(* Title: HOL/Tools/refute.ML
Author: Tjark Weber, TU Muenchen
Finite model generation for HOL formulas, using a SAT solver.
*)
(* ------------------------------------------------------------------------- *)
(* Declares the 'REFUTE' signature as well as a structure 'Refute'. *)
(* Documentation is available in the Isabelle/Isar theory 'HOL/Refute.thy'. *)
(* ------------------------------------------------------------------------- *)
signature REFUTE =
sig
exception REFUTE of string * string
(* ------------------------------------------------------------------------- *)
(* Model/interpretation related code (translation HOL -> propositional logic *)
(* ------------------------------------------------------------------------- *)
type params
type interpretation
type model
type arguments
exception MAXVARS_EXCEEDED
val add_interpreter : string -> (Proof.context -> model -> arguments -> term ->
(interpretation * model * arguments) option) -> theory -> theory
val add_printer : string -> (Proof.context -> model -> typ ->
interpretation -> (int -> bool) -> term option) -> theory -> theory
val interpret : Proof.context -> model -> arguments -> term ->
(interpretation * model * arguments)
val print : Proof.context -> model -> typ -> interpretation -> (int -> bool) -> term
val print_model : Proof.context -> model -> (int -> bool) -> string
(* ------------------------------------------------------------------------- *)
(* Interface *)
(* ------------------------------------------------------------------------- *)
val set_default_param : (string * string) -> theory -> theory
val get_default_param : Proof.context -> string -> string option
val get_default_params : Proof.context -> (string * string) list
val actual_params : Proof.context -> (string * string) list -> params
val find_model : Proof.context -> params -> term list -> term -> bool -> unit
(* tries to find a model for a formula: *)
val satisfy_term :
Proof.context -> (string * string) list -> term list -> term -> unit
(* tries to find a model that refutes a formula: *)
val refute_term :
Proof.context -> (string * string) list -> term list -> term -> unit
val refute_goal :
Proof.context -> (string * string) list -> thm -> int -> unit
val setup : theory -> theory
(* ------------------------------------------------------------------------- *)
(* Additional functions used by Nitpick (to be factored out) *)
(* ------------------------------------------------------------------------- *)
val close_form : term -> term
val get_classdef : theory -> string -> (string * term) option
val norm_rhs : term -> term
val get_def : theory -> string * typ -> (string * term) option
val get_typedef : theory -> typ -> (string * term) option
val is_IDT_constructor : theory -> string * typ -> bool
val is_IDT_recursor : theory -> string * typ -> bool
val is_const_of_class: theory -> string * typ -> bool
val string_of_typ : typ -> string
val typ_of_dtyp : Datatype.descr -> (Datatype.dtyp * typ) list -> Datatype.dtyp -> typ
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" and "expect"): *)
(* *)
(* Name Type Description *)
(* *)
(* "sizes" (string * int) list *)
(* Size of ground types (e.g. 'a=2), or depth of IDTs. *)
(* "minsize" int If >0, minimal size of each ground type/IDT depth. *)
(* "maxsize" int If >0, maximal size of each ground type/IDT depth. *)
(* "maxvars" int If >0, use at most 'maxvars' Boolean variables *)
(* when transforming the term into a propositional *)
(* formula. *)
(* "maxtime" int If >0, terminate after at most 'maxtime' seconds. *)
(* "satsolver" string SAT solver to be used. *)
(* "no_assms" bool If "true", assumptions in structured proofs are *)
(* not considered. *)
(* "expect" string Expected result ("genuine", "potential", "none", or *)
(* "unknown"). *)
(* ------------------------------------------------------------------------- *)
type params =
{
sizes : (string * int) list,
minsize : int,
maxsize : int,
maxvars : int,
maxtime : int,
satsolver: string,
no_assms : bool,
expect : string
};
(* ------------------------------------------------------------------------- *)
(* interpretation: a term's interpretation is given by a variable of type *)
(* 'interpretation' *)
(* ------------------------------------------------------------------------- *)
type interpretation =
prop_formula list tree;
(* ------------------------------------------------------------------------- *)
(* model: a model specifies the size of types and the interpretation of *)
(* terms *)
(* ------------------------------------------------------------------------- *)
type model =
(typ * int) list * (term * interpretation) list;
(* ------------------------------------------------------------------------- *)
(* arguments: additional arguments required during interpretation of terms *)
(* ------------------------------------------------------------------------- *)
type arguments =
{
(* just passed unchanged from 'params': *)
maxvars : int,
(* whether to use 'make_equality' or 'make_def_equality': *)
def_eq : bool,
(* the following may change during the translation: *)
next_idx : int,
bounds : interpretation list,
wellformed: prop_formula
};
structure Data = Theory_Data
(
type T =
{interpreters: (string * (Proof.context -> model -> arguments -> term ->
(interpretation * model * arguments) option)) list,
printers: (string * (Proof.context -> model -> typ -> interpretation ->
(int -> bool) -> term option)) list,
parameters: string Symtab.table};
val empty = {interpreters = [], printers = [], parameters = Symtab.empty};
val extend = I;
fun merge
({interpreters = in1, printers = pr1, parameters = pa1},
{interpreters = in2, printers = pr2, parameters = pa2}) : T =
{interpreters = AList.merge (op =) (K true) (in1, in2),
printers = AList.merge (op =) (K true) (pr1, pr2),
parameters = Symtab.merge (op=) (pa1, pa2)};
);
val get_data = Data.get o ProofContext.theory_of;
(* ------------------------------------------------------------------------- *)
(* 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 *)
(* ------------------------------------------------------------------------- *)
fun interpret ctxt model args t =
case get_first (fn (_, f) => f ctxt model args t)
(#interpreters (get_data ctxt)) of
NONE => raise REFUTE ("interpret",
"no interpreter for term " ^ quote (Syntax.string_of_term ctxt t))
| SOME x => x;
(* ------------------------------------------------------------------------- *)
(* print: converts the interpretation 'intr', which must denote a term of *)
(* type 'T', into a term using a suitable printer *)
(* ------------------------------------------------------------------------- *)
fun print ctxt model T intr assignment =
case get_first (fn (_, f) => f ctxt model T intr assignment)
(#printers (get_data ctxt)) of
NONE => raise REFUTE ("print",
"no printer for type " ^ quote (Syntax.string_of_typ ctxt 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 *)
(* ------------------------------------------------------------------------- *)
fun print_model ctxt model assignment =
let
val (typs, terms) = model
val typs_msg =
if null typs then
"empty universe (no type variables in term)\n"
else
"Size of types: " ^ commas (map (fn (T, i) =>
Syntax.string_of_typ ctxt T ^ ": " ^ string_of_int i) typs) ^ "\n"
val show_consts_msg =
if not (Config.get ctxt show_consts) andalso Library.exists (is_Const o fst) terms then
"enable \"show_consts\" to show the interpretation of constants\n"
else
""
val terms_msg =
if null terms then
"empty interpretation (no free variables in term)\n"
else
cat_lines (map_filter (fn (t, intr) =>
(* print constants only if 'show_consts' is true *)
if Config.get ctxt show_consts orelse not (is_Const t) then
SOME (Syntax.string_of_term ctxt t ^ ": " ^
Syntax.string_of_term ctxt
(print ctxt model (Term.type_of t) intr assignment))
else
NONE) terms) ^ "\n"
in
typs_msg ^ show_consts_msg ^ terms_msg
end;
(* ------------------------------------------------------------------------- *)
(* PARAMETER MANAGEMENT *)
(* ------------------------------------------------------------------------- *)
fun add_interpreter name f = Data.map (fn {interpreters, printers, parameters} =>
case AList.lookup (op =) interpreters name of
NONE => {interpreters = (name, f) :: interpreters,
printers = printers, parameters = parameters}
| SOME _ => error ("Interpreter " ^ name ^ " already declared"));
fun add_printer name f = Data.map (fn {interpreters, printers, parameters} =>
case AList.lookup (op =) printers name of
NONE => {interpreters = interpreters,
printers = (name, f) :: printers, parameters = parameters}
| SOME _ => error ("Printer " ^ name ^ " already declared"));
(* ------------------------------------------------------------------------- *)
(* set_default_param: stores the '(name, value)' pair in Data's *)
(* parameter table *)
(* ------------------------------------------------------------------------- *)
fun set_default_param (name, value) = Data.map
(fn {interpreters, printers, parameters} =>
{interpreters = interpreters, printers = printers,
parameters = Symtab.update (name, value) parameters});
(* ------------------------------------------------------------------------- *)
(* get_default_param: retrieves the value associated with 'name' from *)
(* Data's parameter table *)
(* ------------------------------------------------------------------------- *)
val get_default_param = Symtab.lookup o #parameters o get_data;
(* ------------------------------------------------------------------------- *)
(* get_default_params: returns a list of all '(name, value)' pairs that are *)
(* stored in Data's parameter table *)
(* ------------------------------------------------------------------------- *)
val get_default_params = Symtab.dest o #parameters o get_data;
(* ------------------------------------------------------------------------- *)
(* actual_params: takes a (possibly empty) list 'params' of parameters that *)
(* override the default parameters currently specified, and *)
(* returns a record that can be passed to 'find_model'. *)
(* ------------------------------------------------------------------------- *)
fun actual_params ctxt override =
let
(* (string * string) list * string -> bool *)
fun read_bool (parms, name) =
case AList.lookup (op =) parms name of
SOME "true" => true
| SOME "false" => false
| SOME s => error ("parameter " ^ quote name ^
" (value is " ^ quote s ^ ") must be \"true\" or \"false\"")
| NONE => error ("parameter " ^ quote name ^
" must be assigned a value")
(* (string * string) list * string -> int *)
fun read_int (parms, name) =
case AList.lookup (op =) parms name of
SOME s =>
(case Int.fromString s of
SOME i => i
| NONE => error ("parameter " ^ quote name ^
" (value is " ^ quote s ^ ") must be an integer value"))
| NONE => error ("parameter " ^ quote name ^
" must be assigned a value")
(* (string * string) list * string -> string *)
fun read_string (parms, name) =
case AList.lookup (op =) parms name of
SOME s => s
| NONE => error ("parameter " ^ quote name ^
" must be assigned a value")
(* 'override' first, defaults last: *)
(* (string * string) list *)
val allparams = override @ get_default_params ctxt
(* int *)
val minsize = read_int (allparams, "minsize")
val maxsize = read_int (allparams, "maxsize")
val maxvars = read_int (allparams, "maxvars")
val maxtime = read_int (allparams, "maxtime")
