File refute.ML (Isabelle2005: October 2005)

# File refute.ML

    (*  Title:      HOL/Tools/refute.ML
        ID:         $Id: refute.ML,v 1.40 2005/09/19 21:45:59 webertj Exp $
        Author:     Tjark Weber
        Copyright   2003-2005

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

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

    signature REFUTE =
    sig

    	exception REFUTE of string * string

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

    	type params
    	type interpretation
    	type model
    	type arguments

    	exception MAXVARS_EXCEEDED

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

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

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

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

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

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

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

    	val setup : (theory -> theory) list
    end;

    structure Refute : REFUTE =
    struct

    	open PropLogic;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    	type interpretation =
    		prop_formula list tree;

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

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

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

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

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

    	structure RefuteData = TheoryDataFun(RefuteDataArgs);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    (* ------------------------------------------------------------------------- *)
    (* collect_axioms: collects (monomorphic, universally quantified 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          *)

    	(* 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.                        *)

    	(* When an axiom is added as relevant, further axioms may need to be      *)
    	(* added as well (e.g. when a constant is defined in terms of other       *)
    	(* constants).  To avoid infinite recursion (which should not happen for  *)
    	(* constants anyway, but it could happen for "typedef"-related axioms,    *)
    	(* since they contain the type again), we use an accumulator 'axs' and    *)
    	(* add a relevant axiom only if it is not in 'axs' yet.                   *)

