src/HOL/Tools/refute.ML

renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields

(*  Title:      HOL/Tools/refute.ML
    Author:     Tjark Weber, TU Muenchen

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

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

  val print       : theory -> model -> typ -> interpretation -> (int -> bool) -> 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 list -> term -> bool -> unit

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

  val setup : theory -> theory

(* ------------------------------------------------------------------------- *)
(* Additional functions used by Nitpick (to be factored out)                 *)
(* ------------------------------------------------------------------------- *)

  val close_form : term -> term
  val get_classdef : theory -> string -> (string * term) option
  val norm_rhs : term -> term
  val get_def : theory -> string * typ -> (string * term) option
  val get_typedef : theory -> typ -> (string * term) option
  val is_IDT_constructor : theory -> string * typ -> bool
  val is_IDT_recursor : theory -> string * typ -> bool
  val is_const_of_class: theory -> string * typ -> bool
  val monomorphic_term : Type.tyenv -> term -> term
  val specialize_type : theory -> (string * typ) -> term -> term
  val string_of_typ : typ -> string
  val typ_of_dtyp : Datatype.descr -> (Datatype.dtyp * typ) list -> Datatype.dtyp -> typ
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" and "expect"):                                                    *)
(*                                                                           *)
(* 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.                              *)
(* "no_assms"    bool    If "true", assumptions in structured proofs are     *)
(*                       not considered.                                     *)
(* "expect"      string  Expected result ("genuine", "potential", "none", or *)
(*                       "unknown").                                         *)
(* ------------------------------------------------------------------------- *)

  type params =
    {
      sizes    : (string * int) list,
      minsize  : int,
      maxsize  : int,
      maxvars  : int,
      maxtime  : int,
      satsolver: string,
      no_assms : bool,
      expect   : 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 =
    (typ * int) list * (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 RefuteData = Theory_Data
  (
    type T =
      {interpreters: (string * (theory -> model -> arguments -> term ->
        (interpretation * model * arguments) option)) list,
       printers: (string * (theory -> model -> typ -> interpretation ->
        (int -> bool) -> term option)) list,
       parameters: string Symtab.table};
    val empty = {interpreters = [], printers = [], parameters = Symtab.empty};
    val extend = I;
    fun merge
      ({interpreters = in1, printers = pr1, parameters = pa1},
       {interpreters = in2, printers = pr2, parameters = pa2}) : T =
      {interpreters = AList.merge (op =) (K true) (in1, in2),
       printers = AList.merge (op =) (K true) (pr1, pr2),
       parameters = Symtab.merge (op=) (pa1, pa2)};
  );


(* ------------------------------------------------------------------------- *)
(* 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 (Syntax.string_of_term_global thy t))
    | SOME x => x;

(* ------------------------------------------------------------------------- *)
(* print: converts the interpretation 'intr', which must denote a term of    *)
(*        type 'T', into a term using a suitable printer                     *)
(* ------------------------------------------------------------------------- *)

  (* theory -> model -> Term.typ -> 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 type " ^ quote (Syntax.string_of_typ_global 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) =>
          Syntax.string_of_typ_global 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
        cat_lines (map_filter (fn (t, intr) =>
          (* print constants only if 'show_consts' is true *)
          if (!show_consts) orelse not (is_Const t) then
            SOME (Syntax.string_of_term_global thy t ^ ": " ^
              Syntax.string_of_term_global thy
                (print thy model (Term.type_of 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.typ -> 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) = RefuteData.map 
    (fn {interpreters, printers, parameters} =>
      {interpreters = interpreters, printers = printers,
        parameters = Symtab.update (name, value) parameters});

(* ------------------------------------------------------------------------- *)
(* 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 -> bool *)
    fun read_bool (parms, name) =
      case AList.lookup (op =) parms name of
        SOME "true" => true
      | SOME "false" => false
      | SOME s => error ("parameter " ^ quote name ^
        " (value is " ^ quote s ^ ") must be \"true\" or \"false\"")
      | NONE   => error ("parameter " ^ quote name ^
          " must be assigned a value")
    (* (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")
    val no_assms = read_bool (allparams, "no_assms")
    val expect = the_default "" (AList.lookup (op =) allparams "expect")
    (* 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     = map_filter
      (fn (name, value) => Option.map (pair name) (Int.fromString value))
      (filter (fn (name, _) => name<>"minsize" andalso name<>"maxsize"
        andalso name<>"maxvars" andalso name<>"maxtime"
        andalso name<>"satsolver" andalso name<>"no_assms") allparams)
  in
    {sizes=sizes, minsize=minsize, maxsize=maxsize, maxvars=maxvars,
      maxtime=maxtime, satsolver=satsolver, no_assms=no_assms, expect=expect}
  end;


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

  fun typ_of_dtyp descr typ_assoc (Datatype_Aux.DtTFree a) =
    (* replace a 'DtTFree' variable by the associated type *)
    the (AList.lookup (op =) typ_assoc (Datatype_Aux.DtTFree a))
    | typ_of_dtyp descr typ_assoc (Datatype_Aux.DtType (s, ds)) =
    Type (s, map (typ_of_dtyp descr typ_assoc) ds)
    | typ_of_dtyp descr typ_assoc (Datatype_Aux.DtRec i) =
    let
      val (s, ds, _) = the (AList.lookup (op =) 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 (OldTerm.term_vars t))
  in
    fold (fn ((x, i), T) => fn t' =>
      Term.all T $ Abs (x, T, abstract_over (Var ((x, i), T), t'))) vars t
  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 " ^ Syntax.string_of_term_global Pure.thy 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 Datatype.get_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)
      (Datatype.get_all 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;

(* ------------------------------------------------------------------------- *)
(* norm_rhs: 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