(* string *)
val satsolver = read_string (allparams, "satsolver")
val no_assms = read_bool (allparams, "no_assms")
val expect = the_default "" (AList.lookup (op =) allparams "expect")
(* all remaining parameters of the form "string=int" are collected in *)
(* 'sizes' *)
(* TODO: it is currently not possible to specify a size for a type *)
(* whose name is one of the other parameters (e.g. 'maxvars') *)
(* (string * int) list *)
val sizes = map_filter
(fn (name, value) => Option.map (pair name) (Int.fromString value))
(filter (fn (name, _) => name<>"minsize" andalso name<>"maxsize"
andalso name<>"maxvars" andalso name<>"maxtime"
andalso name<>"satsolver" andalso name<>"no_assms") allparams)
in
{sizes=sizes, minsize=minsize, maxsize=maxsize, maxvars=maxvars,
maxtime=maxtime, satsolver=satsolver, no_assms=no_assms, expect=expect}
end;
(* ------------------------------------------------------------------------- *)
(* TRANSLATION HOL -> PROPOSITIONAL LOGIC, BOOLEAN ASSIGNMENT -> MODEL *)
(* ------------------------------------------------------------------------- *)
fun typ_of_dtyp descr typ_assoc (Datatype_Aux.DtTFree a) =
(* replace a 'DtTFree' variable by the associated type *)
the (AList.lookup (op =) typ_assoc (Datatype_Aux.DtTFree a))
| typ_of_dtyp descr typ_assoc (Datatype_Aux.DtType (s, ds)) =
Type (s, map (typ_of_dtyp descr typ_assoc) ds)
| typ_of_dtyp descr typ_assoc (Datatype_Aux.DtRec i) =
let
val (s, ds, _) = the (AList.lookup (op =) descr i)
in
Type (s, map (typ_of_dtyp descr typ_assoc) ds)
end;
(* ------------------------------------------------------------------------- *)
(* close_form: universal closure over schematic variables in 't' *)
(* ------------------------------------------------------------------------- *)
(* Term.term -> Term.term *)
fun close_form t =
let
(* (Term.indexname * Term.typ) list *)
val vars = sort_wrt (fst o fst) (map dest_Var (OldTerm.term_vars t))
in
fold (fn ((x, i), T) => fn t' =>
Term.all T $ Abs (x, T, abstract_over (Var ((x, i), T), t'))) vars t
end;
val monomorphic_term = Sledgehammer_Util.monomorphic_term
val specialize_type = Sledgehammer_Util.specialize_type
(* ------------------------------------------------------------------------- *)
(* is_const_of_class: returns 'true' iff 'Const (s, T)' is a constant that *)
(* denotes membership to an axiomatic type class *)
(* ------------------------------------------------------------------------- *)
fun is_const_of_class thy (s, T) =
let
val class_const_names = map Logic.const_of_class (Sign.all_classes thy)
in
(* I'm not quite sure if checking the name 's' is sufficient, *)
(* or if we should also check the type 'T'. *)
member (op =) class_const_names s
end;
(* ------------------------------------------------------------------------- *)
(* is_IDT_constructor: returns 'true' iff 'Const (s, T)' is the constructor *)
(* of an inductive datatype in 'thy' *)
(* ------------------------------------------------------------------------- *)
fun is_IDT_constructor thy (s, T) =
(case body_type T of
Type (s', _) =>
(case Datatype.get_constrs thy s' of
SOME constrs =>
List.exists (fn (cname, cty) =>
cname = s andalso Sign.typ_instance thy (T, cty)) constrs
| NONE => false)
| _ => false);
(* ------------------------------------------------------------------------- *)
(* is_IDT_recursor: returns 'true' iff 'Const (s, T)' is the recursion *)
(* operator of an inductive datatype in 'thy' *)
(* ------------------------------------------------------------------------- *)
fun is_IDT_recursor thy (s, T) =
let
val rec_names = Symtab.fold (append o #rec_names o snd)
(Datatype.get_all thy) []
in
(* I'm not quite sure if checking the name 's' is sufficient, *)
(* or if we should also check the type 'T'. *)
member (op =) rec_names s
end;
(* ------------------------------------------------------------------------- *)
(* norm_rhs: maps f ?t1 ... ?tn == rhs to %t1...tn. rhs *)
(* ------------------------------------------------------------------------- *)
fun norm_rhs eqn =
let
fun lambda (v as Var ((x, _), T)) t = Abs (x, T, abstract_over (v, t))
| lambda v t = raise TERM ("lambda", [v, t])
val (lhs, rhs) = Logic.dest_equals eqn
val (_, args) = Term.strip_comb lhs
in
fold lambda (rev args) rhs
end
(* ------------------------------------------------------------------------- *)
(* get_def: looks up the definition of a constant *)
(* ------------------------------------------------------------------------- *)
fun get_def thy (s, T) =
let
(* (string * Term.term) list -> (string * Term.term) option *)
fun get_def_ax [] = NONE
| get_def_ax ((axname, ax) :: axioms) =
(let
val (lhs, _) = Logic.dest_equals ax (* equations only *)
val c = Term.head_of lhs
val (s', T') = Term.dest_Const c
in
if s=s' then
let
val typeSubs = Sign.typ_match thy (T', T) Vartab.empty
val ax' = monomorphic_term typeSubs ax
val rhs = norm_rhs ax'
in
SOME (axname, rhs)
end
else
get_def_ax axioms
end handle ERROR _ => get_def_ax axioms
| TERM _ => get_def_ax axioms
| Type.TYPE_MATCH => get_def_ax axioms)
in
get_def_ax (Theory.all_axioms_of thy)
end;
(* ------------------------------------------------------------------------- *)
(* get_typedef: looks up the definition of a type, as created by "typedef" *)
(* ------------------------------------------------------------------------- *)
fun get_typedef thy T =
let
(* (string * Term.term) list -> (string * Term.term) option *)
fun get_typedef_ax [] = NONE
| get_typedef_ax ((axname, ax) :: axioms) =
(let
(* Term.term -> Term.typ option *)
fun type_of_type_definition (Const (s', T')) =
if s'= @{const_name 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 (domain_type T')
val typeSubs = Sign.typ_match thy (T'', T) Vartab.empty
in
SOME (axname, monomorphic_term typeSubs ax)
end
| NONE => get_typedef_ax axioms
end handle ERROR _ => get_typedef_ax axioms
| TERM _ => get_typedef_ax axioms
| Type.TYPE_MATCH => get_typedef_ax axioms)
in
get_typedef_ax (Theory.all_axioms_of thy)
end;
(* ------------------------------------------------------------------------- *)
(* get_classdef: looks up the defining axiom for an axiomatic type class, as *)
(* created by the "axclass" command *)
(* ------------------------------------------------------------------------- *)
fun get_classdef thy class =
let
val axname = class ^ "_class_def"
in
Option.map (pair axname)
(AList.lookup (op =) (Theory.all_axioms_of thy) axname)
end;
(* ------------------------------------------------------------------------- *)
(* unfold_defs: unfolds all defined constants in a term 't', beta-eta *)
(* normalizes the result term; certain constants are not *)
(* unfolded (cf. 'collect_axioms' and the various interpreters *)
(* below): if the interpretation respects a definition anyway, *)
(* that definition does not need to be unfolded *)
(* ------------------------------------------------------------------------- *)
(* Note: we could intertwine unfolding of constants and beta-(eta-) *)
(* normalization; this would save some unfolding for terms where *)
(* constants are eliminated by beta-reduction (e.g. 'K c1 c2'). On *)
(* the other hand, this would cause additional work for terms where *)
(* constants are duplicated by beta-reduction (e.g. 'S c1 c2 c3'). *)
fun unfold_defs thy t =
let
(* Term.term -> Term.term *)
fun unfold_loop t =
case t of
(* Pure *)
Const (@{const_name all}, _) => t
| Const (@{const_name "=="}, _) => t
| Const (@{const_name "==>"}, _) => t
| Const (@{const_name TYPE}, _) => t (* axiomatic type classes *)
(* HOL *)
| Const (@{const_name Trueprop}, _) => t
| Const (@{const_name Not}, _) => t
| (* redundant, since 'True' is also an IDT constructor *)
Const (@{const_name True}, _) => t
| (* redundant, since 'False' is also an IDT constructor *)
Const (@{const_name False}, _) => t
| Const (@{const_name undefined}, _) => t
| Const (@{const_name The}, _) => t
| Const (@{const_name Hilbert_Choice.Eps}, _) => t
| Const (@{const_name All}, _) => t
| Const (@{const_name Ex}, _) => t
| Const (@{const_name HOL.eq}, _) => t
| Const (@{const_name HOL.conj}, _) => t
| Const (@{const_name HOL.disj}, _) => t
| Const (@{const_name HOL.implies}, _) => t
(* sets *)
| Const (@{const_name Collect}, _) => t
| Const (@{const_name Set.member}, _) => t
(* other optimizations *)
| Const (@{const_name Finite_Set.card}, _) => t
| Const (@{const_name Finite_Set.finite}, _) => t
| Const (@{const_name Orderings.less}, Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ bool}])])) => t
| Const (@{const_name Groups.plus}, Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ nat}])])) => t
| Const (@{const_name Groups.minus}, Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ nat}])])) => t
| Const (@{const_name Groups.times}, Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ nat}])])) => t
| Const (@{const_name List.append}, _) => t
(* UNSOUND
| Const (@{const_name lfp}, _) => t
| Const (@{const_name gfp}, _) => t
*)
| Const (@{const_name fst}, _) => t
| Const (@{const_name snd}, _) => t
(* simply-typed lambda calculus *)
| Const (s, T) =>
(if is_IDT_constructor thy (s, T)
orelse is_IDT_recursor thy (s, T) then
t (* do not unfold IDT constructors/recursors *)
(* unfold the constant if there is a defining equation *)
else
case get_def thy (s, T) of
SOME (axname, rhs) =>
(* Note: if the term to be unfolded (i.e. 'Const (s, T)') *)
(* occurs on the right-hand side of the equation, i.e. in *)
(* 'rhs', we must not use this equation to unfold, because *)
(* that would loop. Here would be the right place to *)
(* check this. However, getting this really right seems *)
(* difficult because the user may state arbitrary axioms, *)
(* which could interact with overloading to create loops. *)
((*tracing (" unfolding: " ^ axname);*)
unfold_loop rhs)
| NONE => t)
| Free _ => t
| Var _ => t
| Bound _ => t
| Abs (s, T, body) => Abs (s, T, unfold_loop body)
| t1 $ t2 => (unfold_loop t1) $ (unfold_loop t2)
val result = Envir.beta_eta_contract (unfold_loop t)
in
result
end;
(* ------------------------------------------------------------------------- *)
(* collect_axioms: collects (monomorphic, universally quantified, unfolded *)
(* versions of) all HOL axioms that are relevant w.r.t 't' *)
(* ------------------------------------------------------------------------- *)
(* Note: to make the collection of axioms more easily extensible, this *)
(* function could be based on user-supplied "axiom collectors", *)
(* similar to 'interpret'/interpreters or 'print'/printers *)
(* Note: currently we use "inverse" functions to the definitional *)
(* mechanisms provided by Isabelle/HOL, e.g. for "axclass", *)
(* "typedef", "definition". A more general approach could consider *)
(* *every* axiom of the theory and collect it if it has a constant/ *)
(* type/typeclass in common with the term 't'. *)
(* Which axioms are "relevant" for a particular term/type goes hand in *)
(* hand with the interpretation of that term/type by its interpreter (see *)
(* way below): if the interpretation respects an axiom anyway, the axiom *)
(* does not need to be added as a constraint here. *)
(* To avoid collecting the same axiom multiple times, we use an *)
(* accumulator 'axs' which contains all axioms collected so far. *)
fun collect_axioms ctxt t =
let
val thy = ProofContext.theory_of ctxt
val _ = tracing "Adding axioms..."
