_of thm, subgoal));


(* ------------------------------------------------------------------------- *)
(* INTERPRETERS: Auxiliary Functions                                         *)
(* ------------------------------------------------------------------------- *)

(* ------------------------------------------------------------------------- *)
(* make_constants: returns all interpretations that have the same tree       *)
(*                 structure as 'intr', but consist of unit vectors with     *)
(*                 'True'/'False' only (no Boolean variables)                *)
(* ------------------------------------------------------------------------- *)

  (* interpretation -> interpretation list *)

  fun make_constants intr =
  let
    (* returns a list with all unit vectors of length n *)
    (* int -> interpretation list *)
    fun unit_vectors n =
    let
      (* returns the k-th unit vector of length n *)
      (* int * int -> interpretation *)
      fun unit_vector (k,n) =
        Leaf ((replicate (k-1) False) @ (True :: (replicate (n-k) False)))
      (* int -> interpretation list -> interpretation list *)
      fun unit_vectors_acc k vs =
        if k>n then [] else (unit_vector (k,n))::(unit_vectors_acc (k+1) vs)
    in
      unit_vectors_acc 1 []
    end
    (* returns a list of lists, each one consisting of n (possibly *)
    (* identical) elements from 'xs'                               *)
    (* int -> 'a list -> 'a list list *)
    fun pick_all 1 xs =
      map single xs
      | pick_all n xs =
      let val rec_pick = pick_all (n-1) xs in
        Library.foldl (fn (acc, x) => map (cons x) rec_pick @ acc) ([], xs)
      end
  in
    case intr of
      Leaf xs => unit_vectors (length xs)
    | Node xs => map (fn xs' => Node xs') (pick_all (length xs)
      (make_constants (hd xs)))
  end;

(* ------------------------------------------------------------------------- *)
(* size_of_type: returns the number of constants in a type (i.e. 'length     *)
(*               (make_constants intr)', but implemented more efficiently)   *)
(* ------------------------------------------------------------------------- *)

  (* interpretation -> int *)

  fun size_of_type intr =
  let
    (* power (a, b) computes a^b, for a>=0, b>=0 *)
    (* int * int -> int *)
    fun power (a, 0) = 1
      | power (a, 1) = a
      | power (a, b) = let val ab = power(a, b div 2) in
        ab * ab * power(a, b mod 2)
      end
  in
    case intr of
      Leaf xs => length xs
    | Node xs => power (size_of_type (hd xs), length xs)
  end;

(* ------------------------------------------------------------------------- *)
(* TT/FF: interpretations that denote "true" or "false", respectively        *)
(* ------------------------------------------------------------------------- *)

  (* interpretation *)

  val TT = Leaf [True, False];

  val FF = Leaf [False, True];

(* ------------------------------------------------------------------------- *)
(* make_equality: returns an interpretation that denotes (extensional)       *)
(*                equality of two interpretations                            *)
(* - two interpretations are 'equal' iff they are both defined and denote    *)
(*   the same value                                                          *)
(* - two interpretations are 'not_equal' iff they are both defined at least  *)
(*   partially, and a defined part denotes different values                  *)
(* - a completely undefined interpretation is neither 'equal' nor            *)
(*   'not_equal' to another interpretation                                   *)
(* ------------------------------------------------------------------------- *)

  (* We could in principle represent '=' on a type T by a particular        *)
  (* interpretation.  However, the size of that interpretation is quadratic *)
  (* in the size of T.  Therefore comparing the interpretations 'i1' and    *)
  (* 'i2' directly is more efficient than constructing the interpretation   *)
  (* for equality on T first, and "applying" this interpretation to 'i1'    *)
  (* and 'i2' in the usual way (cf. 'interpretation_apply') then.           *)

  (* interpretation * interpretation -> interpretation *)

  fun make_equality (i1, i2) =
  let
    (* interpretation * interpretation -> prop_formula *)
    fun equal (i1, i2) =
      (case i1 of
        Leaf xs =>
        (case i2 of
          Leaf ys => PropLogic.dot_product (xs, ys)  (* defined and equal *)
        | Node _  => raise REFUTE ("make_equality",
          "second interpretation is higher"))
      | Node xs =>
        (case i2 of
          Leaf _  => raise REFUTE ("make_equality",
          "first interpretation is higher")
        | Node ys => PropLogic.all (map equal (xs ~~ ys))))
    (* interpretation * interpretation -> prop_formula *)
    fun not_equal (i1, i2) =
      (case i1 of
        Leaf xs =>
        (case i2 of
          (* defined and not equal *)
          Leaf ys => PropLogic.all ((PropLogic.exists xs)
          :: (PropLogic.exists ys)
          :: (map (fn (x,y) => SOr (SNot x, SNot y)) (xs ~~ ys)))
        | Node _  => raise REFUTE ("make_equality",
          "second interpretation is higher"))
      | Node xs =>
        (case i2 of
          Leaf _  => raise REFUTE ("make_equality",
          "first interpretation is higher")
        | Node ys => PropLogic.exists (map not_equal (xs ~~ ys))))
  in
    (* a value may be undefined; therefore 'not_equal' is not just the *)
    (* negation of 'equal'                                             *)
    Leaf [equal (i1, i2), not_equal (i1, i2)]
  end;

(* ------------------------------------------------------------------------- *)
(* make_def_equality: returns an interpretation that denotes (extensional)   *)
(*                    equality of two interpretations                        *)
(* This function treats undefined/partially defined interpretations          *)
(* different from 'make_equality': two undefined interpretations are         *)
(* considered equal, while a defined interpretation is considered not equal  *)
(* to an undefined interpretation.                                           *)
(* ------------------------------------------------------------------------- *)

  (* interpretation * interpretation -> interpretation *)

  fun make_def_equality (i1, i2) =
  let
    (* interpretation * interpretation -> prop_formula *)
    fun equal (i1, i2) =
      (case i1 of
        Leaf xs =>
        (case i2 of
          (* defined and equal, or both undefined *)
          Leaf ys => SOr (PropLogic.dot_product (xs, ys),
          SAnd (PropLogic.all (map SNot xs), PropLogic.all (map SNot ys)))
        | Node _  => raise REFUTE ("make_def_equality",
          "second interpretation is higher"))
      | Node xs =>
        (case i2 of
          Leaf _  => raise REFUTE ("make_def_equality",
          "first interpretation is higher")
        | Node ys => PropLogic.all (map equal (xs ~~ ys))))
    (* interpretation *)
    val eq = equal (i1, i2)
  in
    Leaf [eq, SNot eq]
  end;

(* ------------------------------------------------------------------------- *)
(* interpretation_apply: returns an interpretation that denotes the result   *)
(*                       of applying the function denoted by 'i1' to the     *)
(*                       argument denoted by 'i2'                            *)
(* ------------------------------------------------------------------------- *)

  (* interpretation * interpretation -> interpretation *)

  fun interpretation_apply (i1, i2) =
  let
    (* interpretation * interpretation -> interpretation *)
    fun interpretation_disjunction (tr1,tr2) =
      tree_map (fn (xs,ys) => map (fn (x,y) => SOr(x,y)) (xs ~~ ys))
        (tree_pair (tr1,tr2))
    (* prop_formula * interpretation -> interpretation *)
    fun prop_formula_times_interpretation (fm,tr) =
      tree_map (map (fn x => SAnd (fm,x))) tr
    (* prop_formula list * interpretation list -> interpretation *)
    fun prop_formula_list_dot_product_interpretation_list ([fm],[tr]) =
      prop_formula_times_interpretation (fm,tr)
      | prop_formula_list_dot_product_interpretation_list (fm::fms,tr::trees) =
      interpretation_disjunction (prop_formula_times_interpretation (fm,tr),
        prop_formula_list_dot_product_interpretation_list (fms,trees))
      | prop_formula_list_dot_product_interpretation_list (_,_) =
      raise REFUTE ("interpretation_apply", "empty list (in dot product)")
    (* concatenates 'x' with every list in 'xss', returning a new list of *)
    (* lists                                                              *)
    (* 'a -> 'a list list -> 'a list list *)
    fun cons_list x xss =
      map (cons x) xss
    (* returns a list of lists, each one consisting of one element from each *)
    (* element of 'xss'                                                      *)
    (* 'a list list -> 'a list list *)
    fun pick_all [xs] =
      map single xs
      | pick_all (xs::xss) =
      let val rec_pick = pick_all xss in
        Library.foldl (fn (acc, x) => (cons_list x rec_pick) @ acc) ([], xs)
      end
      | pick_all _ =
      raise REFUTE ("interpretation_apply", "empty list (in pick_all)")
    (* interpretation -> prop_formula list *)
    fun interpretation_to_prop_formula_list (Leaf xs) =
      xs
      | interpretation_to_prop_formula_list (Node trees) =
      map PropLogic.all (pick_all
        (map interpretation_to_prop_formula_list trees))
  in
    case i1 of
      Leaf _ =>
      raise REFUTE ("interpretation_apply", "first interpretation is a leaf")
    | Node xs =>
      prop_formula_list_dot_product_interpretation_list
        (interpretation_to_prop_formula_list i2, xs)
  end;