(* ------------------------------------------------------------------------- *)
(* 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
    (* (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 -> 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 (@{const_name all}, _) => t
      | Const (@{const_name "=="}, _) => t
      | Const (@{const_name "==>"}, _) => t
      | Const (@{const_name TYPE}, _) => t  (* axiomatic type classes *)
      (* HOL *)
      | Const (@{const_name Trueprop}, _) => t
      | Const (@{const_name Not}, _) => t
      | (* redundant, since 'True' is also an IDT constructor *)
        Const (@{const_name True}, _) => t
      | (* redundant, since 'False' is also an IDT constructor *)
        Const (@{const_name False}, _) => t
      | Const (@{const_name undefined}, _) => t
      | Const (@{const_name The}, _) => t
      | Const (@{const_name Hilbert_Choice.Eps}, _) => t
      | Const (@{const_name All}, _) => t
      | Const (@{const_name Ex}, _) => t
      | Const (@{const_name "op ="}, _) => t
      | Const (@{const_name "op &"}, _) => t
      | Const (@{const_name "op |"}, _) => t
      | Const (@{const_name "op -->"}, _) => t
      (* sets *)
      | Const (@{const_name Collect}, _) => t
      | Const (@{const_name "op :"}, _) => t
      (* other optimizations *)
      | Const (@{const_name Finite_Set.card}, _) => t
      | Const (@{const_name Finite_Set.finite}, _) => t
      | Const (@{const_name Algebras.less}, Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("bool", [])])])) => t
      | Const (@{const_name Algebras.plus}, Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => t
      | Const (@{const_name Algebras.minus}, Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => t
      | Const (@{const_name Algebras.times}, Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => t
      | Const (@{const_name List.append}, _) => t
      | Const (@{const_name lfp}, _) => t
      | Const (@{const_name gfp}, _) => t
      | Const (@{const_name fst}, _) => t
      | Const (@{const_name 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.  *)
          ((*tracing (" 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 _ = tracing "Adding axioms..."
    val axioms = Theory.all_axioms_of thy
    fun collect_this_axiom (axname, ax) axs =
      let
        val ax' = unfold_defs thy ax
      in
        if member (op aconv) axs ax' then axs
        else (tracing axname; collect_term_axioms ax' (ax' :: axs))
      end
    and collect_sort_axioms T axs =
      let
        val sort =
          (case T of
            TFree (_, sort) => sort
          | TVar (_, sort)  => sort
          | _ => raise REFUTE ("collect_axioms",
              "type " ^ Syntax.string_of_typ_global 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_info 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.add_tvars ax [] of
            [] => (axname, ax)
          | [(idx, S)] => (axname, monomorphic_term (Vartab.make [(idx, (S, T))]) ax)
          | _ =>
            raise REFUTE ("collect_axioms", "class axiom " ^ axname ^ " (" ^
              Syntax.string_of_term_global thy ax ^
              ") contains more than one type variable")))
          class_axioms
      in
        fold collect_this_axiom monomorphic_class_axioms axs
      end
    and collect_type_axioms T axs =
      case T of
      (* simple types *)
        Type ("prop", []) => axs
      | Type ("fun", [T1, T2]) => collect_type_axioms T2 (collect_type_axioms T1 axs)
      (* axiomatic type classes *)
      | Type ("itself", [T1]) => collect_type_axioms T1 axs
      | Type (s, Ts) =>
        (case Datatype.get_info thy s of
          SOME info =>  (* inductive datatype *)
            (* only collect relevant type axioms for the argument types *)
            fold collect_type_axioms Ts axs
        | 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 *)
            fold collect_type_axioms Ts axs))
      (* axiomatic type classes *)
      | TFree _ => collect_sort_axioms T axs
      (* axiomatic type classes *)
      | TVar _ => collect_sort_axioms T axs
    and collect_term_axioms t axs =
      case t of
      (* Pure *)
        Const (@{const_name all}, _) => axs
      | Const (@{const_name "=="}, _) => axs
      | Const (@{const_name "==>"}, _) => axs
      (* axiomatic type classes *)
      | Const (@{const_name TYPE}, T) => collect_type_axioms T axs
      (* HOL *)
      | Const (@{const_name Trueprop}, _) => axs
      | Const (@{const_name Not}, _) => axs
      (* redundant, since 'True' is also an IDT constructor *)
      | Const (@{const_name True}, _) => axs
      (* redundant, since 'False' is also an IDT constructor *)
      | Const (@{const_name False}, _) => axs
      | Const (@{const_name undefined}, T) => collect_type_axioms T axs
      | Const (@{const_name The}, T) =>
        let
          val ax = specialize_type thy (@{const_name The}, T)
            (the (AList.lookup (op =) axioms "HOL.the_eq_trivial"))
        in
          collect_this_axiom ("HOL.the_eq_trivial", ax) axs
        end
      | Const (@{const_name Hilbert_Choice.Eps}, T) =>
        let
          val ax = specialize_type thy (@{const_name Hilbert_Choice.Eps}, T)
            (the (AList.lookup (op =) axioms "Hilbert_Choice.someI"))
        in
          collect_this_axiom ("Hilbert_Choice.someI", ax) axs
        end
      | Const (@{const_name All}, T) => collect_type_axioms T axs
      | Const (@{const_name Ex}, T) => collect_type_axioms T axs
      | Const (@{const_name "op ="}, T) => collect_type_axioms T axs
      | Const (@{const_name "op &"}, _) => axs
      | Const (@{const_name "op |"}, _) => axs
      | Const (@{const_name "op -->"}, _) => axs
      (* sets *)
      | Const (@{const_name Collect}, T) => collect_type_axioms T axs
      | Const (@{const_name "op :"}, T) => collect_type_axioms T axs
      (* other optimizations *)
      | Const (@{const_name Finite_Set.card}, T) => collect_type_axioms T axs
      | Const (@{const_name Finite_Set.finite}, T) =>
        collect_type_axioms T axs
      | Const (@{const_name Algebras.less}, T as Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("bool", [])])])) =>
          collect_type_axioms T axs
      | Const (@{const_name Algebras.plus}, T as Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
          collect_type_axioms T axs
      | Const (@{const_name Algebras.minus}, T as Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
          collect_type_axioms T axs
      | Const (@{const_name Algebras.times}, T as Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
          collect_type_axioms T axs
      | Const (@{const_name List.append}, T) => collect_type_axioms T axs
      | Const (@{const_name lfp}, T) => collect_type_axioms T axs
      | Const (@{const_name gfp}, T) => collect_type_axioms T axs
      | Const (@{const_name fst}, T) => collect_type_axioms T axs
      | Const (@{const_name snd}, T) => collect_type_axioms T axs
      (* 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 of_class = Logic.mk_of_class (TVar (("'a", 0), [class]), class)
              val ax_in = SOME (specialize_type thy (s, T) of_class)
                (* type match may fail due to sort constraints *)
                handle Type.TYPE_MATCH => NONE
              val ax_1 = Option.map (fn ax => (Syntax.string_of_term_global thy ax, ax)) ax_in
              val ax_2 = Option.map (apsnd (specialize_type thy (s, T))) (get_classdef thy class)
            in
              collect_type_axioms T (fold collect_this_axiom (map_filter I [ax_1, ax_2]) axs)
            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 T axs
          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 T axs
      | Free (_, T) => collect_type_axioms T axs
      | Var (_, T) => collect_type_axioms T axs
      | Bound _ => axs
      | Abs (_, T, body) => collect_term_axioms body (collect_type_axioms T axs)
      | t1 $ t2 => collect_term_axioms t2 (collect_term_axioms t1 axs)
    val result = map close_form (collect_term_axioms t [])
    val _ = tracing " ...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 *)
(*               and all mutually recursive IDTs are considered.             *)
(* ------------------------------------------------------------------------- *)