val axioms = Theory.all_axioms_of thy
fun collect_this_axiom (axname, ax) axs =
let
val ax' = unfold_defs thy ax
in
if member (op aconv) axs ax' then axs
else (tracing axname; collect_term_axioms ax' (ax' :: axs))
end
and collect_sort_axioms T axs =
let
val sort =
(case T of
TFree (_, sort) => sort
| TVar (_, sort) => sort
| _ => raise REFUTE ("collect_axioms",
"type " ^ Syntax.string_of_typ ctxt T ^ " is not a variable"))
(* obtain axioms for all superclasses *)
val superclasses = sort @ maps (Sign.super_classes thy) sort
(* merely an optimization, because 'collect_this_axiom' disallows *)
(* duplicate axioms anyway: *)
val superclasses = distinct (op =) superclasses
val class_axioms = maps (fn class => map (fn ax =>
("<" ^ class ^ ">", Thm.prop_of ax))
(#axioms (AxClass.get_info thy class) handle ERROR _ => []))
superclasses
(* replace the (at most one) schematic type variable in each axiom *)
(* by the actual type 'T' *)
val monomorphic_class_axioms = map (fn (axname, ax) =>
(case Term.add_tvars ax [] of
[] => (axname, ax)
| [(idx, S)] => (axname, monomorphic_term (Vartab.make [(idx, (S, T))]) ax)
| _ =>
raise REFUTE ("collect_axioms", "class axiom " ^ axname ^ " (" ^
Syntax.string_of_term ctxt ax ^
") contains more than one type variable")))
class_axioms
in
fold collect_this_axiom monomorphic_class_axioms axs
end
and collect_type_axioms T axs =
case T of
(* simple types *)
Type ("prop", []) => axs
| Type ("fun", [T1, T2]) => collect_type_axioms T2 (collect_type_axioms T1 axs)
(* axiomatic type classes *)
| Type ("itself", [T1]) => collect_type_axioms T1 axs
| Type (s, Ts) =>
(case Datatype.get_info thy s of
SOME info => (* inductive datatype *)
(* only collect relevant type axioms for the argument types *)
fold collect_type_axioms Ts axs
| NONE =>
(case get_typedef thy T of
SOME (axname, ax) =>
collect_this_axiom (axname, ax) axs
| NONE =>
(* unspecified type, perhaps introduced with "typedecl" *)
(* at least collect relevant type axioms for the argument types *)
fold collect_type_axioms Ts axs))
(* axiomatic type classes *)
| TFree _ => collect_sort_axioms T axs
(* axiomatic type classes *)
| TVar _ => collect_sort_axioms T axs
and collect_term_axioms t axs =
case t of
(* Pure *)
Const (@{const_name all}, _) => axs
| Const (@{const_name "=="}, _) => axs
| Const (@{const_name "==>"}, _) => axs
(* axiomatic type classes *)
| Const (@{const_name TYPE}, T) => collect_type_axioms T axs
(* HOL *)
| Const (@{const_name Trueprop}, _) => axs
| Const (@{const_name Not}, _) => axs
(* redundant, since 'True' is also an IDT constructor *)
| Const (@{const_name True}, _) => axs
(* redundant, since 'False' is also an IDT constructor *)
| Const (@{const_name False}, _) => axs
| Const (@{const_name undefined}, T) => collect_type_axioms T axs
| Const (@{const_name The}, T) =>
let
val ax = specialize_type thy (@{const_name The}, T)
(the (AList.lookup (op =) axioms "HOL.the_eq_trivial"))
in
collect_this_axiom ("HOL.the_eq_trivial", ax) axs
end
| Const (@{const_name Hilbert_Choice.Eps}, T) =>
let
val ax = specialize_type thy (@{const_name Hilbert_Choice.Eps}, T)
(the (AList.lookup (op =) axioms "Hilbert_Choice.someI"))
in
collect_this_axiom ("Hilbert_Choice.someI", ax) axs
end
| Const (@{const_name All}, T) => collect_type_axioms T axs
| Const (@{const_name Ex}, T) => collect_type_axioms T axs
| Const (@{const_name HOL.eq}, T) => collect_type_axioms T axs
| Const (@{const_name HOL.conj}, _) => axs
| Const (@{const_name HOL.disj}, _) => axs
| Const (@{const_name HOL.implies}, _) => axs
(* sets *)
| Const (@{const_name Collect}, T) => collect_type_axioms T axs
| Const (@{const_name Set.member}, T) => collect_type_axioms T axs
(* other optimizations *)
| Const (@{const_name Finite_Set.card}, T) => collect_type_axioms T axs
| Const (@{const_name Finite_Set.finite}, T) =>
collect_type_axioms T axs
| Const (@{const_name Orderings.less}, T as Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ bool}])])) =>
collect_type_axioms T axs
| Const (@{const_name Groups.plus}, T as Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ nat}])])) =>
collect_type_axioms T axs
| Const (@{const_name Groups.minus}, T as Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ nat}])])) =>
collect_type_axioms T axs
| Const (@{const_name Groups.times}, T as Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ nat}])])) =>
collect_type_axioms T axs
| Const (@{const_name List.append}, T) => collect_type_axioms T axs
(* UNSOUND
| Const (@{const_name lfp}, T) => collect_type_axioms T axs
| Const (@{const_name gfp}, T) => collect_type_axioms T axs
*)
| Const (@{const_name fst}, T) => collect_type_axioms T axs
| Const (@{const_name snd}, T) => collect_type_axioms T axs
(* simply-typed lambda calculus *)
| Const (s, T) =>
if is_const_of_class thy (s, T) then
(* axiomatic type classes: add "OFCLASS(?'a::c, c_class)" *)
(* and the class definition *)
let
val class = Logic.class_of_const s
val of_class = Logic.mk_of_class (TVar (("'a", 0), [class]), class)
val ax_in = SOME (specialize_type thy (s, T) of_class)
(* type match may fail due to sort constraints *)
handle Type.TYPE_MATCH => NONE
val ax_1 = Option.map (fn ax => (Syntax.string_of_term ctxt ax, ax)) ax_in
val ax_2 = Option.map (apsnd (specialize_type thy (s, T))) (get_classdef thy class)
in
collect_type_axioms T (fold collect_this_axiom (map_filter I [ax_1, ax_2]) axs)
end
else if is_IDT_constructor thy (s, T)
orelse is_IDT_recursor thy (s, T)
then
(* only collect relevant type axioms *)
collect_type_axioms T axs
else
(* other constants should have been unfolded, with some *)
(* exceptions: e.g. Abs_xxx/Rep_xxx functions for *)
(* typedefs, or type-class related constants *)
(* only collect relevant type axioms *)
collect_type_axioms T axs
| Free (_, T) => collect_type_axioms T axs
| Var (_, T) => collect_type_axioms T axs
| Bound _ => axs
| Abs (_, T, body) => collect_term_axioms body (collect_type_axioms T axs)
| t1 $ t2 => collect_term_axioms t2 (collect_term_axioms t1 axs)
val result = map close_form (collect_term_axioms t [])
val _ = tracing " ...done."
in
result
end;
(* ------------------------------------------------------------------------- *)
(* ground_types: collects all ground types in a term (including argument *)
(* types of other types), suppressing duplicates. Does not *)
(* return function types, set types, non-recursive IDTs, or *)
(* 'propT'. For IDTs, also the argument types of constructors *)
(* and all mutually recursive IDTs are considered. *)
(* ------------------------------------------------------------------------- *)
fun ground_types ctxt t =
let
val thy = ProofContext.theory_of ctxt
fun collect_types T acc =
(case T of
Type ("fun", [T1, T2]) => collect_types T1 (collect_types T2 acc)
| Type ("prop", []) => acc
| Type (s, Ts) =>
(case Datatype.get_info thy s of
SOME info => (* inductive datatype *)
let
val index = #index info
val descr = #descr info
val (_, typs, _) = the (AList.lookup (op =) descr index)
val typ_assoc = typs ~~ Ts
(* sanity check: every element in 'dtyps' must be a *)
(* 'DtTFree' *)
val _ = if Library.exists (fn d =>
case d of Datatype_Aux.DtTFree _ => false | _ => true) typs then
raise REFUTE ("ground_types", "datatype argument (for type "
^ Syntax.string_of_typ ctxt T ^ ") is not a variable")
else ()
(* required for mutually recursive datatypes; those need to *)
(* be added even if they are an instance of an otherwise non- *)
(* recursive datatype *)
fun collect_dtyp d acc =
let
val dT = typ_of_dtyp descr typ_assoc d
in
case d of
Datatype_Aux.DtTFree _ =>
collect_types dT acc
| Datatype_Aux.DtType (_, ds) =>
collect_types dT (fold_rev collect_dtyp ds acc)
| Datatype_Aux.DtRec i =>
if member (op =) acc dT then
acc (* prevent infinite recursion *)
else
let
val (_, dtyps, dconstrs) = the (AList.lookup (op =) descr i)
(* if the current type is a recursive IDT (i.e. a depth *)
(* is required), add it to 'acc' *)
val acc_dT = if Library.exists (fn (_, ds) =>
Library.exists Datatype_Aux.is_rec_type ds) dconstrs then
insert (op =) dT acc
else acc
(* collect argument types *)
val acc_dtyps = fold_rev collect_dtyp dtyps acc_dT
(* collect constructor types *)
val acc_dconstrs = fold_rev collect_dtyp (maps snd dconstrs) acc_dtyps
in
acc_dconstrs
end
end
in
(* argument types 'Ts' could be added here, but they are also *)
(* added by 'collect_dtyp' automatically *)
collect_dtyp (Datatype_Aux.DtRec index) acc
end
| NONE =>
(* not an inductive datatype, e.g. defined via "typedef" or *)
(* "typedecl" *)
insert (op =) T (fold collect_types Ts acc))
| TFree _ => insert (op =) T acc
| TVar _ => insert (op =) T acc)
in
fold_types collect_types t []
end;
(* ------------------------------------------------------------------------- *)
(* string_of_typ: (rather naive) conversion from types to strings, used to *)
(* look up the size of a type in 'sizes'. Parameterized *)
(* types with different parameters (e.g. "'a list" vs. "bool *)
(* list") are identified. *)
(* ------------------------------------------------------------------------- *)
(* Term.typ -> string *)
fun string_of_typ (Type (s, _)) = s
| string_of_typ (TFree (s, _)) = s
| string_of_typ (TVar ((s,_), _)) = s;
(* ------------------------------------------------------------------------- *)
(* first_universe: returns the "first" (i.e. smallest) universe by assigning *)
(* 'minsize' to every type for which no size is specified in *)
(* 'sizes' *)
(* ------------------------------------------------------------------------- *)
(* Term.typ list -> (string * int) list -> int -> (Term.typ * int) list *)
fun first_universe xs sizes minsize =
let
fun size_of_typ T =
case AList.lookup (op =) sizes (string_of_typ T) of
SOME n => n
| NONE => minsize
in
map (fn T => (T, size_of_typ T)) xs
end;
(* ------------------------------------------------------------------------- *)
(* next_universe: enumerates all universes (i.e. assignments of sizes to *)
(* types), where the minimal size of a type is given by *)
(* 'minsize', the maximal size is given by 'maxsize', and a *)
(* type may have a fixed size given in 'sizes' *)
(* ------------------------------------------------------------------------- *)
(* (Term.typ * int) list -> (string * int) list -> int -> int ->
(Term.typ * int) list option *)
fun next_universe xs sizes minsize maxsize =
let
(* creates the "first" list of length 'len', where the sum of all list *)
(* elements is 'sum', and the length of the list is 'len' *)
(* int -> int -> int -> int list option *)
fun make_first _ 0 sum =
if sum = 0 then
SOME []
else
NONE
| make_first max len sum =
if sum <= max orelse max < 0 then
Option.map (fn xs' => sum :: xs') (make_first max (len-1) 0)
else
Option.map (fn xs' => max :: xs') (make_first max (len-1) (sum-max))
(* enumerates all int lists with a fixed length, where 0<=x<='max' for *)
(* all list elements x (unless 'max'<0) *)
(* int -> int -> int -> int list -> int list option *)
fun next max len sum [] =
NONE
| next max len sum [x] =
(* we've reached the last list element, so there's no shift possible *)
make_first max (len+1) (sum+x+1) (* increment 'sum' by 1 *)
| next max len sum (x1::x2::xs) =
if x1>0 andalso (x2<max orelse max<0) then
(* we can shift *)
SOME (the (make_first max (len+1) (sum+x1-1)) @ (x2+1) :: xs)
else
(* continue search *)
next max (len+1) (sum+x1) (x2::xs)
(* only consider those types for which the size is not fixed *)
val mutables = filter_out (AList.defined (op =) sizes o string_of_typ o fst) xs
(* subtract 'minsize' from every size (will be added again at the end) *)
val diffs = map (fn (_, n) => n-minsize) mutables
in
case next (maxsize-minsize) 0 0 diffs of
SOME diffs' =>
(* merge with those types for which the size is fixed *)
SOME (fst (fold_map (fn (T, _) => fn ds =>
case AList.lookup (op =) sizes (string_of_typ T) of
(* return the fixed size *)
SOME n => ((T, n), ds)
(* consume the head of 'ds', add 'minsize' *)
| NONE => ((T, minsize + hd ds), tl ds))
xs diffs'))
| NONE => NONE
end;
(* ------------------------------------------------------------------------- *)
(* toTrue: converts the interpretation of a Boolean value to a propositional *)
(* formula that is true iff the interpretation denotes "true" *)
(* ------------------------------------------------------------------------- *)
(* interpretation -> prop_formula *)
fun toTrue (Leaf [fm, _]) = fm
| toTrue _ = raise REFUTE ("toTrue", "interpretation does not denote a Boolean value");
(* ------------------------------------------------------------------------- *)
(* toFalse: converts the interpretation of a Boolean value to a *)
(* propositional formula that is true iff the interpretation *)
(* denotes "false" *)
(* ------------------------------------------------------------------------- *)
(* interpretation -> prop_formula *)
fun toFalse (Leaf [_, fm]) = fm
| toFalse _ = raise REFUTE ("toFalse", "interpretation does not denote a Boolean value");
(* ------------------------------------------------------------------------- *)
(* find_model: repeatedly calls 'interpret' with appropriate parameters, *)
(* applies a SAT solver, and (in case a model is found) displays *)
(* the model to the user by calling 'print_model' *)
(* {...} : parameters that control the translation/model generation *)
(* assm_ts : assumptions to be considered unless "no_assms" is specified *)
(* t : term to be translated into a propositional formula *)
(* negate : if true, find a model that makes 't' false (rather than true) *)
(* ------------------------------------------------------------------------- *)
fun find_model ctxt
{sizes, minsize, maxsize, maxvars, maxtime, satsolver, no_assms, expect}
assm_ts t negate =
let
val thy = ProofContext.theory_of ctxt
(* string -> unit *)
fun check_expect outcome_code =
if expect = "" orelse outcome_code = expect then ()
else error ("Unexpected outcome: " ^ quote outcome_code ^ ".")
(* unit -> unit *)
fun wrapper () =
let
val timer = Timer.startRealTimer ()
val t =
if no_assms then t
else if negate then Logic.list_implies (assm_ts, t)
else Logic.mk_conjunction_list (t :: assm_ts)
val u = unfold_defs thy t
val _ = tracing ("Unfolded term: " ^ Syntax.string_of_term ctxt u)
val axioms = collect_axioms ctxt u
(* Term.typ list *)
val types = fold (union (op =) o ground_types ctxt) (u :: axioms) []
val _ = tracing ("Ground types: "
^ (if null types then "none."
else commas (map (Syntax.string_of_typ ctxt) types)))
(* we can only consider fragments of recursive IDTs, so we issue a *)
(* warning if the formula contains a recursive IDT *)
(* TODO: no warning needed for /positive/ occurrences of IDTs *)
val maybe_spurious = Library.exists (fn
Type (s, _) =>
(case Datatype.get_info thy s of
SOME info => (* inductive datatype *)
let
val index = #index info
val descr = #descr info
val (_, _, constrs) = the (AList.lookup (op =) descr index)
in
(* recursive datatype? *)
Library.exists (fn (_, ds) =>
Library.exists Datatype_Aux.is_rec_type ds) constrs
end
| NONE => false)
| _ => false) types
val _ =
if maybe_spurious then
warning ("Term contains a recursive datatype; "
^ "countermodel(s) may be spurious!")
else
()
(* (Term.typ * int) list -> string *)
fun find_model_loop universe =
let
val msecs_spent = Time.toMilliseconds (Timer.checkRealTimer timer)
val _ = maxtime = 0 orelse msecs_spent < 1000 * maxtime
orelse raise TimeLimit.TimeOut
val init_model = (universe, [])
val init_args = {maxvars = maxvars, def_eq = false, next_idx = 1,
bounds = [], wellformed = True}
val _ = tracing ("Translating term (sizes: "
^ commas (map (fn (_, n) => string_of_int n) universe) ^ ") ...")