(* ------------------------------------------------------------------------- *)
(* eta_expand: eta-expands a term 't' by adding 'i' lambda abstractions      *)
(* ------------------------------------------------------------------------- *)

  (* Term.term -> int -> Term.term *)

  fun eta_expand t i =
  let
    val Ts = Term.binder_types (Term.fastype_of t)
    val t' = Term.incr_boundvars i t
  in
    foldr (fn (T, term) => Abs ("<eta_expand>", T, term))
      (Term.list_comb (t', map Bound (i-1 downto 0))) (List.take (Ts, i))
  end;

(* ------------------------------------------------------------------------- *)
(* sum: returns the sum of a list 'xs' of integers                           *)
(* ------------------------------------------------------------------------- *)

  (* int list -> int *)

  fun sum xs = foldl op+ 0 xs;

(* ------------------------------------------------------------------------- *)
(* product: returns the product of a list 'xs' of integers                   *)
(* ------------------------------------------------------------------------- *)

  (* int list -> int *)

  fun product xs = foldl op* 1 xs;

(* ------------------------------------------------------------------------- *)
(* size_of_dtyp: the size of (an initial fragment of) an inductive data type *)
(*               is the sum (over its constructors) of the product (over     *)
(*               their arguments) of the size of the argument types          *)
(* ------------------------------------------------------------------------- *)

  (* theory -> (Term.typ * int) list -> DatatypeAux.descr ->
    (DatatypeAux.dtyp * Term.typ) list ->
    (string * DatatypeAux.dtyp list) list -> int *)

  fun size_of_dtyp thy typ_sizes descr typ_assoc constructors =
    sum (map (fn (_, dtyps) =>
      product (map (fn dtyp =>
        let
          val T         = typ_of_dtyp descr typ_assoc dtyp
          val (i, _, _) = interpret thy (typ_sizes, [])
            {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
            (Free ("dummy", T))
        in
          size_of_type i
        end) dtyps)) constructors);


(* ------------------------------------------------------------------------- *)
(* INTERPRETERS: Actual Interpreters                                         *)
(* ------------------------------------------------------------------------- *)

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* simply typed lambda calculus: Isabelle's basic term syntax, with type *)
  (* variables, function types, and propT                                  *)