  fun ground_types thy t =
  let
    fun collect_types T acc =
      (case T of
        Type ("fun", [T1, T2]) => collect_types T1 (collect_types T2 acc)
      | Type ("prop", [])      => acc
      | Type (s, Ts)           =>
        (case Datatype.get_info thy s of
          SOME info =>  (* inductive datatype *)
          let
            val index        = #index info
            val descr        = #descr info
            val (_, typs, _) = the (AList.lookup (op =) descr index)
            val typ_assoc    = typs ~~ Ts
            (* sanity check: every element in 'dtyps' must be a *)
            (* 'DtTFree'                                        *)
            val _ = if Library.exists (fn d =>
              case d of Datatype_Aux.DtTFree _ => false | _ => true) typs then
              raise REFUTE ("ground_types", "datatype argument (for type "
                ^ Syntax.string_of_typ_global thy T ^ ") is not a variable")
            else ()
            (* required for mutually recursive datatypes; those need to   *)
            (* be added even if they are an instance of an otherwise non- *)
            (* recursive datatype                                         *)
            fun collect_dtyp d acc =
            let
              val dT = typ_of_dtyp descr typ_assoc d
            in
              case d of
                Datatype_Aux.DtTFree _ =>
                collect_types dT acc
              | Datatype_Aux.DtType (_, ds) =>
                collect_types dT (fold_rev collect_dtyp ds acc)
              | Datatype_Aux.DtRec i =>
                if dT mem acc then
                  acc  (* prevent infinite recursion *)
                else
                  let
                    val (_, dtyps, dconstrs) = the (AList.lookup (op =) descr i)
                    (* if the current type is a recursive IDT (i.e. a depth *)
                    (* is required), add it to 'acc'                        *)
                    val acc_dT = if Library.exists (fn (_, ds) =>
                      Library.exists Datatype_Aux.is_rec_type ds) dconstrs then
                        insert (op =) dT acc
                      else acc
                    (* collect argument types *)
                    val acc_dtyps = fold_rev collect_dtyp dtyps acc_dT
                    (* collect constructor types *)
                    val acc_dconstrs = fold_rev collect_dtyp (maps snd dconstrs) acc_dtyps
                  in
                    acc_dconstrs
                  end
            end
          in
            (* argument types 'Ts' could be added here, but they are also *)
            (* added by 'collect_dtyp' automatically                      *)
            collect_dtyp (Datatype_Aux.DtRec index) acc
          end
        | NONE =>
          (* not an inductive datatype, e.g. defined via "typedef" or *)
          (* "typedecl"                                               *)
          insert (op =) T (fold collect_types Ts acc))
      | TFree _                => insert (op =) T acc
      | TVar _                 => insert (op =) T acc)
  in
    fold_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 (the (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 = filter_out (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 (fst (fold_map (fn (T, _) => fn ds =>
        case AList.lookup (op =) sizes (string_of_typ T) of
        (* return the fixed size *)
          SOME n => ((T, n), ds)
        (* consume the head of 'ds', add 'minsize' *)
        | NONE   => ((T, minsize + hd ds), tl ds))
        xs diffs'))
    | 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      *)
(* assm_ts   : assumptions to be considered unless "no_assms" is specified   *)
(* 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,
    no_assms, expect} assm_ts t negate =
  let
    (* string -> unit *)
    fun check_expect outcome_code =
      if expect = "" orelse outcome_code = expect then ()
      else error ("Unexpected outcome: " ^ quote outcome_code ^ ".")
    (* unit -> unit *)
    fun wrapper () =
    let
      val timer  = Timer.startRealTimer ()
      val t = if no_assms then t
              else if negate then Logic.list_implies (assm_ts, t)
              else Logic.mk_conjunction_list (t :: assm_ts)
      val u      = unfold_defs thy t
      val _      = tracing ("Unfolded term: " ^ Syntax.string_of_term_global thy u)
      val axioms = collect_axioms thy u
      (* Term.typ list *)
      val types = fold (union (op =) o ground_types thy) (u :: axioms) []
      val _     = tracing ("Ground types: "
        ^ (if null types then "none."
           else commas (map (Syntax.string_of_typ_global 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 maybe_spurious = Library.exists (fn
          Type (s, _) =>
          (case Datatype.get_info thy s of
            SOME info =>  (* inductive datatype *)
            let
              val index           = #index info
              val descr           = #descr info
              val (_, _, constrs) = the (AList.lookup (op =) descr index)
            in
              (* recursive datatype? *)
              Library.exists (fn (_, ds) =>
                Library.exists Datatype_Aux.is_rec_type ds) constrs
            end
          | NONE => false)
        | _ => false) types
      val _ = if maybe_spurious then
          warning ("Term contains a recursive datatype; "
            ^ "countermodel(s) may be spurious!")
        else
          ()
      (* (Term.typ * int) list -> string *)
      fun find_model_loop universe =
      let
        val msecs_spent = Time.toMilliseconds (Timer.checkRealTimer timer)
        val _ = maxtime = 0 orelse msecs_spent < 1000 * maxtime
                orelse raise TimeLimit.TimeOut
        val init_model = (universe, [])
        val init_args  = {maxvars = maxvars, def_eq = false, next_idx = 1,
          bounds = [], wellformed = True}
        val _ = tracing ("Translating term (sizes: "
          ^ commas (map (fn (_, n) => string_of_int n) universe) ^ ") ...")
        (* translate 'u' and all axioms *)
        val (intrs, (model, args)) = fold_map (fn t' => fn (m, a) =>
          let
            val (i, m', a') = interpret thy m a t'
          in
            (* set 'def_eq' to 'true' *)
            (i, (m', {maxvars = #maxvars a', def_eq = true,
              next_idx = #next_idx a', bounds = #bounds a',
              wellformed = #wellformed a'}))
          end) (u :: axioms) (init_model, init_args)
        (* 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]
        val solver =
          SatSolver.invoke_solver satsolver
          handle Option.Option =>
                 error ("Unknown SAT solver: " ^ quote satsolver ^
                        ". Available solvers: " ^
                        commas (map (quote o fst) (!SatSolver.solvers)) ^ ".")
      in
        priority "Invoking SAT solver...";
        (case solver fm of
          SatSolver.SATISFIABLE assignment =>
          (priority ("*** Model found: ***\n" ^ print_model thy model
            (fn i => case assignment i of SOME b => b | NONE => true));
           if maybe_spurious then "potential" else "genuine")
        | SatSolver.UNSATISFIABLE _ =>
          (priority "No model exists.";
          case next_universe universe sizes minsize maxsize of
            SOME universe' => find_model_loop universe'
          | NONE           => (priority
            "Search terminated, no larger universe within the given limits.";
            "none"))
        | SatSolver.UNKNOWN =>
          (priority "No model found.";
          case next_universe universe sizes minsize maxsize of
            SOME universe' => find_model_loop universe'
          | NONE           => (priority
            "Search terminated, no larger universe within the given limits.";
            "unknown"))
        ) handle SatSolver.NOT_CONFIGURED =>
          (error ("SAT solver " ^ quote satsolver ^ " is not configured.");
           "unknown")
      end handle MAXVARS_EXCEEDED =>
        (priority ("Search terminated, number of Boolean variables ("
          ^ string_of_int maxvars ^ " allowed) exceeded.");
          "unknown")
        val outcome_code = find_model_loop (first_universe types sizes minsize)
      in
        check_expect outcome_code
      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 *)
      priority ("Trying to find a model that "
        ^ (if negate then "refutes" else "satisfies") ^ ": "
        ^ Syntax.string_of_term_global thy t);
      if maxtime>0 then (
        TimeLimit.timeLimit (Time.fromSeconds maxtime)
          wrapper ()
        handle TimeLimit.TimeOut =>
          (priority ("Search terminated, time limit (" ^
              string_of_int maxtime
              ^ (if maxtime=1 then " second" else " seconds") ^ ") exceeded.");
           check_expect "unknown")
      ) 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 list -> Term.term -> unit *)

  fun satisfy_term thy params assm_ts t =
    find_model thy (actual_params thy params) assm_ts 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 list -> Term.term -> unit *)

  fun refute_term thy params assm_ts 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.add_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 (OldTerm.term_vars t))
    (* Term.term *)
    val ex_closure = fold (fn ((x, i), T) => fn t' =>
      HOLogic.exists_const T $
        Abs (x, T, abstract_over (Var ((x, i), T), t'))) vars t
    (* 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 (@{const_name all}, _) $ Abs (_, _, t)) =
        strip_all_body t
      | strip_all_body (Const (@{const_name Trueprop}, _) $ t) =
        strip_all_body t
      | strip_all_body (Const (@{const_name 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 (@{const_name all}, _) $ Abs (a, T, t)) =
      (a, T) :: strip_all_vars t
      | strip_all_vars (Const (@{const_name Trueprop}, _) $ t) =
      strip_all_vars t
      | strip_all_vars (Const (@{const_name 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) assm_ts subst_t true
    handle REFUTE (s, s') => error ("REFUTE " ^ s ^ " " ^ s') (* ### *)
  end;

(* ------------------------------------------------------------------------- *)
(* refute_goal                                                               *)
(* ------------------------------------------------------------------------- *)

  fun refute_goal ctxt params th i =
  let
    val t = th |> prop_of
  in
    if Logic.count_prems t = 0 then
      priority "No subgoal!"
    else
      let
        val assms = map term_of (Assumption.all_assms_of ctxt)
        val (t, frees) = Logic.goal_params t i
      in
        refute_term (ProofContext.theory_of ctxt) params assms
        (subst_bounds (frees, t))
      end
  end


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

(* ------------------------------------------------------------------------- *)
(* make_constants: returns all interpretations for type 'T' that consist of  *)
(*                 unit vectors with 'True'/'False' only (no Boolean         *)
(*                 variables)                                                *)
(* ------------------------------------------------------------------------- *)

  (* theory -> model -> Term.typ -> interpretation list *)

  fun make_constants thy model T =
  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 *)
      fun unit_vectors_loop k =
        if k>n then [] else unit_vector (k,n) :: unit_vectors_loop (k+1)
    in
      unit_vectors_loop 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
        maps (fn x => map (cons x) rec_pick) xs
      end
    (* returns all constant interpretations that have the same tree *)
    (* structure as the interpretation argument                     *)
    (* interpretation -> interpretation list *)
    fun make_constants_intr (Leaf xs) = unit_vectors (length xs)
      | make_constants_intr (Node xs) = map Node (pick_all (length xs)
      (make_constants_intr (hd xs)))
    (* obtain the interpretation for a variable of type 'T' *)
    val (i, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1,
      bounds=[], wellformed=True} (Free ("dummy", T))
  in
    make_constants_intr i
  end;