    	fun collect_axioms thy t =
    	let
    		val _ = immediate_output "Adding axioms..."
    		(* (string * Term.term) list *)
    		val axioms = Theory.all_axioms_of thy;
    		(* string list *)
    		val rec_names = Symtab.foldl (fn (acc, (_, info)) =>
    			#rec_names info @ acc) ([], DatatypePackage.get_datatypes thy)
    		(* string list *)
    		val const_of_class_names = map Sign.const_of_class (Sign.classes (sign_of thy))
    		(* 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)                                                    *)
    		(* (string * Term.typ) * Term.term -> Term.term *)
    		fun specialize_type ((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)
    			val typeSubs = (case find_typeSubs t of
    				  SOME x => x
    				| NONE   => raise Type.TYPE_MATCH (* no match found - perhaps due to sort constraints *))
    		in
    			map_term_types
    				(map_type_tvar
    					(fn v =>
    						case Type.lookup (typeSubs, v) of
    						  NONE =>
    							(* schematic type variable not instantiated *)
    							raise REFUTE ("collect_axioms", "term " ^ Sign.string_of_term (sign_of thy) t ^ " still has a polymorphic type (after instantiating type of " ^ quote s ^ ")")
    						| SOME typ =>
    							typ))
    					t
    		end
    		(* applies a type substitution 'typeSubs' for all type variables in a  *)
    		(* term 't'                                                            *)
    		(* (Term.sort * Term.typ) Term.Vartab.table -> Term.term -> Term.term *)
    		fun monomorphic_term typeSubs t =
    			map_term_types (map_type_tvar
    				(fn v =>
    					case Type.lookup (typeSubs, v) of
    					  NONE =>
    						(* schematic type variable not instantiated *)
    						raise ERROR
    					| SOME typ =>
    						typ)) t
    		(* Term.term list * Term.typ -> Term.term list *)
    		fun collect_sort_axioms (axs, T) =
    			let
    				(* collect the axioms for a single 'class' (but not for its superclasses) *)
    				(* Term.term list * string -> Term.term list *)
    				fun collect_class_axioms (axs, class) =
    					let
    						(* obtain the axioms generated by the "axclass" command *)
    						(* (string * Term.term) list *)
    						val class_axioms             = List.filter (fn (s, _) => String.isPrefix (Sign.const_of_class class ^ ".axioms_") s) axioms
    						(* replace the one schematic type variable in each axiom by the actual type 'T' *)
    						(* (string * Term.term) list *)
    						val monomorphic_class_axioms = map (fn (axname, ax) =>
    							let
    								val (idx, S) = (case term_tvars ax of
    									  [is] => is
    									| _    => raise REFUTE ("collect_axioms", "class axiom " ^ axname ^ " (" ^ Sign.string_of_term (sign_of thy) ax ^ ") does not contain exactly one type variable"))
    							in
    								(axname, monomorphic_term (Vartab.make [(idx, (S, T))]) ax)
    							end) class_axioms
    						(* Term.term list * (string * Term.term) list -> Term.term list *)
    						fun collect_axiom (axs, (axname, ax)) =
    							if mem_term (ax, axs) then
    								axs
    							else (
    								immediate_output (" " ^ axname);
    								collect_term_axioms (ax :: axs, ax)
    							)
    					in
    						Library.foldl collect_axiom (axs, monomorphic_class_axioms)
    					end
    				(* string list *)
    				val sort = (case T of
    					  TFree (_, sort) => sort
    					| TVar (_, sort)  => sort
    					| _               => raise REFUTE ("collect_axioms", "type " ^ Sign.string_of_typ (sign_of thy) T ^ " is not a variable"))
    				(* obtain all superclasses of classes in 'sort' *)
    				(* string list *)
    				val superclasses = Graph.all_succs ((#2 o #classes o Type.rep_tsig o Sign.tsig_of o sign_of) thy) sort
    			in
    				Library.foldl collect_class_axioms (axs, superclasses)
    			end
    		(* Term.term list * Term.typ -> Term.term list *)
    		and collect_type_axioms (axs, T) =
    			case T of
    			(* simple types *)
    			  Type ("prop", [])      => axs
    			| Type ("fun", [T1, T2]) => collect_type_axioms (collect_type_axioms (axs, T1), T2)
    			| Type ("set", [T1])     => collect_type_axioms (axs, T1)
    			| Type ("itself", [T1])  => collect_type_axioms (axs, T1)  (* axiomatic type classes *)
    			| Type (s, Ts)           =>
    				let
    					(* look up the definition of a type, as created by "typedef" *)
    					(* (string * Term.term) list -> (string * Term.term) option *)
    					fun get_typedefn [] =
    						NONE
    					  | get_typedefn ((axname,ax)::axms) =
    						(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_typedefn axms
    						end
    						handle ERROR           => get_typedefn axms
    						     | MATCH           => get_typedefn axms
    						     | Type.TYPE_MATCH => get_typedefn axms)
    				in
    					case DatatypePackage.datatype_info thy s of
    					  SOME info =>  (* inductive datatype *)
    							(* only collect relevant type axioms for the argument types *)
    							Library.foldl collect_type_axioms (axs, Ts)
    					| NONE =>
    						(case get_typedefn axioms of
    						  SOME (axname, ax) =>
    							if mem_term (ax, axs) then
    								(* only collect relevant type axioms for the argument types *)
    								Library.foldl collect_type_axioms (axs, Ts)
    							else
    								(immediate_output (" " ^ axname);
    								collect_term_axioms (ax :: axs, ax))
    						| NONE =>
    							(* unspecified type, perhaps introduced with 'typedecl' *)
    							(* at least collect relevant type axioms for the argument types *)
    							Library.foldl collect_type_axioms (axs, Ts))
    				end
    			| TFree _                => collect_sort_axioms (axs, T)  (* axiomatic type classes *)
    			| TVar _                 => collect_sort_axioms (axs, T)  (* axiomatic type classes *)
    		(* Term.term list * Term.term -> Term.term list *)
    		and collect_term_axioms (axs, t) =
    			case t of
    			(* Pure *)
    			  Const ("all", _)                => axs
    			| Const ("==", _)                 => axs
    			| Const ("==>", _)                => axs
    			| Const ("TYPE", T)               => collect_type_axioms (axs, T)  (* axiomatic type classes *)
    			(* HOL *)
    			| Const ("Trueprop", _)           => axs
    			| Const ("Not", _)                => axs
    			| Const ("True", _)               => axs  (* redundant, since 'True' is also an IDT constructor *)
    			| Const ("False", _)              => axs  (* redundant, since 'False' is also an IDT constructor *)
    			| Const ("arbitrary", T)          => collect_type_axioms (axs, T)
    			| Const ("The", T)                =>
    				let
    					val ax = specialize_type (("The", T), (the o AList.lookup (op =) axioms) "HOL.the_eq_trivial")
    				in
    					if mem_term (ax, axs) then
    						collect_type_axioms (axs, T)
    					else
    						(immediate_output " HOL.the_eq_trivial";