(* translate 'u' and all axioms *)
val (intrs, (model, args)) = fold_map (fn t' => fn (m, a) =>
let
val (i, m', a') = interpret ctxt m a t'
in
(* set 'def_eq' to 'true' *)
(i, (m', {maxvars = #maxvars a', def_eq = true,
next_idx = #next_idx a', bounds = #bounds a',
wellformed = #wellformed a'}))
end) (u :: axioms) (init_model, init_args)
(* make 'u' either true or false, and make all axioms true, and *)
(* add the well-formedness side condition *)
val fm_u = (if negate then toFalse else toTrue) (hd intrs)
val fm_ax = PropLogic.all (map toTrue (tl intrs))
val fm = PropLogic.all [#wellformed args, fm_ax, fm_u]
val _ =
(if satsolver = "dpll" orelse satsolver = "enumerate" then
warning ("Using SAT solver " ^ quote satsolver ^
"; for better performance, consider installing an \
\external solver.")
else ());
val solver =
SatSolver.invoke_solver satsolver
handle Option.Option =>
error ("Unknown SAT solver: " ^ quote satsolver ^
". Available solvers: " ^
commas (map (quote o fst) (!SatSolver.solvers)) ^ ".")
in
Output.urgent_message "Invoking SAT solver...";
(case solver fm of
SatSolver.SATISFIABLE assignment =>
(Output.urgent_message ("*** Model found: ***\n" ^ print_model ctxt model
(fn i => case assignment i of SOME b => b | NONE => true));
if maybe_spurious then "potential" else "genuine")
| SatSolver.UNSATISFIABLE _ =>
(Output.urgent_message "No model exists.";
case next_universe universe sizes minsize maxsize of
SOME universe' => find_model_loop universe'
| NONE => (Output.urgent_message
"Search terminated, no larger universe within the given limits.";
"none"))
| SatSolver.UNKNOWN =>
(Output.urgent_message "No model found.";
case next_universe universe sizes minsize maxsize of
SOME universe' => find_model_loop universe'
| NONE => (Output.urgent_message
"Search terminated, no larger universe within the given limits.";
"unknown"))) handle SatSolver.NOT_CONFIGURED =>
(error ("SAT solver " ^ quote satsolver ^ " is not configured.");
"unknown")
end
handle MAXVARS_EXCEEDED =>
(Output.urgent_message ("Search terminated, number of Boolean variables ("
^ string_of_int maxvars ^ " allowed) exceeded.");
"unknown")
val outcome_code = find_model_loop (first_universe types sizes minsize)
in
check_expect outcome_code
end
in
(* some parameter sanity checks *)
minsize>=1 orelse
error ("\"minsize\" is " ^ string_of_int minsize ^ ", must be at least 1");
maxsize>=1 orelse
error ("\"maxsize\" is " ^ string_of_int maxsize ^ ", must be at least 1");
maxsize>=minsize orelse
error ("\"maxsize\" (=" ^ string_of_int maxsize ^
") is less than \"minsize\" (=" ^ string_of_int minsize ^ ").");
maxvars>=0 orelse
error ("\"maxvars\" is " ^ string_of_int maxvars ^ ", must be at least 0");
maxtime>=0 orelse
error ("\"maxtime\" is " ^ string_of_int maxtime ^ ", must be at least 0");
(* enter loop with or without time limit *)
Output.urgent_message ("Trying to find a model that "
^ (if negate then "refutes" else "satisfies") ^ ": "
^ Syntax.string_of_term ctxt t);
if maxtime > 0 then (
TimeLimit.timeLimit (Time.fromSeconds maxtime)
wrapper ()
handle TimeLimit.TimeOut =>
(Output.urgent_message ("Search terminated, time limit (" ^
string_of_int maxtime
^ (if maxtime=1 then " second" else " seconds") ^ ") exceeded.");
check_expect "unknown")
) else wrapper ()
end;
(* ------------------------------------------------------------------------- *)
(* INTERFACE, PART 2: FINDING A MODEL *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* satisfy_term: calls 'find_model' to find a model that satisfies 't' *)
(* params : list of '(name, value)' pairs used to override default *)
(* parameters *)
(* ------------------------------------------------------------------------- *)
fun satisfy_term ctxt params assm_ts t =
find_model ctxt (actual_params ctxt params) assm_ts t false;
(* ------------------------------------------------------------------------- *)
(* refute_term: calls 'find_model' to find a model that refutes 't' *)
(* params : list of '(name, value)' pairs used to override default *)
(* parameters *)
(* ------------------------------------------------------------------------- *)
fun refute_term ctxt params assm_ts t =
let
(* disallow schematic type variables, since we cannot properly negate *)
(* terms containing them (their logical meaning is that there EXISTS a *)
(* type s.t. ...; to refute such a formula, we would have to show that *)
(* for ALL types, not ...) *)
val _ = null (Term.add_tvars t []) orelse
error "Term to be refuted contains schematic type variables"
(* existential closure over schematic variables *)
(* (Term.indexname * Term.typ) list *)
val vars = sort_wrt (fst o fst) (map dest_Var (OldTerm.term_vars t))
(* Term.term *)
val ex_closure = fold (fn ((x, i), T) => fn t' =>
HOLogic.exists_const T $
Abs (x, T, abstract_over (Var ((x, i), T), t'))) vars t
(* Note: If 't' is of type 'propT' (rather than 'boolT'), applying *)
(* 'HOLogic.exists_const' is not type-correct. However, this is not *)
(* really a problem as long as 'find_model' still interprets the *)
(* resulting term correctly, without checking its type. *)
(* replace outermost universally quantified variables by Free's: *)
(* refuting a term with Free's is generally faster than refuting a *)
(* term with (nested) quantifiers, because quantifiers are expanded, *)
(* while the SAT solver searches for an interpretation for Free's. *)
(* Also we get more information back that way, namely an *)
(* interpretation which includes values for the (formerly) *)
(* quantified variables. *)
(* maps !!x1...xn. !xk...xm. t to t *)
fun strip_all_body (Const (@{const_name all}, _) $ Abs (_, _, t)) =
strip_all_body t
| strip_all_body (Const (@{const_name Trueprop}, _) $ t) =
strip_all_body t
| strip_all_body (Const (@{const_name All}, _) $ Abs (_, _, t)) =
strip_all_body t
| strip_all_body t = t
(* maps !!x1...xn. !xk...xm. t to [x1, ..., xn, xk, ..., xm] *)
fun strip_all_vars (Const (@{const_name all}, _) $ Abs (a, T, t)) =
(a, T) :: strip_all_vars t
| strip_all_vars (Const (@{const_name Trueprop}, _) $ t) =
strip_all_vars t
| strip_all_vars (Const (@{const_name All}, _) $ Abs (a, T, t)) =
(a, T) :: strip_all_vars t
| strip_all_vars t = [] : (string * typ) list
val strip_t = strip_all_body ex_closure
val frees = Term.rename_wrt_term strip_t (strip_all_vars ex_closure)
val subst_t = Term.subst_bounds (map Free frees, strip_t)
in
find_model ctxt (actual_params ctxt params) assm_ts subst_t true
end;
(* ------------------------------------------------------------------------- *)
(* refute_goal *)
(* ------------------------------------------------------------------------- *)
fun refute_goal ctxt params th i =
let
val t = th |> prop_of
in
if Logic.count_prems t = 0 then
Output.urgent_message "No subgoal!"
else
let
val assms = map term_of (Assumption.all_assms_of ctxt)
val (t, frees) = Logic.goal_params t i
in
refute_term ctxt params assms (subst_bounds (frees, t))
end
end
(* ------------------------------------------------------------------------- *)
(* INTERPRETERS: Auxiliary Functions *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* make_constants: returns all interpretations for type 'T' that consist of *)
(* unit vectors with 'True'/'False' only (no Boolean *)
(* variables) *)
(* ------------------------------------------------------------------------- *)
fun make_constants ctxt model T =
let
(* returns a list with all unit vectors of length n *)
(* int -> interpretation list *)
fun unit_vectors n =
let
(* returns the k-th unit vector of length n *)
(* int * int -> interpretation *)
fun unit_vector (k, n) =
Leaf ((replicate (k-1) False) @ (True :: (replicate (n-k) False)))
(* int -> interpretation list *)
fun unit_vectors_loop k =
if k>n then [] else unit_vector (k,n) :: unit_vectors_loop (k+1)
in
unit_vectors_loop 1
end
(* returns a list of lists, each one consisting of n (possibly *)
(* identical) elements from 'xs' *)
(* int -> 'a list -> 'a list list *)
fun pick_all 1 xs = map single xs
| pick_all n xs =
let val rec_pick = pick_all (n - 1) xs in
maps (fn x => map (cons x) rec_pick) xs
end
(* returns all constant interpretations that have the same tree *)
(* structure as the interpretation argument *)
(* interpretation -> interpretation list *)
fun make_constants_intr (Leaf xs) = unit_vectors (length xs)
| make_constants_intr (Node xs) = map Node (pick_all (length xs)
(make_constants_intr (hd xs)))
(* obtain the interpretation for a variable of type 'T' *)
val (i, _, _) = interpret ctxt model {maxvars=0, def_eq=false, next_idx=1,
bounds=[], wellformed=True} (Free ("dummy", T))
in
make_constants_intr i
end;
(* ------------------------------------------------------------------------- *)
(* size_of_type: returns the number of elements in a type 'T' (i.e. 'length *)
(* (make_constants T)', but implemented more efficiently) *)
(* ------------------------------------------------------------------------- *)
(* returns 0 for an empty ground type or a function type with empty *)
(* codomain, but fails for a function type with empty domain -- *)
(* admissibility of datatype constructor argument types (see "Inductive *)
(* datatypes in HOL - lessons learned ...", S. Berghofer, M. Wenzel, *)
(* TPHOLs 99) ensures that recursive, possibly empty, datatype fragments *)
(* never occur as the domain of a function type that is the type of a *)
(* constructor argument *)
fun size_of_type ctxt model T =
let
(* returns the number of elements that have the same tree structure as a *)
(* given interpretation *)
fun size_of_intr (Leaf xs) = length xs
| size_of_intr (Node xs) = Integer.pow (length xs) (size_of_intr (hd xs))
(* obtain the interpretation for a variable of type 'T' *)
val (i, _, _) = interpret ctxt model {maxvars=0, def_eq=false, next_idx=1,
bounds=[], wellformed=True} (Free ("dummy", T))
in
size_of_intr i
end;
(* ------------------------------------------------------------------------- *)
(* TT/FF: interpretations that denote "true" or "false", respectively *)
(* ------------------------------------------------------------------------- *)
(* interpretation *)
val TT = Leaf [True, False];
val FF = Leaf [False, True];
(* ------------------------------------------------------------------------- *)
(* make_equality: returns an interpretation that denotes (extensional) *)
(* equality of two interpretations *)
(* - two interpretations are 'equal' iff they are both defined and denote *)
(* the same value *)
(* - two interpretations are 'not_equal' iff they are both defined at least *)
(* partially, and a defined part denotes different values *)
(* - a completely undefined interpretation is neither 'equal' nor *)
(* 'not_equal' to another interpretation *)
(* ------------------------------------------------------------------------- *)
(* We could in principle represent '=' on a type T by a particular *)
(* interpretation. However, the size of that interpretation is quadratic *)
(* in the size of T. Therefore comparing the interpretations 'i1' and *)
(* 'i2' directly is more efficient than constructing the interpretation *)
(* for equality on T first, and "applying" this interpretation to 'i1' *)
(* and 'i2' in the usual way (cf. 'interpretation_apply') then. *)
(* interpretation * interpretation -> interpretation *)
fun make_equality (i1, i2) =
let
(* interpretation * interpretation -> prop_formula *)
fun equal (i1, i2) =
(case i1 of
Leaf xs =>
(case i2 of