(* ------------------------------------------------------------------------- *)
(* power: '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;

(* ------------------------------------------------------------------------- *)
(* size_of_type: returns the number of elements in a type 'T' (i.e. 'length  *)
(*               (make_constants T)', but implemented more efficiently)      *)
(* ------------------------------------------------------------------------- *)

  (* theory -> model -> Term.typ -> int *)

  (* returns 0 for an empty ground type or a function type with empty      *)
  (* codomain, but fails for a function type with empty domain --          *)
  (* admissibility of datatype constructor argument types (see "Inductive  *)
  (* datatypes in HOL - lessons learned ...", S. Berghofer, M. Wenzel,     *)
  (* TPHOLs 99) ensures that recursive, possibly empty, datatype fragments *)
  (* never occur as the domain of a function type that is the type of a    *)
  (* constructor argument                                                  *)

  fun size_of_type thy model T =
  let
    (* returns the number of elements that have the same tree structure as a *)
    (* given interpretation                                                  *)
    fun size_of_intr (Leaf xs) = length xs
      | size_of_intr (Node xs) = power (size_of_intr (hd xs), length xs)
    (* obtain the interpretation for a variable of type 'T' *)
    val (i, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1,
      bounds=[], wellformed=True} (Free ("dummy", T))
  in
    size_of_intr i
  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
        maps (fn x => map (cons x) rec_pick) 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
    fold_rev (fn T => fn term => Abs ("<eta_expand>", T, term))
      (List.take (Ts, i))
      (Term.list_comb (t', map Bound (i-1 downto 0)))
  end;

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

  fun size_of_dtyp thy typ_sizes descr typ_assoc constructors =
    Integer.sum (map (fn (_, dtyps) =>
      Integer.prod (map (size_of_type thy (typ_sizes, []) o
        (typ_of_dtyp descr typ_assoc)) 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 the (AList.lookup (op =) 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 ... T1' different copies of the        *)
          (* interpretation for 'T2', which are then combined into a single *)
          (* new interpretation                                             *)
          (* 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 thy model T1)
          (* 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 constants = make_constants thy model T
          (* interpret the 'body' separately for each constant *)
          val (bodies, (model', args')) = fold_map
            (fn c => fn (m, a) =>
              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'                                   *)
                (i', (m', {maxvars = maxvars, def_eq = def_eq,
                  next_idx = #next_idx a', bounds = bounds,
                  wellformed = #wellformed a'}))
              end)
            constants (model, args)
        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 (@{const_name 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 (@{const_name all}, _) =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const (@{const_name "=="}, _) $ 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 (@{const_name "=="}, _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const (@{const_name "=="}, _) =>
      SOME (interpret thy model args (eta_expand t 2))
    | Const (@{const_name "==>"}, _) $ 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 (@{const_name "==>"}, _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const (@{const_name "==>"}, _) =>
      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 (@{const_name Trueprop}, _) =>
      SOME (Node [TT, FF], model, args)
    | Const (@{const_name Not}, _) =>
      SOME (Node [FF, TT], model, args)
    (* redundant, since 'True' is also an IDT constructor *)
    | Const (@{const_name True}, _) =>
      SOME (TT, model, args)
    (* redundant, since 'False' is also an IDT constructor *)
    | Const (@{const_name False}, _) =>
      SOME (FF, model, args)
    | Const (@{const_name 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 (@{const_name All}, _) =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const (@{const_name 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 (@{const_name Ex}, _) =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const (@{const_name "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 (@{const_name "op ="}, _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const (@{const_name "op ="}, _) =>
      SOME (interpret thy model args (eta_expand t 2))
    | Const (@{const_name "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 (@{const_name "op &"}, _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const (@{const_name "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 (@{const_name "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 (@{const_name "op |"}, _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const (@{const_name "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 (@{const_name "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 (@{const_name "op -->"}, _) $ t1 =>
      SOME (interpret thy model args (eta_expand t 1))
    | Const (@{const_name "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 *)

  (* interprets variables and constants whose type is an IDT (this is        *)
  (* relatively easy and merely requires us to compute the size of the 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 Datatype.get_info 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))
          (* sanity check: depth must be at least 0 *)
          val _ = (case depth of SOME n =>
            if n<0 then
              raise REFUTE ("IDT_interpreter", "negative depth")
            else ()
            | _ => ())
        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) = the (AList.lookup (op =) 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 Datatype_Aux.DtTFree _ => false | _ => true) dtyps
                then
                  raise REFUTE ("IDT_interpreter",
                    "datatype argument (for type "
                    ^ Syntax.string_of_typ_global 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 *)

  (* This function imposes an order on the elements of a datatype fragment  *)
  (* as follows: C_i x_1 ... x_n < C_j y_1 ... y_m iff i < j or             *)
  (* (x_1, ..., x_n) < (y_1, ..., y_m).  With this order, a constructor is  *)
  (* a function C_i that maps some argument indices x_1, ..., x_n to the    *)
  (* datatype element given by index C_i x_1 ... x_n.  The idea remains the *)
  (* same for recursive datatypes, although the computation of indices gets *)
  (* a little tricky.                                                       *)