    						collect_term_axioms (ax :: axs, ax))
    				end
    			| Const ("Hilbert_Choice.Eps", T) =>
    				let
    					val ax = specialize_type (("Hilbert_Choice.Eps", T),
                          (the o AList.lookup (op =) axioms) "Hilbert_Choice.someI")
    				in
    					if mem_term (ax, axs) then
    						collect_type_axioms (axs, T)
    					else
    						(immediate_output " Hilbert_Choice.someI";
    						collect_term_axioms (ax :: axs, ax))
    				end
    			| Const ("All", _) $ t1           => collect_term_axioms (axs, t1)
    			| Const ("Ex", _) $ t1            => collect_term_axioms (axs, t1)
    			| Const ("op =", T)               => collect_type_axioms (axs, T)
    			| Const ("op &", _)               => axs
    			| Const ("op |", _)               => axs
    			| Const ("op -->", _)             => axs
    			(* sets *)
    			| Const ("Collect", T)            => collect_type_axioms (axs, T)
    			| Const ("op :", T)               => collect_type_axioms (axs, T)
    			(* other optimizations *)
    			| Const ("Finite_Set.card", T)    => collect_type_axioms (axs, T)
    			| Const ("Finite_Set.Finites", T) => collect_type_axioms (axs, T)
    			| Const ("op <", T as Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("bool", [])])])) => collect_type_axioms (axs, T)
    			| Const ("op +", T as Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => collect_type_axioms (axs, T)
    			| Const ("op -", T as Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => collect_type_axioms (axs, T)
    			| Const ("op *", T as Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => collect_type_axioms (axs, T)
    			| Const ("List.op @", T)          => collect_type_axioms (axs, T)
    			| Const ("Lfp.lfp", T)            => collect_type_axioms (axs, T)
    			| Const ("Gfp.gfp", T)            => collect_type_axioms (axs, T)
    			(* simply-typed lambda calculus *)
    			| Const (s, T)                    =>
    				let
    					(* look up the definition of a constant, as created by "constdefs" *)
    					(* string -> Term.typ -> (string * Term.term) list -> (string * Term.term) option *)
    					fun get_defn [] =
    						NONE
    					  | get_defn ((axname, ax)::axms) =
    						(let
    							val (lhs, _) = Logic.dest_equals ax  (* equations only *)
    							val c        = head_of lhs
    							val (s', T') = dest_Const c
    						in
    							if s=s' then
    								let
    									val typeSubs = Sign.typ_match thy (T', T) Vartab.empty
    								in
    									SOME (axname, monomorphic_term typeSubs ax)
    								end
    							else
    								get_defn axms
    						end
    						handle ERROR           => get_defn axms
    						     | TERM _          => get_defn axms
    						     | Type.TYPE_MATCH => get_defn axms)
    					(* axiomatic type classes *)
    					(* unit -> bool *)
    					fun is_const_of_class () =
    						(* I'm not quite sure if checking the name 's' is sufficient, *)
    						(* or if we should also check the type 'T'                    *)
    						s mem const_of_class_names
    					(* inductive data types *)
    					(* unit -> bool *)
    					fun is_IDT_constructor () =
    						(case body_type T of
    						  Type (s', _) =>
    							(case DatatypePackage.constrs_of thy s' of
    							  SOME constrs =>
    								Library.exists (fn c =>
    									(case c of
    									  Const (cname, ctype) =>
    										cname = s andalso Sign.typ_instance thy (T, ctype)
    									| _ =>
    										raise REFUTE ("collect_axioms", "IDT constructor is not a constant")))
    									constrs
    							| NONE =>
    								false)
    						| _  =>
    							false)
    					(* unit -> bool *)
    					fun is_IDT_recursor () =
    						(* I'm not quite sure if checking the name 's' is sufficient, *)
    						(* or if we should also check the type 'T'                    *)
    						s mem rec_names
    				in
    					if is_const_of_class () then
    						(* axiomatic type classes: add "OFCLASS(?'a::c, c_class)" and *)
    						(* the introduction rule "class.intro" as axioms              *)
    						let
    							val class   = Sign.class_of_const s
    							val inclass = Logic.mk_inclass (TVar (("'a", 0), [class]), class)
    							(* Term.term option *)
    							val ofclass_ax = (SOME (specialize_type ((s, T), inclass)) handle Type.TYPE_MATCH => NONE)
    							val intro_ax   = (Option.map (fn t => specialize_type ((s, T), t))
    								(AList.lookup (op =) axioms (class ^ ".intro")) handle Type.TYPE_MATCH => NONE)
    							val axs'       = (case ofclass_ax of NONE => axs | SOME ax => if mem_term (ax, axs) then
    									(* collect relevant type axioms *)
    									collect_type_axioms (axs, T)
    								else
    									(immediate_output (" " ^ Sign.string_of_term (sign_of thy) ax);
    									collect_term_axioms (ax :: axs, ax)))
    							val axs''      = (case intro_ax of NONE => axs' | SOME ax => if mem_term (ax, axs') then
    									(* collect relevant type axioms *)
    									collect_type_axioms (axs', T)
    								else
    									(immediate_output (" " ^ s ^ ".intro");
    									collect_term_axioms (ax :: axs', ax)))
    						in
    							axs''
    						end
    					else if is_IDT_constructor () then
    						(* only collect relevant type axioms *)
    						collect_type_axioms (axs, T)
    					else if is_IDT_recursor () then
    						(* only collect relevant type axioms *)
    						collect_type_axioms (axs, T)
    					else (
    						case get_defn axioms of
    						  SOME (axname, ax) =>
    							if mem_term (ax, axs) then
    								(* collect relevant type axioms *)
    								collect_type_axioms (axs, T)
    							else
    								(immediate_output (" " ^ axname);
    								collect_term_axioms (ax :: axs, ax))
    						| NONE =>
    							(* collect relevant type axioms *)
    							collect_type_axioms (axs, T)
    					)
    				end
    			| Free (_, T)                     => collect_type_axioms (axs, T)
    			| Var (_, T)                      => collect_type_axioms (axs, T)
    			| Bound i                         => axs
    			| Abs (_, T, body)                => collect_term_axioms (collect_type_axioms (axs, T), body)
    			| t1 $ t2                         => collect_term_axioms (collect_term_axioms (axs, t1), t2)
    		(* universal closure over schematic variables *)
    		(* Term.term -> Term.term *)
    		fun close_form t =
    		let
    			(* (Term.indexname * Term.typ) list *)
    			val vars = sort_wrt (fst o fst) (map dest_Var (term_vars t))
    		in
    			Library.foldl
    				(fn (t', ((x, i), T)) => (Term.all T) $ Abs (x, T, abstract_over (Var((x, i), T), t')))
    				(t, vars)
    		end
    		(* Term.term list *)
    		val result = map close_form (collect_term_axioms ([], t))
    		val _ = writeln " ...done."
    	in
    		result
    	end;