Leaf ys => PropLogic.dot_product (xs, ys) (* defined and equal *)
| Node _ => raise REFUTE ("make_equality",
"second interpretation is higher"))
| Node xs =>
(case i2 of
Leaf _ => raise REFUTE ("make_equality",
"first interpretation is higher")
| Node ys => PropLogic.all (map equal (xs ~~ ys))))
(* interpretation * interpretation -> prop_formula *)
fun not_equal (i1, i2) =
(case i1 of
Leaf xs =>
(case i2 of
(* defined and not equal *)
Leaf ys => PropLogic.all ((PropLogic.exists xs)
:: (PropLogic.exists ys)
:: (map (fn (x,y) => SOr (SNot x, SNot y)) (xs ~~ ys)))
| Node _ => raise REFUTE ("make_equality",
"second interpretation is higher"))
| Node xs =>
(case i2 of
Leaf _ => raise REFUTE ("make_equality",
"first interpretation is higher")
| Node ys => PropLogic.exists (map not_equal (xs ~~ ys))))
in
(* a value may be undefined; therefore 'not_equal' is not just the *)
(* negation of 'equal' *)
Leaf [equal (i1, i2), not_equal (i1, i2)]
end;
(* ------------------------------------------------------------------------- *)
(* make_def_equality: returns an interpretation that denotes (extensional) *)
(* equality of two interpretations *)
(* This function treats undefined/partially defined interpretations *)
(* different from 'make_equality': two undefined interpretations are *)
(* considered equal, while a defined interpretation is considered not equal *)
(* to an undefined interpretation. *)
(* ------------------------------------------------------------------------- *)
(* interpretation * interpretation -> interpretation *)
fun make_def_equality (i1, i2) =
let
(* interpretation * interpretation -> prop_formula *)
fun equal (i1, i2) =
(case i1 of
Leaf xs =>
(case i2 of
(* defined and equal, or both undefined *)
Leaf ys => SOr (PropLogic.dot_product (xs, ys),
SAnd (PropLogic.all (map SNot xs), PropLogic.all (map SNot ys)))
| Node _ => raise REFUTE ("make_def_equality",
"second interpretation is higher"))
| Node xs =>
(case i2 of
Leaf _ => raise REFUTE ("make_def_equality",
"first interpretation is higher")
| Node ys => PropLogic.all (map equal (xs ~~ ys))))
(* interpretation *)
val eq = equal (i1, i2)
in
Leaf [eq, SNot eq]
end;
(* ------------------------------------------------------------------------- *)
(* interpretation_apply: returns an interpretation that denotes the result *)
(* of applying the function denoted by 'i1' to the *)
(* argument denoted by 'i2' *)
(* ------------------------------------------------------------------------- *)
(* interpretation * interpretation -> interpretation *)
fun interpretation_apply (i1, i2) =
let
(* interpretation * interpretation -> interpretation *)
fun interpretation_disjunction (tr1,tr2) =
tree_map (fn (xs,ys) => map (fn (x,y) => SOr(x,y)) (xs ~~ ys))
(tree_pair (tr1,tr2))
(* prop_formula * interpretation -> interpretation *)
fun prop_formula_times_interpretation (fm,tr) =
tree_map (map (fn x => SAnd (fm,x))) tr
(* prop_formula list * interpretation list -> interpretation *)
fun prop_formula_list_dot_product_interpretation_list ([fm],[tr]) =
prop_formula_times_interpretation (fm,tr)
| prop_formula_list_dot_product_interpretation_list (fm::fms,tr::trees) =
interpretation_disjunction (prop_formula_times_interpretation (fm,tr),
prop_formula_list_dot_product_interpretation_list (fms,trees))
| prop_formula_list_dot_product_interpretation_list (_,_) =
raise REFUTE ("interpretation_apply", "empty list (in dot product)")
(* concatenates 'x' with every list in 'xss', returning a new list of *)
(* lists *)
(* 'a -> 'a list list -> 'a list list *)
fun cons_list x xss = map (cons x) xss
(* returns a list of lists, each one consisting of one element from each *)
(* element of 'xss' *)
(* 'a list list -> 'a list list *)
fun pick_all [xs] = map single xs
| pick_all (xs::xss) =
let val rec_pick = pick_all xss in
maps (fn x => map (cons x) rec_pick) xs
end
| pick_all _ = raise REFUTE ("interpretation_apply", "empty list (in pick_all)")
(* interpretation -> prop_formula list *)
fun interpretation_to_prop_formula_list (Leaf xs) = xs
| interpretation_to_prop_formula_list (Node trees) =
map PropLogic.all (pick_all
(map interpretation_to_prop_formula_list trees))
in
case i1 of
Leaf _ =>
raise REFUTE ("interpretation_apply", "first interpretation is a leaf")
| Node xs =>
prop_formula_list_dot_product_interpretation_list
(interpretation_to_prop_formula_list i2, xs)
end;
(* ------------------------------------------------------------------------- *)
(* eta_expand: eta-expands a term 't' by adding 'i' lambda abstractions *)
(* ------------------------------------------------------------------------- *)
(* Term.term -> int -> Term.term *)
fun eta_expand t i =
let
val Ts = Term.binder_types (Term.fastype_of t)
val t' = Term.incr_boundvars i t
in
fold_rev (fn T => fn term => Abs ("<eta_expand>", T, term))
(List.take (Ts, i))
(Term.list_comb (t', map Bound (i-1 downto 0)))
end;
(* ------------------------------------------------------------------------- *)
(* size_of_dtyp: the size of (an initial fragment of) an inductive data type *)
(* is the sum (over its constructors) of the product (over *)
(* their arguments) of the size of the argument types *)
(* ------------------------------------------------------------------------- *)
fun size_of_dtyp ctxt typ_sizes descr typ_assoc constructors =
Integer.sum (map (fn (_, dtyps) =>
Integer.prod (map (size_of_type ctxt (typ_sizes, []) o
(typ_of_dtyp descr typ_assoc)) dtyps))
constructors);
(* ------------------------------------------------------------------------- *)
(* INTERPRETERS: Actual Interpreters *)
(* ------------------------------------------------------------------------- *)
(* simply typed lambda calculus: Isabelle's basic term syntax, with type *)
(* variables, function types, and propT *)
fun stlc_interpreter ctxt model args t =
let
val thy = ProofContext.theory_of ctxt
val (typs, terms) = model
val {maxvars, def_eq, next_idx, bounds, wellformed} = args
(* Term.typ -> (interpretation * model * arguments) option *)
fun interpret_groundterm T =
let
(* unit -> (interpretation * model * arguments) option *)
fun interpret_groundtype () =
let
(* the model must specify a size for ground types *)
val size =
if T = Term.propT then 2
else the (AList.lookup (op =) typs T)
val next = next_idx + size
(* check if 'maxvars' is large enough *)
val _ = (if next - 1 > maxvars andalso maxvars > 0 then
raise MAXVARS_EXCEEDED else ())
(* prop_formula list *)
val fms = map BoolVar (next_idx upto (next_idx + size - 1))
(* interpretation *)
val intr = Leaf fms
(* prop_formula list -> prop_formula *)
fun one_of_two_false [] = True
| one_of_two_false (x::xs) = SAnd (PropLogic.all (map (fn x' =>
SOr (SNot x, SNot x')) xs), one_of_two_false xs)
(* prop_formula *)
val wf = one_of_two_false fms
in
(* extend the model, increase 'next_idx', add well-formedness *)
(* condition *)
SOME (intr, (typs, (t, intr)::terms), {maxvars = maxvars,
def_eq = def_eq, next_idx = next, bounds = bounds,
wellformed = SAnd (wellformed, wf)})
end
in
case T of
Type ("fun", [T1, T2]) =>
let
(* we create 'size_of_type ... T1' different copies of the *)
(* interpretation for 'T2', which are then combined into a single *)
(* new interpretation *)
(* make fresh copies, with different variable indices *)
(* 'idx': next variable index *)
(* 'n' : number of copies *)
(* int -> int -> (int * interpretation list * prop_formula *)
fun make_copies idx 0 = (idx, [], True)
| make_copies idx n =
let
val (copy, _, new_args) = interpret ctxt (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 ctxt model T1)
(* combine copies into a single interpretation *)
val intr = Node copies
in
(* extend the model, increase 'next_idx', add well-formedness *)
(* condition *)
SOME (intr, (typs, (t, intr)::terms), {maxvars = maxvars,
def_eq = def_eq, next_idx = next, bounds = bounds,
wellformed = SAnd (wellformed, wf)})
end
| Type _ => interpret_groundtype ()
| TFree _ => interpret_groundtype ()
| TVar _ => interpret_groundtype ()
end
in
case AList.lookup (op =) terms t of
SOME intr =>
(* return an existing interpretation *)
SOME (intr, model, args)
| NONE =>
(case t of
Const (_, T) => interpret_groundterm T
| Free (_, T) => interpret_groundterm T
| Var (_, T) => interpret_groundterm T
| Bound i => SOME (List.nth (#bounds args, i), model, args)
| Abs (x, T, body) =>
let
(* create all constants of type 'T' *)
val constants = make_constants ctxt model T
(* interpret the 'body' separately for each constant *)
val (bodies, (model', args')) = fold_map
(fn c => fn (m, a) =>
let
(* add 'c' to 'bounds' *)
val (i', m', a') = interpret ctxt m {maxvars = #maxvars a,
def_eq = #def_eq a, next_idx = #next_idx a,
bounds = (c :: #bounds a), wellformed = #wellformed a} body
in
(* keep the new model m' and 'next_idx' and 'wellformed', *)
(* but use old 'bounds' *)
(i', (m', {maxvars = maxvars, def_eq = def_eq,
next_idx = #next_idx a', bounds = bounds,
wellformed = #wellformed a'}))
end)
constants (model, args)
in
SOME (Node bodies, model', args')
end
| t1 $ t2 =>
let
(* interpret 't1' and 't2' separately *)
val (intr1, model1, args1) = interpret ctxt model args t1
val (intr2, model2, args2) = interpret ctxt model1 args1 t2
in
SOME (interpretation_apply (intr1, intr2), model2, args2)
end)
end;
fun Pure_interpreter ctxt model args t =
case t of
Const (@{const_name all}, _) $ t1 =>
let
val (i, m, a) = interpret ctxt model args t1
in
case i of
Node xs =>
(* 3-valued logic *)
let
val fmTrue = PropLogic.all (map toTrue xs)
val fmFalse = PropLogic.exists (map toFalse xs)
in
SOME (Leaf [fmTrue, fmFalse], m, a)
end
| _ =>
raise REFUTE ("Pure_interpreter",
"\"all\" is followed by a non-function")
end
| Const (@{const_name all}, _) =>
SOME (interpret ctxt model args (eta_expand t 1))
| Const (@{const_name "=="}, _) $ t1 $ t2 =>
let
val (i1, m1, a1) = interpret ctxt model args t1
val (i2, m2, a2) = interpret ctxt m1 a1 t2
in
(* we use either 'make_def_equality' or 'make_equality' *)
SOME ((if #def_eq args then make_def_equality else make_equality)
(i1, i2), m2, a2)
end
| Const (@{const_name "=="}, _) $ t1 =>
SOME (interpret ctxt model args (eta_expand t 1))
| Const (@{const_name "=="}, _) =>
SOME (interpret ctxt model args (eta_expand t 2))
| Const (@{const_name "==>"}, _) $ t1 $ t2 =>
(* 3-valued logic *)
let
val (i1, m1, a1) = interpret ctxt model args t1
val (i2, m2, a2) = interpret ctxt m1 a1 t2
val fmTrue = PropLogic.SOr (toFalse i1, toTrue i2)
val fmFalse = PropLogic.SAnd (toTrue i1, toFalse i2)
in
SOME (Leaf [fmTrue, fmFalse], m2, a2)
end
| Const (@{const_name "==>"}, _) $ t1 =>
SOME (interpret ctxt model args (eta_expand t 1))
| Const (@{const_name "==>"}, _) =>
SOME (interpret ctxt model args (eta_expand t 2))
| _ => NONE;
fun HOLogic_interpreter ctxt model args t =
(* Providing interpretations directly is more efficient than unfolding the *)
(* logical constants. In HOL however, logical constants can themselves be *)
(* arguments. They are then translated using eta-expansion. *)
case t of
Const (@{const_name Trueprop}, _) =>
SOME (Node [TT, FF], model, args)
| Const (@{const_name Not}, _) =>
SOME (Node [FF, TT], model, args)
(* redundant, since 'True' is also an IDT constructor *)
| Const (@{const_name True}, _) =>
SOME (TT, model, args)
(* redundant, since 'False' is also an IDT constructor *)
| Const (@{const_name False}, _) =>
SOME (FF, model, args)
| Const (@{const_name All}, _) $ t1 => (* similar to "all" (Pure) *)
let
