(* Title: HOL/Tools/refute.ML ID: $Id: refute.ML,v 1.40 2005/09/19 21:45:59 webertj Exp $ Author: Tjark Weber Copyright 2003-2005 Finite model generation for HOL formulas, using a SAT solver. *) (* ------------------------------------------------------------------------- *) (* Declares the 'REFUTE' signature as well as a structure 'Refute'. *) (* Documentation is available in the Isabelle/Isar theory 'HOL/Refute.thy'. *) (* ------------------------------------------------------------------------- *) signature REFUTE = sig exception REFUTE of string * string (* ------------------------------------------------------------------------- *) (* Model/interpretation related code (translation HOL -> propositional logic *) (* ------------------------------------------------------------------------- *) type params type interpretation type model type arguments exception MAXVARS_EXCEEDED val add_interpreter : string -> (theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option) -> theory -> theory val add_printer : string -> (theory -> model -> Term.term -> interpretation -> (int -> bool) -> Term.term option) -> theory -> theory val interpret : theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) val print : theory -> model -> Term.term -> interpretation -> (int -> bool) -> Term.term val print_model : theory -> model -> (int -> bool) -> string (* ------------------------------------------------------------------------- *) (* Interface *) (* ------------------------------------------------------------------------- *) val set_default_param : (string * string) -> theory -> theory val get_default_param : theory -> string -> string option val get_default_params : theory -> (string * string) list val actual_params : theory -> (string * string) list -> params val find_model : theory -> params -> Term.term -> bool -> unit val satisfy_term : theory -> (string * string) list -> Term.term -> unit (* tries to find a model for a formula *) val refute_term : theory -> (string * string) list -> Term.term -> unit (* tries to find a model that refutes a formula *) val refute_subgoal : theory -> (string * string) list -> Thm.thm -> int -> unit val setup : (theory -> theory) list end; structure Refute : REFUTE = struct open PropLogic; (* We use 'REFUTE' only for internal error conditions that should *) (* never occur in the first place (i.e. errors caused by bugs in our *) (* code). Otherwise (e.g. to indicate invalid input data) we use *) (* 'error'. *) exception REFUTE of string * string; (* ("in function", "cause") *) (* should be raised by an interpreter when more variables would be *) (* required than allowed by 'maxvars' *) exception MAXVARS_EXCEEDED; (* ------------------------------------------------------------------------- *) (* TREES *) (* ------------------------------------------------------------------------- *) (* ------------------------------------------------------------------------- *) (* tree: implements an arbitrarily (but finitely) branching tree as a list *) (* of (lists of ...) elements *) (* ------------------------------------------------------------------------- *) datatype 'a tree = Leaf of 'a | Node of ('a tree) list; (* ('a -> 'b) -> 'a tree -> 'b tree *) fun tree_map f tr = case tr of Leaf x => Leaf (f x) | Node xs => Node (map (tree_map f) xs); (* ('a * 'b -> 'a) -> 'a * ('b tree) -> 'a *) fun tree_foldl f = let fun itl (e, Leaf x) = f(e,x) | itl (e, Node xs) = Library.foldl (tree_foldl f) (e,xs) in itl end; (* 'a tree * 'b tree -> ('a * 'b) tree *) fun tree_pair (t1, t2) = case t1 of Leaf x => (case t2 of Leaf y => Leaf (x,y) | Node _ => raise REFUTE ("tree_pair", "trees are of different height (second tree is higher)")) | Node xs => (case t2 of (* '~~' will raise an exception if the number of branches in *) (* both trees is different at the current node *) Node ys => Node (map tree_pair (xs ~~ ys)) | Leaf _ => raise REFUTE ("tree_pair", "trees are of different height (first tree is higher)")); (* ------------------------------------------------------------------------- *) (* params: parameters that control the translation into a propositional *) (* formula/model generation *) (* *) (* The following parameters are supported (and required (!), except for *) (* "sizes"): *) (* *) (* Name Type Description *) (* *) (* "sizes" (string * int) list *) (* Size of ground types (e.g. 'a=2), or depth of IDTs. *) (* "minsize" int If >0, minimal size of each ground type/IDT depth. *) (* "maxsize" int If >0, maximal size of each ground type/IDT depth. *) (* "maxvars" int If >0, use at most 'maxvars' Boolean variables *) (* when transforming the term into a propositional *) (* formula. *) (* "maxtime" int If >0, terminate after at most 'maxtime' seconds. *) (* "satsolver" string SAT solver to be used. *) (* ------------------------------------------------------------------------- *) type params = { sizes : (string * int) list, minsize : int, maxsize : int, maxvars : int, maxtime : int, satsolver: string }; (* ------------------------------------------------------------------------- *) (* interpretation: a term's interpretation is given by a variable of type *) (* 'interpretation' *) (* ------------------------------------------------------------------------- *) type interpretation = prop_formula list tree; (* ------------------------------------------------------------------------- *) (* model: a model specifies the size of types and the interpretation of *) (* terms *) (* ------------------------------------------------------------------------- *) type model = (Term.typ * int) list * (Term.term * interpretation) list; (* ------------------------------------------------------------------------- *) (* arguments: additional arguments required during interpretation of terms *) (* ------------------------------------------------------------------------- *) type arguments = { maxvars : int, (* just passed unchanged from 'params' *) def_eq : bool, (* whether to use 'make_equality' or 'make_def_equality' *) (* the following may change during the translation *) next_idx : int, bounds : interpretation list, wellformed: prop_formula }; structure RefuteDataArgs = struct val name = "HOL/refute"; type T = {interpreters: (string * (theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option)) list, printers: (string * (theory -> model -> Term.term -> interpretation -> (int -> bool) -> Term.term option)) list, parameters: string Symtab.table}; val empty = {interpreters = [], printers = [], parameters = Symtab.empty}; val copy = I; val extend = I; fun merge _ ({interpreters = in1, printers = pr1, parameters = pa1}, {interpreters = in2, printers = pr2, parameters = pa2}) = {interpreters = rev (merge_alists (rev in1) (rev in2)), printers = rev (merge_alists (rev pr1) (rev pr2)), parameters = Symtab.merge (op=) (pa1, pa2)}; fun print sg {interpreters, printers, parameters} = Pretty.writeln (Pretty.chunks [Pretty.strs ("default parameters:" :: List.concat (map (fn (name, value) => [name, "=", value]) (Symtab.dest parameters))), Pretty.strs ("interpreters:" :: map fst interpreters), Pretty.strs ("printers:" :: map fst printers)]); end; structure RefuteData = TheoryDataFun(RefuteDataArgs); (* ------------------------------------------------------------------------- *) (* interpret: interprets the term 't' using a suitable interpreter; returns *) (* the interpretation and a (possibly extended) model that keeps *) (* track of the interpretation of subterms *) (* ------------------------------------------------------------------------- *) (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) *) fun interpret thy model args t = (case get_first (fn (_, f) => f thy model args t) (#interpreters (RefuteData.get thy)) of NONE => raise REFUTE ("interpret", "no interpreter for term " ^ quote (Sign.string_of_term (sign_of thy) t)) | SOME x => x); (* ------------------------------------------------------------------------- *) (* print: converts the constant denoted by the term 't' into a term using a *) (* suitable printer *) (* ------------------------------------------------------------------------- *) (* theory -> model -> Term.term -> interpretation -> (int -> bool) -> Term.term *) fun print thy model t intr assignment = (case get_first (fn (_, f) => f thy model t intr assignment) (#printers (RefuteData.get thy)) of NONE => raise REFUTE ("print", "no printer for term " ^ quote (Sign.string_of_term (sign_of thy) t)) | SOME x => x); (* ------------------------------------------------------------------------- *) (* print_model: turns the model into a string, using a fixed interpretation *) (* (given by an assignment for Boolean variables) and suitable *) (* printers *) (* ------------------------------------------------------------------------- *) (* theory -> model -> (int -> bool) -> string *) fun print_model thy model assignment = let val (typs, terms) = model val typs_msg = if null typs then "empty universe (no type variables in term)\n" else "Size of types: " ^ commas (map (fn (T, i) => Sign.string_of_typ (sign_of thy) T ^ ": " ^ string_of_int i) typs) ^ "\n" val show_consts_msg = if not (!show_consts) andalso Library.exists (is_Const o fst) terms then "set \"show_consts\" to show the interpretation of constants\n" else "" val terms_msg = if null terms then "empty interpretation (no free variables in term)\n" else space_implode "\n" (List.mapPartial (fn (t, intr) => (* print constants only if 'show_consts' is true *) if (!show_consts) orelse not (is_Const t) then SOME (Sign.string_of_term (sign_of thy) t ^ ": " ^ Sign.string_of_term (sign_of thy) (print thy model t intr assignment)) else NONE) terms) ^ "\n" in typs_msg ^ show_consts_msg ^ terms_msg end; (* ------------------------------------------------------------------------- *) (* PARAMETER MANAGEMENT *) (* ------------------------------------------------------------------------- *) (* string -> (theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option) -> theory -> theory *) fun add_interpreter name f thy = let val {interpreters, printers, parameters} = RefuteData.get thy in case AList.lookup (op =) interpreters name of NONE => RefuteData.put {interpreters = (name, f) :: interpreters, printers = printers, parameters = parameters} thy | SOME _ => error ("Interpreter " ^ name ^ " already declared") end; (* string -> (theory -> model -> Term.term -> interpretation -> (int -> bool) -> Term.term option) -> theory -> theory *) fun add_printer name f thy = let val {interpreters, printers, parameters} = RefuteData.get thy in case AList.lookup (op =) printers name of NONE => RefuteData.put {interpreters = interpreters, printers = (name, f) :: printers, parameters = parameters} thy | SOME _ => error ("Printer " ^ name ^ " already declared") end; (* ------------------------------------------------------------------------- *) (* set_default_param: stores the '(name, value)' pair in RefuteData's *) (* parameter table *) (* ------------------------------------------------------------------------- *) (* (string * string) -> theory -> theory *) fun set_default_param (name, value) thy = let val {interpreters, printers, parameters} = RefuteData.get thy in case Symtab.lookup parameters name of NONE => RefuteData.put {interpreters = interpreters, printers = printers, parameters = Symtab.extend (parameters, [(name, value)])} thy | SOME _ => RefuteData.put {interpreters = interpreters, printers = printers, parameters = Symtab.update (name, value) parameters} thy end; (* ------------------------------------------------------------------------- *) (* get_default_param: retrieves the value associated with 'name' from *) (* RefuteData's parameter table *) (* ------------------------------------------------------------------------- *) (* theory -> string -> string option *) val get_default_param = Symtab.lookup o #parameters o RefuteData.get; (* ------------------------------------------------------------------------- *) (* get_default_params: returns a list of all '(name, value)' pairs that are *) (* stored in RefuteData's parameter table *) (* ------------------------------------------------------------------------- *) (* theory -> (string * string) list *) val get_default_params = Symtab.dest o #parameters o RefuteData.get; (* ------------------------------------------------------------------------- *) (* actual_params: takes a (possibly empty) list 'params' of parameters that *) (* override the default parameters currently specified in 'thy', and *) (* returns a record that can be passed to 'find_model'. *) (* ------------------------------------------------------------------------- *) (* theory -> (string * string) list -> params *) fun actual_params thy override = let (* (string * string) list * string -> int *) fun read_int (parms, name) = case AList.lookup (op =) parms name of SOME s => (case Int.fromString s of SOME i => i | NONE => error ("parameter " ^ quote name ^ " (value is " ^ quote s ^ ") must be an integer value")) | NONE => error ("parameter " ^ quote name ^ " must be assigned a value") (* (string * string) list * string -> string *) fun read_string (parms, name) = case AList.lookup (op =) parms name of SOME s => s | NONE => error ("parameter " ^ quote name ^ " must be assigned a value") (* (string * string) list *) val allparams = override @ (get_default_params thy) (* 'override' first, defaults last *) (* int *) val minsize = read_int (allparams, "minsize") val maxsize = read_int (allparams, "maxsize") val maxvars = read_int (allparams, "maxvars") val maxtime = read_int (allparams, "maxtime") (* string *) val satsolver = read_string (allparams, "satsolver") (* all remaining parameters of the form "string=int" are collected in *) (* 'sizes' *) (* TODO: it is currently not possible to specify a size for a type *) (* whose name is one of the other parameters (e.g. 