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

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

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

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

    	(* Term.typ -> string *)

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

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

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

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

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

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

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

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

    	(* interpretation -> prop_formula *)

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

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

    	(* interpretation -> prop_formula *)

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

    (* ------------------------------------------------------------------------- *)
    (* find_model: repeatedly calls 'interpret' with appropriate parameters,     *)
    (*             applies a SAT solver, and (in case a model is found) displays *)
    (*             the model to the user by calling 'print_model'                *)
    (* thy       : the current theory                                            *)
    (* {...}     : parameters that control the translation/model generation      *)
    (* t         : term to be translated into a propositional formula            *)
    (* negate    : if true, find a model that makes 't' false (rather than true) *)
    (* Note: exception 'TimeOut' is raised if the algorithm does not terminate   *)
    (*       within 'maxtime' seconds (if 'maxtime' >0)                          *)
    (* ------------------------------------------------------------------------- *)

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

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

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

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

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

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

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

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

    	fun refute_term thy params t =
    	let
    		(* disallow schematic type variables, since we cannot properly negate  *)
    		(* terms containing them (their logical meaning is that there EXISTS a *)
    		(* type s.t. ...; to refute such a formula, we would have to show that *)
    		(* for ALL types, not ...)                                             *)
    		val _ = assert (null (term_tvars t)) "Term to be refuted contains schematic type variables"
    		(* existential closure over schematic variables *)
    		(* (Term.indexname * Term.typ) list *)
    		val vars = sort_wrt (fst o fst) (map dest_Var (term_vars t))
    		(* Term.term *)
    		val ex_closure = Library.foldl
    			(fn (t', ((x, i), T)) => (HOLogic.exists_const T) $ Abs (x, T, abstract_over (Var ((x, i), T), t')))
    			(t, vars)
    		(* 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.                                                  *)
    	in
    		find_model thy (actual_params thy params) ex_closure true
    	end;

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

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

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

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

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

    	(* interpretation -> interpretation list *)

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

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

    	(* interpretation -> int *)

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

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

    	(* interpretation *)

    	val TT = Leaf [True, False];

    	val FF = Leaf [False, True];