val (i, m, a) = interpret ctxt model args t1
in
case i of
Node xs =>
(* 3-valued logic *)
let
val fmTrue = PropLogic.all (map toTrue xs)
val fmFalse = PropLogic.exists (map toFalse xs)
in
SOME (Leaf [fmTrue, fmFalse], m, a)
end
| _ =>
raise REFUTE ("HOLogic_interpreter",
"\"All\" is followed by a non-function")
end
| Const (@{const_name All}, _) =>
SOME (interpret ctxt model args (eta_expand t 1))
| Const (@{const_name Ex}, _) $ t1 =>
let
val (i, m, a) = interpret ctxt model args t1
in
case i of
Node xs =>
(* 3-valued logic *)
let
val fmTrue = PropLogic.exists (map toTrue xs)
val fmFalse = PropLogic.all (map toFalse xs)
in
SOME (Leaf [fmTrue, fmFalse], m, a)
end
| _ =>
raise REFUTE ("HOLogic_interpreter",
"\"Ex\" is followed by a non-function")
end
| Const (@{const_name Ex}, _) =>
SOME (interpret ctxt model args (eta_expand t 1))
| Const (@{const_name HOL.eq}, _) $ t1 $ t2 => (* similar to "==" (Pure) *)
let
val (i1, m1, a1) = interpret ctxt model args t1
val (i2, m2, a2) = interpret ctxt m1 a1 t2
in
SOME (make_equality (i1, i2), m2, a2)
end
| Const (@{const_name HOL.eq}, _) $ t1 =>
SOME (interpret ctxt model args (eta_expand t 1))
| Const (@{const_name HOL.eq}, _) =>
SOME (interpret ctxt model args (eta_expand t 2))
| Const (@{const_name HOL.conj}, _) $ t1 $ t2 =>
(* 3-valued logic *)
let
val (i1, m1, a1) = interpret ctxt model args t1
val (i2, m2, a2) = interpret ctxt m1 a1 t2
val fmTrue = PropLogic.SAnd (toTrue i1, toTrue i2)
val fmFalse = PropLogic.SOr (toFalse i1, toFalse i2)
in
SOME (Leaf [fmTrue, fmFalse], m2, a2)
end
| Const (@{const_name HOL.conj}, _) $ t1 =>
SOME (interpret ctxt model args (eta_expand t 1))
| Const (@{const_name HOL.conj}, _) =>
SOME (interpret ctxt model args (eta_expand t 2))
(* this would make "undef" propagate, even for formulae like *)
(* "False & undef": *)
(* SOME (Node [Node [TT, FF], Node [FF, FF]], model, args) *)
| Const (@{const_name HOL.disj}, _) $ t1 $ t2 =>
(* 3-valued logic *)
let
val (i1, m1, a1) = interpret ctxt model args t1
val (i2, m2, a2) = interpret ctxt m1 a1 t2
val fmTrue = PropLogic.SOr (toTrue i1, toTrue i2)
val fmFalse = PropLogic.SAnd (toFalse i1, toFalse i2)
in
SOME (Leaf [fmTrue, fmFalse], m2, a2)
end
| Const (@{const_name HOL.disj}, _) $ t1 =>
SOME (interpret ctxt model args (eta_expand t 1))
| Const (@{const_name HOL.disj}, _) =>
SOME (interpret ctxt model args (eta_expand t 2))
(* this would make "undef" propagate, even for formulae like *)
(* "True | undef": *)
(* SOME (Node [Node [TT, TT], Node [TT, FF]], model, args) *)
| Const (@{const_name HOL.implies}, _) $ t1 $ t2 => (* similar to "==>" (Pure) *)
(* 3-valued logic *)
let
val (i1, m1, a1) = interpret ctxt model args t1
val (i2, m2, a2) = interpret ctxt m1 a1 t2
val fmTrue = PropLogic.SOr (toFalse i1, toTrue i2)
val fmFalse = PropLogic.SAnd (toTrue i1, toFalse i2)
in
SOME (Leaf [fmTrue, fmFalse], m2, a2)
end
| Const (@{const_name HOL.implies}, _) $ t1 =>
SOME (interpret ctxt model args (eta_expand t 1))
| Const (@{const_name HOL.implies}, _) =>
SOME (interpret ctxt model args (eta_expand t 2))
(* this would make "undef" propagate, even for formulae like *)
(* "False --> undef": *)
(* SOME (Node [Node [TT, FF], Node [TT, TT]], model, args) *)
| _ => NONE;
(* interprets variables and constants whose type is an IDT (this is *)
(* relatively easy and merely requires us to compute the size of the IDT); *)
(* constructors of IDTs however are properly interpreted by *)
(* 'IDT_constructor_interpreter' *)
fun IDT_interpreter ctxt model args t =
let
val thy = ProofContext.theory_of ctxt
val (typs, terms) = model
(* Term.typ -> (interpretation * model * arguments) option *)
fun interpret_term (Type (s, Ts)) =
(case Datatype.get_info thy s of
SOME info => (* inductive datatype *)
let
(* int option -- only recursive IDTs have an associated depth *)
val depth = AList.lookup (op =) typs (Type (s, Ts))
(* sanity check: depth must be at least 0 *)
val _ =
(case depth of SOME n =>
if n < 0 then
raise REFUTE ("IDT_interpreter", "negative depth")
else ()
| _ => ())
in
(* termination condition to avoid infinite recursion *)
if depth = (SOME 0) then
(* return a leaf of size 0 *)
SOME (Leaf [], model, args)
else
let
val index = #index info
val descr = #descr info
val (_, dtyps, constrs) = the (AList.lookup (op =) descr index)
val typ_assoc = dtyps ~~ Ts
(* sanity check: every element in 'dtyps' must be a 'DtTFree' *)
val _ =
if Library.exists (fn d =>
case d of Datatype_Aux.DtTFree _ => false | _ => true) dtyps
then
raise REFUTE ("IDT_interpreter",
"datatype argument (for type "
^ Syntax.string_of_typ ctxt (Type (s, Ts))
^ ") is not a variable")
else ()
(* if the model specifies a depth for the current type, *)
(* decrement it to avoid infinite recursion *)
val typs' = case depth of NONE => typs | SOME n =>
AList.update (op =) (Type (s, Ts), n-1) typs
(* recursively compute the size of the datatype *)
val size = size_of_dtyp ctxt typs' descr typ_assoc constrs
val next_idx = #next_idx args
val next = next_idx+size
(* check if 'maxvars' is large enough *)
val _ = (if next-1 > #maxvars args andalso
#maxvars args > 0 then raise MAXVARS_EXCEEDED else ())
(* prop_formula list *)
val fms = map BoolVar (next_idx upto (next_idx+size-1))
(* interpretation *)
val intr = Leaf fms
(* prop_formula list -> prop_formula *)
fun one_of_two_false [] = True
| one_of_two_false (x::xs) = SAnd (PropLogic.all (map (fn x' =>
SOr (SNot x, SNot x')) xs), one_of_two_false xs)
(* prop_formula *)
val wf = one_of_two_false fms
in
(* extend the model, increase 'next_idx', add well-formedness *)
(* condition *)
SOME (intr, (typs, (t, intr)::terms), {maxvars = #maxvars args,
def_eq = #def_eq args, next_idx = next, bounds = #bounds args,
wellformed = SAnd (#wellformed args, wf)})
end
end
| NONE => (* not an inductive datatype *)
NONE)
| interpret_term _ = (* a (free or schematic) type variable *)
NONE
in
case AList.lookup (op =) terms t of
SOME intr =>
(* return an existing interpretation *)
SOME (intr, model, args)
| NONE =>
(case t of
Free (_, T) => interpret_term T
| Var (_, T) => interpret_term T
| Const (_, T) => interpret_term T
| _ => NONE)
end;
(* This function imposes an order on the elements of a datatype fragment *)
(* as follows: C_i x_1 ... x_n < C_j y_1 ... y_m iff i < j or *)
(* (x_1, ..., x_n) < (y_1, ..., y_m). With this order, a constructor is *)
(* a function C_i that maps some argument indices x_1, ..., x_n to the *)
(* datatype element given by index C_i x_1 ... x_n. The idea remains the *)
(* same for recursive datatypes, although the computation of indices gets *)
(* a little tricky. *)
fun IDT_constructor_interpreter ctxt model args t =
let
val thy = ProofContext.theory_of ctxt
(* returns a list of canonical representations for terms of the type 'T' *)
(* It would be nice if we could just use 'print' for this, but 'print' *)
(* for IDTs calls 'IDT_constructor_interpreter' again, and this could *)
(* lead to infinite recursion when we have (mutually) recursive IDTs. *)
(* (Term.typ * int) list -> Term.typ -> Term.term list *)
fun canonical_terms typs T =
(case T of
Type ("fun", [T1, T2]) =>
(* 'T2' might contain a recursive IDT, so we cannot use 'print' (at *)
(* least not for 'T2' *)
let
(* returns a list of lists, each one consisting of n (possibly *)
(* identical) elements from 'xs' *)
(* int -> 'a list -> 'a list list *)
fun pick_all 1 xs = map single xs
| pick_all n xs =
let val rec_pick = pick_all (n-1) xs in
maps (fn x => map (cons x) rec_pick) xs
end
(* ["x1", ..., "xn"] *)
val terms1 = canonical_terms typs T1
(* ["y1", ..., "ym"] *)
val terms2 = canonical_terms typs T2
(* [[("x1", "y1"), ..., ("xn", "y1")], ..., *)
(* [("x1", "ym"), ..., ("xn", "ym")]] *)
val functions = map (curry (op ~~) terms1)
(pick_all (length terms1) terms2)
(* [["(x1, y1)", ..., "(xn, y1)"], ..., *)
(* ["(x1, ym)", ..., "(xn, ym)"]] *)
val pairss = map (map HOLogic.mk_prod) functions
(* Term.typ *)
val HOLogic_prodT = HOLogic.mk_prodT (T1, T2)
val HOLogic_setT = HOLogic.mk_setT HOLogic_prodT
(* Term.term *)
val HOLogic_empty_set = Const (@{const_abbrev Set.empty}, HOLogic_setT)
val HOLogic_insert =
Const (@{const_name insert}, HOLogic_prodT --> HOLogic_setT --> HOLogic_setT)
in
(* functions as graphs, i.e. as a (HOL) set of pairs "(x, y)" *)
map (fn ps => fold_rev (fn pair => fn acc => HOLogic_insert $ pair $ acc) ps
HOLogic_empty_set) pairss
end
| Type (s, Ts) =>
(case Datatype.get_info thy s of
SOME info =>
(case AList.lookup (op =) typs T of
SOME 0 =>
(* termination condition to avoid infinite recursion *)
[] (* at depth 0, every IDT is empty *)
| _ =>
let
val index = #index info
val descr = #descr info
val (_, dtyps, constrs) = the (AList.lookup (op =) descr index)
val typ_assoc = dtyps ~~ Ts
(* sanity check: every element in 'dtyps' must be a 'DtTFree' *)
val _ =
if Library.exists (fn d =>
case d of Datatype_Aux.DtTFree _ => false | _ => true) dtyps
then
raise REFUTE ("IDT_constructor_interpreter",
"datatype argument (for type "
^ Syntax.string_of_typ ctxt T
^ ") is not a variable")
else ()
(* decrement depth for the IDT 'T' *)
val typs' =
(case AList.lookup (op =) typs T of NONE => typs
| SOME n => AList.update (op =) (T, n-1) typs)
fun constructor_terms terms [] = terms
| constructor_terms terms (d::ds) =
let
val dT = typ_of_dtyp descr typ_assoc d
val d_terms = canonical_terms typs' dT
in
(* C_i x_1 ... x_n < C_i y_1 ... y_n if *)
(* (x_1, ..., x_n) < (y_1, ..., y_n) *)
constructor_terms
(map_product (curry op $) terms d_terms) ds
end
in
(* C_i ... < C_j ... if i < j *)
maps (fn (cname, ctyps) =>
let
val cTerm = Const (cname,
map (typ_of_dtyp descr typ_assoc) ctyps ---> T)
in
constructor_terms [cTerm] ctyps
end) constrs
end)
| NONE =>
(* not an inductive datatype; in this case the argument types in *)
(* 'Ts' may not be IDTs either, so 'print' should be safe *)
map (fn intr => print ctxt (typs, []) T intr (K false))
(make_constants ctxt (typs, []) T))
| _ => (* TFree ..., TVar ... *)
map (fn intr => print ctxt (typs, []) T intr (K false))
(make_constants ctxt (typs, []) T))
val (typs, terms) = model
in
case AList.lookup (op =) terms t of
SOME intr =>
(* return an existing interpretation *)
SOME (intr, model, args)
| NONE =>
(case t of
Const (s, T) =>
(case body_type T of
Type (s', Ts') =>
(case Datatype.get_info thy s' of
SOME info => (* body type is an inductive datatype *)
let
val index = #index info
val descr = #descr info
val (_, dtyps, constrs) = the (AList.lookup (op =) descr index)
val typ_assoc = dtyps ~~ Ts'
(* sanity check: every element in 'dtyps' must be a 'DtTFree' *)
val _ = if Library.exists (fn d =>
case d of Datatype_Aux.DtTFree _ => false | _ => true) dtyps
then
raise REFUTE ("IDT_constructor_interpreter",
"datatype argument (for type "
^ Syntax.string_of_typ ctxt (Type (s', Ts'))
^ ") is not a variable")
else ()
(* split the constructors into those occuring before/after *)
(* 'Const (s, T)' *)
val (constrs1, constrs2) = take_prefix (fn (cname, ctypes) =>
not (cname = s andalso Sign.typ_instance thy (T,
map (typ_of_dtyp descr typ_assoc) ctypes
---> Type (s', Ts')))) constrs
in
case constrs2 of
[] =>
(* 'Const (s, T)' is not a constructor of this datatype *)
NONE
| (_, ctypes)::cs =>
let
(* int option -- only /recursive/ IDTs have an associated *)
(* depth *)
val depth = AList.lookup (op =) typs (Type (s', Ts'))
(* this should never happen: at depth 0, this IDT fragment *)