  fun stlc_interpreter thy model args t =
  let
    val (typs, terms)                                   = model
    val {maxvars, def_eq, next_idx, bounds, wellformed} = args
    (* Term.typ -> (interpretation * model * arguments) option *)
    fun interpret_groundterm T =
    let
      (* unit -> (interpretation * model * arguments) option *)
      fun interpret_groundtype () =
      let
        (* the model must specify a size for ground types *)
        val size = (if T = Term.propT then 2 else lookup typs T)
        val next = next_idx+size
        (* check if 'maxvars' is large enough *)
        val _    = (if next-1>maxvars andalso maxvars>0 then
          raise MAXVARS_EXCEEDED else ())
        (* prop_formula list *)
        val fms  = map BoolVar (next_idx upto (next_idx+size-1))
        (* interpretation *)
        val intr = Leaf fms
        (* prop_formula list -> prop_formula *)
        fun one_of_two_false []      = True
          | one_of_two_false (x::xs) = SAnd (PropLogic.all (map (fn x' =>
          SOr (SNot x, SNot x')) xs), one_of_two_false xs)
        (* prop_formula *)
        val wf   = one_of_two_false fms
      in
        (* extend the model, increase 'next_idx', add well-formedness *)
        (* condition                                                  *)
        SOME (intr, (typs, (t, intr)::terms), {maxvars = maxvars,
          def_eq = def_eq, next_idx = next, bounds = bounds,
          wellformed = SAnd (wellformed, wf)})
      end
    in
      case T of
        Type ("fun", [T1, T2]) =>
        let
          (* we create 'size_of_type (interpret (... T1))' different copies *)
          (* of the interpretation for 'T2', which are then combined into a *)
          (* single new interpretation                                      *)
          val (i1, _, _) = interpret thy model {maxvars=0, def_eq=false,
            next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T1))
          (* make fresh copies, with different variable indices *)
          (* 'idx': next variable index                         *)
          (* 'n'  : number of copies                            *)
          (* int -> int -> (int * interpretation list * prop_formula *)
          fun make_copies idx 0 =
            (idx, [], True)
            | make_copies idx n =
            let
              val (copy, _, new_args) = interpret thy (typs, [])
                {maxvars = maxvars, def_eq = false, next_idx = idx,
                bounds = [], wellformed = True} (Free ("dummy", T2))
              val (idx', copies, wf') = make_copies (#next_idx new_args) (n-1)
            in
              (idx', copy :: copies, SAnd (#wellformed new_args, wf'))
            end
          val (next, copies, wf) = make_copies next_idx (size_of_type i1)
          (* combine copies into a single interpretation *)
          val intr = Node copies
        in
          (* extend the model, increase 'next_idx', add well-formedness *)
          (* condition                                                  *)
          SOME (intr, (typs, (t, intr)::terms), {maxvars = maxvars,
            def_eq = def_eq, next_idx = next, bounds = bounds,
            wellformed = SAnd (wellformed, wf)})
        end
      | Type _  => interpret_groundtype ()
      | TFree _ => interpret_groundtype ()
      | TVar  _ => interpret_groundtype ()
    end
  in
    case AList.lookup (op =) terms t of
      SOME intr =>
      (* return an existing interpretation *)
      SOME (intr, model, args)
    | NONE =>
      (case t of
        Const (_, T)     =>
        interpret_groundterm T
      | Free (_, T)      =>
        interpret_groundterm T
      | Var (_, T)       =>
        interpret_groundterm T
      | Bound i          =>
        SOME (List.nth (#bounds args, i), model, args)
      | Abs (x, T, body) =>
        let
          (* create all constants of type 'T' *)
          val (i, _, _) = interpret thy model {maxvars=0, def_eq=false,
            next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T))
          val constants = make_constants i
          (* interpret the 'body' separately for each constant *)
          val ((model', args'), bodies) = foldl_map
            (fn ((m, a), c) =>
              let
                (* add 'c' to 'bounds' *)
                val (i', m', a') = interpret thy m {maxvars = #maxvars a,
                  def_eq = #def_eq a, next_idx = #next_idx a,
                  bounds = (c :: #bounds a), wellformed = #wellformed a} body
              in
                (* keep the new model m' and 'next_idx' and 'wellformed', *)
                (* but use old 'bounds'                                   *)
                ((m', {maxvars = maxvars, def_eq = def_eq,
                  next_idx = #next_idx a', bounds = bounds,
                  wellformed = #wellformed a'}), i')
              end)
            ((model, args), constants)
        in
          SOME (Node bodies, model', args')
        end
      | t1 $ t2          =>
        let
          (* interpret 't1' and 't2' separately *)
          val (intr1, model1, args1) = interpret thy model args t1
          val (intr2, model2, args2) = interpret thy model1 args1 t2
        in
          SOME (interpretation_apply (intr1, intr2), model2, args2)
        end)
  end;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  fun Pure_interpreter thy model args t =
    case t of
      Const ("all", _) $ t1 =>
      let
        val (i, m, a) = interpret thy model args t1
      in
        case i of
          Node xs =>
          (* 3-valued logic *)
          let
            val fmTrue  = PropLogic.all (map toTrue xs)
            val fmFalse = PropLogic.exists (map toFalse xs)
          in
            SOME (Leaf [fmTrue, fmFalse], m, a)
          end
        | _ =>
          raise REFUTE ("Pure_interpreter",
            "\"all\" is followed by a non-function")
      end
    | Const ("all", _) =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const ("==", _) $ t1 $ t2 =>
      let
        val (i1, m1, a1) = interpret thy model args t1
        val (i2, m2, a2) = interpret thy m1 a1 t2
      in
        (* we use either 'make_def_equality' or 'make_equality' *)
        SOME ((if #def_eq args then make_def_equality else make_equality)
          (i1, i2), m2, a2)
      end
    | Const ("==", _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const ("==", _) =>
      SOME (interpret thy model args (eta_expand t 2))
    | Const ("==>", _) $ t1 $ t2 =>
      (* 3-valued logic *)
      let
        val (i1, m1, a1) = interpret thy model args t1
        val (i2, m2, a2) = interpret thy m1 a1 t2
        val fmTrue       = PropLogic.SOr (toFalse i1, toTrue i2)
        val fmFalse      = PropLogic.SAnd (toTrue i1, toFalse i2)
      in
        SOME (Leaf [fmTrue, fmFalse], m2, a2)
      end
    | Const ("==>", _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const ("==>", _) =>
      SOME (interpret thy model args (eta_expand t 2))
    | _ => NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  fun HOLogic_interpreter thy model args t =
  (* Providing interpretations directly is more efficient than unfolding the *)
  (* logical constants.  In HOL however, logical constants can themselves be *)
  (* arguments.  They are then translated using eta-expansion.               *)
    case t of
      Const ("Trueprop", _) =>
      SOME (Node [TT, FF], model, args)
    | Const ("Not", _) =>
      SOME (Node [FF, TT], model, args)
    (* redundant, since 'True' is also an IDT constructor *)
    | Const ("True", _) =>
      SOME (TT, model, args)
    (* redundant, since 'False' is also an IDT constructor *)
    | Const ("False", _) =>
      SOME (FF, model, args)
    | Const ("All", _) $ t1 =>  (* similar to "all" (Pure) *)
      let
        val (i, m, a) = interpret thy model args t1
      in
        case i of
          Node xs =>
          (* 3-valued logic *)
          let
            val fmTrue  = PropLogic.all (map toTrue xs)
            val fmFalse = PropLogic.exists (map toFalse xs)
          in
            SOME (Leaf [fmTrue, fmFalse], m, a)
          end
        | _ =>
          raise REFUTE ("HOLogic_interpreter",
            "\"All\" is followed by a non-function")
      end
    | Const ("All", _) =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const ("Ex", _) $ t1 =>
      let
        val (i, m, a) = interpret thy model args t1
      in
        case i of
          Node xs =>
          (* 3-valued logic *)
          let
            val fmTrue  = PropLogic.exists (map toTrue xs)
            val fmFalse = PropLogic.all (map toFalse xs)
          in
            SOME (Leaf [fmTrue, fmFalse], m, a)
          end
        | _ =>
          raise REFUTE ("HOLogic_interpreter",
            "\"Ex\" is followed by a non-function")
      end
    | Const ("Ex", _) =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const ("op =", _) $ t1 $ t2 =>  (* similar to "==" (Pure) *)
      let
        val (i1, m1, a1) = interpret thy model args t1
        val (i2, m2, a2) = interpret thy m1 a1 t2
      in
        SOME (make_equality (i1, i2), m2, a2)
      end
    | Const ("op =", _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const ("op =", _) =>
      SOME (interpret thy model args (eta_expand t 2))
    | Const ("op &", _) $ t1 $ t2 =>
      (* 3-valued logic *)
      let
        val (i1, m1, a1) = interpret thy model args t1
        val (i2, m2, a2) = interpret thy m1 a1 t2
        val fmTrue       = PropLogic.SAnd (toTrue i1, toTrue i2)
        val fmFalse      = PropLogic.SOr (toFalse i1, toFalse i2)
      in
        SOME (Leaf [fmTrue, fmFalse], m2, a2)
      end
    | Const ("op &", _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const ("op &", _) =>
      SOME (interpret thy model args (eta_expand t 2))
      (* this would make "undef" propagate, even for formulae like *)
      (* "False & undef":                                          *)
      (* SOME (Node [Node [TT, FF], Node [FF, FF]], model, args) *)
    | Const ("op |", _) $ t1 $ t2 =>
      (* 3-valued logic *)
      let
        val (i1, m1, a1) = interpret thy model args t1
        val (i2, m2, a2) = interpret thy m1 a1 t2
        val fmTrue       = PropLogic.SOr (toTrue i1, toTrue i2)
        val fmFalse      = PropLogic.SAnd (toFalse i1, toFalse i2)
      in
        SOME (Leaf [fmTrue, fmFalse], m2, a2)
      end
    | Const ("op |", _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const ("op |", _) =>
      SOME (interpret thy model args (eta_expand t 2))
      (* this would make "undef" propagate, even for formulae like *)
      (* "True | undef":                                           *)
      (* SOME (Node [Node [TT, TT], Node [TT, FF]], model, args) *)
    | Const ("op -->", _) $ t1 $ t2 =>  (* similar to "==>" (Pure) *)
      (* 3-valued logic *)
      let
        val (i1, m1, a1) = interpret thy model args t1
        val (i2, m2, a2) = interpret thy m1 a1 t2
        val fmTrue       = PropLogic.SOr (toFalse i1, toTrue i2)
        val fmFalse      = PropLogic.SAnd (toTrue i1, toFalse i2)
      in
        SOME (Leaf [fmTrue, fmFalse], m2, a2)
      end
    | Const ("op -->", _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const ("op -->", _) =>
      SOME (interpret thy model args (eta_expand t 2))
      (* this would make "undef" propagate, even for formulae like *)
      (* "False --> undef":                                        *)
      (* SOME (Node [Node [TT, FF], Node [TT, TT]], model, args) *)
    | _ => NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  fun set_interpreter thy model args t =
  (* "T set" is isomorphic to "T --> bool" *)
  let
    val (typs, terms) = model
  in
    case AList.lookup (op =) terms t of
      SOME intr =>
      (* return an existing interpretation *)
      SOME (intr, model, args)
    | NONE =>
      (case t of
        Free (x, Type ("set", [T])) =>
        let
          val (intr, _, args') =
            interpret thy (typs, []) args (Free (x, T --> HOLogic.boolT))
        in
          SOME (intr, (typs, (t, intr)::terms), args')
        end
      | Var ((x, i), Type ("set", [T])) =>
        let
          val (intr, _, args') =
            interpret thy (typs, []) args (Var ((x,i), T --> HOLogic.boolT))
        in
          SOME (intr, (typs, (t, intr)::terms), args')
        end
      | Const (s, Type ("set", [T])) =>
        let
          val (intr, _, args') =
            interpret thy (typs, []) args (Const (s, T --> HOLogic.boolT))
        in
          SOME (intr, (typs, (t, intr)::terms), args')
        end
      (* 'Collect' == identity *)
      | Const ("Collect", _) $ t1 =>
        SOME (interpret thy model args t1)
      | Const ("Collect", _) =>
        SOME (interpret thy model args (eta_expand t 1))
      (* 'op :' == application *)
      | Const ("op :", _) $ t1 $ t2 =>
        SOME (interpret thy model args (t2 $ t1))
      | Const ("op :", _) $ t1 =>
        SOME (interpret thy model args (eta_expand t 1))
      | Const ("op :", _) =>
        SOME (interpret thy model args (eta_expand t 2))
      | _ => NONE)
  end;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* interprets variables and constants whose type is an IDT; *)
  (* constructors of IDTs however are properly interpreted by *)
  (* 'IDT_constructor_interpreter'                            *)