'maxvars') *) (* (string * int) list *) val sizes = List.mapPartial (fn (name,value) => (case Int.fromString value of SOME i => SOME (name, i) | NONE => NONE)) (List.filter (fn (name,_) => name<>"minsize" andalso name<>"maxsize" andalso name<>"maxvars" andalso name<>"maxtime" andalso name<>"satsolver") allparams) in {sizes=sizes, minsize=minsize, maxsize=maxsize, maxvars=maxvars, maxtime=maxtime, satsolver=satsolver} end; (* ------------------------------------------------------------------------- *) (* TRANSLATION HOL -> PROPOSITIONAL LOGIC, BOOLEAN ASSIGNMENT -> MODEL *) (* ------------------------------------------------------------------------- *) (* ------------------------------------------------------------------------- *) (* typ_of_dtyp: converts a data type ('DatatypeAux.dtyp') into a type *) (* ('Term.typ'), given type parameters for the data type's type *) (* arguments *) (* ------------------------------------------------------------------------- *) (* DatatypeAux.descr -> (DatatypeAux.dtyp * Term.typ) list -> DatatypeAux.dtyp -> Term.typ *) fun typ_of_dtyp descr typ_assoc (DatatypeAux.DtTFree a) = (* replace a 'DtTFree' variable by the associated type *) (the o AList.lookup (op =) typ_assoc) (DatatypeAux.DtTFree a) | typ_of_dtyp descr typ_assoc (DatatypeAux.DtType (s, ds)) = Type (s, map (typ_of_dtyp descr typ_assoc) ds) | typ_of_dtyp descr typ_assoc (DatatypeAux.DtRec i) = let val (s, ds, _) = (the o AList.lookup (op =) descr) i in Type (s, map (typ_of_dtyp descr typ_assoc) ds) end; (* ------------------------------------------------------------------------- *) (* collect_axioms: collects (monomorphic, universally quantified versions *) (* of) all HOL axioms that are relevant w.r.t 't' *) (* ------------------------------------------------------------------------- *) (* Note: to make the collection of axioms more easily extensible, this *) (* function could be based on user-supplied "axiom collectors", *) (* similar to 'interpret'/interpreters or 'print'/printers *) (* theory -> Term.term -> Term.term list *) (* Which axioms are "relevant" for a particular term/type goes hand in *) (* hand with the interpretation of that term/type by its interpreter (see *) (* way below): if the interpretation respects an axiom anyway, the axiom *) (* does not need to be added as a constraint here. *) (* When an axiom is added as relevant, further axioms may need to be *) (* added as well (e.g. when a constant is defined in terms of other *) (* constants). To avoid infinite recursion (which should not happen for *) (* constants anyway, but it could happen for "typedef"-related axioms, *) (* since they contain the type again), we use an accumulator 'axs' and *) (* add a relevant axiom only if it is not in 'axs' yet. *) fun collect_axioms thy t = let val _ = immediate_output "Adding axioms..." (* (string * Term.term) list *) val axioms = Theory.all_axioms_of thy; (* string list *) val rec_names = Symtab.foldl (fn (acc, (_, info)) => #rec_names info @ acc) ([], DatatypePackage.get_datatypes thy) (* string list *) val const_of_class_names = map Sign.const_of_class (Sign.classes (sign_of thy)) (* given a constant 's' of type 'T', which is a subterm of 't', where *) (* 't' has a (possibly) more general type, the schematic type *) (* variables in 't' are instantiated to match the type 'T' (may raise *) (* Type.TYPE_MATCH) *) (* (string * Term.typ) * Term.term -> Term.term *) fun specialize_type ((s, T), t) = let fun find_typeSubs (Const (s', T')) = (if s=s' then SOME (Sign.typ_match thy (T', T) Vartab.empty) handle Type.TYPE_MATCH => NONE else NONE) | find_typeSubs (Free _) = NONE | find_typeSubs (Var _) = NONE | find_typeSubs (Bound _) = NONE | find_typeSubs (Abs (_, _, body)) = find_typeSubs body | find_typeSubs (t1 $ t2) = (case find_typeSubs t1 of SOME x => SOME x | NONE => find_typeSubs t2) val typeSubs = (case find_typeSubs t of SOME x => x | NONE => raise Type.TYPE_MATCH (* no match found - perhaps due to sort constraints *)) in map_term_types (map_type_tvar (fn v => case Type.lookup (typeSubs, v) of NONE => (* schematic type variable not instantiated *) raise REFUTE ("collect_axioms", "term " ^ Sign.string_of_term (sign_of thy) t ^ " still has a polymorphic type (after instantiating type of " ^ quote s ^ ")") | SOME typ => typ)) t end (* applies a type substitution 'typeSubs' for all type variables in a *) (* term 't' *) (* (Term.sort * Term.typ) Term.Vartab.table -> Term.term -> Term.term *) fun monomorphic_term typeSubs t = map_term_types (map_type_tvar (fn v => case Type.lookup (typeSubs, v) of NONE => (* schematic type variable not instantiated *) raise ERROR | SOME typ => typ)) t (* Term.term list * Term.typ -> Term.term list *) fun collect_sort_axioms (axs, T) = let (* collect the axioms for a single 'class' (but not for its superclasses) *) (* Term.term list * string -> Term.term list *) fun collect_class_axioms (axs, class) = let (* obtain the axioms generated by the "axclass" command *) (* (string * Term.term) list *) val class_axioms = List.filter (fn (s, _) => String.isPrefix (Sign.const_of_class class ^ ".axioms_") s) axioms (* replace the one schematic type variable in each axiom by the actual type 'T' *) (* (string * Term.term) list *) val monomorphic_class_axioms = map (fn (axname, ax) => let val (idx, S) = (case term_tvars ax of [is] => is | _ => raise REFUTE ("collect_axioms", "class axiom " ^ axname ^ " (" ^ Sign.string_of_term (sign_of thy) ax ^ ") does not contain exactly one type variable")) in (axname, monomorphic_term (Vartab.make [(idx, (S, T))]) ax) end) class_axioms (* Term.term list * (string * Term.term) list -> Term.term list *) fun collect_axiom (axs, (axname, ax)) = if mem_term (ax, axs) then axs else ( immediate_output (" " ^ axname); collect_term_axioms (ax :: axs, ax) ) in Library.foldl collect_axiom (axs, monomorphic_class_axioms) end (* string list *) val sort = (case T of TFree (_, sort) => sort | TVar (_, sort) => sort | _ => raise REFUTE ("collect_axioms", "type " ^ Sign.string_of_typ (sign_of thy) T ^ " is not a variable")) (* obtain all superclasses of classes in 'sort' *) (* string list *) val superclasses = Graph.all_succs ((#2 o #classes o Type.rep_tsig o Sign.tsig_of o sign_of) thy) sort in Library.foldl collect_class_axioms (axs, superclasses) end (* Term.term list * Term.typ -> Term.term list *) and collect_type_axioms (axs, T) = case T of (* simple types *) Type ("prop", []) => axs | Type ("fun", [T1, T2]) => collect_type_axioms (collect_type_axioms (axs, T1), T2) | Type ("set", [T1]) => collect_type_axioms (axs, T1) | Type ("itself", [T1]) => collect_type_axioms (axs, T1) (* axiomatic type classes *) | Type (s, Ts) => let (* look up the definition of a type, as created by "typedef" *) (* (string * Term.term) list -> (string * Term.term) option *) fun get_typedefn [] = NONE | get_typedefn ((axname,ax)::axms) = (let (* Term.term -> Term.typ option *) fun type_of_type_definition (Const (s', T')) = if s'="Typedef.type_definition" then SOME T' else NONE | type_of_type_definition (Free _) = NONE | type_of_type_definition (Var _) = NONE | type_of_type_definition (Bound _) = NONE | type_of_type_definition (Abs (_, _, body)) = type_of_type_definition body | type_of_type_definition (t1 $ t2) = (case type_of_type_definition t1 of SOME x => SOME x | NONE => type_of_type_definition t2) in case type_of_type_definition ax of SOME T' => let val T'' = (domain_type o domain_type) T' val typeSubs = Sign.typ_match thy (T'', T) Vartab.empty in SOME (axname, monomorphic_term typeSubs ax) end | NONE => get_typedefn axms end handle ERROR => get_typedefn axms | MATCH => get_typedefn axms | Type.TYPE_MATCH => get_typedefn axms) in case DatatypePackage.datatype_info thy s of SOME info => (* inductive datatype *) (* only collect relevant type axioms for the argument types *) Library.foldl collect_type_axioms (axs, Ts) | NONE => (case get_typedefn axioms of SOME (axname, ax) => if mem_term (ax, axs) then (* only collect relevant type axioms for the argument types *) Library.foldl collect_type_axioms (axs, Ts) else (immediate_output (" " ^ axname); collect_term_axioms (ax :: axs, ax)) | NONE => (* unspecified type, perhaps introduced with 'typedecl' *) (* at least collect relevant type axioms for the argument types *) Library.foldl collect_type_axioms (axs, Ts)) end | TFree _ => collect_sort_axioms (axs, T) (* axiomatic type classes *) | TVar _ => collect_sort_axioms (axs, T) (* axiomatic type classes *) (* Term.term list * Term.term -> Term.term list *) and collect_term_axioms (axs, t) = case t of (* Pure *) Const ("all", _) => axs | Const ("==", _) => axs | Const ("==>", _) => axs | Const ("TYPE", T) => collect_type_axioms (axs, T) (* axiomatic type classes *) (* HOL *) | Const ("Trueprop", _) => axs | Const ("Not", _) => axs | Const ("True", _) => axs (* redundant, since 'True' is also an IDT constructor *) | Const ("False", _) => axs (* redundant, since 'False' is also an IDT constructor *) | Const ("arbitrary", T) => collect_type_axioms (axs, T) | Const ("The", T) => let val ax = specialize_type (("The", T), (the o AList.lookup (op =) axioms) "HOL.the_eq_trivial") in if mem_term (ax, axs) then collect_type_axioms (axs, T) else (immediate_output " HOL.the_eq_trivial"; collect_term_axioms (ax :: axs, ax)) end | Const ("Hilbert_Choice.Eps", T) => let val ax = specialize_type (("Hilbert_Choice.Eps", T), (the o AList.lookup (op =) axioms) "Hilbert_Choice.someI") in if mem_term (ax, axs) then collect_type_axioms (axs, T) else (immediate_output " Hilbert_Choice.someI"; collect_term_axioms (ax :: axs, ax)) end | Const ("All", _) $ t1 => collect_term_axioms (axs, t1) | Const ("Ex", _) $ t1 => collect_term_axioms (axs, t1) | Const ("op =", T) => collect_type_axioms (axs, T) | Const ("op &", _) => axs | Const ("op |", _) => axs | Const ("op -->", _) => axs (* sets *) | Const ("Collect", T) => collect_type_axioms (axs, T) | Const ("op :", T) => collect_type_axioms (axs, T) (* other optimizations *) | Const ("Finite_Set.card", T) => collect_type_axioms (axs, T) | Const ("Finite_Set.Finites", T) => collect_type_axioms (axs, T) | Const ("op <", T as Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("bool", [])])])) => collect_type_axioms (axs, T) | Const ("op +", T as Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => collect_type_axioms (axs, T) | Const ("op -", T as Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => collect_type_axioms (axs, T) | Const ("op *", T as Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => collect_type_axioms (axs, T) | Const ("List.op @", T) => collect_type_axioms (axs, T) | Const ("Lfp.lfp", T) => collect_type_axioms (axs, T) | Const ("Gfp.gfp", T) => collect_type_axioms (axs, T) (* simply-typed