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

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

    	(* interpretation * interpretation -> interpretation *)

    	fun make_equality (i1, i2) =
    	let
    		(* interpretation * interpretation -> prop_formula *)
    		fun equal (i1, i2) =
    			(case i1 of
    			  Leaf xs =>
    				(case i2 of
    				  Leaf ys => PropLogic.dot_product (xs, ys)  (* defined and equal *)
    				| Node _  => raise REFUTE ("make_equality", "second interpretation is higher"))
    			| Node xs =>
    				(case i2 of
    				  Leaf _  => raise REFUTE ("make_equality", "first interpretation is higher")
    				| Node ys => PropLogic.all (map equal (xs ~~ ys))))
    		(* interpretation * interpretation -> prop_formula *)
    		fun not_equal (i1, i2) =
    			(case i1 of
    			  Leaf xs =>
    				(case i2 of
    				  Leaf ys => PropLogic.all ((PropLogic.exists xs) :: (PropLogic.exists ys) ::
    					(map (fn (x,y) => SOr (SNot x, SNot y)) (xs ~~ ys)))  (* defined and not 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.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
    				  Leaf ys => SOr (PropLogic.dot_product (xs, ys),  (* defined and equal, or both undefined *)
    					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 'i2' 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 (fn xs => x::xs) 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 (fn x => [x]) xs
    		  | pick_all (xs::xss) =
    			let val rec_pick = pick_all xss in
    				Library.foldl (fn (acc, x) => (cons_list x rec_pick) @ acc) ([], xs)
    			end
    		  | pick_all _ =
    			raise REFUTE ("interpretation_apply", "empty list (in pick_all)")
    		(* interpretation -> prop_formula list *)
    		fun interpretation_to_prop_formula_list (Leaf xs) =
    			xs
    		  | interpretation_to_prop_formula_list (Node trees) =
    			map PropLogic.all (pick_all (map interpretation_to_prop_formula_list trees))
    	in
    		case i1 of
    		  Leaf _ =>
    			raise REFUTE ("interpretation_apply", "first interpretation is a leaf")
    		| Node xs =>
    			prop_formula_list_dot_product_interpretation_list (interpretation_to_prop_formula_list i2, xs)
    	end;

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

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

    	fun eta_expand t i =
    	let
    		val Ts = binder_types (fastype_of t)
    	in
    		foldr (fn (T, t) => Abs ("<eta_expand>", T, t))
    			(list_comb (t, map Bound (i-1 downto 0))) (Library.take (i, Ts))
    	end;

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

    	(* int list -> int *)

    	fun sum xs = foldl op+ 0 xs;

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

    	(* int list -> int *)

    	fun product xs = foldl op* 1 xs;