  fun IDT_interpreter thy model args t =
  let
    val (typs, terms) = model
    (* Term.typ -> (interpretation * model * arguments) option *)
    fun interpret_term (Type (s, Ts)) =
      (case DatatypePackage.get_datatype thy s of
        SOME info =>  (* inductive datatype *)
        let
          (* int option -- only recursive IDTs have an associated depth *)
          val depth = AList.lookup (op =) typs (Type (s, Ts))
        in
          (* termination condition to avoid infinite recursion *)
          if depth = (SOME 0) then
            (* return a leaf of size 0 *)
            SOME (Leaf [], model, args)
          else
            let
              val index               = #index info
              val descr               = #descr info
              val (_, dtyps, constrs) = lookup descr index
              val typ_assoc           = dtyps ~~ Ts
              (* sanity check: every element in 'dtyps' must be a 'DtTFree' *)
              val _ = (if Library.exists (fn d =>
                  case d of DatatypeAux.DtTFree _ => false | _ => true) dtyps
                then
                  raise REFUTE ("IDT_interpreter",
                    "datatype argument (for type "
                    ^ Sign.string_of_typ thy (Type (s, Ts))
                    ^ ") is not a variable")
                else
                  ())
              (* if the model specifies a depth for the current type, *)
              (* decrement it to avoid infinite recursion             *)
              val typs'    = case depth of NONE => typs | SOME n =>
                AList.update (op =) (Type (s, Ts), n-1) typs
              (* recursively compute the size of the datatype *)
              val size     = size_of_dtyp thy typs' descr typ_assoc constrs
              val next_idx = #next_idx args
              val next     = next_idx+size
              (* check if 'maxvars' is large enough *)
              val _        = (if next-1 > #maxvars args andalso
                #maxvars args > 0 then raise MAXVARS_EXCEEDED else ())
              (* prop_formula list *)
              val fms      = map BoolVar (next_idx upto (next_idx+size-1))
              (* interpretation *)
              val intr     = Leaf fms
              (* prop_formula list -> prop_formula *)
              fun one_of_two_false []      = True
                | one_of_two_false (x::xs) = SAnd (PropLogic.all (map (fn x' =>
                SOr (SNot x, SNot x')) xs), one_of_two_false xs)
              (* prop_formula *)
              val wf       = one_of_two_false fms
            in
              (* extend the model, increase 'next_idx', add well-formedness *)
              (* condition                                                  *)
              SOME (intr, (typs, (t, intr)::terms), {maxvars = #maxvars args,
                def_eq = #def_eq args, next_idx = next, bounds = #bounds args,
                wellformed = SAnd (#wellformed args, wf)})
            end
        end
      | NONE =>  (* not an inductive datatype *)
        NONE)
      | interpret_term _ =  (* a (free or schematic) type variable *)
      NONE
  in
    case AList.lookup (op =) terms t of
      SOME intr =>
      (* return an existing interpretation *)
      SOME (intr, model, args)
    | NONE =>
      (case t of
        Free (_, T)  => interpret_term T
      | Var (_, T)   => interpret_term T
      | Const (_, T) => interpret_term T
      | _            => NONE)
  end;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  fun IDT_constructor_interpreter thy model args t =
  let
    val (typs, terms) = model
  in
    case AList.lookup (op =) terms t of
      SOME intr =>
      (* return an existing interpretation *)
      SOME (intr, model, args)
    | NONE =>
      (case t of
        Const (s, T) =>
        (case body_type T of
          Type (s', Ts') =>
          (case DatatypePackage.get_datatype thy s' of
            SOME info =>  (* body type is an inductive datatype *)
            let
              val index               = #index info
              val descr               = #descr info
              val (_, dtyps, constrs) = lookup descr index
              val typ_assoc           = dtyps ~~ Ts'
              (* sanity check: every element in 'dtyps' must be a 'DtTFree' *)
              val _ = (if Library.exists (fn d =>
                  case d of DatatypeAux.DtTFree _ => false | _ => true) dtyps
                then
                  raise REFUTE ("IDT_constructor_interpreter",
                    "datatype argument (for type "
                    ^ Sign.string_of_typ thy (Type (s', Ts'))
                    ^ ") is not a variable")
                else
                  ())
              (* split the constructors into those occuring before/after *)
              (* 'Const (s, T)'                                          *)
              val (constrs1, constrs2) = take_prefix (fn (cname, ctypes) =>
                not (cname = s andalso Sign.typ_instance thy (T,
                  map (typ_of_dtyp descr typ_assoc) ctypes
                    ---> Type (s', Ts')))) constrs
            in
              case constrs2 of
                [] =>
                (* 'Const (s, T)' is not a constructor of this datatype *)
                NONE
              | (_, ctypes)::cs =>
                let
                  (* compute the total size of the datatype (with the *)
                  (* current depth)                                   *)
                  val (i, _, _) = interpret thy (typs, []) {maxvars=0,
                    def_eq=false, next_idx=1, bounds=[], wellformed=True}
                    (Free ("dummy", Type (s', Ts')))
                  val total     = size_of_type i
                  (* int option -- only /recursive/ IDTs have an associated *)
                  (*               depth                                    *)
                  val depth = AList.lookup (op =) typs (Type (s', Ts'))
                  val typs' = (case depth of NONE => typs | SOME n =>
                    AList.update (op =) (Type (s', Ts'), n-1) typs)
                  (* returns an interpretation where everything is mapped to *)
                  (* "undefined"                                             *)
                  (* DatatypeAux.dtyp list -> interpretation *)
                  fun make_undef [] =
                    Leaf (replicate total False)
                    | make_undef (d::ds) =
                    let
                      (* compute the current size of the type 'd' *)
                      val T           = typ_of_dtyp descr typ_assoc d
                      val (i, _, _)   = interpret thy (typs, []) {maxvars=0,
                        def_eq=false, next_idx=1, bounds=[], wellformed=True}
                        (Free ("dummy", T))
                      val size        = size_of_type i
                    in
                      Node (replicate size (make_undef ds))
                    end
                  (* returns the interpretation for a constructor at depth 1 *)
                  (* int * DatatypeAux.dtyp list -> int * interpretation *)
                  fun make_constr (offset, []) =
                    if offset<total then
                      (offset+1, Leaf ((replicate offset False) @ True ::
                        (replicate (total-offset-1) False)))
                    else
                      raise REFUTE ("IDT_constructor_interpreter",
                        "offset >= total")
                    | make_constr (offset, d::ds) =
                    let
                      (* compute the current and the old size of the type 'd' *)
                      val T           = typ_of_dtyp descr typ_assoc d
                      val (i, _, _)   = interpret thy (typs, []) {maxvars=0,
                        def_eq=false, next_idx=1, bounds=[], wellformed=True}
                        (Free ("dummy", T))
                      val size        = size_of_type i
                      val (i', _, _)  = interpret thy (typs', []) {maxvars=0,
                        def_eq=false, next_idx=1, bounds=[], wellformed=True}
                        (Free ("dummy", T))
                      val size'       = size_of_type i'
                      (* sanity check *)
                      val _           = if size < size' then
                          raise REFUTE ("IDT_constructor_interpreter",
                            "current size is less than old size")
                        else ()
                      (* int * interpretation list *)
                      val (new_offset, intrs) = foldl_map make_constr
                        (offset, replicate size' ds)
                      (* interpretation list *)
                      val undefs = replicate (size - size') (make_undef ds)
                    in
                      (* elements that exist at the previous depth are      *)
                      (* mapped to a defined value, while new elements are  *)
                      (* mapped to "undefined" by the recursive constructor *)
                      (new_offset, Node (intrs @ undefs))
                    end
                  (* extends the interpretation for a constructor (both      *)
                  (* recursive and non-recursive) obtained at depth n (n>=1) *)
                  (* to depth n+1                                            *)
                  (* int * DatatypeAux.dtyp list * interpretation
                    -> int * interpretation *)
                  fun extend_constr (offset, [], Leaf xs) =
                    let
                      (* returns the k-th unit vector of length n *)
                      (* int * int -> interpretation *)
                      fun unit_vector (k, n) =
                        Leaf ((replicate (k-1) False) @ True ::
                          (replicate (n-k) False))
                      (* int *)
                      val k = find_index_eq True xs
                    in
                      if k=(~1) then
                        (* if the element was mapped to "undefined" before, *)
                        (* map it to the value given by 'offset' now (and   *)
                        (* extend the length of the leaf)                   *)
                        (offset+1, unit_vector (offset+1, total))
                      else
                        (* if the element was already mapped to a defined  *)
                        (* value, map it to the same value again, just     *)
                        (* extend the length of the leaf, do not increment *)
                        (* the 'offset'                                    *)
                        (offset, unit_vector (k+1, total))
                    end
                    | extend_constr (_, [], Node _) =
                    raise REFUTE ("IDT_constructor_interpreter",
                      "interpretation for constructor (with no arguments left)"
                      ^ " is a node")
                    | extend_constr (offset, d::ds, Node xs) =
                    let
                      (* compute the size of the type 'd' *)
                      val T          = typ_of_dtyp descr typ_assoc d
                      val (i, _, _)  = interpret thy (typs, []) {maxvars=0,
                        def_eq=false, next_idx=1, bounds=[], wellformed=True}
                        (Free ("dummy", T))
                      val size       = size_of_type i
                      (* sanity check *)
                      val _          = if size < length xs then
                          raise REFUTE ("IDT_constructor_interpreter",
                            "new size of type is less than old size")
                        else ()
                      (* extend the existing interpretations *)
                      (* int * interpretation list *)
                      val (new_offset, intrs) = foldl_map (fn (off, i) =>
                        extend_constr (off, ds, i)) (offset, xs)
                      (* new elements of the type 'd' are mapped to *)
                      (* "undefined"                                *)
                      val undefs = replicate (size - length xs) (make_undef ds)
                    in
                      (new_offset, Node (intrs @ undefs))
                    end
                    | extend_constr (_, d::ds, Leaf _) =
                    raise REFUTE ("IDT_constructor_interpreter",
                      "interpretation for constructor (with arguments left)"
                      ^ " is a leaf")
                  (* returns 'true' iff the constructor has a recursive *)
                  (* argument                                           *)
                  (* DatatypeAux.dtyp list -> bool *)
                  fun is_rec_constr ds =
                    Library.exists DatatypeAux.is_rec_type ds
                  (* constructors before 'Const (s, T)' generate elements of *)
                  (* the datatype                                            *)
                  val offset = size_of_dtyp thy typs' descr typ_assoc constrs1
                in
                  case depth of
                    NONE =>  (* equivalent to a depth of 1 *)
                    SOME (snd (make_constr (offset, ctypes)), model, args)
                  | SOME 0 =>
                    raise REFUTE ("IDT_constructor_interpreter", "depth is 0")
                  | SOME 1 =>
                    SOME (snd (make_constr (offset, ctypes)), model, args)
                  | SOME n =>  (* n > 1 *)
                    let
                      (* interpret the constructor at depth-1 *)
                      val (iC, _, _) = interpret thy (typs', []) {maxvars=0,
                        def_eq=false, next_idx=1, bounds=[], wellformed=True}
                        (Const (s, T))
                      (* elements generated by the constructor at depth-1 *)
                      (* must be added to 'offset'                        *)
                      (* interpretation -> int *)
                      fun number_of_defined_elements (Leaf xs) =
                        if find_index_eq True xs = (~1) then 0 else 1
                        | number_of_defined_elements (Node xs) =
                        sum (map number_of_defined_elements xs)
                      (* int *)
                      val offset' = offset + number_of_defined_elements iC
                    in
                      SOME (snd (extend_constr (offset', ctypes, iC)), model,
                        args)
                    end
                end
            end
          | NONE =>  (* body type is not an inductive datatype *)
            NONE)
        | _ =>  (* body type is a (free or schematic) type variable *)
          NONE)
      | _ =>  (* term is not a constant *)
        NONE)
  end;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* Difficult code ahead.  Make sure you understand the                *)
  (* 'IDT_constructor_interpreter' and the order in which it enumerates *)
  (* elements of an IDT before you try to understand this function.     *)