lambda calculus *) | Const (s, T) => let (* look up the definition of a constant, as created by "constdefs" *) (* string -> Term.typ -> (string * Term.term) list -> (string * Term.term) option *) fun get_defn [] = NONE | get_defn ((axname, ax)::axms) = (let val (lhs, _) = Logic.dest_equals ax (* equations only *) val c = head_of lhs val (s', T') = dest_Const c in if s=s' then let val typeSubs = Sign.typ_match thy (T', T) Vartab.empty in SOME (axname, monomorphic_term typeSubs ax) end else get_defn axms end handle ERROR => get_defn axms | TERM _ => get_defn axms | Type.TYPE_MATCH => get_defn axms) (* axiomatic type classes *) (* unit -> bool *) fun is_const_of_class () = (* I'm not quite sure if checking the name 's' is sufficient, *) (* or if we should also check the type 'T' *) s mem const_of_class_names (* inductive data types *) (* unit -> bool *) fun is_IDT_constructor () = (case body_type T of Type (s', _) => (case DatatypePackage.constrs_of thy s' of SOME constrs => Library.exists (fn c => (case c of Const (cname, ctype) => cname = s andalso Sign.typ_instance thy (T, ctype) | _ => raise REFUTE ("collect_axioms", "IDT constructor is not a constant"))) constrs | NONE => false) | _ => false) (* unit -> bool *) fun is_IDT_recursor () = (* I'm not quite sure if checking the name 's' is sufficient, *) (* or if we should also check the type 'T' *) s mem rec_names in if is_const_of_class () then (* axiomatic type classes: add "OFCLASS(?'a::c, c_class)" and *) (* the introduction rule "class.intro" as axioms *) let val class = Sign.class_of_const s val inclass = Logic.mk_inclass (TVar (("'a", 0), [class]), class) (* Term.term option *) val ofclass_ax = (SOME (specialize_type ((s, T), inclass)) handle Type.TYPE_MATCH => NONE) val intro_ax = (Option.map (fn t => specialize_type ((s, T), t)) (AList.lookup (op =) axioms (class ^ ".intro")) handle Type.TYPE_MATCH => NONE) val axs' = (case ofclass_ax of NONE => axs | SOME ax => if mem_term (ax, axs) then (* collect relevant type axioms *) collect_type_axioms (axs, T) else (immediate_output (" " ^ Sign.string_of_term (sign_of thy) ax); collect_term_axioms (ax :: axs, ax))) val axs'' = (case intro_ax of NONE => axs' | SOME ax => if mem_term (ax, axs') then (* collect relevant type axioms *) collect_type_axioms (axs', T) else (immediate_output (" " ^ s ^ ".intro"); collect_term_axioms (ax :: axs', ax))) in axs'' end else if is_IDT_constructor () then (* only collect relevant type axioms *) collect_type_axioms (axs, T) else if is_IDT_recursor () then (* only collect relevant type axioms *) collect_type_axioms (axs, T) else ( case get_defn axioms of SOME (axname, ax) => if mem_term (ax, axs) then (* collect relevant type axioms *) collect_type_axioms (axs, T) else (immediate_output (" " ^ axname); collect_term_axioms (ax :: axs, ax)) | NONE => (* collect relevant type axioms *) collect_type_axioms (axs, T) ) end | Free (_, T) => collect_type_axioms (axs, T) | Var (_, T) => collect_type_axioms (axs, T) | Bound i => axs | Abs (_, T, body) => collect_term_axioms (collect_type_axioms (axs, T), body) | t1 $ t2 => collect_term_axioms (collect_term_axioms (axs, t1), t2) (* universal closure over schematic variables *) (* Term.term -> Term.term *) fun close_form t = let (* (Term.indexname * Term.typ) list *) val vars = sort_wrt (fst o fst) (map dest_Var (term_vars t)) in Library.foldl (fn (t', ((x, i), T)) => (Term.all T) $ Abs (x, T, abstract_over (Var((x, i), T), t'))) (t, vars) end (* Term.term list *) val result = map close_form (collect_term_axioms ([], t)) val _ = writeln " ...done." in result end; (* ------------------------------------------------------------------------- *) (* ground_types: collects all ground types in a term (including argument *) (* types of other types), suppressing duplicates. Does not *) (* return function types, set types, non-recursive IDTs, or *) (* 'propT'. For IDTs, also the argument types of constructors *) (* are considered. *) (* ------------------------------------------------------------------------- *) (* theory -> Term.term -> Term.typ list *) fun ground_types thy t = let (* Term.typ * Term.typ list -> Term.typ list *) fun collect_types (T, acc) = if T mem acc then acc (* prevent infinite recursion (for IDTs) *) else (case T of Type ("fun", [T1, T2]) => collect_types (T1, collect_types (T2, acc)) | Type ("prop", []) => acc | Type ("set", [T1]) => collect_types (T1, acc) | Type (s, Ts) => (case DatatypePackage.datatype_info thy s of SOME info => (* inductive datatype *) let val index = #index info val descr = #descr info val (_, dtyps, constrs) = (the o AList.lookup (op =) descr) index val typ_assoc = dtyps ~~ Ts (* sanity check: every element in 'dtyps' must be a 'DtTFree' *) val _ = (if Library.exists (fn d => case d of DatatypeAux.DtTFree _ => false | _ => true) dtyps then raise REFUTE ("ground_types", "datatype argument (for type " ^ Sign.string_of_typ (sign_of thy) (Type (s, Ts)) ^ ") is not a variable") else ()) (* if the current type is a recursive IDT (i.e. a depth is required), add it to 'acc' *) val acc' = (if Library.exists (fn (_, ds) => Library.exists DatatypeAux.is_rec_type ds) constrs then T ins acc else acc) (* collect argument types *) val acc_args = foldr collect_types acc' Ts (* collect constructor types *) val acc_constrs = foldr collect_types acc_args (List.concat (map (fn (_, ds) => map (typ_of_dtyp descr typ_assoc) ds) constrs)) in acc_constrs end | NONE => (* not an inductive datatype, e.g. defined via "typedef" or "typedecl" *) T ins (foldr collect_types acc Ts)) | TFree _ => T ins acc | TVar _ => T ins acc) in it_term_types collect_types (t, []) end; (* ------------------------------------------------------------------------- *) (* string_of_typ: (rather naive) conversion from types to strings, used to *) (* look up the size of a type in 'sizes'. Parameterized *) (* types with different parameters (e.g. "'a list" vs. "bool *) (* list") are identified. *) (* ------------------------------------------------------------------------- *) (* Term.typ -> string *) fun string_of_typ (Type (s, _)) = s | string_of_typ (TFree (s, _)) = s | string_of_typ (TVar ((s,_), _)) = s; (* ------------------------------------------------------------------------- *) (* first_universe: returns the "first" (i.e. smallest) universe by assigning *) (* 'minsize' to every type for which no size is specified in *) (* 'sizes' *) (* ------------------------------------------------------------------------- *) (* Term.typ list -> (string * int) list -> int -> (Term.typ * int) list *) fun first_universe xs sizes minsize = let fun size_of_typ T = case AList.lookup (op =) sizes (string_of_typ T) of SOME n => n | NONE => minsize in map (fn T => (T, size_of_typ T)) xs end; (* ------------------------------------------------------------------------- *) (* next_universe: enumerates all universes (i.e. assignments of sizes to *) (* types), where the minimal size of a type is given by *) (* 'minsize', the maximal size is given by 'maxsize', and a *) (* type may have a fixed size given in 'sizes' *) (* ------------------------------------------------------------------------- *) (* (Term.typ * int) list -> (string * int) list -> int -> int -> (Term.typ * int) list option *) fun next_universe xs sizes minsize maxsize = let (* creates the "first" list of length 'len', where the sum of all list *) (* elements is 'sum', and the length of the list is 'len' *) (* int -> int -> int -> int list option *) fun make_first _ 0 sum = if sum=0 then SOME [] else NONE | make_first max len sum = if sum<=max orelse max<0 then Option.map (fn xs' => sum :: xs') (make_first max (len-1) 0) else Option.map (fn xs' => max :: xs') (make_first max (len-1) (sum-max)) (* enumerates all int lists with a fixed length, where 0<=x<='max' for *) (* all list elements x (unless 'max'<0) *) (* int -> int -> int -> int list -> int list option *) fun next max len sum [] = NONE | next max len sum [x] = (* we've reached the last list element, so there's no shift possible *) make_first max (len+1) (sum+x+1) (* increment 'sum' by 1 *) | next max len sum (x1::x2::xs) = if x1>0 andalso (x2<max orelse max<0) then (* we can shift *) SOME (valOf (make_first max (len+1) (sum+x1-1)) @ (x2+1) :: xs) else (* continue search *) next max (len+1) (sum+x1) (x2::xs) (* only consider those types for which the size is not fixed *) val mutables = List.filter (not o (AList.defined (op =) sizes) o string_of_typ o fst) xs (* subtract 'minsize' from every size (will be added again at the end) *) val diffs = map (fn (_, n) => n-minsize) mutables in case next (maxsize-minsize) 0 0 diffs of SOME diffs' => (* merge with those types for which the size is fixed *) SOME (snd (foldl_map (fn (ds, (T, _)) => case AList.lookup (op =) sizes (string_of_typ T) of SOME n => (ds, (T, n)) (* return the fixed size *) | NONE => (tl ds, (T, minsize + hd ds))) (* consume the head of 'ds', add 'minsize' *) (diffs', xs))) | NONE => NONE end; (* ------------------------------------------------------------------------- *) (* toTrue: converts the interpretation of a Boolean value to a propositional *) (* formula that is true iff the interpretation denotes "true" *) (* ------------------------------------------------------------------------- *) (* interpretation -> prop_formula *) fun toTrue (Leaf [fm, _]) = fm | toTrue _ = raise REFUTE ("toTrue", "interpretation does not denote a Boolean value"); (* ------------------------------------------------------------------------- *) (* toFalse: converts the interpretation of a Boolean value to a *) (* propositional formula that is true iff the interpretation *) (* denotes "false" *) (* ------------------------------------------------------------------------- *) (* interpretation -> prop_formula *) fun toFalse (Leaf [_, fm]) = fm | toFalse _ = raise REFUTE ("toFalse", "interpretation does not denote a Boolean value"); (* ------------------------------------------------------------------------- *) (* find_model: repeatedly calls 'interpret' with appropriate parameters, *) (* applies a SAT solver, and (in case a model is found) displays *) (* the model to the user by calling 'print_model' *) (* thy : the current theory *) (* {...} : parameters that control the translation/model generation *) (* t : term to be translated into a propositional formula *) (* negate : if true, find a model that makes 't' false (rather than true) *) (* Note: exception 'TimeOut' is raised if the algorithm does not terminate *) (* within 'maxtime' seconds (if 'maxtime' >0) *) (* ------------------------------------------------------------------------- *) (* theory -> params -> Term.term -> bool -> unit *) fun find_model thy {sizes, minsize, maxsize, maxvars, maxtime, satsolver} t negate = let (* unit -> unit *) fun wrapper () = let (* Term.term list *) val axioms = collect_axioms thy t (* Term.typ list *) val types = Library.foldl (fn (acc, t') => acc union (ground_types thy t')) ([], t :: axioms) val _ = writeln ("Ground types: " ^ (if null types then "none." else commas (map (Sign.string_of_typ (sign_of thy)) types))) (* we can only consider fragments of recursive IDTs, so we issue a *) (* warning if the formula contains a recursive IDT *) (* TODO: no warning needed for /positive/ occurrences of IDTs *) val _ = if Library.exists (fn Type (s, _) => (case DatatypePackage.datatype_info thy s of SOME info => (* inductive datatype *) let val index = #index info val descr = #descr info val (_, _, constrs) = (the o AList.lookup (op =) descr) index in (* recursive datatype? *) Library.exists (fn (_, ds) => Library.exists DatatypeAux.is_rec_type ds) constrs end | NONE => false) | _ => false) types then warning "Term contains a recursive datatype; countermodel(s) may be spurious!" else () (* (Term.typ * int) list -> unit *) fun find_model_loop universe = let val init_model = (universe, []) val init_args = {maxvars = maxvars, def_eq = false, next_idx = 1, bounds = [], wellformed = True} val _ = immediate_output ("Translating term (sizes: " ^ commas (map (fn (_, n) => string_of_int n) universe) ^ ") ...") (* translate 't' and all axioms *) val ((model, args), intrs) = foldl_map (fn ((m, a), t') => let val (i, m', a') = interpret thy m a t' in (* set 'def_eq' to 'true' *) ((m', {maxvars = #maxvars a', def_eq = true, next_idx = #next_idx a', bounds = #bounds a', wellformed = #wellformed a'}), i) end) ((init_model, init_args), t :: axioms) (* make 't' either true or false, and make all axioms true, and *) (* add the well-formedness side condition *) val fm_t = (if negate then toFalse else toTrue) (hd intrs) val fm_ax = PropLogic.all (map toTrue (tl intrs)) val fm = PropLogic.all [#wellformed args, fm_ax, fm_t] in immediate_output " invoking SAT solver..."; (case SatSolver.invoke_solver satsolver fm of SatSolver.SATISFIABLE assignment => (writeln " model found!"; writeln ("*** Model found: ***\n" ^ print_model thy model (fn i => case assignment i of SOME b => b | NONE => true))) | SatSolver.UNSATISFIABLE _ => (immediate_output " no model exists.