  fun IDT_constructor_interpreter thy model args t =
  let
    (* returns a list of canonical representations for terms of the type 'T' *)
    (* It would be nice if we could just use 'print' for this, but 'print'   *)
    (* for IDTs calls 'IDT_constructor_interpreter' again, and this could    *)
    (* lead to infinite recursion when we have (mutually) recursive IDTs.    *)
    (* (Term.typ * int) list -> Term.typ -> Term.term list *)
    fun canonical_terms typs T =
      (case T of
        Type ("fun", [T1, T2]) =>
        (* 'T2' might contain a recursive IDT, so we cannot use 'print' (at *)
        (* least not for 'T2'                                               *)
        let
          (* 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
              maps (fn x => map (cons x) rec_pick) xs
            end
          (* ["x1", ..., "xn"] *)
          val terms1 = canonical_terms typs T1
          (* ["y1", ..., "ym"] *)
          val terms2 = canonical_terms typs T2
          (* [[("x1", "y1"), ..., ("xn", "y1")], ..., *)
          (*   [("x1", "ym"), ..., ("xn", "ym")]]     *)
          val functions = map (curry (op ~~) terms1)
            (pick_all (length terms1) terms2)
          (* [["(x1, y1)", ..., "(xn, y1)"], ..., *)
          (*   ["(x1, ym)", ..., "(xn, ym)"]]     *)
          val pairss = map (map HOLogic.mk_prod) functions
          (* Term.typ *)
          val HOLogic_prodT = HOLogic.mk_prodT (T1, T2)
          val HOLogic_setT  = HOLogic.mk_setT HOLogic_prodT
          (* Term.term *)
          val HOLogic_empty_set = HOLogic.mk_set HOLogic_prodT []
          val HOLogic_insert    =
            Const (@{const_name insert}, HOLogic_prodT --> HOLogic_setT --> HOLogic_setT)
        in
          (* functions as graphs, i.e. as a (HOL) set of pairs "(x, y)" *)
          map (fn ps => fold_rev (fn pair => fn acc => HOLogic_insert $ pair $ acc) ps
            HOLogic_empty_set) pairss
        end
      | Type (s, Ts) =>
        (case Datatype.get_info thy s of
          SOME info =>
          (case AList.lookup (op =) typs T of
            SOME 0 =>
            (* termination condition to avoid infinite recursion *)
            []  (* at depth 0, every IDT is empty *)
          | _ =>
            let
              val index               = #index info
              val descr               = #descr info
              val (_, dtyps, constrs) = the (AList.lookup (op =) 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 Datatype_Aux.DtTFree _ => false | _ => true) dtyps
                then
                  raise REFUTE ("IDT_constructor_interpreter",
                    "datatype argument (for type "
                    ^ Syntax.string_of_typ_global thy T
                    ^ ") is not a variable")
                else ()
              (* decrement depth for the IDT 'T' *)
              val typs' = (case AList.lookup (op =) typs T of NONE => typs
                | SOME n => AList.update (op =) (T, n-1) typs)
              fun constructor_terms terms [] = terms
                | constructor_terms terms (d::ds) =
                let
                  val dT = typ_of_dtyp descr typ_assoc d
                  val d_terms = canonical_terms typs' dT
                in
                  (* C_i x_1 ... x_n < C_i y_1 ... y_n if *)
                  (* (x_1, ..., x_n) < (y_1, ..., y_n)    *)
                  constructor_terms
                    (map_product (curry op $) terms d_terms) ds
                end
            in
              (* C_i ... < C_j ... if i < j *)
              maps (fn (cname, ctyps) =>
                let
                  val cTerm = Const (cname,
                    map (typ_of_dtyp descr typ_assoc) ctyps ---> T)
                in
                  constructor_terms [cTerm] ctyps
                end) constrs
            end)
        | NONE =>
          (* not an inductive datatype; in this case the argument types in *)
          (* 'Ts' may not be IDTs either, so 'print' should be safe        *)
          map (fn intr => print thy (typs, []) T intr (K false))
            (make_constants thy (typs, []) T))
      | _ =>  (* TFree ..., TVar ... *)
        map (fn intr => print thy (typs, []) T intr (K false))
          (make_constants thy (typs, []) T))
    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 Datatype.get_info thy s' of
            SOME info =>  (* body type is an inductive datatype *)
            let
              val index               = #index info
              val descr               = #descr info
              val (_, dtyps, constrs) = the (AList.lookup (op =) 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 Datatype_Aux.DtTFree _ => false | _ => true) dtyps
                then
                  raise REFUTE ("IDT_constructor_interpreter",
                    "datatype argument (for type "
                    ^ Syntax.string_of_typ_global 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
                  (* int option -- only /recursive/ IDTs have an associated *)
                  (*               depth                                    *)
                  val depth = AList.lookup (op =) typs (Type (s', Ts'))
                  (* this should never happen: at depth 0, this IDT fragment *)
                  (* is definitely empty, and in this case we don't need to  *)
                  (* interpret its constructors                              *)
                  val _ = (case depth of SOME 0 =>
                      raise REFUTE ("IDT_constructor_interpreter",
                        "depth is 0")
                    | _ => ())
                  val typs' = (case depth of NONE => typs | SOME n =>
                    AList.update (op =) (Type (s', Ts'), n-1) typs)
                  (* elements of the datatype come before elements generated *)
                  (* by 'Const (s, T)' iff they are generated by a           *)
                  (* constructor in constrs1                                 *)
                  val offset = size_of_dtyp thy typs' descr typ_assoc constrs1
                  (* compute the total (current) size of the datatype *)
                  val total = offset +
                    size_of_dtyp thy typs' descr typ_assoc constrs2
                  (* sanity check *)
                  val _ = if total <> size_of_type thy (typs, [])
                    (Type (s', Ts')) then
                      raise REFUTE ("IDT_constructor_interpreter",
                        "total is not equal to current size")
                    else ()
                  (* returns an interpretation where everything is mapped to *)
                  (* an "undefined" element of the datatype                  *)
                  fun make_undef [] =
                    Leaf (replicate total False)
                    | make_undef (d::ds) =
                    let
                      (* compute the current size of the type 'd' *)
                      val dT   = typ_of_dtyp descr typ_assoc d
                      val size = size_of_type thy (typs, []) dT
                    in
                      Node (replicate size (make_undef ds))
                    end
                  (* returns the interpretation for a constructor *)
                  fun make_constr [] offset =
                    if offset < total then
                      (Leaf (replicate offset False @ True ::
                        (replicate (total - offset - 1) False)), offset + 1)
                    else
                      raise REFUTE ("IDT_constructor_interpreter",
                        "offset >= total")
                    | make_constr (d::ds) offset =
                    let
                      (* Term.typ *)
                      val dT = typ_of_dtyp descr typ_assoc d
                      (* compute canonical term representations for all   *)
                      (* elements of the type 'd' (with the reduced depth *)
                      (* for the IDT)                                     *)
                      val terms' = canonical_terms typs' dT
                      (* sanity check *)
                      val _ =
                        if length terms' <> size_of_type thy (typs', []) dT
                        then
                          raise REFUTE ("IDT_constructor_interpreter",
                            "length of terms' is not equal to old size")
                        else ()
                      (* compute canonical term representations for all   *)
                      (* elements of the type 'd' (with the current depth *)
                      (* for the IDT)                                     *)
                      val terms = canonical_terms typs dT
                      (* sanity check *)
                      val _ =
                        if length terms <> size_of_type thy (typs, []) dT
                        then
                          raise REFUTE ("IDT_constructor_interpreter",
                            "length of terms is not equal to current size")
                        else ()
                      (* sanity check *)
                      val _ =
                        if length terms < length terms' then
                          raise REFUTE ("IDT_constructor_interpreter",
                            "current size is less than old size")
                        else ()
                      (* sanity check: every element of terms' must also be *)
                      (*               present in terms                     *)
                      val _ =
                        if List.all (member (op =) terms) terms' then ()
                        else
                          raise REFUTE ("IDT_constructor_interpreter",
                            "element has disappeared")
                      (* sanity check: the order on elements of terms' is    *)
                      (*               the same in terms, for those elements *)
                      val _ =
                        let
                          fun search (x::xs) (y::ys) =
                                if x = y then search xs ys else search (x::xs) ys
                            | search (x::xs) [] =
                                raise REFUTE ("IDT_constructor_interpreter",
                                  "element order not preserved")
                            | search [] _ = ()
                        in  search terms' terms  end
                      (* int * interpretation list *)
                      val (intrs, new_offset) =
                        fold_map (fn t_elem => fn off =>
                          (* if 't_elem' existed at the previous depth,    *)
                          (* proceed recursively, otherwise map the entire *)
                          (* subtree to "undefined"                        *)
                          if t_elem mem terms' then
                            make_constr ds off
                          else
                            (make_undef ds, off))
                        terms offset
                    in
                      (Node intrs, new_offset)
                    end
                in
                  SOME (fst (make_constr ctypes offset), model, args)
                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 one of the (mutually *)
            (* recursive) datatypes given by 'info'                    *)
            let
              (* number of all constructors, including those of different  *)
              (* (mutually recursive) datatypes within the same descriptor *)
              val mconstrs_count =
                Integer.sum (map (fn (_, (_, _, cs)) => length cs) (#descr info))
            in
              if mconstrs_count < length params then
                (* too many actual parameters; for now we'll use the *)
                (* 'stlc_interpreter' to strip off one application   *)
                NONE
              else if mconstrs_count > length params 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 - length params)))
              else  (* mconstrs_count = length params *)
                let
                  (* interpret each parameter separately *)
                  val (p_intrs, (model', args')) = fold_map (fn p => fn (m, a) =>
                    let
                      val (i, m', a') = interpret thy m a p
                    in
                      (i, (m', a'))
                    end) params (model, args)
                  val (typs, _) = model'
                  (* 'index' is /not/ necessarily the index of the IDT that *)
                  (* the recursion operator is associated with, but merely  *)
                  (* the index of some mutually recursive IDT               *)
                  val index         = #index info
                  val descr         = #descr info
                  val (_, dtyps, _) = the (AList.lookup (op =) descr index)
                  (* sanity check: we assume that the order of constructors *)
                  (*               in 'descr' is the same as the order of   *)
                  (*               corresponding parameters, otherwise the  *)
                  (*               association code below won't match the   *)
                  (*               right constructors/parameters; we also   *)
                  (*               assume that the order of recursion       *)
                  (*               operators in '#rec_names info' is the    *)
                  (*               same as the order of corresponding       *)
                  (*               datatypes in 'descr'                     *)
