(*  Title:      HOL/Tools/refute.ML

     ID:         $Id$

     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 prep_ext = 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 assoc (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 assoc (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 *)

 	fun get_default_param thy name = Symtab.lookup ((#parameters o RefuteData.get) thy, name);

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

 (* get_default_params: returns a list of all '(name, value)' pairs that are  *)

 (*                     stored in RefuteData's parameter table                *)

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

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

 	fun get_default_params thy = (Symtab.dest o #parameters o RefuteData.get) thy;

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

 (* 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 assoc_string (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 assoc_string (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 *)

 		(valOf o assoc) (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, _) = (valOf o assoc) (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 = List.concat (map (Symtab.dest o #axioms o Theory.rep_theory) (thy :: Theory.ancestors_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 (Type.typ_match (Sign.tsig_of (sign_of thy)) (Vartab.empty, (T', T))) 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 (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 ((#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 = Type.typ_match (Sign.tsig_of (sign_of thy)) (Vartab.empty, (T'', T))

 								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), (valOf o assoc) (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), (valOf o assoc) (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)

 			(* 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 = Type.typ_match (Sign.tsig_of (sign_of thy)) (Vartab.empty, (T', T))

 								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 Type.typ_instance (Sign.tsig_of (sign_of 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)) (assoc (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 (" " ^ class ^ ".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) = (valOf o assoc) (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 assoc (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 (fn (T, _) => assoc (sizes, string_of_typ T) = NONE) 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 assoc (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) = (valOf o assoc) (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 (valOf o assoc) (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 assoc (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 assoc (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 = assoc (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) = (valOf o assoc) (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 => overwrite (typs, (Type (s, Ts), n-1)))

 							(* 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 assoc (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 assoc (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) = (valOf o assoc) (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 Type.typ_instance (Sign.tsig_of (sign_of 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 = assoc (typs, Type (s', Ts'))

 									val typs' = (case depth of NONE => typs | SOME n => overwrite (typs, (Type (s', Ts'), n-1)))

 									(* 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, _) = (valOf o assoc) (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, (valOf o assoc) (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 ((valOf o assoc) (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) = (valOf o assoc) (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 ((valOf o assoc) (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)) ((valOf o assoc) (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 valOf) o toList o valOf o assoc) (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;

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

 (* 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) = (valOf o assoc) (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_printer "stlc" stlc_printer,

 		 add_printer "set"  set_printer,

 		 add_printer "IDT"  IDT_printer];

 end