\n"; case next_universe universe sizes minsize maxsize of SOME universe' => find_model_loop universe' | NONE => writeln "Search terminated, no larger universe within the given limits.") | SatSolver.UNKNOWN => (immediate_output " no model found.\n"; case next_universe universe sizes minsize maxsize of SOME universe' => find_model_loop universe' | NONE => writeln "Search terminated, no larger universe within the given limits.") ) handle SatSolver.NOT_CONFIGURED => error ("SAT solver " ^ quote satsolver ^ " is not configured.") end handle MAXVARS_EXCEEDED => writeln ("\nSearch terminated, number of Boolean variables (" ^ string_of_int maxvars ^ " allowed) exceeded.") in find_model_loop (first_universe types sizes minsize) end in (* some parameter sanity checks *) assert (minsize>=1) ("\"minsize\" is " ^ string_of_int minsize ^ ", must be at least 1"); assert (maxsize>=1) ("\"maxsize\" is " ^ string_of_int maxsize ^ ", must be at least 1"); assert (maxsize>=minsize) ("\"maxsize\" (=" ^ string_of_int maxsize ^ ") is less than \"minsize\" (=" ^ string_of_int minsize ^ ")."); assert (maxvars>=0) ("\"maxvars\" is " ^ string_of_int maxvars ^ ", must be at least 0"); assert (maxtime>=0) ("\"maxtime\" is " ^ string_of_int maxtime ^ ", must be at least 0"); (* enter loop with or without time limit *) writeln ("Trying to find a model that " ^ (if negate then "refutes" else "satisfies") ^ ": " ^ Sign.string_of_term (sign_of thy) t); if maxtime>0 then ( TimeLimit.timeLimit (Time.fromSeconds (Int.toLarge maxtime)) wrapper () handle TimeLimit.TimeOut => writeln ("\nSearch terminated, time limit (" ^ string_of_int maxtime ^ (if maxtime=1 then " second" else " seconds") ^ ") exceeded.") ) else wrapper () end; (* ------------------------------------------------------------------------- *) (* INTERFACE, PART 2: FINDING A MODEL *) (* ------------------------------------------------------------------------- *) (* ------------------------------------------------------------------------- *) (* satisfy_term: calls 'find_model' to find a model that satisfies 't' *) (* params : list of '(name, value)' pairs used to override default *) (* parameters *) (* ------------------------------------------------------------------------- *) (* theory -> (string * string) list -> Term.term -> unit *) fun satisfy_term thy params t = find_model thy (actual_params thy params) t false; (* ------------------------------------------------------------------------- *) (* refute_term: calls 'find_model' to find a model that refutes 't' *) (* params : list of '(name, value)' pairs used to override default *) (* parameters *) (* ------------------------------------------------------------------------- *) (* theory -> (string * string) list -> Term.term -> unit *) fun refute_term thy params t = let (* disallow schematic type variables, since we cannot properly negate *) (* terms containing them (their logical meaning is that there EXISTS a *) (* type s.t. ...; to refute such a formula, we would have to show that *) (* for ALL types, not ...) *) val _ = assert (null (term_tvars t)) "Term to be refuted contains schematic type variables" (* existential closure over schematic variables *) (* (Term.indexname * Term.typ) list *) val vars = sort_wrt (fst o fst) (map dest_Var (term_vars t)) (* Term.term *) val ex_closure = Library.foldl (fn (t', ((x, i), T)) => (HOLogic.exists_const T) $ Abs (x, T, abstract_over (Var ((x, i), T), t'))) (t, vars) (* If 't' is of type 'propT' (rather than 'boolT'), applying *) (* 'HOLogic.exists_const' is not type-correct. However, this *) (* is not really a problem as long as 'find_model' still *) (* interprets the resulting term correctly, without checking *) (* its type. *) in find_model thy (actual_params thy params) ex_closure true end; (* ------------------------------------------------------------------------- *) (* refute_subgoal: calls 'refute_term' on a specific subgoal *) (* params : list of '(name, value)' pairs used to override default *) (* parameters *) (* subgoal : 0-based index specifying the subgoal number *) (* ------------------------------------------------------------------------- *) (* theory -> (string * string) list -> Thm.thm -> int -> unit *) fun refute_subgoal thy params thm subgoal = refute_term thy params (List.nth (prems_of thm, subgoal)); (* ------------------------------------------------------------------------- *) (* INTERPRETERS: Auxiliary Functions *) (* ------------------------------------------------------------------------- *) (* ------------------------------------------------------------------------- *) (* make_constants: returns all interpretations that have the same tree *) (* structure as 'intr', but consist of unit vectors with *) (* 'True'/'False' only (no Boolean variables) *) (* ------------------------------------------------------------------------- *) (* interpretation -> interpretation list *) fun make_constants intr = let (* returns a list with all unit vectors of length n *) (* int -> interpretation list *) fun unit_vectors n = let (* returns the k-th unit vector of length n *) (* int * int -> interpretation *) fun unit_vector (k,n) = Leaf ((replicate (k-1) False) @ (True :: (replicate (n-k) False))) (* int -> interpretation list -> interpretation list *) fun unit_vectors_acc k vs = if k>n then [] else (unit_vector (k,n))::(unit_vectors_acc (k+1) vs) in unit_vectors_acc 1 [] end (* concatenates 'x' with every list in 'xss', returning a new list of lists *) (* 'a -> 'a list list -> 'a list list *) fun cons_list x xss = map (fn xs => x::xs) xss (* returns a list of lists, each one consisting of n (possibly identical) elements from 'xs' *) (* int -> 'a list -> 'a list list *) fun pick_all 1 xs = map (fn x => [x]) xs | pick_all n xs = let val rec_pick = pick_all (n-1) xs in Library.foldl (fn (acc, x) => (cons_list x rec_pick) @ acc) ([], xs) end in case intr of Leaf xs => unit_vectors (length xs) | Node xs => map (fn xs' => Node xs') (pick_all (length xs) (make_constants (hd xs))) end; (* ------------------------------------------------------------------------- *) (* size_of_type: returns the number of constants in a type (i.e. 'length *) (* (make_constants intr)', but implemented more efficiently) *) (* ------------------------------------------------------------------------- *) (* interpretation -> int *) fun size_of_type intr = let (* power (a, b) computes a^b, for a>=0, b>=0 *) (* int * int -> int *) fun power (a, 0) = 1 | power (a, 1) = a | power (a, b) = let val ab = power(a, b div 2) in ab * ab * power(a, b mod 2) end in case intr of Leaf xs => length xs | Node xs => power (size_of_type (hd xs), length xs) end; (* ------------------------------------------------------------------------- *) (* TT/FF: interpretations that denote "true" or "false", respectively *) (* ------------------------------------------------------------------------- *) (* interpretation *) val TT = Leaf [True, False]; val FF = Leaf [False, True]; (* ------------------------------------------------------------------------- *) (* make_equality: returns an interpretation that denotes (extensional) *) (* equality of two interpretations *) (* - two interpretations are 'equal' iff they are both defined and denote *) (* the same value *) (* - two interpretations are 'not_equal' iff they are both defined at least *) (* partially, and a defined part denotes different values *) (* - a completely undefined interpretation is neither 'equal' nor *) (* 'not_equal' to another interpretation *) (* ------------------------------------------------------------------------- *) (* We could in principle represent '=' on a type T by a particular *) (* interpretation. However, the size of that interpretation is quadratic *) (* in the size of T. Therefore comparing the interpretations 'i1' and *) (* 'i2' directly is more efficient than constructing the interpretation *) (* for equality on T first, and "applying" this interpretation to 'i1' *) (* and 'i2' in the usual way (cf. 'interpretation_apply') then. *) (* interpretation * interpretation -> interpretation *) fun make_equality (i1, i2) = let (* interpretation * interpretation -> prop_formula *) fun equal (i1, i2) = (case i1 of Leaf xs => (case i2 of Leaf ys => PropLogic.dot_product (xs, ys) (* defined and equal *) | Node _ => raise REFUTE ("make_equality", "second interpretation is higher")) | Node xs => (case i2 of Leaf _ => raise REFUTE ("make_equality", "first interpretation is higher") | Node ys => PropLogic.all (map equal (xs ~~ ys)))) (* interpretation * interpretation -> prop_formula *) fun not_equal (i1, i2) = (case i1 of Leaf xs => (case i2 of Leaf ys => PropLogic.all ((PropLogic.exists xs) :: (PropLogic.exists ys) :: (map (fn (x,y) => SOr (SNot x, SNot y)) (xs ~~ ys))) (* defined and not equal *) | Node _ => raise REFUTE ("make_equality", "second interpretation is higher")) | Node xs => (case i2 of Leaf _ => raise REFUTE ("make_equality", "first interpretation is higher") | Node ys => PropLogic.exists (map not_equal (xs ~~ ys)))) in (* a value may be undefined; therefore 'not_equal' is not just the *) (* negation of 'equal' *) Leaf [equal (i1, i2), not_equal (i1, i2)] end; (* ------------------------------------------------------------------------- *) (* make_def_equality: returns an interpretation that denotes (extensional) *) (* equality of two interpretations *) (* This function treats undefined/partially defined interpretations *) (* different from 'make_equality': two undefined interpretations are *) (* considered equal, while a defined interpretation is considered not equal *) (* to an undefined interpretation. *) (* ------------------------------------------------------------------------- *) (* interpretation * interpretation -> interpretation *) fun make_def_equality (i1, i2) = let (* interpretation * interpretation -> prop_formula *) fun equal (i1, i2) = (case i1 of Leaf xs => (case i2 of Leaf ys => SOr (PropLogic.dot_product (xs, ys), (* defined and equal, or both undefined *) SAnd (PropLogic.all (map SNot xs), PropLogic.all (map SNot ys))) | Node _ => raise REFUTE ("make_def_equality", "second interpretation is higher")) | Node xs => (case i2 of Leaf _ => raise REFUTE ("make_def_equality", "first interpretation is higher") | Node ys => PropLogic.all (map equal (xs ~~ ys)))) (* interpretation *) val eq = equal (i1, i2) in Leaf [eq, SNot eq] end; (* ------------------------------------------------------------------------- *) (* interpretation_apply: returns an interpretation that denotes the result *) (* of applying the function denoted by 'i2' to the *) (* argument denoted by 'i2' *) (* ------------------------------------------------------------------------- *) (* interpretation * interpretation -> interpretation *) fun interpretation_apply (i1, i2) = let (* interpretation * interpretation -> interpretation *) fun interpretation_disjunction (tr1,tr2) = tree_map (fn (xs,ys) => map (fn (x,y) => SOr(x,y)) (xs ~~ ys)) (tree_pair (tr1,tr2)) (* prop_formula * interpretation -> interpretation *) fun prop_formula_times_interpretation (fm,tr) = tree_map (map (fn x => SAnd (fm,x))) tr (* prop_formula list * interpretation list -> interpretation *) fun prop_formula_list_dot_product_interpretation_list ([fm],[tr]) = prop_formula_times_interpretation (fm,tr) | prop_formula_list_dot_product_interpretation_list (fm::fms,tr::trees) = interpretation_disjunction (prop_formula_times_interpretation (fm,tr), prop_formula_list_dot_product_interpretation_list (fms,trees)) | prop_formula_list_dot_product_interpretation_list (_,_) = raise REFUTE ("interpretation_apply", "empty list (in dot product)") (* concatenates 'x' with every list in 'xss', returning a new list of lists *) (* 'a -> 'a list list -> 'a list list *) fun cons_list x xss = map (fn xs => x::xs) xss (* returns a list of lists, each one consisting of one element from each element of 'xss' *) (* 'a list list -> 'a list list *) fun pick_all [xs] = map (fn x => [x]) xs | pick_all (xs::xss) = let val rec_pick = pick_all xss in Library.foldl (fn (acc, x) => (cons_list x rec_pick) @ acc) ([], xs) end | pick_all _ = raise REFUTE ("interpretation_apply", "empty list (in pick_all)") (* interpretation -> prop_formula list *) fun interpretation_to_prop_formula_list (Leaf xs) = xs | interpretation_to_prop_formula_list (Node trees) = map PropLogic.all (pick_all (map interpretation_to_prop_formula_list trees)) in case i1 of Leaf _ => raise REFUTE ("interpretation_apply", "first interpretation is a leaf") | Node xs => prop_formula_list_dot_product_interpretation_list (interpretation_to_prop_formula_list i2, xs) end; (* ------------------------------------------------------------------------- *) (* eta_expand: eta-expands a term 't' by adding 'i' lambda abstractions *) (* ------------------------------------------------------------------------- *) (* Term.term -> int -> Term.term *) fun eta_expand t i = let val Ts = binder_types (fastype_of t) in foldr (fn (T, t) => Abs ("<eta_expand>", T, t)) (list_comb (t, map Bound (i-1 downto 0))) (Library.take (i, Ts)) end; (* ------------------------------------------------------------------------- *) (* sum: returns the sum of a list 'xs' of integers *) (* ------------------------------------------------------------------------- *) (* int list -> int *) fun sum xs = foldl op+ 0 xs; (* ------------------------------------------------------------------------- *) (* product: returns the product of a list 'xs' of integers *) (* ------------------------------------------------------------------------- *) (* int list -> int *) fun product xs = foldl op* 1 xs; (* ------------------------------------------------------------------------- *) (* size_of_dtyp: the size of (an initial fragment of) an inductive data type *) (* is the sum (over its constructors) of the product (over *) (* their arguments) of the size of the argument types *) (* ------------------------------------------------------------------------- *) (* theory -> (Term.typ * int) list -> DatatypeAux.descr -> (DatatypeAux.dtyp * Term.typ) list -> (string * DatatypeAux.dtyp list) list -> int *) fun size_of_dtyp thy typ_sizes descr typ_assoc constructors = sum (map (fn (_, dtyps) => product (map (fn dtyp => let val T = typ_of_dtyp descr typ_assoc dtyp val (i, _, _) = interpret thy (typ_sizes, []) {maxvars=0, def_eq = false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) in size_of_type i end) dtyps)) constructors); (* ------------------------------------------------------------------------- *) (* INTERPRETERS: Actual Interpreters *) (* ------------------------------------------------------------------------- *) (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* simply typed lambda calculus: Isabelle's basic term syntax, with type *) (* variables, function types, and propT *) fun stlc_interpreter thy model args t = let val (typs, terms) = model val {maxvars, def_eq, next_idx, bounds, wellformed} = args (* Term.typ -> (interpretation * model * arguments) option *) fun interpret_groundterm T = let (* unit -> (interpretation * model * arguments) option *) fun interpret_groundtype () = let val size = (if T = Term.propT then 2 else (the o AList.lookup (op =) typs) T) (* the model MUST specify a size for ground types *) val next = next_idx+size val _ = (if next-1>maxvars andalso maxvars>0 then raise MAXVARS_EXCEEDED else ()) (* check if 'maxvars' is large enough *) (* prop_formula list *) val fms = map BoolVar (next_idx upto (next_idx+size-1)) (* interpretation *) val intr = Leaf fms (* prop_formula list -> prop_formula *) fun one_of_two_false [] = True | one_of_two_false (x::xs) = SAnd (PropLogic.all (map (fn x' => SOr (SNot x, SNot x')) xs), one_of_two_false xs) (* prop_formula *) val wf = one_of_two_false fms in (* extend the model, increase 'next_idx', add well-formedness condition *) SOME (intr, (typs, (t, intr)::terms), {maxvars = maxvars, def_eq = def_eq, next_idx = next, bounds = bounds, wellformed = SAnd (wellformed, wf)}) end in case T of Type ("fun", [T1, T2]) => let (* we create 'size_of_type (interpret (... T1))' different copies *) (* of the interpretation for 'T2', which are then combined into a *) (* single new interpretation *) val (i1, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T1)) (* make fresh copies, with different variable indices *) (* 'idx': next variable index *) (* 'n' : number of copies *) (* int -> int -> (int * interpretation list * prop_formula *) fun make_copies idx 0 = (idx, [], True) | make_copies idx n = let val (copy, _, new_args) = interpret thy (typs, []) {maxvars = maxvars, def_eq = false, next_idx = idx, bounds = [], wellformed = True} (Free ("dummy", T2)) val (idx', copies, wf') = make_copies (#next_idx new_args) (n-1) in (idx', copy :: copies, SAnd (#wellformed new_args, wf')) end val (next, copies, wf) = make_copies next_idx (size_of_type i1) (* combine copies into a single interpretation *) val intr = Node copies in (* extend the model, increase 'next_idx', add well-formedness condition *) SOME (intr, (typs, (t, intr)::terms), {maxvars = maxvars, def_eq = def_eq, next_idx = next, bounds = bounds, wellformed = SAnd (wellformed, wf)}) end | Type _ => interpret_groundtype () | TFree _ => interpret_groundtype () | TVar _ => interpret_groundtype () end in case AList.lookup (op =) terms t of SOME intr => (* return an existing interpretation *) SOME (intr, model, args) | NONE => (case t of Const (_, T) => interpret_groundterm T | Free (_, T) => interpret_groundterm T | Var (_, T) => interpret_groundterm T | Bound i => SOME (List.nth (#bounds args, i), model, args) | Abs (x, T, body) => let (* create all constants of type 'T' *) val (i, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) val constants = make_constants i (* interpret the 'body' separately for each constant *) val ((model', args'), bodies) = foldl_map (fn ((m, a), c) => let (* add 'c' to 'bounds' *) val (i', m', a') = interpret thy m {maxvars = #maxvars a, def_eq = #def_eq a, next_idx = #next_idx a, bounds = (c :: #bounds a), wellformed = #wellformed a} body in (* keep the new model m' and 'next_idx' and 'wellformed', but use old 'bounds' *) ((m', {maxvars = maxvars, def_eq = def_eq, next_idx = #next_idx a', bounds = bounds, wellformed = #wellformed a'}), i') end) ((model, args), constants) in SOME (Node bodies, model', args') end | t1 $ t2 => let (* interpret 't1' and 't2' separately *) val (intr1, model1, args1) = interpret thy model args t1 val (intr2, model2, args2) = interpret thy model1 args1 t2 in SOME (interpretation_apply (intr1, intr2), model2, args2) end) end; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) fun Pure_interpreter thy model args t = case t of Const ("all", _) $ t1 => (* in the meta-logic, 'all' MUST be followed by an argument term *) let val (i, m, a) = interpret thy model args t1 in case i of Node xs => let val fmTrue = PropLogic.all (map toTrue xs) val fmFalse = PropLogic.exists (map toFalse xs) in SOME (Leaf [fmTrue, fmFalse], m, a) end | _ => raise REFUTE ("Pure_interpreter", "\"all\" is not followed by a function") end | Const ("==", _) $ t1 $ t2 => let val (i1, m1, a1) = interpret thy model args t1 val (i2, m2, a2) = interpret thy m1 a1 t2 in (* we use either 'make_def_equality' or 'make_equality' *) SOME ((if #def_eq args then make_def_equality else make_equality) (i1, i2), m2, a2) end | Const ("==>", _) => (* simpler than translating 'Const ("==>", _) $ t1 $ t2' *) SOME (Node [Node [TT, FF], Node [TT, TT]], model, args) | _ => NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) fun HOLogic_interpreter thy model args t = (* ------------------------------------------------------------------------- *) (* Providing interpretations directly is more efficient than unfolding the *) (* logical constants. In HOL however, logical constants can themselves be *) (* arguments. "All" and "Ex" are then translated just like any other *) (* constant, with the relevant axiom being added by 'collect_axioms'. *) (* ------------------------------------------------------------------------- *) case t of Const ("Trueprop", _) => SOME (Node [TT, FF], model, args) | Const ("Not", _) => SOME (Node [FF, TT], model, args) | Const ("True", _) => (* redundant, since 'True' is also an IDT constructor *) SOME (TT, model, args) | Const ("False", _) => (* redundant, since 'False' is also an IDT constructor *) SOME (FF, model, args) | Const ("All", _) $ t1 => (* if "All" occurs without an argument (i.e. as argument to a higher-order *) (* function or predicate), it is handled by the 'stlc_interpreter' (i.e. *) (* by unfolding its definition) *) let val (i, m, a) = interpret thy model args t1 in case i of Node xs => let val fmTrue = PropLogic.all (map toTrue xs) val fmFalse = PropLogic.exists (map toFalse xs) in SOME (Leaf [fmTrue, fmFalse], m, a) end | _ => raise REFUTE ("HOLogic_interpreter", "\"All\" is followed by a non-function") end | Const ("Ex", _) $ t1 => (* if "Ex" occurs without an argument (i.e. as argument to a higher-order *) (* function or predicate), it is handled by the 'stlc_interpreter' (i.e. *) (* by unfolding its definition) *) let val (i, m, a) = interpret thy model args t1 in case i of Node xs => let val fmTrue = PropLogic.exists (map toTrue xs) val fmFalse = PropLogic.all (map toFalse xs) in SOME (Leaf [fmTrue, fmFalse], m, a) end | _ => raise REFUTE ("HOLogic_interpreter", "\"Ex\" is followed by a non-function") end | Const ("op =", _) $ t1 $ t2 => let val (i1, m1, a1) = interpret thy model args t1 val (i2, m2, a2) = interpret thy m1 a1 t2 in SOME (make_equality (i1, i2), m2, a2) end | Const ("op =", _) $ t1 => SOME (interpret thy model args (eta_expand t 1)) | Const ("op =", _) => SOME (interpret thy model args (eta_expand t 2)) | Const ("op &", _) $ t1 $ t2 => (* 3-valued logic *) let val (i1, m1, a1) = interpret thy model args t1 val (i2, m2, a2) = interpret thy m1 a1 t2 val fmTrue = PropLogic.SAnd (toTrue i1, toTrue i2) val fmFalse = PropLogic.SOr (toFalse i1, toFalse i2) in SOME (Leaf [fmTrue, fmFalse], m2, a2) end | Const ("op &", _) $ t1 => SOME (interpret thy model args (eta_expand t 1)) | Const ("op &", _) => SOME (interpret thy model args (eta_expand t 2)) (* SOME (Node [Node [TT, FF], Node [FF, FF]], model, args) *) | Const ("op |", _) $ t1 $ t2 => (* 3-valued logic *) let val (i1, m1, a1) = interpret thy model args t1 val (i2, m2, a2) = interpret thy m1 a1 t2 val fmTrue = PropLogic.SOr (toTrue i1, toTrue i2) val fmFalse = PropLogic.SAnd (toFalse i1, toFalse i2) in SOME (Leaf [fmTrue, fmFalse], m2, a2) end | Const ("op |", _) $ t1 => SOME (interpret thy model args (eta_expand t 1)) | Const ("op |", _) => SOME (interpret thy model args (eta_expand t 2)) (* SOME (Node [Node [TT, TT], Node [TT, FF]], model, args) *) | Const ("op -->", _) $ t1 $ t2 => (* 3-valued logic *) let val (i1, m1, a1) = interpret thy model args t1 val (i2, m2, a2) = interpret thy m1 a1 t2 val fmTrue = PropLogic.SOr (toFalse i1, toTrue i2) val fmFalse = PropLogic.SAnd (toTrue i1, toFalse i2) in SOME (Leaf [fmTrue, fmFalse], m2, a2) end | Const ("op -->", _) => (* SOME (Node [Node [TT, FF], Node [TT, TT]], model, args) *) SOME (interpret thy model args (eta_expand t 2)) | _ => NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) fun set_interpreter thy model args t = (* "T set" is isomorphic to "T --> bool" *) let val (typs, terms) = model in case AList.lookup (op =) terms t of SOME intr => (* return an existing interpretation *) SOME (intr, model, args) | NONE => (case t of Free (x, Type ("set", [T])) => let val (intr, _, args') = interpret thy (typs, []) args (Free (x, T --> HOLogic.boolT)) in SOME (intr, (typs, (t, intr)::terms), args') end | Var ((x, i), Type ("set", [T])) => let val (intr, _, args') = interpret thy (typs, []) args (Var ((x,i), T --> HOLogic.boolT)) in SOME (intr, (typs, (t, intr)::terms), args') end | Const (s, Type ("set", [T])) => let val (intr, _, args') = interpret thy (typs, []) args (Const (s, T --> HOLogic.boolT)) in SOME (intr, (typs, (t, intr)::terms), args') end (* 'Collect' == identity *) | Const ("Collect", _) $ t1 => SOME (interpret thy model args t1) | Const ("Collect", _) => SOME (interpret thy model args (eta_expand t 1)) (* 'op :' == application *) | Const ("op :", _) $ t1 $ t2 => SOME (interpret thy model args (t2 $ t1)) | Const ("op :", _) $ t1 => SOME (interpret thy model args (eta_expand t 1)) | Const ("op :", _) => SOME (interpret thy model args (eta_expand t 2)) | _ => NONE) end; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* interprets variables and constants whose type is an IDT; constructors of *) (* IDTs are properly interpreted by 'IDT_constructor_interpreter' however *) fun IDT_interpreter thy model args t = let val (typs, terms) = model (* Term.typ -> (interpretation * model * arguments) option *) fun interpret_term (Type (s, Ts)) = (case DatatypePackage.datatype_info thy s of SOME info => (* inductive datatype *) let (* int option -- only recursive IDTs have an associated depth *) val depth = AList.lookup (op =) typs (Type (s, Ts)) in if depth = (SOME 0) then (* termination condition to avoid infinite recursion *) (* return a leaf of size 0 *) SOME (Leaf [], model, args) else let val index = #index info val descr = #descr info val (_, dtyps, constrs) = (the o AList.lookup (op =) descr) index val typ_assoc = dtyps ~~ Ts (* sanity check: every element in 'dtyps' must be a 'DtTFree' *) val _ = (if Library.exists (fn d => case d of DatatypeAux.DtTFree _ => false | _ => true) dtyps then raise REFUTE ("IDT_interpreter", "datatype argument (for type " ^ Sign.string_of_typ (sign_of thy) (Type (s, Ts)) ^ ") is not a variable") else ()) (* if the model specifies a depth for the current type, decrement it to avoid infinite recursion *) val typs' = case depth of NONE => typs | SOME n => AList.update (op =) (Type (s, Ts), n-1) typs (* recursively compute the size of the datatype *) val size = size_of_dtyp thy typs' descr typ_assoc constrs val next_idx = #next_idx args val next = next_idx+size val _ = (if next-1>(#maxvars args) andalso (#maxvars args)>0 then raise MAXVARS_EXCEEDED else ()) (* check if 'maxvars' is large enough *) (* prop_formula list *) val fms = map BoolVar (next_idx upto (next_idx+size-1)) (* interpretation *) val intr = Leaf fms (* prop_formula list -> prop_formula *) fun one_of_two_false [] = True | one_of_two_false (x::xs) = SAnd (PropLogic.all (map (fn x' => SOr (SNot x, SNot x')) xs), one_of_two_false xs) (* prop_formula *) val wf = one_of_two_false fms in (* extend the model, increase 'next_idx', add well-formedness condition *) SOME (intr, (typs, (t, intr)::terms), {maxvars = #maxvars args, def_eq = #def_eq args, next_idx = next, bounds = #bounds args, wellformed = SAnd (#wellformed args, wf)}) end end | NONE => (* not an inductive datatype *) NONE) | interpret_term _ = (* a (free or schematic) type variable *) NONE in case AList.lookup (op =) terms t of SOME intr => (* return an existing interpretation *) SOME (intr, model, args) | NONE => (case t of Free (_, T) => interpret_term T | Var (_, T) => interpret_term T | Const (_, T) => interpret_term T | _ => NONE) end; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) fun IDT_constructor_interpreter thy model args t = let val (typs, terms) = model in case AList.lookup (op =) terms t of SOME intr => (* return an existing interpretation *) SOME (intr, model, args) | NONE => (case t of Const (s, T) => (case body_type T of Type (s', Ts') => (case DatatypePackage.datatype_info thy s' of SOME info => (* body type is an inductive datatype *) let val index = #index info val descr = #descr info val (_, dtyps, constrs) = (the o AList.lookup (op =) descr) index val typ_assoc = dtyps ~~ Ts' (* sanity check: every element in 'dtyps' must be a 'DtTFree' *) val _ = (if Library.exists (fn d => case d of DatatypeAux.DtTFree _ => false | _ => true) dtyps then raise REFUTE ("IDT_constructor_interpreter", "datatype argument (for type " ^ Sign.string_of_typ (sign_of thy) (Type (s', Ts')) ^ ") is not a variable") else ()) (* split the constructors into those occuring before/after 'Const (s, T)' *) val (constrs1, constrs2) = take_prefix (fn (cname, ctypes) => not (cname = s andalso Sign.typ_instance thy (T, map (typ_of_dtyp descr typ_assoc) ctypes ---> Type (s', Ts')))) constrs in case constrs2 of [] => (* 'Const (s, T)' is not a constructor of this datatype *) NONE | (_, ctypes)::cs => let (* compute the total size of the datatype (with the current depth) *) val (i, _, _) = interpret thy (typs, []) {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type (s', Ts'))) val total = size_of_type i (* int option -- only recursive IDTs have an associated depth *) val depth = AList.lookup (op =) typs (Type (s', Ts')) val typs' = (case depth of NONE => typs | SOME n => AList.update (op =) (Type (s', Ts'), n-1) typs) (* returns an interpretation where everything is mapped to "undefined" *) (* DatatypeAux.dtyp list -> interpretation *) fun make_undef [] = Leaf (replicate total False) | make_undef (d::ds) = let (* compute the current size of the type 'd' *) val T = typ_of_dtyp descr typ_assoc d val (i, _, _) = interpret thy (typs, []) {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) val size = size_of_type i in Node (replicate size (make_undef ds)) end (* returns the interpretation for a constructor at depth 1 *) (* int * DatatypeAux.dtyp list -> int * interpretation *) fun make_constr (offset, []) = if offset<total then (offset+1, Leaf ((replicate offset False) @ True :: (replicate (total-offset-1) False))) else raise REFUTE ("IDT_constructor_interpreter", "offset >= total") | make_constr (offset, d::ds) = let (* compute the current and the old size of the type 'd' *) val T = typ_of_dtyp descr typ_assoc d val (i, _, _) = interpret thy (typs, []) {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) val size = size_of_type i val (i', _, _) = interpret thy (typs', []) {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) val size' = size_of_type i' (* sanity check *) val _ = if size < size' then raise REFUTE ("IDT_constructor_interpreter", "current size is less than old size") else () (* int * interpretation list *) val (new_offset, intrs) = foldl_map make_constr (offset, replicate size' ds) (* interpretation list *) val undefs = replicate (size - size') (make_undef ds) in (* elements that exist at the previous depth are mapped to a defined *) (* value, while new elements are mapped to "undefined" by the *) (* recursive constructor *) (new_offset, Node (intrs @ undefs)) end (* extends the interpretation for a constructor (both recursive *) (* and non-recursive) obtained at depth n (n>=1) to depth n+1 *) (* int * DatatypeAux.dtyp list * interpretation -> int * interpretation *) fun extend_constr (offset, [], Leaf xs) = let (* returns the k-th unit vector of length n *) (* int * int -> interpretation *) fun unit_vector (k, n) = Leaf ((replicate (k-1) False) @ (True :: (replicate (n-k) False))) (* int *) val k = find_index_eq True xs in if k=(~1) then (* if the element was mapped to "undefined" before, map it to *) (* the value given by 'offset' now (and extend the length of *) (* the leaf) *) (offset+1, unit_vector (offset+1, total)) else (* if the element was already mapped to a defined value, map it *) (* to the same value again, just extend the length of the leaf, *) (* do not increment the 'offset' *) (offset, unit_vector (k+1, total)) end | extend_constr (_, [], Node _) = raise REFUTE ("IDT_constructor_interpreter", "interpretation for constructor (with no arguments left) is a node") | extend_constr (offset, d::ds, Node xs) = let (* compute the size of the type 'd' *) val T = typ_of_dtyp descr typ_assoc d val (i, _, _) = interpret thy (typs, []) {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) val size = size_of_type i (* sanity check *) val _ = if size < length xs then raise REFUTE ("IDT_constructor_interpreter", "new size of type is less than old size") else () (* extend the existing interpretations *) (* int * interpretation list *) val (new_offset, intrs) = foldl_map (fn (off, i) => extend_constr (off, ds, i)) (offset, xs) (* new elements of the type 'd' are mapped to "undefined" *) val undefs = replicate (size - length xs) (make_undef ds) in (new_offset, Node (intrs @ undefs)) end | extend_constr (_, d::ds, Leaf _) = raise REFUTE ("IDT_constructor_interpreter", "interpretation for constructor (with arguments left) is a leaf") (* returns 'true' iff the constructor has a recursive argument *) (* DatatypeAux.dtyp list -> bool *) fun is_rec_constr ds = Library.exists DatatypeAux.is_rec_type ds (* constructors before 'Const (s, T)' generate elements of the datatype *) val offset = size_of_dtyp thy typs' descr typ_assoc constrs1 in case depth of NONE => (* equivalent to a depth of 1 *) SOME (snd (make_constr (offset, ctypes)), model, args) | SOME 0 => raise REFUTE ("IDT_constructor_interpreter", "depth is 0") | SOME 1 => SOME (snd (make_constr (offset, ctypes)), model, args) | SOME n => (* n > 1 *) let (* interpret the constructor at depth-1 *) val (iC, _, _) = interpret thy (typs', []) {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Const (s, T)) (* elements generated by the constructor at depth-1 must be added to 'offset' *) (* interpretation -> int *) fun number_of_defined_elements (Leaf xs) = if find_index_eq True xs = (~1) then 0 else 1 | number_of_defined_elements (Node xs) = sum (map number_of_defined_elements xs) (* int *) val offset' = offset + number_of_defined_elements iC in SOME (snd (extend_constr (offset', ctypes, iC)), model, args) end end end | NONE => (* body type is not an inductive datatype *) NONE) | _ => (* body type is a (free or schematic) type variable *) NONE) | _ => (* term is not a constant *) NONE) end; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* Difficult code ahead. Make sure you understand the 'IDT_constructor_interpreter' *) (* and the order in which it enumerates elements of an IDT before you try to *) (* understand this function. *) fun IDT_recursion_interpreter thy model args t = case strip_comb t of (* careful: here we descend arbitrarily deep into 't', *) (* possibly before any other interpreter for atomic *) (* terms has had a chance to look at 't' *) (Const (s, T), params) => (* iterate over all datatypes in 'thy' *) Symtab.foldl (fn (result, (_, info)) => case result of SOME _ => result (* just keep 'result' *) | NONE => if s mem (#rec_names info) then (* we do have a recursion operator of the datatype given by 'info', *) (* or of a mutually recursive datatype *) let val index = #index info val descr = #descr info val (dtname, dtyps, _) = (the o AList.lookup (op =) descr) index (* number of all constructors, including those of different *) (* (mutually recursive) datatypes within the same descriptor 'descr' *) val mconstrs_count = sum (map (fn (_, (_, _, cs)) => length cs) descr) val params_count = length params (* the type of a recursion operator: [T1, ..., Tn, IDT] ---> Tresult *) val IDT = List.nth (binder_types T, mconstrs_count) in if (fst o dest_Type) IDT <> dtname then (* recursion operator of a mutually recursive datatype *) NONE else if mconstrs_count < params_count then (* too many actual parameters; for now we'll use the *) (* 'stlc_interpreter' to strip off one application *) NONE else if mconstrs_count > params_count then (* too few actual parameters; we use eta expansion *) (* Note that the resulting expansion of lambda abstractions *) (* by the 'stlc_interpreter' may be rather slow (depending on *) (* the argument types and the size of the IDT, of course). *) SOME (interpret thy model args (eta_expand t (mconstrs_count - params_count))) else (* mconstrs_count = params_count *) let (* interpret each parameter separately *) val ((model', args'), p_intrs) = foldl_map (fn ((m, a), p) => let val (i, m', a') = interpret thy m a p in ((m', a'), i) end) ((model, args), params) val (typs, _) = model' val typ_assoc = dtyps ~~ (snd o dest_Type) IDT (* interpret each constructor in the descriptor (including *) (* those of mutually recursive datatypes) *) (* (int * interpretation