                  val _ = if map fst descr <> (0 upto (length descr - 1)) then
                      raise REFUTE ("IDT_recursion_interpreter",
                        "order of constructors and corresponding parameters/" ^
                          "recursion operators and corresponding datatypes " ^
                          "different?")
                    else ()
                  (* sanity check: every element in 'dtyps' must be a *)
                  (*               'DtTFree'                          *)
                  val _ = if Library.exists (fn d =>
                    case d of Datatype_Aux.DtTFree _ => false
                            | _ => true) dtyps
                    then
                      raise REFUTE ("IDT_recursion_interpreter",
                        "datatype argument is not a variable")
                    else ()
                  (* the type of a recursion operator is *)
                  (* [T1, ..., Tn, IDT] ---> Tresult     *)
                  val IDT = List.nth (binder_types T, mconstrs_count)
                  (* by our assumption on the order of recursion operators *)
                  (* and datatypes, this is the index of the datatype      *)
                  (* corresponding to the given recursion operator         *)
                  val idt_index = find_index (fn s' => s' = s) (#rec_names info)
                  (* mutually recursive types must have the same type   *)
                  (* parameters, unless the mutual recursion comes from *)
                  (* indirect recursion                                 *)
                  fun rec_typ_assoc acc [] =
                    acc
                    | rec_typ_assoc acc ((d, T)::xs) =
                    (case AList.lookup op= acc d of
                      NONE =>
                      (case d of
                        Datatype_Aux.DtTFree _ =>
                        (* add the association, proceed *)
                        rec_typ_assoc ((d, T)::acc) xs
                      | Datatype_Aux.DtType (s, ds) =>
                        let
                          val (s', Ts) = dest_Type T
                        in
                          if s=s' then
                            rec_typ_assoc ((d, T)::acc) ((ds ~~ Ts) @ xs)
                          else
                            raise REFUTE ("IDT_recursion_interpreter",
                              "DtType/Type mismatch")
                        end
                      | Datatype_Aux.DtRec i =>
                        let
                          val (_, ds, _) = the (AList.lookup (op =) descr i)
                          val (_, Ts)    = dest_Type T
                        in
                          rec_typ_assoc ((d, T)::acc) ((ds ~~ Ts) @ xs)
                        end)
                    | SOME T' =>
                      if T=T' then
                        (* ignore the association since it's already *)
                        (* present, proceed                          *)
                        rec_typ_assoc acc xs
                      else
                        raise REFUTE ("IDT_recursion_interpreter",
                          "different type associations for the same dtyp"))
                  val typ_assoc = filter
                    (fn (Datatype_Aux.DtTFree _, _) => true | (_, _) => false)
                    (rec_typ_assoc []
                      (#2 (the (AList.lookup (op =) descr idt_index)) ~~ (snd o dest_Type) IDT))
                  (* sanity check: typ_assoc must associate types to the   *)
                  (*               elements of 'dtyps' (and only to those) *)
                  val _ = if not (eq_set (op =) (dtyps, map fst typ_assoc))
                    then
                      raise REFUTE ("IDT_recursion_interpreter",
                        "type association has extra/missing elements")
                    else ()
                  (* 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
                        (Datatype_Aux.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
                  (* associate constructors with corresponding parameters *)
                  (* (int * (interpretation * interpretation) list) list *)
                  val (mc_p_intrs, p_intrs') = fold_map
                    (fn (idx, c_intrs) => fn p_intrs' =>
                      let
                        val len = length c_intrs
                      in
                        ((idx, c_intrs ~~ List.take (p_intrs', len)),
                          List.drop (p_intrs', len))
                      end) mc_intrs p_intrs
                  (* sanity check: no 'p_intr' may be left afterwards *)
                  val _ = if p_intrs' <> [] then
                      raise REFUTE ("IDT_recursion_interpreter",
                        "more parameter than constructor interpretations")
                    else ()
                  (* The recursion operator, applied to 'mconstrs_count'     *)
                  (* arguments, is a function that maps every element of the *)
                  (* inductive datatype to an element of some result type.   *)
                  (* Recursion operators for mutually recursive IDTs are     *)
                  (* translated simultaneously.                              *)
                  (* Since the order on datatype elements is given by an     *)
                  (* order on constructors (and then by the order on         *)
                  (* argument tuples), we can simply copy corresponding      *)
                  (* subtrees from 'p_intrs', in the order in which they are *)
                  (* given.                                                  *)
                  (* interpretation * interpretation -> interpretation list *)
                  fun ci_pi (Leaf xs, pi) =
                    (* if the constructor does not match the arguments to a *)
                    (* defined element of the IDT, the corresponding value  *)
                    (* of the parameter must be ignored                     *)
                    if List.exists (equal True) xs then [pi] else []
                    | ci_pi (Node xs, Node ys) =
                    maps ci_pi (xs ~~ ys)
                    | ci_pi (Node _, Leaf _) =
                    raise REFUTE ("IDT_recursion_interpreter",
                      "constructor takes more arguments than the " ^
                        "associated parameter")
                  (* (int * interpretation list) list *)
                  val rec_operators = map (fn (idx, c_p_intrs) =>
                    (idx, maps ci_pi c_p_intrs)) mc_p_intrs
                  (* sanity check: every recursion operator must provide as  *)
                  (*               many values as the corresponding datatype *)
                  (*               has elements                              *)
                  val _ = map (fn (idx, intrs) =>
                    let
                      val T = typ_of_dtyp descr typ_assoc
                        (Datatype_Aux.DtRec idx)
                    in
                      if length intrs <> size_of_type thy (typs, []) T then
                        raise REFUTE ("IDT_recursion_interpreter",
                          "wrong number of interpretations for rec. operator")
                      else ()
                    end) rec_operators
                  (* For non-recursive datatypes, we are pretty much done at *)
                  (* this point.  For recursive datatypes however, we still  *)
                  (* need to apply the interpretations in 'rec_operators' to *)
                  (* (recursively obtained) interpretations for recursive    *)
                  (* constructor arguments.  To do so more efficiently, we   *)
                  (* copy 'rec_operators' into arrays first.  Each Boolean   *)
                  (* indicates whether the recursive arguments have been     *)
                  (* considered already.                                     *)
                  (* (int * (bool * interpretation) Array.array) list *)
                  val REC_OPERATORS = map (fn (idx, intrs) =>
                    (idx, Array.fromList (map (pair false) intrs)))
                    rec_operators
                  (* 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 (fn x => x = True) xs = elem then
                      SOME []
                    else
                      NONE
                    | get_args (Node xs) elem =
                    let
                      (* interpretation list * 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 datatype element")
                      | 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, the (AList.lookup (op =) mc_intrs idx))
                    end
                  (* computes one entry in 'REC_OPERATORS', 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 =
                  let
                    val arr          = the (AList.lookup (op =) REC_OPERATORS idx)
                    val (flag, intr) = Array.sub (arr, elem)
                  in
                    if flag then
                      (* simply return the previously computed result *)
                      intr
                    else
                      (* we have to apply 'intr' to interpretations for all *)
                      (* recursive arguments                                *)
                      let
                        (* int * int list *)
                        val (c, args) = get_cargs idx elem
                        (* find the indices of the constructor's /recursive/ *)
                        (* arguments                                         *)
                        val (_, _, constrs) = the (AList.lookup (op =) descr idx)
                        val (_, dtyps)      = List.nth (constrs, c)
                        val rec_dtyps_args  = filter
                          (Datatype_Aux.is_rec_type o fst) (dtyps ~~ args)
                        (* map those indices to interpretations *)
                        val rec_dtyps_intrs = map (fn (dtyp, arg) =>
                          let
                            val dT     = typ_of_dtyp descr typ_assoc dtyp
                            val consts = make_constants thy (typs, []) dT
                            val arg_i  = List.nth (consts, arg)
                          in
                            (dtyp, arg_i)
                          end) rec_dtyps_args
                        (* takes the dtyp and interpretation of an element, *)
                        (* and computes the interpretation for the          *)
                        (* corresponding recursive argument                 *)
                        fun rec_intr (Datatype_Aux.DtRec i) (Leaf xs) =
                          (* recursive argument is "rec_i params elem" *)
                          compute_array_entry i (find_index (fn x => x = True) xs)
                          | rec_intr (Datatype_Aux.DtRec _) (Node _) =
                          raise REFUTE ("IDT_recursion_interpreter",
                            "interpretation for IDT is a node")
                          | rec_intr (Datatype_Aux.DtType ("fun", [dt1, dt2]))
                            (Node xs) =
                          (* recursive argument is something like     *)
                          (* "\<lambda>x::dt1. rec_? params (elem x)" *)
                          Node (map (rec_intr dt2) xs)
                          | rec_intr (Datatype_Aux.DtType ("fun", [_, _]))
                            (Leaf _) =
                          raise REFUTE ("IDT_recursion_interpreter",
                            "interpretation for function dtyp is a leaf")
                          | rec_intr _ _ =
                          (* admissibility ensures that every recursive type *)
                          (* is of the form 'Dt_1 -> ... -> Dt_k ->          *)
                          (* (DtRec i)'                                      *)
                          raise REFUTE ("IDT_recursion_interpreter",
                            "non-recursive codomain in recursive dtyp")
                        (* obtain interpretations for recursive arguments *)
                        (* interpretation list *)
                        val arg_intrs = map (uncurry rec_intr) rec_dtyps_intrs
                        (* apply 'intr' to all recursive arguments *)
                        val result = fold (fn arg_i => fn i =>
                          interpretation_apply (i, arg_i)) arg_intrs intr
                        (* update 'REC_OPERATORS' *)
                        val _ = Array.update (arr, elem, (true, result))
                      in
                        result
                      end
                  end
                  val idt_size = Array.length (the (AList.lookup (op =) REC_OPERATORS idt_index))
                  (* sanity check: the size of 'IDT' should be 'idt_size' *)
                  val _ = if idt_size <> size_of_type thy (typs, []) IDT then
                        raise REFUTE ("IDT_recursion_interpreter",
                          "unexpected size of IDT (wrong type associated?)")
                      else ()
                  (* interpretation *)
                  val rec_op = Node (map_range (compute_array_entry idt_index) idt_size)
                in
                  SOME (rec_op, model', args')
                end
            end
          else
            NONE  (* not a recursion operator of this datatype *)
        ) (Datatype.get_all thy) NONE
    | _ =>  (* head of term is not a constant *)
      NONE;