  fun IDT_recursion_interpreter thy model args t =
    (* careful: here we descend arbitrarily deep into 't', possibly before *)
    (* any other interpreter for atomic terms has had a chance to look at  *)
    (* 't'                                                                 *)
    case strip_comb t of
      (Const (s, T), params) =>
      (* iterate over all datatypes in 'thy' *)
      Symtab.fold (fn (_, info) => fn result =>
        case result of
          SOME _ =>
          result  (* just keep 'result' *)
        | NONE =>
          if member (op =) (#rec_names info) s then
            (* we do have a recursion operator of the datatype given by *)
            (* 'info', or of a mutually recursive datatype              *)
            let
              val index              = #index info
              val descr              = #descr info
              val (dtname, dtyps, _) = lookup descr index
              (* number of all constructors, including those of different  *)
              (* (mutually recursive) datatypes within the same descriptor *)
              (* 'descr'                                                   *)
              val mconstrs_count = sum (map (fn (_, (_, _, cs)) => length cs)
                descr)
              val params_count   = length params
              (* the type of a recursion operator: *)
              (* [T1, ..., Tn, IDT] ---> Tresult   *)
              val IDT = List.nth (binder_types T, mconstrs_count)
            in
              if (fst o dest_Type) IDT <> dtname then
                (* recursion operator of a mutually recursive datatype *)
                NONE
              else if mconstrs_count < params_count then
                (* too many actual parameters; for now we'll use the *)
                (* 'stlc_interpreter' to strip off one application   *)
                NONE
              else if mconstrs_count > params_count then
                (* too few actual parameters; we use eta expansion          *)
                (* Note that the resulting expansion of lambda abstractions *)
                (* by the 'stlc_interpreter' may be rather slow (depending  *)
                (* on the argument types and the size of the IDT, of        *)
                (* course).                                                 *)
                SOME (interpret thy model args (eta_expand t
                  (mconstrs_count - params_count)))
              else  (* mconstrs_count = params_count *)
                let
                  (* interpret each parameter separately *)
                  val ((model', args'), p_intrs) = foldl_map (fn ((m, a), p) =>
                    let
                      val (i, m', a') = interpret thy m a p
                    in
                      ((m', a'), i)
                    end) ((model, args), params)
                  val (typs, _) = model'
                  val typ_assoc = dtyps ~~ (snd o dest_Type) IDT
                  (* interpret each constructor in the descriptor (including *)
                  (* those of mutually recursive datatypes)                  *)
                  (* (int * interpretation list) list *)
                  val mc_intrs = map (fn (idx, (_, _, cs)) =>
                    let
                      val c_return_typ = typ_of_dtyp descr typ_assoc
                        (DatatypeAux.DtRec idx)
                    in
                      (idx, map (fn (cname, cargs) =>
                        (#1 o interpret thy (typs, []) {maxvars=0,
                          def_eq=false, next_idx=1, bounds=[],
                          wellformed=True}) (Const (cname, map (typ_of_dtyp
                          descr typ_assoc) cargs ---> c_return_typ))) cs)
                    end) descr
                  (* the recursion operator is a function that maps every   *)
                  (* element of the inductive datatype (and of mutually     *)
                  (* recursive types) to an element of some result type; an *)
                  (* array entry of NONE means that the actual result has   *)
                  (* not been computed yet                                  *)
                  (* (int * interpretation option Array.array) list *)
                  val INTRS = map (fn (idx, _) =>
                    let
                      val T         = typ_of_dtyp descr typ_assoc
                        (DatatypeAux.DtRec idx)
                      val (i, _, _) = interpret thy (typs, []) {maxvars=0,
                        def_eq=false, next_idx=1, bounds=[], wellformed=True}
                        (Free ("dummy", T))
                      val size      = size_of_type i
                    in
                      (idx, Array.array (size, NONE))
                    end) descr
                  (* takes an interpretation, and if some leaf of this     *)
                  (* interpretation is the 'elem'-th element of the type,  *)
                  (* the indices of the arguments leading to this leaf are *)
                  (* returned                                              *)
                  (* interpretation -> int -> int list option *)
                  fun get_args (Leaf xs) elem =
                    if find_index_eq True xs = elem then
                      SOME []
                    else
                      NONE
                    | get_args (Node xs) elem =
                    let
                      (* interpretation * int -> int list option *)
                      fun search ([], _) =
                        NONE
                        | search (x::xs, n) =
                        (case get_args x elem of
                          SOME result => SOME (n::result)
                        | NONE        => search (xs, n+1))
                    in
                      search (xs, 0)
                    end
                  (* returns the index of the constructor and indices for *)
                  (* its arguments that generate the 'elem'-th element of *)
                  (* the datatype given by 'idx'                          *)
                  (* int -> int -> int * int list *)
                  fun get_cargs idx elem =
                    let
                      (* int * interpretation list -> int * int list *)
                      fun get_cargs_rec (_, []) =
                        raise REFUTE ("IDT_recursion_interpreter",
                          "no matching constructor found for element "
                          ^ string_of_int elem ^ " in datatype "
                          ^ Sign.string_of_typ thy IDT ^ " (datatype index "
                          ^ string_of_int idx ^ ")")
                        | get_cargs_rec (n, x::xs) =
                        (case get_args x elem of
                          SOME args => (n, args)
                        | NONE      => get_cargs_rec (n+1, xs))
                    in
                      get_cargs_rec (0, lookup mc_intrs idx)
                    end
                  (* returns the number of constructors in datatypes that *)
                  (* occur in the descriptor 'descr' before the datatype  *)
                  (* given by 'idx'                                       *)
                  fun get_coffset idx =
                    let
                      fun get_coffset_acc _ [] =
                        raise REFUTE ("IDT_recursion_interpreter", "index "
                          ^ string_of_int idx ^ " not found in descriptor")
                        | get_coffset_acc sum ((i, (_, _, cs))::descr') =
                        if i=idx then
                          sum
                        else
                          get_coffset_acc (sum + length cs) descr'
                    in
                      get_coffset_acc 0 descr
                    end
                  (* computes one entry in INTRS, and recursively all      *)
                  (* entries needed for it, where 'idx' gives the datatype *)
                  (* and 'elem' the element of it                          *)
                  (* int -> int -> interpretation *)
                  fun compute_array_entry idx elem =
                    case Array.sub (lookup INTRS idx, elem) of
                      SOME result =>
                      (* simply return the previously computed result *)
                      result
                    | NONE =>
                      let
                        (* int * int list *)
                        val (c, args) = get_cargs idx elem
                        (* interpretation * int list -> interpretation *)
                        fun select_subtree (tr, []) =
                          tr  (* return the whole tree *)
                          | select_subtree (Leaf _, _) =
                          raise REFUTE ("IDT_recursion_interpreter",
                            "interpretation for parameter is a leaf; "
                            ^ "cannot select a subtree")
                          | select_subtree (Node tr, x::xs) =
                          select_subtree (List.nth (tr, x), xs)
                        (* select the correct subtree of the parameter *)
                        (* corresponding to constructor 'c'            *)
                        val p_intr = select_subtree (List.nth
                          (p_intrs, get_coffset idx + c), args)
                        (* find the indices of the constructor's recursive *)
                        (* arguments                                       *)
                        val (_, _, constrs) = lookup descr idx
                        val constr_args     = (snd o List.nth) (constrs, c)
                        val rec_args        = List.filter
                          (DatatypeAux.is_rec_type o fst) (constr_args ~~ args)
                        val rec_args'       = map (fn (dtyp, elem) =>
                          (DatatypeAux.dest_DtRec dtyp, elem)) rec_args
                        (* apply 'p_intr' to recursively computed results *)
                        val result = foldl (fn ((idx, elem), intr) =>
                          interpretation_apply (intr,
                          compute_array_entry idx elem)) p_intr rec_args'
                        (* update 'INTRS' *)
                        val _ = Array.update (lookup INTRS idx, elem,
                          SOME result)
                      in
                        result
                      end
                  (* compute all entries in INTRS for the current datatype *)
                  (* (given by 'index')                                    *)
                  (* TODO: we can use Array.modifyi instead once PolyML's *)
                  (*       Array signature conforms to the ML standard    *)
                  (* (int * 'a -> 'a) -> 'a array -> unit *)
                  fun modifyi f arr =
                    let
                      val size = Array.length arr
                      fun modifyi_loop i =
                        if i < size then (
                          Array.update (arr, i, f (i, Array.sub (arr, i)));
                          modifyi_loop (i+1)
                        ) else
                          ()
                    in
                      modifyi_loop 0
                    end
                  val _ = modifyi (fn (i, _) =>
                    SOME (compute_array_entry index i)) (lookup INTRS index)
                  (* 'a Array.array -> 'a list *)
                  fun toList arr =
                    Array.foldr op:: [] arr
                in
                  (* return the part of 'INTRS' that corresponds to the *)
                  (* current datatype                                   *)
                  SOME ((Node o map Option.valOf o toList o lookup INTRS)
                    index, model', args')
                end
            end
          else
            NONE  (* not a recursion operator of this datatype *)
        ) (DatatypePackage.get_datatypes thy) NONE
    | _ =>  (* head of term is not a constant *)
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'card' could in principle be interpreted with *)
  (* interpreters available already (using its definition), but the code *)
  (* below is more efficient                                             *)