list) list *) val mc_intrs = map (fn (idx, (_, _, cs)) => let val c_return_typ = typ_of_dtyp descr typ_assoc (DatatypeAux.DtRec idx) in (idx, map (fn (cname, cargs) => (#1 o interpret thy (typs, []) {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True}) (Const (cname, map (typ_of_dtyp descr typ_assoc) cargs ---> c_return_typ))) cs) end) descr (* the recursion operator is a function that maps every element of *) (* the inductive datatype (and of mutually recursive types) to an *) (* element of some result type; an array entry of NONE means that *) (* the actual result has not been computed yet *) (* (int * interpretation option Array.array) list *) val INTRS = map (fn (idx, _) => let val T = typ_of_dtyp descr typ_assoc (DatatypeAux.DtRec idx) val (i, _, _) = interpret thy (typs, []) {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) val size = size_of_type i in (idx, Array.array (size, NONE)) end) descr (* takes an interpretation, and if some leaf of this interpretation *) (* is the 'elem'-th element of the type, the indices of the arguments *) (* leading to this leaf are returned *) (* interpretation -> int -> int list option *) fun get_args (Leaf xs) elem = if find_index_eq True xs = elem then SOME [] else NONE | get_args (Node xs) elem = let (* interpretation * int -> int list option *) fun search ([], _) = NONE | search (x::xs, n) = (case get_args x elem of SOME result => SOME (n::result) | NONE => search (xs, n+1)) in search (xs, 0) end (* returns the index of the constructor and indices for its *) (* arguments that generate the 'elem'-th element of the datatype *) (* given by 'idx' *) (* int -> int -> int * int list *) fun get_cargs idx elem = let (* int * interpretation list -> int * int list *) fun get_cargs_rec (_, []) = raise REFUTE ("IDT_recursion_interpreter", "no matching constructor found for element " ^ string_of_int elem ^ " in datatype " ^ Sign.string_of_typ (sign_of thy) IDT ^ " (datatype index " ^ string_of_int idx ^ ")") | get_cargs_rec (n, x::xs) = (case get_args x elem of SOME args => (n, args) | NONE => get_cargs_rec (n+1, xs)) in get_cargs_rec (0, (the o AList.lookup (op =) mc_intrs) idx) end (* returns the number of constructors in datatypes that occur in *) (* the descriptor 'descr' before the datatype given by 'idx' *) fun get_coffset idx = let fun get_coffset_acc _ [] = raise REFUTE ("IDT_recursion_interpreter", "index " ^ string_of_int idx ^ " not found in descriptor") | get_coffset_acc sum ((i, (_, _, cs))::descr') = if i=idx then sum else get_coffset_acc (sum + length cs) descr' in get_coffset_acc 0 descr end (* computes one entry in INTRS, and recursively all entries needed for it, *) (* where 'idx' gives the datatype and 'elem' the element of it *) (* int -> int -> interpretation *) fun compute_array_entry idx elem = case Array.sub ((the o AList.lookup (op =) INTRS) idx, elem) of SOME result => (* simply return the previously computed result *) result | NONE => let (* int * int list *) val (c, args) = get_cargs idx elem (* interpretation * int list -> interpretation *) fun select_subtree (tr, []) = tr (* return the whole tree *) | select_subtree (Leaf _, _) = raise REFUTE ("IDT_recursion_interpreter", "interpretation for parameter is a leaf; cannot select a subtree") | select_subtree (Node tr, x::xs) = select_subtree (List.nth (tr, x), xs) (* select the correct subtree of the parameter corresponding to constructor 'c' *) val p_intr = select_subtree (List.nth (p_intrs, get_coffset idx + c), args) (* find the indices of the constructor's recursive arguments *) val (_, _, constrs) = (the o AList.lookup (op =) descr) idx val constr_args = (snd o List.nth) (constrs, c) val rec_args = List.filter (DatatypeAux.is_rec_type o fst) (constr_args ~~ args) val rec_args' = map (fn (dtyp, elem) => (DatatypeAux.dest_DtRec dtyp, elem)) rec_args (* apply 'p_intr' to recursively computed results *) val result = foldl (fn ((idx, elem), intr) => interpretation_apply (intr, compute_array_entry idx elem)) p_intr rec_args' (* update 'INTRS' *) val _ = Array.update ((the o AList.lookup (op =) INTRS) idx, elem, SOME result) in result end (* compute all entries in INTRS for the current datatype (given by 'index') *) (* TODO: we can use Array.modify instead once PolyML conforms to the ML standard *) (* (int * 'a -> 'a) -> 'a array -> unit *) fun modifyi f arr = let val size = Array.length arr fun modifyi_loop i = if i < size then ( Array.update (arr, i, f (i, Array.sub (arr, i))); modifyi_loop (i+1) ) else () in modifyi_loop 0 end val _ = modifyi (fn (i, _) => SOME (compute_array_entry index i)) ((the o AList.lookup (op =) INTRS) index) (* 'a Array.array -> 'a list *) fun toList arr = Array.foldr op:: [] arr in (* return the part of 'INTRS' that corresponds to the current datatype *) SOME ((Node o map the o toList o the o AList.lookup (op =) INTRS) index, model', args') end end else NONE (* not a recursion operator of this datatype *) ) (NONE, DatatypePackage.get_datatypes thy) | _ => (* head of term is not a constant *) NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* only an optimization: 'card' could in principle be interpreted with *) (* interpreters available already (using its definition), but the code *) (* below is more efficient *) fun Finite_Set_card_interpreter thy model args t = case t of Const ("Finite_Set.card", Type ("fun", [Type ("set", [T]), Type ("nat", [])])) => let val (i_nat, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("nat", []))) val size_nat = size_of_type i_nat val (i_set, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("set", [T]))) val constants = make_constants i_set (* interpretation -> int *) fun number_of_elements (Node xs) = Library.foldl (fn (n, x) => if x=TT then n+1 else if x=FF then n else raise REFUTE ("Finite_Set_card_interpreter", "interpretation for set type does not yield a Boolean")) (0, xs) | number_of_elements (Leaf _) = raise REFUTE ("Finite_Set_card_interpreter", "interpretation for set type is a leaf") (* takes an interpretation for a set and returns an interpretation for a 'nat' *) (* interpretation -> interpretation *) fun card i = let val n = number_of_elements i in if n<size_nat then Leaf ((replicate n False) @ True :: (replicate (size_nat-n-1) False)) else Leaf (replicate size_nat False) end in SOME (Node (map card constants), model, args) end | _ => NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* only an optimization: 'Finites' could in principle be interpreted with *) (* interpreters available already (using its definition), but the code *) (* below is more efficient *) fun Finite_Set_Finites_interpreter thy model args t = case t of Const ("Finite_Set.Finites", Type ("set", [Type ("set", [T])])) => let val (i_set, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("set", [T]))) val size_set = size_of_type i_set in (* we only consider finite models anyway, hence EVERY set is in "Finites" *) SOME (Node (replicate size_set TT), model, args) end | _ => NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* only an optimization: 'op <' could in principle be interpreted with *) (* interpreters available already (using its definition), but the code *) (* below is more efficient *) fun Nat_less_interpreter thy model args t = case t of Const ("op <", Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("bool", [])])])) => let val (i_nat, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("nat", []))) val size_nat = size_of_type i_nat (* int -> interpretation *) (* the 'n'-th nat is not less than the first 'n' nats, while it *) (* is less than the remaining 'size_nat - n' nats *) fun less n = Node ((replicate n FF) @ (replicate (size_nat - n) TT)) in SOME (Node (map less (1 upto size_nat)), model, args) end | _ => NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* only an optimization: 'op +' could in principle be interpreted with *) (* interpreters available already (using its definition), but the code *) (* below is more efficient *) fun Nat_plus_interpreter thy model args t = case t of Const ("op +", Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => let val (i_nat, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("nat", []))) val size_nat = size_of_type i_nat (* int -> int -> interpretation *) fun plus m n = let val element = (m+n)+1 in if element > size_nat then Leaf (replicate size_nat False) else Leaf ((replicate (element-1) False) @ True :: (replicate (size_nat - element) False)) end in SOME (Node (map (fn m => Node (map (plus m) (0 upto size_nat-1))) (0 upto size_nat-1)), model, args) end | _ => NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* only an optimization: 'op -' could in principle be interpreted with *) (* interpreters available already (using its definition), but the code *) (* below is more efficient *) fun Nat_minus_interpreter thy model args t = case t of Const ("op -", Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => let val (i_nat, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("nat", []))) val size_nat = size_of_type i_nat (* int -> int -> interpretation *) fun minus m n = let val element = Int.max (m-n, 0) + 1 in Leaf ((replicate (element-1) False) @ True :: (replicate (size_nat - element) False)) end in SOME (Node (map (fn m => Node (map (minus m) (0 upto size_nat-1))) (0 upto size_nat-1)), model, args) end | _ => NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* only an optimization: 'op *' could in principle be interpreted with *) (* interpreters available already (using its definition), but the code *) (* below is more efficient *) fun Nat_mult_interpreter thy model args t = case t of Const ("op *", Type ("fun", [Type ("nat", []), Type ("fun", [Type ("nat", []), Type ("nat", [])])])) => let val (i_nat, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("nat", []))) val size_nat = size_of_type i_nat (* nat -> nat -> interpretation *) fun mult m n = let val element = (m*n)+1 in if element > size_nat then Leaf (replicate size_nat False) else Leaf ((replicate (element-1) False) @ True :: (replicate (size_nat - element) False)) end in SOME (Node (map (fn m => Node (map (mult m) (0 upto size_nat-1))) (0 upto size_nat-1)), model, args) end | _ => NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* only an optimization: 'op @' could in principle be interpreted with *) (* interpreters available already (using its definition), but the code *) (* below is more efficient *) fun List_append_interpreter thy model args t = case t of Const ("List.op @", Type ("fun", [Type ("List.list", [T]), Type ("fun", [Type ("List.list", [_]), Type ("List.list", [_])])])) => let val (i_elem, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) val size_elem = size_of_type i_elem val (i_list, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("List.list", [T]))) val size_list = size_of_type i_list (* power (a, b) computes a^b, for a>=0, b>=0 *) (* int * int -> int *) fun power (a, 0) = 1 | power (a, 1) = a | power (a, b) = let val ab = power(a, b div 2) in ab * ab * power(a, b mod 2) end (* log (a, b) computes floor(log_a(b)), i.e. the largest integer x s.t. a^x <= b, for a>=2, b>=1 *) (* int * int -> int *) fun log (a, b) = let fun logloop (ax, x) = if ax > b then x-1 else logloop (a * ax, x+1) in logloop (1, 0) end (* nat -> nat -> interpretation *) fun append m n = let (* The following formula depends on the order in which lists are *) (* enumerated by the 'IDT_constructor_interpreter'. It took me *) (* a while to come up with this formula. *) val element = n + m * (if size_elem = 1 then 1 else power (size_elem, log (size_elem, n+1))) + 1 in if element > size_list then Leaf (replicate size_list False) else Leaf ((replicate (element-1) False) @ True :: (replicate (size_list - element) False)) end in SOME (Node (map (fn m => Node (map (append m) (0 upto size_list-1))) (0 upto size_list-1)), model, args) end | _ => NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* only an optimization: 'lfp' could in principle be interpreted with *) (* interpreters available already (using its definition), but the code *) (* below is more efficient *) fun Lfp_lfp_interpreter thy model args t = case t of Const ("Lfp.lfp", Type ("fun", [Type ("fun", [Type ("set", [T]), Type ("set", [_])]), Type ("set", [_])])) => let val (i_elem, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) val size_elem = size_of_type i_elem (* the universe (i.e. the set that contains every element) *) val i_univ = Node (replicate size_elem TT) (* all sets with elements from type 'T' *) val (i_set, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("set", [T]))) val i_sets = make_constants i_set (* all functions that map sets to sets *) val (i_fun, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("fun", [Type ("set", [T]), Type ("set", [T])]))) val i_funs = make_constants i_fun (* "lfp(f) == Inter({u. f(u) <= u})" *) (* interpretation * interpretation -> bool *) fun is_subset (Node subs, Node sups) = List.all (fn (sub, sup) => (sub = FF) orelse (sup = TT)) (subs ~~ sups) | is_subset (_, _) = raise REFUTE ("Lfp_lfp_interpreter", "is_subset: interpretation for set is not a node") (* interpretation * interpretation -> interpretation *) fun intersection (Node xs, Node ys) = Node (map (fn (x, y) => if (x = TT) andalso (y = TT) then TT else FF) (xs ~~ ys)) | intersection (_, _) = raise REFUTE ("Lfp_lfp_interpreter", "intersection: interpretation for set is not a node") (* interpretation -> interpretaion *) fun lfp (Node resultsets) = foldl (fn ((set, resultset), acc) => if is_subset (resultset, set) then intersection (acc, set) else acc) i_univ (i_sets ~~ resultsets) | lfp _ = raise REFUTE ("Lfp_lfp_interpreter", "lfp: interpretation for function is not a node") in SOME (Node (map lfp i_funs), model, args) end | _ => NONE; (* theory -> model -> arguments -> Term.term -> (interpretation * model * arguments) option *) (* only an optimization: 'gfp' could in principle be interpreted with *) (* interpreters available already (using its definition), but the code *) (* below is more efficient *) fun Gfp_gfp_interpreter thy model args t = case t of Const ("Gfp.gfp", Type ("fun", [Type ("fun", [Type ("set", [T]), Type ("set", [_])]), Type ("set", [_])])) => let nonfix union (*because "union" is used below*) val (i_elem, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) val size_elem = size_of_type i_elem (* the universe (i.e. the set that contains every element) *) val i_univ = Node (replicate size_elem TT) (* all sets with elements from type 'T' *) val (i_set, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("set", [T]))) val i_sets = make_constants i_set (* all functions that map sets to sets *) val (i_fun, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", Type ("fun", [Type ("set", [T]), Type ("set", [T])]))) val i_funs = make_constants i_fun (* "gfp(f) == Union({u. u <= f(u)})" *) (* interpretation * interpretation -> bool *) fun is_subset (Node subs, Node sups) = List.all (fn (sub, sup) => (sub = FF) orelse (sup = TT)) (subs ~~ sups) | is_subset (_, _) = raise REFUTE ("Gfp_gfp_interpreter", "is_subset: interpretation for set is not a node") (* interpretation * interpretation -> interpretation *) fun union (Node xs, Node ys) = Node (map (fn (x,y) => if x=TT orelse y=TT then TT else FF) (xs ~~ ys)) | union (_, _) = raise REFUTE ("Gfp_gfp_interpreter", "union: interpretation for set is not a node") (* interpretation -> interpretaion *) fun gfp (Node resultsets) = foldl (fn ((set, resultset), acc) => if is_subset (set, resultset) then union (acc, set) else acc) i_univ (i_sets ~~ resultsets) | gfp _ = raise REFUTE ("Gfp_gfp_interpreter", "gfp: interpretation for function is not a node") in SOME (Node (map gfp i_funs), model, args) end | _ => NONE; (* ------------------------------------------------------------------------- *) (* PRINTERS *) (* ------------------------------------------------------------------------- *) (* theory -> model -> Term.term -> interpretation -> (int -> bool) -> Term.term option *) fun stlc_printer thy model t intr assignment = let (* Term.term -> Term.typ option *) fun typeof (Free (_, T)) = SOME T | typeof (Var (_, T)) = SOME T | typeof (Const (_, T)) = SOME T | typeof _ = NONE (* string -> string *) fun strip_leading_quote s = (implode o (fn ss => case ss of [] => [] | x::xs => if x="'" then xs else ss) o explode) s (* Term.typ -> string *) fun string_of_typ (Type (s, _)) = s | string_of_typ (TFree (x, _)) = strip_leading_quote x | string_of_typ (TVar ((x,i), _)) = strip_leading_quote x ^ string_of_int i (* interpretation -> int *) fun index_from_interpretation (Leaf xs) = find_index (PropLogic.eval assignment) xs | index_from_interpretation _ = raise REFUTE ("stlc_printer", "interpretation for ground type is not a leaf") in case typeof t of SOME T => (case T of Type ("fun", [T1, T2]) => let (* create all constants of type 'T1' *) val (i, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T1)) val constants = make_constants i (* interpretation list *) val results = (case intr of Node xs => xs | _ => raise REFUTE ("stlc_printer", "interpretation for function type is a leaf")) (* Term.term list *) val pairs = map (fn (arg, result) => HOLogic.mk_prod (print thy model (Free ("dummy", T1)) arg assignment, print thy model (Free ("dummy", T2)) result assignment)) (constants ~~ results) (* Term.typ *) val HOLogic_prodT = HOLogic.mk_prodT (T1, T2) val HOLogic_setT = HOLogic.mk_setT HOLogic_prodT (* Term.term *) val HOLogic_empty_set = Const ("{}", HOLogic_setT) val HOLogic_insert = Const ("insert", HOLogic_prodT --> HOLogic_setT --> HOLogic_setT) in SOME (foldr (fn (pair, acc) => HOLogic_insert $ pair $ acc) HOLogic_empty_set pairs) end | Type ("prop", []) => (case index_from_interpretation intr of (~1) => SOME (HOLogic.mk_Trueprop (Const ("arbitrary", HOLogic.boolT))) | 0 => SOME (HOLogic.mk_Trueprop HOLogic.true_const) | 1 => SOME (HOLogic.mk_Trueprop HOLogic.false_const) | _ => raise REFUTE ("stlc_interpreter", "illegal interpretation for a propositional value")) | Type _ => if index_from_interpretation intr = (~1) then SOME (Const ("arbitrary", T)) else SOME (Const (string_of_typ T ^ string_of_int (index_from_interpretation intr), T)) | TFree _ => if index_from_interpretation intr = (~1) then SOME (Const ("arbitrary", T)) else SOME (Const (string_of_typ T ^ string_of_int (index_from_interpretation intr), T)) | TVar _ => if index_from_interpretation intr = (~1) then SOME (Const ("arbitrary", T)) else SOME (Const (string_of_typ T ^ string_of_int (index_from_interpretation intr), T))) | NONE => NONE end; (* theory -> model -> Term.term -> interpretation -> (int -> bool) -> string option *) fun set_printer thy model t intr assignment = let (* Term.term -> Term.typ option *) fun typeof (Free (_, T)) = SOME T | typeof (Var (_, T)) = SOME T | typeof (Const (_, T)) = SOME T | typeof _ = NONE in case typeof t of SOME (Type ("set", [T])) => let (* create all constants of type 'T' *) val (i, _, _) = interpret thy model {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", T)) val constants = make_constants i (* interpretation list *) val results = (case intr of Node xs => xs | _ => raise REFUTE ("set_printer", "interpretation for set type is a leaf")) (* Term.term list *) val elements = List.mapPartial (fn (arg, result) => case result of Leaf [fmTrue, fmFalse] => if PropLogic.eval assignment fmTrue then SOME (print thy model (Free ("dummy", T)) arg assignment) else (*if PropLogic.eval assignment fmFalse then*) NONE | _ => raise REFUTE ("set_printer", "illegal interpretation for a Boolean value")) (constants ~~ results) (* Term.typ *) val HOLogic_setT = HOLogic.mk_setT T (* Term.term *) val HOLogic_empty_set = Const ("{}", HOLogic_setT) val HOLogic_insert = Const ("insert", T --> HOLogic_setT --> HOLogic_setT) in SOME (Library.foldl (fn (acc, elem) => HOLogic_insert $ elem $ acc) (HOLogic_empty_set, elements)) end | _ => NONE end; (* theory -> model -> Term.term -> interpretation -> (int -> bool) -> Term.term option *) fun IDT_printer thy model t intr assignment = let (* Term.term -> Term.typ option *) fun typeof (Free (_, T)) = SOME T | typeof (Var (_, T)) = SOME T | typeof (Const (_, T)) = SOME T | typeof _ = NONE in case typeof t of SOME (Type (s, Ts)) => (case DatatypePackage.datatype_info thy s of SOME info => (* inductive datatype *) let val (typs, _) = model val index = #index info val descr = #descr info val (_, dtyps, constrs) = (the o AList.lookup (op =) descr) index val typ_assoc = dtyps ~~ Ts (* sanity check: every element in 'dtyps' must be a 'DtTFree' *) val _ = (if Library.exists (fn d => case d of DatatypeAux.DtTFree _ => false | _ => true) dtyps then raise REFUTE ("IDT_printer", "datatype argument (for type " ^ Sign.string_of_typ (sign_of thy) (Type (s, Ts)) ^ ") is not a variable") else ()) (* the index of the element in the datatype *) val element = (case intr of Leaf xs => find_index (PropLogic.eval assignment) xs | Node _ => raise REFUTE ("IDT_printer", "interpretation is not a leaf")) in if element < 0 then SOME (Const ("arbitrary", Type (s, Ts))) else let (* takes a datatype constructor, and if for some arguments this constructor *) (* generates the datatype's element that is given by 'element', returns the *) (* constructor (as a term) as well as the indices of the arguments *) (* string * DatatypeAux.dtyp list -> (Term.term * int list) option *) fun get_constr_args (cname, cargs) = let val cTerm = Const (cname, (map (typ_of_dtyp descr typ_assoc) cargs) ---> Type (s, Ts)) val (iC, _, _) = interpret thy (typs, []) {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} cTerm (* interpretation -> int list option *) fun get_args (Leaf xs) = if find_index_eq True xs = element then SOME [] else NONE | get_args (Node xs) = let (* interpretation * int -> int list option *) fun search ([], _) = NONE | search (x::xs, n) = (case get_args x of SOME result => SOME (n::result) | NONE => search (xs, n+1)) in search (xs, 0) end in Option.map (fn args => (cTerm, cargs, args)) (get_args iC) end (* Term.term * DatatypeAux.dtyp list * int list *) val (cTerm, cargs, args) = (case get_first get_constr_args constrs of SOME x => x | NONE => raise REFUTE ("IDT_printer", "no matching constructor found for element " ^ string_of_int element)) val argsTerms = map (fn (d, n) => let val dT = typ_of_dtyp descr typ_assoc d val (i, _, _) = interpret thy (typs, []) {maxvars=0, def_eq=false, next_idx=1, bounds=[], wellformed=True} (Free ("dummy", dT)) val consts = make_constants i (* we only need the n-th element of this *) (* list, so there might be a more efficient implementation that does *) (* not generate all constants *) in print thy (typs, []) (Free ("dummy", dT)) (List.nth (consts, n)) assignment end) (cargs ~~ args) in SOME (Library.foldl op$ (cTerm, argsTerms)) end end | NONE => (* not an inductive datatype *) NONE) | _ => (* a (free or schematic) type variable *) NONE end; (* ------------------------------------------------------------------------- *) (* use 'setup Refute.setup' in an Isabelle theory to initialize the 'Refute' *) (* structure *) (* ------------------------------------------------------------------------- *) (* ------------------------------------------------------------------------- *) (* Note: the interpreters and printers are used in reverse order; however, *) (* an interpreter that can handle non-atomic terms ends up being *) (* applied before the 'stlc_interpreter' breaks the term apart into *) (* subterms that are then passed to other interpreters! *) (* ------------------------------------------------------------------------- *) (* (theory -> theory) list *) val setup = [RefuteData.init, add_interpreter "stlc" stlc_interpreter, add_interpreter "Pure" Pure_interpreter, add_interpreter "HOLogic" HOLogic_interpreter, add_interpreter "set" set_interpreter, add_interpreter "IDT" IDT_interpreter, add_interpreter "IDT_constructor" IDT_constructor_interpreter, add_interpreter "IDT_recursion" IDT_recursion_interpreter, add_interpreter "Finite_Set.card" Finite_Set_card_interpreter, add_interpreter "Finite_Set.Finites" Finite_Set_Finites_interpreter, add_interpreter "Nat.op <" Nat_less_interpreter, add_interpreter "Nat.op +" Nat_plus_interpreter, add_interpreter "Nat.op -" Nat_minus_interpreter, add_interpreter "Nat.op *" Nat_mult_interpreter, add_interpreter "List.op @" List_append_interpreter, add_interpreter "Lfp.lfp" Lfp_lfp_interpreter, add_interpreter "Gfp.gfp" Gfp_gfp_interpreter, add_printer "stlc" stlc_printer, add_printer "set" set_printer, add_printer "IDT" IDT_printer]; end