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

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

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

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

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

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

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

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

    	fun Pure_interpreter thy model args t =
    		case t of
    		  Const ("all", _) $ t1 =>  (* in the meta-logic, 'all' MUST be followed by an argument term *)
    			let
    				val (i, m, a) = interpret thy model args t1
    			in
    				case i of
    				  Node xs =>
    					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 not followed by a function")
    			end
    		| Const ("==", _) $ t1 $ t2 =>
    			let
    				val (i1, m1, a1) = interpret thy model args t1
    				val (i2, m2, a2) = interpret thy m1 a1 t2
    			in
    				(* we use either 'make_def_equality' or 'make_equality' *)
    				SOME ((if #def_eq args then make_def_equality else make_equality) (i1, i2), m2, a2)
    			end
    		| Const ("==>", _) =>  (* simpler than translating 'Const ("==>", _) $ t1 $ t2' *)
    			SOME (Node [Node [TT, FF], Node [TT, TT]], model, args)
    		| _ => 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.  "All" and "Ex" are then translated just like any other        *)
    	(* constant, with the relevant axiom being added by 'collect_axioms'.        *)
    	(* ------------------------------------------------------------------------- *)
    		case t of
    		  Const ("Trueprop", _) =>
    			SOME (Node [TT, FF], model, args)
    		| Const ("Not", _) =>
    			SOME (Node [FF, TT], model, args)
    		| Const ("True", _) =>  (* redundant, since 'True' is also an IDT constructor *)
    			SOME (TT, model, args)
    		| Const ("False", _) =>  (* redundant, since 'False' is also an IDT constructor *)
    			SOME (FF, model, args)
    		| Const ("All", _) $ t1 =>
    		(* if "All" occurs without an argument (i.e. as argument to a higher-order *)
    		(* function or predicate), it is handled by the 'stlc_interpreter' (i.e.   *)
    		(* by unfolding its definition)                                            *)
    			let
    				val (i, m, a) = interpret thy model args t1
    			in
    				case i of
    				  Node xs =>
    					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 ("Ex", _) $ t1 =>
    		(* if "Ex" occurs without an argument (i.e. as argument to a higher-order  *)
    		(* function or predicate), it is handled by the 'stlc_interpreter' (i.e.   *)
    		(* by unfolding its definition)                                            *)
    			let
    				val (i, m, a) = interpret thy model args t1
    			in
    				case i of
    				  Node xs =>
    					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 ("op =", _) $ t1 $ t2 =>
    			let
    				val (i1, m1, a1) = interpret thy model args t1
    				val (i2, m2, a2) = interpret thy m1 a1 t2
    			in
    				SOME (make_equality (i1, i2), m2, a2)
    			end
    		| Const ("op =", _) $ t1 =>
    			SOME (interpret thy model args (eta_expand t 1))
    		| Const ("op =", _) =>
    			SOME (interpret thy model args (eta_expand t 2))
    		| Const ("op &", _) $ t1 $ t2 =>
    			(* 3-valued logic *)
    			let
    				val (i1, m1, a1) = interpret thy model args t1
    				val (i2, m2, a2) = interpret thy m1 a1 t2
    				val fmTrue       = PropLogic.SAnd (toTrue i1, toTrue i2)
    				val fmFalse      = PropLogic.SOr (toFalse i1, toFalse i2)
    			in
    				SOME (Leaf [fmTrue, fmFalse], m2, a2)
    			end
    		| Const ("op &", _) $ t1 =>
    			SOME (interpret thy model args (eta_expand t 1))
    		| Const ("op &", _) =>
    			SOME (interpret thy model args (eta_expand t 2))
    			(* SOME (Node [Node [TT, FF], Node [FF, FF]], model, args) *)
    		| Const ("op |", _) $ t1 $ t2 =>
    			(* 3-valued logic *)
    			let
    				val (i1, m1, a1) = interpret thy model args t1
    				val (i2, m2, a2) = interpret thy m1 a1 t2
    				val fmTrue       = PropLogic.SOr (toTrue i1, toTrue i2)
    				val fmFalse      = PropLogic.SAnd (toFalse i1, toFalse i2)
    			in
    				SOME (Leaf [fmTrue, fmFalse], m2, a2)
    			end
    		| Const ("op |", _) $ t1 =>
    			SOME (interpret thy model args (eta_expand t 1))
    		| Const ("op |", _) =>
    			SOME (interpret thy model args (eta_expand t 2))
    			(* SOME (Node [Node [TT, TT], Node [TT, FF]], model, args) *)
    		| Const ("op -->", _) $ t1 $ t2 =>
    			(* 3-valued logic *)
    			let
    				val (i1, m1, a1) = interpret thy model args t1
    				val (i2, m2, a2) = interpret thy m1 a1 t2
    				val fmTrue       = PropLogic.SOr (toFalse i1, toTrue i2)
    				val fmFalse      = PropLogic.SAnd (toTrue i1, toFalse i2)
    			in
    				SOME (Leaf [fmTrue, fmFalse], m2, a2)
    			end
    		| Const ("op -->", _) =>
    			(* SOME (Node [Node [TT, FF], Node [TT, TT]], model, args) *)
    			SOME (interpret thy model args (eta_expand t 2))
    		| _ => NONE;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    	(* (theory -> theory) list *)

    	val setup =
    		[RefuteData.init,
    		 add_interpreter "stlc"    stlc_interpreter,
    		 add_interpreter "Pure"    Pure_interpreter,
    		 add_interpreter "HOLogic" HOLogic_interpreter,
    		 add_interpreter "set"     set_interpreter,
    		 add_interpreter "IDT"             IDT_interpreter,
    		 add_interpreter "IDT_constructor" IDT_constructor_interpreter,
    		 add_interpreter "IDT_recursion"   IDT_recursion_interpreter,
    		 add_interpreter "Finite_Set.card"    Finite_Set_card_interpreter,
    		 add_interpreter "Finite_Set.Finites" Finite_Set_Finites_interpreter,
    		 add_interpreter "Nat.op <" Nat_less_interpreter,
    		 add_interpreter "Nat.op +" Nat_plus_interpreter,
    		 add_interpreter "Nat.op -" Nat_minus_interpreter,
    		 add_interpreter "Nat.op *" Nat_mult_interpreter,
    		 add_interpreter "List.op @" List_append_interpreter,
    		 add_interpreter "Lfp.lfp" Lfp_lfp_interpreter,
    		 add_interpreter "Gfp.gfp" Gfp_gfp_interpreter,
    		 add_printer "stlc" stlc_printer,
    		 add_printer "set"  set_printer,
    		 add_printer "IDT"  IDT_printer];

    end