(* is definitely empty, and in this case we don't need to *)
(* interpret its constructors *)
val _ = (case depth of SOME 0 =>
raise REFUTE ("IDT_constructor_interpreter",
"depth is 0")
| _ => ())
val typs' = (case depth of NONE => typs | SOME n =>
AList.update (op =) (Type (s', Ts'), n-1) typs)
(* elements of the datatype come before elements generated *)
(* by 'Const (s, T)' iff they are generated by a *)
(* constructor in constrs1 *)
val offset = size_of_dtyp ctxt typs' descr typ_assoc constrs1
(* compute the total (current) size of the datatype *)
val total = offset +
size_of_dtyp ctxt typs' descr typ_assoc constrs2
(* sanity check *)
val _ = if total <> size_of_type ctxt (typs, [])
(Type (s', Ts')) then
raise REFUTE ("IDT_constructor_interpreter",
"total is not equal to current size")
else ()
(* returns an interpretation where everything is mapped to *)
(* an "undefined" element of the datatype *)
fun make_undef [] = Leaf (replicate total False)
| make_undef (d::ds) =
let
(* compute the current size of the type 'd' *)
val dT = typ_of_dtyp descr typ_assoc d
val size = size_of_type ctxt (typs, []) dT
in
Node (replicate size (make_undef ds))
end
(* returns the interpretation for a constructor *)
fun make_constr [] offset =
if offset < total then
(Leaf (replicate offset False @ True ::
(replicate (total - offset - 1) False)), offset + 1)
else
raise REFUTE ("IDT_constructor_interpreter",
"offset >= total")
| make_constr (d::ds) offset =
let
(* Term.typ *)
val dT = typ_of_dtyp descr typ_assoc d
(* compute canonical term representations for all *)
(* elements of the type 'd' (with the reduced depth *)
(* for the IDT) *)
val terms' = canonical_terms typs' dT
(* sanity check *)
val _ =
if length terms' <> size_of_type ctxt (typs', []) dT
then
raise REFUTE ("IDT_constructor_interpreter",
"length of terms' is not equal to old size")
else ()
(* compute canonical term representations for all *)
(* elements of the type 'd' (with the current depth *)
(* for the IDT) *)
val terms = canonical_terms typs dT
(* sanity check *)
val _ =
if length terms <> size_of_type ctxt (typs, []) dT
then
raise REFUTE ("IDT_constructor_interpreter",
"length of terms is not equal to current size")
else ()
(* sanity check *)
val _ =
if length terms < length terms' then
raise REFUTE ("IDT_constructor_interpreter",
"current size is less than old size")
else ()
(* sanity check: every element of terms' must also be *)
(* present in terms *)
val _ =
if forall (member (op =) terms) terms' then ()
else
raise REFUTE ("IDT_constructor_interpreter",
"element has disappeared")
(* sanity check: the order on elements of terms' is *)
(* the same in terms, for those elements *)
val _ =
let
fun search (x::xs) (y::ys) =
if x = y then search xs ys else search (x::xs) ys
| search (x::xs) [] =
raise REFUTE ("IDT_constructor_interpreter",
"element order not preserved")
| search [] _ = ()
in search terms' terms end
(* int * interpretation list *)
val (intrs, new_offset) =
fold_map (fn t_elem => fn off =>
(* if 't_elem' existed at the previous depth, *)
(* proceed recursively, otherwise map the entire *)
(* subtree to "undefined" *)
if member (op =) terms' t_elem then
make_constr ds off
else
(make_undef ds, off))
terms offset
in
(Node intrs, new_offset)
end
in
SOME (fst (make_constr ctypes offset), model, args)
end
end
| NONE => (* body type is not an inductive datatype *)
NONE)
| _ => (* body type is a (free or schematic) type variable *)
NONE)
| _ => (* term is not a constant *)
NONE)
end;
(* 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 ctxt model args t =
let
val thy = ProofContext.theory_of ctxt
in
(* careful: here we descend arbitrarily deep into 't', possibly before *)
(* any other interpreter for atomic terms has had a chance to look at *)
(* 't' *)
case strip_comb t of
(Const (s, T), params) =>
(* iterate over all datatypes in 'thy' *)
Symtab.fold (fn (_, info) => fn result =>
case result of
SOME _ =>
result (* just keep 'result' *)
| NONE =>
if member (op =) (#rec_names info) s then
(* we do have a recursion operator of one of the (mutually *)
(* recursive) datatypes given by 'info' *)
let
(* number of all constructors, including those of different *)
(* (mutually recursive) datatypes within the same descriptor *)
val mconstrs_count =
Integer.sum (map (fn (_, (_, _, cs)) => length cs) (#descr info))
in
if mconstrs_count < length params then
(* too many actual parameters; for now we'll use the *)
(* 'stlc_interpreter' to strip off one application *)
NONE
else if mconstrs_count > length params then
(* too few actual parameters; we use eta expansion *)
(* Note that the resulting expansion of lambda abstractions *)
(* by the 'stlc_interpreter' may be rather slow (depending *)
(* on the argument types and the size of the IDT, of *)
(* course). *)
SOME (interpret ctxt model args (eta_expand t
(mconstrs_count - length params)))
else (* mconstrs_count = length params *)
let
(* interpret each parameter separately *)
val (p_intrs, (model', args')) = fold_map (fn p => fn (m, a) =>
let
val (i, m', a') = interpret ctxt m a p
in
(i, (m', a'))
end) params (model, args)
val (typs, _) = model'
(* 'index' is /not/ necessarily the index of the IDT that *)
(* the recursion operator is associated with, but merely *)
(* the index of some mutually recursive IDT *)
val index = #index info
val descr = #descr info
val (_, dtyps, _) = the (AList.lookup (op =) descr index)
(* sanity check: we assume that the order of constructors *)
(* in 'descr' is the same as the order of *)
(* corresponding parameters, otherwise the *)
(* association code below won't match the *)
(* right constructors/parameters; we also *)
(* assume that the order of recursion *)
(* operators in '#rec_names info' is the *)
(* same as the order of corresponding *)
(* datatypes in 'descr' *)
val _ = if map fst descr <> (0 upto (length descr - 1)) then
raise REFUTE ("IDT_recursion_interpreter",
"order of constructors and corresponding parameters/" ^
"recursion operators and corresponding datatypes " ^
"different?")
else ()
(* sanity check: every element in 'dtyps' must be a *)
(* 'DtTFree' *)
val _ =
if Library.exists (fn d =>
case d of Datatype_Aux.DtTFree _ => false
| _ => true) dtyps
then
raise REFUTE ("IDT_recursion_interpreter",
"datatype argument is not a variable")
else ()
(* the type of a recursion operator is *)
(* [T1, ..., Tn, IDT] ---> Tresult *)
val IDT = List.nth (binder_types T, mconstrs_count)
(* by our assumption on the order of recursion operators *)
(* and datatypes, this is the index of the datatype *)
(* corresponding to the given recursion operator *)
val idt_index = find_index (fn s' => s' = s) (#rec_names info)
(* mutually recursive types must have the same type *)
(* parameters, unless the mutual recursion comes from *)
(* indirect recursion *)
fun rec_typ_assoc acc [] = acc
| rec_typ_assoc acc ((d, T)::xs) =
(case AList.lookup op= acc d of
NONE =>
(case d of
Datatype_Aux.DtTFree _ =>
(* add the association, proceed *)
rec_typ_assoc ((d, T)::acc) xs
| Datatype_Aux.DtType (s, ds) =>
let
val (s', Ts) = dest_Type T
in
if s=s' then
rec_typ_assoc ((d, T)::acc) ((ds ~~ Ts) @ xs)
else
raise REFUTE ("IDT_recursion_interpreter",
"DtType/Type mismatch")
end
| Datatype_Aux.DtRec i =>
let
val (_, ds, _) = the (AList.lookup (op =) descr i)
val (_, Ts) = dest_Type T
in
rec_typ_assoc ((d, T)::acc) ((ds ~~ Ts) @ xs)
end)
| SOME T' =>
if T=T' then
(* ignore the association since it's already *)
(* present, proceed *)
rec_typ_assoc acc xs
else
raise REFUTE ("IDT_recursion_interpreter",
"different type associations for the same dtyp"))
val typ_assoc = filter
(fn (Datatype_Aux.DtTFree _, _) => true | (_, _) => false)
(rec_typ_assoc []
(#2 (the (AList.lookup (op =) descr idt_index)) ~~ (snd o dest_Type) IDT))
(* sanity check: typ_assoc must associate types to the *)
(* elements of 'dtyps' (and only to those) *)
val _ =
if not (eq_set (op =) (dtyps, map fst typ_assoc))
then
raise REFUTE ("IDT_recursion_interpreter",
"type association has extra/missing elements")
else ()
(* interpret each constructor in the descriptor (including *)
(* those of mutually recursive datatypes) *)
(* (int * interpretation list) list *)
val mc_intrs = map (fn (idx, (_, _, cs)) =>
let
val c_return_typ = typ_of_dtyp descr typ_assoc
(Datatype_Aux.DtRec idx)
in
(idx, map (fn (cname, cargs) =>
(#1 o interpret ctxt (typs, []) {maxvars=0,
def_eq=false, next_idx=1, bounds=[],
wellformed=True}) (Const (cname, map (typ_of_dtyp
descr typ_assoc) cargs ---> c_return_typ))) cs)
end) descr
(* associate constructors with corresponding parameters *)
(* (int * (interpretation * interpretation) list) list *)
val (mc_p_intrs, p_intrs') = fold_map
(fn (idx, c_intrs) => fn p_intrs' =>
let
val len = length c_intrs
in
((idx, c_intrs ~~ List.take (p_intrs', len)),
List.drop (p_intrs', len))
end) mc_intrs p_intrs
(* sanity check: no 'p_intr' may be left afterwards *)
val _ =
if p_intrs' <> [] then
raise REFUTE ("IDT_recursion_interpreter",
"more parameter than constructor interpretations")
else ()
(* The recursion operator, applied to 'mconstrs_count' *)
(* arguments, is a function that maps every element of the *)
(* inductive datatype to an element of some result type. *)
(* Recursion operators for mutually recursive IDTs are *)
(* translated simultaneously. *)
(* Since the order on datatype elements is given by an *)
(* order on constructors (and then by the order on *)
(* argument tuples), we can simply copy corresponding *)
(* subtrees from 'p_intrs', in the order in which they are *)
(* given. *)
(* interpretation * interpretation -> interpretation list *)
fun ci_pi (Leaf xs, pi) =
(* if the constructor does not match the arguments to a *)
(* defined element of the IDT, the corresponding value *)
(* of the parameter must be ignored *)
if List.exists (equal True) xs then [pi] else []
| ci_pi (Node xs, Node ys) = maps ci_pi (xs ~~ ys)
| ci_pi (Node _, Leaf _) =
raise REFUTE ("IDT_recursion_interpreter",
"constructor takes more arguments than the " ^
"associated parameter")
(* (int * interpretation list) list *)
val rec_operators = map (fn (idx, c_p_intrs) =>
(idx, maps ci_pi c_p_intrs)) mc_p_intrs
(* sanity check: every recursion operator must provide as *)
(* many values as the corresponding datatype *)
(* has elements *)
val _ = map (fn (idx, intrs) =>
let
val T = typ_of_dtyp descr typ_assoc
(Datatype_Aux.DtRec idx)
in
if length intrs <> size_of_type ctxt (typs, []) T then
raise REFUTE ("IDT_recursion_interpreter",
"wrong number of interpretations for rec. operator")
else ()
end) rec_operators
(* For non-recursive datatypes, we are pretty much done at *)
(* this point. For recursive datatypes however, we still *)
(* need to apply the interpretations in 'rec_operators' to *)
(* (recursively obtained) interpretations for recursive *)
(* constructor arguments. To do so more efficiently, we *)
(* copy 'rec_operators' into arrays first. Each Boolean *)
(* indicates whether the recursive arguments have been *)
(* considered already. *)
(* (int * (bool * interpretation) Array.array) list *)
val REC_OPERATORS = map (fn (idx, intrs) =>
(idx, Array.fromList (map (pair false) intrs)))
rec_operators
(* takes an interpretation, and if some leaf of this *)
(* interpretation is the 'elem'-th element of the type, *)
(* the indices of the arguments leading to this leaf are *)
(* returned *)
(* interpretation -> int -> int list option *)
fun get_args (Leaf xs) elem =