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

  fun set_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
      (* 'Collect' == identity *)
        Const (@{const_name Collect}, _) $ t1 =>
        SOME (interpret thy model args t1)
      | Const (@{const_name Collect}, _) =>
        SOME (interpret thy model args (eta_expand t 1))
      (* 'op :' == application *)
      | Const (@{const_name "op :"}, _) $ t1 $ t2 =>
        SOME (interpret thy model args (t2 $ t1))
      | Const (@{const_name "op :"}, _) $ t1 =>
        SOME (interpret thy model args (eta_expand t 1))
      | Const (@{const_name "op :"}, _) =>
        SOME (interpret thy model args (eta_expand t 2))
      | _ => NONE)
  end;

  (* 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 (@{const_name Finite_Set.card},
        Type ("fun", [Type ("fun", [T, Type ("bool", [])]),
                      Type ("nat", [])])) =>
      let
        (* interpretation -> int *)
        fun number_of_elements (Node xs) =
            fold (fn x => fn n =>
              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"))
              xs 0
          | number_of_elements (Leaf _) =
          raise REFUTE ("Finite_Set_card_interpreter",
            "interpretation for set type is a leaf")
        val size_of_nat = size_of_type thy model (Type ("nat", []))
        (* takes an interpretation for a set and returns an interpretation *)
        (* for a 'nat' denoting the set's cardinality                      *)
        (* interpretation -> interpretation *)
        fun card i =
          let
            val n = number_of_elements i
          in
            if n<size_of_nat then
              Leaf ((replicate n False) @ True ::
                (replicate (size_of_nat-n-1) False))
            else
              Leaf (replicate size_of_nat False)
          end
        val set_constants =
          make_constants thy model (Type ("fun", [T, Type ("bool", [])]))
      in
        SOME (Node (map card set_constants), 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 (@{const_name Finite_Set.finite},
        Type ("fun", [Type ("fun", [T, Type ("bool", [])]),
                      Type ("bool", [])])) $ _ =>
        (* we only consider finite models anyway, hence EVERY set is *)
        (* "finite"                                                  *)
        SOME (TT, model, args)
    | Const (@{const_name Finite_Set.finite},
        Type ("fun", [Type ("fun", [T, Type ("bool", [])]),
                      Type ("bool", [])])) =>
      let
        val size_of_set =
          size_of_type thy model (Type ("fun", [T, Type ("bool", [])]))
      in
        (* we only consider finite models anyway, hence EVERY set is *)
        (* "finite"                                                  *)
        SOME (Node (replicate size_of_set TT), model, args)
      end
    | _ =>
      NONE;

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

  (* only an optimization: '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 (@{const_name Algebras.less}, Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("bool", [])])])) =>
      let
        val size_of_nat = size_of_type thy model (Type ("nat", []))
        (* the 'n'-th nat is not less than the first 'n' nats, while it *)
        (* is less than the remaining 'size_of_nat - n' nats            *)
        (* int -> interpretation *)
        fun less n = Node ((replicate n FF) @ (replicate (size_of_nat - n) TT))
      in
        SOME (Node (map less (1 upto size_of_nat)), model, args)
      end
    | _ =>
      NONE;

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

  (* only an optimization: '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 (@{const_name Algebras.plus}, Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
      let
        val size_of_nat = size_of_type thy model (Type ("nat", []))
        (* int -> int -> interpretation *)
        fun plus m n =
          let
            val element = m + n
          in
            if element > size_of_nat - 1 then
              Leaf (replicate size_of_nat False)
            else
              Leaf ((replicate element False) @ True ::
                (replicate (size_of_nat - element - 1) False))
          end
      in
        SOME (Node (map_range (fn m => Node (map_range (plus m) size_of_nat)) size_of_nat),
          model, args)
      end
    | _ =>
      NONE;

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

  (* only an optimization: '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 (@{const_name Algebras.minus}, Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
      let
        val size_of_nat = size_of_type thy model (Type ("nat", []))
        (* int -> int -> interpretation *)
        fun minus m n =
          let
            val element = Int.max (m-n, 0)
          in
            Leaf ((replicate element False) @ True ::
              (replicate (size_of_nat - element - 1) False))
          end
      in
        SOME (Node (map_range (fn m => Node (map_range (minus m) size_of_nat)) size_of_nat),
          model, args)
      end
    | _ =>
      NONE;

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

  (* only an optimization: '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 (@{const_name Algebras.times}, Type ("fun", [Type ("nat", []),
        Type ("fun", [Type ("nat", []), Type ("nat", [])])])) =>
      let
        val size_of_nat = size_of_type thy model (Type ("nat", []))
        (* nat -> nat -> interpretation *)
        fun mult m n =
          let
            val element = m * n
          in
            if element > size_of_nat - 1 then
              Leaf (replicate size_of_nat False)
            else
              Leaf ((replicate element False) @ True ::
                (replicate (size_of_nat - element - 1) False))
          end
      in
        SOME (Node (map_range (fn m => Node (map_range (mult m) size_of_nat)) size_of_nat),
          model, args)
      end
    | _ =>
      NONE;