  fun Finite_Set_card_interpreter thy model args t =
    case t of
      Const ("Finite_Set.card",
        Type ("fun", [Type ("set", [T]), Type ("nat", [])])) =>
      let
        val (i_nat, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("nat", [])))
        val size_nat      = size_of_type i_nat
        val (i_set, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("set", [T])))
        val constants     = make_constants i_set
        (* interpretation -> int *)
        fun number_of_elements (Node xs) =
          Library.foldl (fn (n, x) =>
            if x=TT then
              n+1
            else if x=FF then
              n
            else
              raise REFUTE ("Finite_Set_card_interpreter",
                "interpretation for set type does not yield a Boolean"))
            (0, xs)
          | number_of_elements (Leaf _) =
          raise REFUTE ("Finite_Set_card_interpreter",
            "interpretation for set type is a leaf")
        (* takes an interpretation for a set and returns an interpretation *)
        (* for a 'nat'                                                     *)
        (* interpretation -> interpretation *)
        fun card i =
          let
            val n = number_of_elements i
          in
            if n<size_nat then
              Leaf ((replicate n False) @ True ::
                (replicate (size_nat-n-1) False))
            else
              Leaf (replicate size_nat False)
          end
      in
        SOME (Node (map card constants), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'Finites' could in principle be interpreted with *)
  (* interpreters available already (using its definition), but the code    *)
  (* below is more efficient                                                *)

  fun Finite_Set_Finites_interpreter thy model args t =
    case t of
      Const ("Finite_Set.Finites", Type ("set", [Type ("set", [T])])) =>
      let
        val (i_set, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("set", [T])))
        val size_set      = size_of_type i_set
      in
        (* we only consider finite models anyway, hence EVERY set is in *)
        (* "Finites"                                                    *)
        SOME (Node (replicate size_set TT), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'finite' could in principle be interpreted with  *)
  (* interpreters available already (using its definition), but the code    *)
  (* below is more efficient                                                *)

  fun Finite_Set_finite_interpreter thy model args t =
    case t of
      Const ("Finite_Set.finite",
        Type ("fun", [Type ("set", [T]), Type ("bool", [])])) $ _ =>
        (* we only consider finite models anyway, hence EVERY set is *)
        (* "finite"                                                  *)
        SOME (TT, model, args)
    | Const ("Finite_Set.finite",
        Type ("fun", [Type ("set", [T]), Type ("bool", [])])) =>
      let
        val (i_set, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("set", [T])))
        val size_set      = size_of_type i_set
      in
        (* we only consider finite models anyway, hence EVERY set is *)
        (* "finite"                                                  *)
        SOME (Node (replicate size_set TT), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'Orderings.less' could in principle be            *)
  (* interpreted with interpreters available already (using its definition), *)
  (* but the code below is more efficient                                    *)

  fun Nat_less_interpreter thy model args t =
    case t of
      Const ("Orderings.less", Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("bool", [])])])) =>
      let
        val (i_nat, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("nat", [])))
        val size_nat      = size_of_type i_nat
        (* int -> interpretation *)
        (* the 'n'-th nat is not less than the first 'n' nats, while it *)
        (* is less than the remaining 'size_nat - n' nats               *)
        fun less n = Node ((replicate n FF) @ (replicate (size_nat - n) TT))
      in
        SOME (Node (map less (1 upto size_nat)), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'HOL.plus' could in principle be interpreted with *)
  (* interpreters available already (using its definition), but the code     *)
  (* below is more efficient                                                 *)

  fun Nat_plus_interpreter thy model args t =
    case t of
      Const ("HOL.plus", Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
      let
        val (i_nat, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("nat", [])))
        val size_nat      = size_of_type i_nat
        (* int -> int -> interpretation *)
        fun plus m n =
          let
            val element = (m+n)+1
          in
            if element > size_nat then
              Leaf (replicate size_nat False)
            else
              Leaf ((replicate (element-1) False) @ True ::
                (replicate (size_nat - element) False))
          end
      in
        SOME (Node (map (fn m => Node (map (plus m) (0 upto size_nat-1)))
          (0 upto size_nat-1)), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'HOL.minus' could in principle be interpreted *)
  (* with interpreters available already (using its definition), but the *)
  (* code below is more efficient                                        *)

  fun Nat_minus_interpreter thy model args t =
    case t of
      Const ("HOL.minus", Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
      let
        val (i_nat, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("nat", [])))
        val size_nat      = size_of_type i_nat
        (* int -> int -> interpretation *)
        fun minus m n =
          let
            val element = Int.max (m-n, 0) + 1
          in
            Leaf ((replicate (element-1) False) @ True ::
              (replicate (size_nat - element) False))
          end
      in
        SOME (Node (map (fn m => Node (map (minus m) (0 upto size_nat-1)))
          (0 upto size_nat-1)), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'HOL.times' could in principle be interpreted with *)
  (* interpreters available already (using its definition), but the code      *)
  (* below is more efficient                                                  *)

  fun Nat_times_interpreter thy model args t =
    case t of
      Const ("HOL.times", Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
      let
        val (i_nat, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("nat", [])))
        val size_nat      = size_of_type i_nat
        (* nat -> nat -> interpretation *)
        fun mult m n =
          let
            val element = (m*n)+1
          in
            if element > size_nat then
              Leaf (replicate size_nat False)
            else
              Leaf ((replicate (element-1) False) @ True ::
                (replicate (size_nat - element) False))
          end
      in
        SOME (Node (map (fn m => Node (map (mult m) (0 upto size_nat-1)))
          (0 upto size_nat-1)), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'op @' could in principle be interpreted with *)
  (* interpreters available already (using its definition), but the code *)
  (* below is more efficient                                             *)

  fun List_append_interpreter thy model args t =
    case t of
      Const ("List.op @", Type ("fun", [Type ("List.list", [T]), Type ("fun",
        [Type ("List.list", [_]), Type ("List.list", [_])])])) =>
      let
        val (i_elem, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", T))
        val size_elem      = size_of_type i_elem
        val (i_list, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("List.list", [T])))
        val size_list      = size_of_type i_list
        (* power (a, b) computes a^b, for a>=0, b>=0 *)
        (* int * int -> int *)
        fun power (a, 0) = 1
          | power (a, 1) = a
          | power (a, b) =
          let val ab = power(a, b div 2) in ab * ab * power(a, b mod 2) end
        (* log (a, b) computes floor(log_a(b)), i.e. the largest integer x *)
        (* s.t. a^x <= b, for a>=2, b>=1                                   *)
        (* int * int -> int *)
        fun log (a, b) =
          let
            fun logloop (ax, x) =
              if ax > b then x-1 else logloop (a * ax, x+1)
          in
            logloop (1, 0)
          end
        (* nat -> nat -> interpretation *)
        fun append m n =
          let
            (* The following formula depends on the order in which lists are *)
            (* enumerated by the 'IDT_constructor_interpreter'.  It took me  *)
            (* a little while to come up with this formula.                  *)
            val element = n + m * (if size_elem = 1 then 1
              else power (size_elem, log (size_elem, n+1))) + 1
          in
            if element > size_list then
              Leaf (replicate size_list False)
            else
              Leaf ((replicate (element-1) False) @ True ::
                (replicate (size_list - element) False))
          end
      in
        SOME (Node (map (fn m => Node (map (append m) (0 upto size_list-1)))
          (0 upto size_list-1)), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'lfp' could in principle be interpreted with  *)
  (* interpreters available already (using its definition), but the code *)
  (* below is more efficient                                             *)

