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