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

  (* only an optimization: 'append' 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 (@{const_name List.append}, Type ("fun", [Type ("List.list", [T]), Type ("fun",
        [Type ("List.list", [_]), Type ("List.list", [_])])])) =>
      let
        val size_elem   = size_of_type thy model T
        val size_list   = size_of_type thy model (Type ("List.list", [T]))
        (* maximal length of lists; 0 if we only consider the empty list *)
        val list_length = let
            (* int -> int -> int -> int *)
            fun list_length_acc len lists total =
              if lists = total then
                len
              else if lists < total then
                list_length_acc (len+1) (lists*size_elem) (total-lists)
              else
                raise REFUTE ("List_append_interpreter",
                  "size_list not equal to 1 + size_elem + ... + " ^
                    "size_elem^len, for some len")
          in
            list_length_acc 0 1 size_list
          end
        val elements = 0 upto (size_list-1)
        (* FIXME: there should be a nice formula, which computes the same as *)
        (*        the following, but without all this intermediate tree      *)
        (*        length/offset stuff                                        *)
        (* associate each list with its length and offset in a complete tree *)
        (* of width 'size_elem' and depth 'length_list' (with 'size_list'    *)
        (* nodes total)                                                      *)
        (* (int * (int * int)) list *)
        val (lenoff_lists, _) = fold_map (fn elem => fn (offsets, off) =>
          (* corresponds to a pre-order traversal of the tree *)
          let
            val len = length offsets
            (* associate the given element with len/off *)
            val assoc = (elem, (len, off))
          in
            if len < list_length then
              (* go to first child node *)
              (assoc, (off :: offsets, off * size_elem))
            else if off mod size_elem < size_elem - 1 then
              (* go to next sibling node *)
              (assoc, (offsets, off + 1))
            else
              (* go back up the stack until we find a level where we can go *)
              (* to the next sibling node                                   *)
              let
                val offsets' = dropwhile
                  (fn off' => off' mod size_elem = size_elem - 1) offsets
              in
                case offsets' of
                  [] =>
                  (* we're at the last node in the tree; the next value *)
                  (* won't be used anyway                               *)
                  (assoc, ([], 0))
                | off'::offs' =>
                  (* go to next sibling node *)
                  (assoc, (offs', off' + 1))
              end
          end) elements ([], 0)
        (* we also need the reverse association (from length/offset to *)
        (* index)                                                      *)
        val lenoff'_lists = map Library.swap lenoff_lists
        (* returns the interpretation for "(list no. m) @ (list no. n)" *)
        (* nat -> nat -> interpretation *)
        fun append m n =
          let
            val (len_m, off_m) = the (AList.lookup (op =) lenoff_lists m)
            val (len_n, off_n) = the (AList.lookup (op =) lenoff_lists n)
            val len_elem = len_m + len_n
            val off_elem = off_m * power (size_elem, len_n) + off_n
          in
            case AList.lookup op= lenoff'_lists (len_elem, off_elem)  of
              NONE =>
              (* undefined *)
              Leaf (replicate size_list False)
            | SOME element =>
              Leaf ((replicate element False) @ True ::
                (replicate (size_list - element - 1) False))
          end
      in
        SOME (Node (map (fn m => Node (map (append m) elements)) elements),
          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_interpreter thy model args t =
    case t of
      Const (@{const_name lfp}, Type ("fun", [Type ("fun",
        [Type ("fun", [T, Type ("bool", [])]),
         Type ("fun", [_, Type ("bool", [])])]),
         Type ("fun", [_, Type ("bool", [])])])) =>
      let
        val size_elem = size_of_type thy model T
        (* 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_sets =
          make_constants thy model (Type ("fun", [T, Type ("bool", [])]))
        (* all functions that map sets to sets *)
        val i_funs = make_constants thy model (Type ("fun",
          [Type ("fun", [T, Type ("bool", [])]),
           Type ("fun", [T, Type ("bool", [])])]))
        (* "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_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_interpreter",
            "intersection: interpretation for set is not a node")
        (* interpretation -> interpretaion *)
        fun lfp (Node resultsets) =
          fold (fn (set, resultset) => fn acc =>
            if is_subset (resultset, set) then
              intersection (acc, set)
            else
              acc) (i_sets ~~ resultsets) i_univ
          | lfp _ =
            raise REFUTE ("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_interpreter thy model args t =
    case t of
      Const (@{const_name gfp}, Type ("fun", [Type ("fun",
        [Type ("fun", [T, Type ("bool", [])]),
         Type ("fun", [_, Type ("bool", [])])]),
         Type ("fun", [_, Type ("bool", [])])])) =>
      let
        val size_elem = size_of_type thy model T
        (* 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_sets =
          make_constants thy model (Type ("fun", [T, Type ("bool", [])]))
        (* all functions that map sets to sets *)
        val i_funs = make_constants thy model (Type ("fun",
          [Type ("fun", [T, Type ("bool", [])]),
           Type ("fun", [T, Type ("bool", [])])]))
        (* "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_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_interpreter",
            "union: interpretation for set is not a node")
        (* interpretation -> interpretaion *)
        fun gfp (Node resultsets) =
          fold (fn (set, resultset) => fn acc =>
            if is_subset (set, resultset) then
              union (acc, set)
            else
              acc) (i_sets ~~ resultsets) i_univ
          | gfp _ =
            raise REFUTE ("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 (@{const_name fst}, Type ("fun", [Type ("*", [T, U]), _])) =>
      let
        val constants_T = make_constants thy model T
        val size_U      = size_of_type thy model U
      in
        SOME (Node (maps (replicate size_U) constants_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 (@{const_name snd}, Type ("fun", [Type ("*", [T, U]), _])) =>
      let
        val size_T      = size_of_type thy model T
        val constants_U = make_constants thy model U
      in
        SOME (Node (flat (replicate size_T constants_U)), model, args)
      end
    | _ =>
      NONE;


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

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

  fun stlc_printer thy model T intr assignment =
  let
    (* 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 T of
      Type ("fun", [T1, T2]) =>
      let
        (* create all constants of type 'T1' *)
        val constants = make_constants thy model T1
        (* 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 T1 arg assignment,
             print thy model 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 = HOLogic.mk_set HOLogic_prodT []
        val HOLogic_insert    =
          Const (@{const_name insert}, HOLogic_prodT --> HOLogic_setT --> HOLogic_setT)
      in
        SOME (fold_rev (fn pair => fn acc => HOLogic_insert $ pair $ acc) pairs HOLogic_empty_set)
      end
    | Type ("prop", [])      =>
      (case index_from_interpretation intr of
        ~1 => SOME (HOLogic.mk_Trueprop (Const (@{const_name undefined}, 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 (@{const_name undefined}, 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 (@{const_name undefined}, 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 (@{const_name undefined}, T))
      else
        SOME (Const (string_of_typ T ^
          string_of_int (index_from_interpretation intr), T))
  end;

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

  fun IDT_printer thy model T intr assignment =
    (case T of
      Type (s, Ts) =>
      (case Datatype.get_info thy s of
        SOME info =>  (* inductive datatype *)
        let
          val (typs, _)           = model
          val index               = #index info
          val descr               = #descr info
          val (_, dtyps, constrs) = the (AList.lookup (op =) 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 Datatype_Aux.DtTFree _ => false | _ => true) dtyps
            then
              raise REFUTE ("IDT_printer", "datatype argument (for type " ^
                Syntax.string_of_typ_global 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 (@{const_name undefined}, 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                                      *)
            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 (fn x => x = 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
            val (cTerm, cargs, args) =
              (* we could speed things up by computing the correct          *)
              (* constructor directly (rather than testing all              *)
              (* constructors), based on the order in which constructors    *)
              (* generate elements of datatypes; the current implementation *)
              (* of 'IDT_printer' however is independent of the internals   *)
              (* of 'IDT_constructor_interpreter'                           *)
              (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
                (* 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 thy (typs, []) dT
              in
                print thy (typs, []) dT (List.nth (consts, n)) assignment
              end) (cargs ~~ args)
          in
            SOME (list_comb (cTerm, argsTerms))
          end
        end
      | NONE =>  (* not an inductive datatype *)
        NONE)
    | _ =>  (* a (free or schematic) type variable *)
      NONE);


(* ------------------------------------------------------------------------- *)
(* 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!                *)
(* ------------------------------------------------------------------------- *)

  val setup =
     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.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.append" List_append_interpreter #>
     add_interpreter "lfp" lfp_interpreter #>
     add_interpreter "gfp" gfp_interpreter #>
     add_interpreter "fst" Product_Type_fst_interpreter #>
     add_interpreter "snd" Product_Type_snd_interpreter #>
     add_printer "stlc" stlc_printer #>
     add_printer "IDT"  IDT_printer;

end  (* structure Refute *)