  fun Lfp_lfp_interpreter thy model args t =
    case t of
      Const ("Lfp.lfp", Type ("fun", [Type ("fun",
        [Type ("set", [T]), Type ("set", [_])]), Type ("set", [_])])) =>
      let
        val (i_elem, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", T))
        val size_elem      = size_of_type i_elem
        (* the universe (i.e. the set that contains every element) *)
        val i_univ         = Node (replicate size_elem TT)
        (* all sets with elements from type 'T' *)
        val (i_set, _, _)  = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("set", [T])))
        val i_sets         = make_constants i_set
        (* all functions that map sets to sets *)
        val (i_fun, _, _)  = interpret thy model {maxvars=0, def_eq=false,
          next_idx=1, bounds=[], wellformed=True} (Free ("dummy",
          Type ("fun", [Type ("set", [T]), Type ("set", [T])])))
        val i_funs         = make_constants i_fun
        (* "lfp(f) == Inter({u. f(u) <= u})" *)
        (* interpretation * interpretation -> bool *)
        fun is_subset (Node subs, Node sups) =
          List.all (fn (sub, sup) => (sub = FF) orelse (sup = TT))
            (subs ~~ sups)
          | is_subset (_, _) =
          raise REFUTE ("Lfp_lfp_interpreter",
            "is_subset: interpretation for set is not a node")
        (* interpretation * interpretation -> interpretation *)
        fun intersection (Node xs, Node ys) =
          Node (map (fn (x, y) => if x=TT andalso y=TT then TT else FF)
            (xs ~~ ys))
          | intersection (_, _) =
          raise REFUTE ("Lfp_lfp_interpreter",
            "intersection: interpretation for set is not a node")
        (* interpretation -> interpretaion *)
        fun lfp (Node resultsets) =
          foldl (fn ((set, resultset), acc) =>
            if is_subset (resultset, set) then
              intersection (acc, set)
            else
              acc) i_univ (i_sets ~~ resultsets)
          | lfp _ =
            raise REFUTE ("Lfp_lfp_interpreter",
              "lfp: interpretation for function is not a node")
      in
        SOME (Node (map lfp i_funs), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'gfp' could in principle be interpreted with  *)
  (* interpreters available already (using its definition), but the code *)
  (* below is more efficient                                             *)

  fun Gfp_gfp_interpreter thy model args t =
    case t of
      Const ("Gfp.gfp", Type ("fun", [Type ("fun",
        [Type ("set", [T]), Type ("set", [_])]), Type ("set", [_])])) =>
      let nonfix union (* because "union" is used below *)
        val (i_elem, _, _) = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", T))
        val size_elem      = size_of_type i_elem
        (* the universe (i.e. the set that contains every element) *)
        val i_univ         = Node (replicate size_elem TT)
        (* all sets with elements from type 'T' *)
        val (i_set, _, _)  = interpret thy model
          {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}
          (Free ("dummy", Type ("set", [T])))
        val i_sets         = make_constants i_set
        (* all functions that map sets to sets *)
        val (i_fun, _, _)  = interpret thy model {maxvars=0, def_eq=false,
          next_idx=1, bounds=[], wellformed=True} (Free ("dummy",
          Type ("fun", [Type ("set", [T]), Type ("set", [T])])))
        val i_funs         = make_constants i_fun
        (* "gfp(f) == Union({u. u <= f(u)})" *)
        (* interpretation * interpretation -> bool *)
        fun is_subset (Node subs, Node sups) =
          List.all (fn (sub, sup) => (sub = FF) orelse (sup = TT))
            (subs ~~ sups)
          | is_subset (_, _) =
          raise REFUTE ("Gfp_gfp_interpreter",
            "is_subset: interpretation for set is not a node")
        (* interpretation * interpretation -> interpretation *)
        fun union (Node xs, Node ys) =
            Node (map (fn (x,y) => if x=TT orelse y=TT then TT else FF)
                 (xs ~~ ys))
          | union (_, _) =
          raise REFUTE ("Gfp_gfp_interpreter",
            "union: interpretation for set is not a node")
        (* interpretation -> interpretaion *)
        fun gfp (Node resultsets) =
          foldl (fn ((set, resultset), acc) =>
            if is_subset (set, resultset) then
              union (acc, set)
            else
              acc) i_univ (i_sets ~~ resultsets)
          | gfp _ =
            raise REFUTE ("Gfp_gfp_interpreter",
              "gfp: interpretation for function is not a node")
      in
        SOME (Node (map gfp i_funs), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'fst' could in principle be interpreted with  *)
  (* interpreters available already (using its definition), but the code *)
  (* below is more efficient                                             *)

  fun Product_Type_fst_interpreter thy model args t =
    case t of
      Const ("fst", Type ("fun", [Type ("*", [T, U]), _])) =>
      let
        val (iT, _, _) = interpret thy model {maxvars=0, def_eq=false,
          next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T))
        val is_T       = make_constants iT
        val (iU, _, _) = interpret thy model {maxvars=0, def_eq=false,
          next_idx=1, bounds=[], wellformed=True} (Free ("dummy", U))
        val size_U     = size_of_type iU
      in
        SOME (Node (List.concat (map (replicate size_U) is_T)), model, args)
      end
    | _ =>
      NONE;

  (* theory -> model -> arguments -> Term.term ->
    (interpretation * model * arguments) option *)

  (* only an optimization: 'snd' could in principle be interpreted with  *)
  (* interpreters available already (using its definition), but the code *)
  (* below is more efficient                                             *)

  fun Product_Type_snd_interpreter thy model args t =
    case t of
      Const ("snd", Type ("fun", [Type ("*", [T, U]), _])) =>
      let
        val (iT, _, _) = interpret thy model {maxvars=0, def_eq=false,
          next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T))
        val size_T     = size_of_type iT
        val (iU, _, _) = interpret thy model {maxvars=0, def_eq=false,
          next_idx=1, bounds=[], wellformed=True} (Free ("dummy", U))
        val is_U       = make_constants iU
      in
        SOME (Node (List.concat (replicate size_T is_U)), model, args)
      end
    | _ =>
      NONE;


(* ------------------------------------------------------------------------- *)
(* PRINTERS                                                                  *)
(* ------------------------------------------------------------------------- *)

  (* theory -> model -> Term.term -> interpretation -> (int -> bool) ->
    Term.term option *)

  fun stlc_printer thy model t intr assignment =
  let
    (* Term.term -> Term.typ option *)
    fun typeof (Free (_, T))  = SOME T
      | typeof (Var (_, T))   = SOME T
      | typeof (Const (_, T)) = SOME T
      | typeof _              = NONE
    (* string -> string *)
    fun strip_leading_quote s =
      (implode o (fn [] => [] | x::xs => if x="'" then xs else x::xs)
        o explode) s
    (* Term.typ -> string *)
    fun string_of_typ (Type (s, _))     = s
      | string_of_typ (TFree (x, _))    = strip_leading_quote x
      | string_of_typ (TVar ((x,i), _)) =
      strip_leading_quote x ^ string_of_int i
    (* interpretation -> int *)
    fun index_from_interpretation (Leaf xs) =
      find_index (PropLogic.eval assignment) xs
      | index_from_interpretation _ =
      raise REFUTE ("stlc_printer",
        "interpretation for ground type is not a leaf")
  in
    case typeof t of
      SOME T =>
      (case T of
        Type ("fun", [T1, T2]) =>
        let
          (* create all constants of type 'T1' *)
          val (i, _, _) = interpret thy model {maxvars=0, def_eq=false,
            next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T1))
          val constants = make_constants i
          (* interpretation list *)
          val results = (case intr of
              Node xs => xs
            | _       => raise REFUTE ("stlc_printer",
              "interpretation for function type is a leaf"))
          (* Term.term list *)
          val pairs = map (fn (arg, result) =>
            HOLogic.mk_prod
              (print thy model (Free ("dummy", T1)) arg assignment,
               print thy model (Free ("dummy", T2)) result assignment))
            (constants ~~ results)
          (* Term.typ *)
          val HOLogic_prodT = HOLogic.mk_prodT (T1, T2)
          val HOLogic_setT  = HOLogic.mk_setT HOLogic_prodT
          (* Term.term *)
          val HOLogic_empty_set = Const ("{}", HOLogic_setT)
          val HOLogic_insert    =
            Const ("insert", HOLogic_prodT --> HOLogic_setT --> HOLogic_setT)
        in
          SOME (foldr (fn (pair, acc) => HOLogic_insert $ pair $ acc)
            HOLogic_empty_set pairs)
        end
      | Type ("prop", [])      =>
        (case index_from_interpretation intr of
          ~1 => SOME (HOLogic.mk_Trueprop (Const ("arbitrary", HOLogic.boolT)))
        | 0  => SOME (HOLogic.mk_Trueprop HOLogic.true_const)
        | 1  => SOME (HOLogic.mk_Trueprop HOLogic.false_const)
        | _  => raise REFUTE ("stlc_interpreter",
          "illegal interpretation for a propositional value"))
      | Type _  => if index_from_interpretation intr = (~1) then
          SOME (Const ("arbitrary", T))
        else
          SOME (Const (string_of_typ T ^
            string_of_int (index_from_interpretation intr), T))
      | TFree _ => if index_from_interpretation intr = (~1) then
          SOME (Const ("arbitrary", T))
        else
          SOME (Const (string_of_typ T ^
            string_of_int (index_from_interpretation intr), T))
      | TVar _  => if index_from_interpretation intr = (~1) then
          SOME (Const ("arbitrary", T))
        else
          SOME (Const (string_of_typ T ^
            string_of_int (index_from_interpretation intr), T)))
    | NONE =>
      NONE
  end;

  (* theory -> model -> Term.term -> interpretation -> (int -> bool) ->
    string option *)

  fun set_printer thy model t intr assignment =
  let
    (* Term.term -> Term.typ option *)
    fun typeof (Free (_, T))  = SOME T
      | typeof (Var (_, T))   = SOME T
      | typeof (Const (_, T)) = SOME T
      | typeof _              = NONE
  in
    case typeof t of
      SOME (Type ("set", [T])) =>
      let
        (* create all constants of type 'T' *)
        val (i, _, _) = interpret thy model {maxvars=0, def_eq=false,
          next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T))
        val constants = make_constants i
        (* interpretation list *)
        val results = (case intr of
            Node xs => xs
          | _       => raise REFUTE ("set_printer",
            "interpretation for set type is a leaf"))
        (* Term.term list *)
        val elements = List.mapPartial (fn (arg, result) =>
          case result of
            Leaf [fmTrue, fmFalse] =>
            if PropLogic.eval assignment fmTrue then
              SOME (print thy model (Free ("dummy", T)) arg assignment)
            else (* if PropLogic.eval assignment fmFalse then *)
              NONE
          | _ =>
            raise REFUTE ("set_printer",
              "illegal interpretation for a Boolean value"))
          (constants ~~ results)
        (* Term.typ *)
        val HOLogic_setT  = HOLogic.mk_setT T
        (* Term.term *)
        val HOLogic_empty_set = Const ("{}", HOLogic_setT)
        val HOLogic_insert    =
          Const ("insert", T --> HOLogic_setT --> HOLogic_setT)
      in
        SOME (Library.foldl (fn (acc, elem) => HOLogic_insert $ elem $ acc)
          (HOLogic_empty_set, elements))
      end
    | _ =>
      NONE
  end;

  (* theory -> model -> Term.term -> interpretation -> (int -> bool) ->
    Term.term option *)

  fun IDT_printer thy model t intr assignment =
  let
    (* Term.term -> Term.typ option *)
    fun typeof (Free (_, T))  = SOME T
      | typeof (Var (_, T))   = SOME T
      | typeof (Const (_, T)) = SOME T
      | typeof _              = NONE
  in
    case typeof t of
      SOME (Type (s, Ts)) =>
      (case DatatypePackage.get_datatype thy s of
        SOME info =>  (* inductive datatype *)
        let
          val (typs, _)           = model
          val index               = #index info
          val descr               = #descr info
          val (_, dtyps, constrs) = lookup descr index
          val typ_assoc           = dtyps ~~ Ts
          (* sanity check: every element in 'dtyps' must be a 'DtTFree' *)
          val _ = (if Library.exists (fn d =>
              case d of DatatypeAux.DtTFree _ => false | _ => true) dtyps
            then
              raise REFUTE ("IDT_printer", "datatype argument (for type " ^
                Sign.string_of_typ thy (Type (s, Ts)) ^ ") is not a variable")
            else
              ())
          (* the index of the element in the datatype *)
          val element = (case intr of
              Leaf xs => find_index (PropLogic.eval assignment) xs
            | Node _  => raise REFUTE ("IDT_printer",
              "interpretation is not a leaf"))
        in
          if element < 0 then
            SOME (Const ("arbitrary", Type (s, Ts)))
          else let
            (* takes a datatype constructor, and if for some arguments this  *)
            (* constructor generates the datatype's element that is given by *)
            (* 'element', returns the constructor (as a term) as well as the *)
            (* indices of the arguments                                      *)
            (* string * DatatypeAux.dtyp list ->
              (Term.term * int list) option *)
            fun get_constr_args (cname, cargs) =
              let
                val cTerm      = Const (cname,
                  map (typ_of_dtyp descr typ_assoc) cargs ---> Type (s, Ts))
                val (iC, _, _) = interpret thy (typs, []) {maxvars=0,
                  def_eq=false, next_idx=1, bounds=[], wellformed=True} cTerm
                (* interpretation -> int list option *)
                fun get_args (Leaf xs) =
                  if find_index_eq True xs = element then
                    SOME []
                  else
                    NONE
                  | get_args (Node xs) =
                  let
                    (* interpretation * int -> int list option *)
                    fun search ([], _) =
                      NONE
                      | search (x::xs, n) =
                      (case get_args x of
                        SOME result => SOME (n::result)
                      | NONE        => search (xs, n+1))
                  in
                    search (xs, 0)
                  end
              in
                Option.map (fn args => (cTerm, cargs, args)) (get_args iC)
              end
            (* Term.term * DatatypeAux.dtyp list * int list *)
            val (cTerm, cargs, args) =
              (case get_first get_constr_args constrs of
                SOME x => x
              | NONE   => raise REFUTE ("IDT_printer",
                "no matching constructor found for element " ^
                string_of_int element))
            val argsTerms = map (fn (d, n) =>
              let
                val dT        = typ_of_dtyp descr typ_assoc d
                val (i, _, _) = interpret thy (typs, []) {maxvars=0,
                  def_eq=false, next_idx=1, bounds=[], wellformed=True}
                  (Free ("dummy", dT))
                (* we only need the n-th element of this list, so there   *)
                (* might be a more efficient implementation that does not *)
                (* generate all constants                                 *)
                val consts    = make_constants i
              in
                print thy (typs, []) (Free ("dummy", dT))
                  (List.nth (consts, n)) assignment
              end) (cargs ~~ args)
          in
            SOME (Library.foldl op$ (cTerm, argsTerms))
          end
        end
      | NONE =>  (* not an inductive datatype *)
        NONE)
    | _ =>  (* a (free or schematic) type variable *)
      NONE
  end;


(* ------------------------------------------------------------------------- *)
(* use 'setup Refute.setup' in an Isabelle theory to initialize the 'Refute' *)
(* structure                                                                 *)
(* ------------------------------------------------------------------------- *)

(* ------------------------------------------------------------------------- *)
(* Note: the interpreters and printers are used in reverse order; however,   *)
(*       an interpreter that can handle non-atomic terms ends up being       *)
(*       applied before the 'stlc_interpreter' breaks the term apart into    *)
(*       subterms that are then passed to other interpreters!                *)
(* ------------------------------------------------------------------------- *)

  (* (theory -> theory) list *)

  val setup =
     RefuteData.init #>
     add_interpreter "stlc"    stlc_interpreter #>
     add_interpreter "Pure"    Pure_interpreter #>
     add_interpreter "HOLogic" HOLogic_interpreter #>
     add_interpreter "set"     set_interpreter #>
     add_interpreter "IDT"             IDT_interpreter #>
     add_interpreter "IDT_constructor" IDT_constructor_interpreter #>
     add_interpreter "IDT_recursion"   IDT_recursion_interpreter #>
     add_interpreter "Finite_Set.card"    Finite_Set_card_interpreter #>
     add_interpreter "Finite_Set.Finites" Finite_Set_Finites_interpreter #>
     add_interpreter "Finite_Set.finite"  Finite_Set_finite_interpreter #>
     add_interpreter "Nat_Orderings.less" Nat_less_interpreter #>
     add_interpreter "Nat_HOL.plus"       Nat_plus_interpreter #>
     add_interpreter "Nat_HOL.minus"      Nat_minus_interpreter #>
     add_interpreter "Nat_HOL.times"      Nat_times_interpreter #>
     add_interpreter "List.op @" List_append_interpreter #>
     add_interpreter "Lfp.lfp" Lfp_lfp_interpreter #>
     add_interpreter "Gfp.gfp" Gfp_gfp_interpreter #>
     add_interpreter "fst" Product_Type_fst_interpreter #>
     add_interpreter "snd" Product_Type_snd_interpreter #>
     add_printer "stlc" stlc_printer #>
     add_printer "set"  set_printer #>
     add_printer "IDT"  IDT_printer;

end  (* structure Refute *)