if find_index (fn x => x = True) xs = elem then
SOME []
else
NONE
| get_args (Node xs) elem =
let
(* interpretation list * int -> int list option *)
fun search ([], _) =
NONE
| search (x::xs, n) =
(case get_args x elem of
SOME result => SOME (n::result)
| NONE => search (xs, n+1))
in
search (xs, 0)
end
(* returns the index of the constructor and indices for *)
(* its arguments that generate the 'elem'-th element of *)
(* the datatype given by 'idx' *)
(* int -> int -> int * int list *)
fun get_cargs idx elem =
let
(* int * interpretation list -> int * int list *)
fun get_cargs_rec (_, []) =
raise REFUTE ("IDT_recursion_interpreter",
"no matching constructor found for datatype element")
| get_cargs_rec (n, x::xs) =
(case get_args x elem of
SOME args => (n, args)
| NONE => get_cargs_rec (n+1, xs))
in
get_cargs_rec (0, the (AList.lookup (op =) mc_intrs idx))
end
(* computes one entry in 'REC_OPERATORS', and recursively *)
(* all entries needed for it, where 'idx' gives the *)
(* datatype and 'elem' the element of it *)
(* int -> int -> interpretation *)
fun compute_array_entry idx elem =
let
val arr = the (AList.lookup (op =) REC_OPERATORS idx)
val (flag, intr) = Array.sub (arr, elem)
in
if flag then
(* simply return the previously computed result *)
intr
else
(* we have to apply 'intr' to interpretations for all *)
(* recursive arguments *)
let
(* int * int list *)
val (c, args) = get_cargs idx elem
(* find the indices of the constructor's /recursive/ *)
(* arguments *)
val (_, _, constrs) = the (AList.lookup (op =) descr idx)
val (_, dtyps) = List.nth (constrs, c)
val rec_dtyps_args = filter
(Datatype_Aux.is_rec_type o fst) (dtyps ~~ args)
(* map those indices to interpretations *)
val rec_dtyps_intrs = map (fn (dtyp, arg) =>
let
val dT = typ_of_dtyp descr typ_assoc dtyp
val consts = make_constants ctxt (typs, []) dT
val arg_i = List.nth (consts, arg)
in
(dtyp, arg_i)
end) rec_dtyps_args
(* takes the dtyp and interpretation of an element, *)
(* and computes the interpretation for the *)
(* corresponding recursive argument *)
fun rec_intr (Datatype_Aux.DtRec i) (Leaf xs) =
(* recursive argument is "rec_i params elem" *)
compute_array_entry i (find_index (fn x => x = True) xs)
| rec_intr (Datatype_Aux.DtRec _) (Node _) =
raise REFUTE ("IDT_recursion_interpreter",
"interpretation for IDT is a node")
| rec_intr (Datatype_Aux.DtType ("fun", [dt1, dt2])) (Node xs) =
(* recursive argument is something like *)
(* "\<lambda>x::dt1. rec_? params (elem x)" *)
Node (map (rec_intr dt2) xs)
| rec_intr (Datatype_Aux.DtType ("fun", [_, _])) (Leaf _) =
raise REFUTE ("IDT_recursion_interpreter",
"interpretation for function dtyp is a leaf")
| rec_intr _ _ =
(* admissibility ensures that every recursive type *)
(* is of the form 'Dt_1 -> ... -> Dt_k -> *)
(* (DtRec i)' *)
raise REFUTE ("IDT_recursion_interpreter",
"non-recursive codomain in recursive dtyp")
(* obtain interpretations for recursive arguments *)
(* interpretation list *)
val arg_intrs = map (uncurry rec_intr) rec_dtyps_intrs
(* apply 'intr' to all recursive arguments *)
val result = fold (fn arg_i => fn i =>
interpretation_apply (i, arg_i)) arg_intrs intr
(* update 'REC_OPERATORS' *)
val _ = Array.update (arr, elem, (true, result))
in
result
end
end
val idt_size = Array.length (the (AList.lookup (op =) REC_OPERATORS idt_index))
(* sanity check: the size of 'IDT' should be 'idt_size' *)
val _ =
if idt_size <> size_of_type ctxt (typs, []) IDT then
raise REFUTE ("IDT_recursion_interpreter",
"unexpected size of IDT (wrong type associated?)")
else ()
(* interpretation *)
val rec_op = Node (map_range (compute_array_entry idt_index) idt_size)
in
SOME (rec_op, model', args')
end
end
else
NONE (* not a recursion operator of this datatype *)
) (Datatype.get_all thy) NONE
| _ => (* head of term is not a constant *)
NONE
end;
fun set_interpreter ctxt model args t =
let
val (typs, terms) = model
in
case AList.lookup (op =) terms t of
SOME intr =>
(* return an existing interpretation *)
SOME (intr, model, args)
| NONE =>
(case t of
(* 'Collect' == identity *)
Const (@{const_name Collect}, _) $ t1 =>
SOME (interpret ctxt model args t1)
| Const (@{const_name Collect}, _) =>
SOME (interpret ctxt model args (eta_expand t 1))
(* 'op :' == application *)
| Const (@{const_name Set.member}, _) $ t1 $ t2 =>
SOME (interpret ctxt model args (t2 $ t1))
| Const (@{const_name Set.member}, _) $ t1 =>
SOME (interpret ctxt model args (eta_expand t 1))
| Const (@{const_name Set.member}, _) =>
SOME (interpret ctxt model args (eta_expand t 2))
| _ => NONE)
end;
(* 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 ctxt model args t =
case t of
Const (@{const_name Finite_Set.card},
Type ("fun", [Type ("fun", [T, @{typ bool}]), @{typ nat}])) =>
let
(* interpretation -> int *)
fun number_of_elements (Node xs) =
fold (fn x => fn n =>
if x = TT then
n + 1
else if x = FF then
n
else
raise REFUTE ("Finite_Set_card_interpreter",
"interpretation for set type does not yield a Boolean"))
xs 0
| number_of_elements (Leaf _) =
raise REFUTE ("Finite_Set_card_interpreter",
"interpretation for set type is a leaf")
val size_of_nat = size_of_type ctxt model (@{typ nat})
(* takes an interpretation for a set and returns an interpretation *)
(* for a 'nat' denoting the set's cardinality *)
(* interpretation -> interpretation *)
fun card i =
let
val n = number_of_elements i
in
if n < size_of_nat then
Leaf ((replicate n False) @ True ::
(replicate (size_of_nat-n-1) False))
else
Leaf (replicate size_of_nat False)
end
val set_constants =
make_constants ctxt model (Type ("fun", [T, HOLogic.boolT]))
in
SOME (Node (map card set_constants), model, args)
end
| _ => NONE;
(* only an optimization: 'finite' could in principle be interpreted with *)
(* interpreters available already (using its definition), but the code *)
(* below is more efficient *)
fun Finite_Set_finite_interpreter ctxt model args t =
case t of
Const (@{const_name Finite_Set.finite},
Type ("fun", [Type ("fun", [T, @{typ bool}]),
@{typ bool}])) $ _ =>
(* we only consider finite models anyway, hence EVERY set is *)
(* "finite" *)
SOME (TT, model, args)
| Const (@{const_name Finite_Set.finite},
Type ("fun", [Type ("fun", [T, @{typ bool}]),
@{typ bool}])) =>
let
val size_of_set =
size_of_type ctxt model (Type ("fun", [T, HOLogic.boolT]))
in
(* we only consider finite models anyway, hence EVERY set is *)
(* "finite" *)
SOME (Node (replicate size_of_set TT), model, args)
end
| _ => NONE;
(* only an optimization: 'less' could in principle be interpreted with *)
(* interpreters available already (using its definition), but the code *)
(* below is more efficient *)
fun Nat_less_interpreter ctxt model args t =
case t of
Const (@{const_name Orderings.less}, Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ bool}])])) =>
let
val size_of_nat = size_of_type ctxt model (@{typ nat})
(* the 'n'-th nat is not less than the first 'n' nats, while it *)
(* is less than the remaining 'size_of_nat - n' nats *)
(* int -> interpretation *)
fun less n = Node ((replicate n FF) @ (replicate (size_of_nat - n) TT))
in
SOME (Node (map less (1 upto size_of_nat)), model, args)
end
| _ => NONE;
(* only an optimization: 'plus' could in principle be interpreted with *)
(* interpreters available already (using its definition), but the code *)
(* below is more efficient *)
fun Nat_plus_interpreter ctxt model args t =
case t of
Const (@{const_name Groups.plus}, Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ nat}])])) =>
let
val size_of_nat = size_of_type ctxt model (@{typ nat})
(* int -> int -> interpretation *)
fun plus m n =
let
val element = m + n
in
if element > size_of_nat - 1 then
Leaf (replicate size_of_nat False)
else
Leaf ((replicate element False) @ True ::
(replicate (size_of_nat - element - 1) False))
end
in
SOME (Node (map_range (fn m => Node (map_range (plus m) size_of_nat)) size_of_nat),
model, args)
end
| _ => NONE;
(* only an optimization: 'minus' could in principle be interpreted *)
(* with interpreters available already (using its definition), but the *)
(* code below is more efficient *)
fun Nat_minus_interpreter ctxt model args t =
case t of
Const (@{const_name Groups.minus}, Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ nat}])])) =>
let
val size_of_nat = size_of_type ctxt model (@{typ nat})
(* int -> int -> interpretation *)
fun minus m n =
let
val element = Int.max (m-n, 0)
in
Leaf ((replicate element False) @ True ::
(replicate (size_of_nat - element - 1) False))
end
in
SOME (Node (map_range (fn m => Node (map_range (minus m) size_of_nat)) size_of_nat),
model, args)
end
| _ => NONE;
(* only an optimization: 'times' could in principle be interpreted *)
(* with interpreters available already (using its definition), but the *)
(* code below is more efficient *)
fun Nat_times_interpreter ctxt model args t =
case t of
Const (@{const_name Groups.times}, Type ("fun", [@{typ nat},
Type ("fun", [@{typ nat}, @{typ nat}])])) =>
let
val size_of_nat = size_of_type ctxt model (@{typ nat})
(* nat -> nat -> interpretation *)
fun mult m n =
let
val element = m * n
in
if element > size_of_nat - 1 then
Leaf (replicate size_of_nat False)
else
Leaf ((replicate element False) @ True ::
(replicate (size_of_nat - element - 1) False))
end
in
SOME (Node (map_range (fn m => Node (map_range (mult m) size_of_nat)) size_of_nat),
model, args)
end
| _ => NONE;
(* only an optimization: 'append' could in principle be interpreted with *)
(* interpreters available already (using its definition), but the code *)
(* below is more efficient *)
fun List_append_interpreter ctxt model args t =
case t of
Const (@{const_name List.append}, Type ("fun", [Type ("List.list", [T]), Type ("fun",
[Type ("List.list", [_]), Type ("List.list", [_])])])) =>
let
val size_elem = size_of_type ctxt model T
val size_list = size_of_type ctxt model (Type ("List.list", [T]))
(* maximal length of lists; 0 if we only consider the empty list *)
val list_length =
let
(* int -> int -> int -> int *)
fun list_length_acc len lists total =
if lists = total then
len
else if lists < total then
list_length_acc (len+1) (lists*size_elem) (total-lists)
else
raise REFUTE ("List_append_interpreter",
"size_list not equal to 1 + size_elem + ... + " ^
"size_elem^len, for some len")
in
list_length_acc 0 1 size_list
end
val elements = 0 upto (size_list-1)
(* FIXME: there should be a nice formula, which computes the same as *)
(* the following, but without all this intermediate tree *)
(* length/offset stuff *)
(* associate each list with its length and offset in a complete tree *)
(* of width 'size_elem' and depth 'length_list' (with 'size_list' *)
(* nodes total) *)
(* (int * (int * int)) list *)
val (lenoff_lists, _) = fold_map (fn elem => fn (offsets, off) =>
(* corresponds to a pre-order traversal of the tree *)
let
val len = length offsets
(* associate the given element with len/off *)
val assoc = (elem, (len, off))
in
if len < list_length then
(* go to first child node *)
(assoc, (off :: offsets, off * size_elem))
else if off mod size_elem < size_elem - 1 then
(* go to next sibling node *)
(assoc, (offsets, off + 1))
else
(* go back up the stack until we find a level where we can go *)
(* to the next sibling node *)
let
val offsets' = snd (chop_while
(fn off' => off' mod size_elem = size_elem - 1) offsets)
in
case offsets' of
[] =>
(* we're at the last node in the tree; the next value *)
(* won't be used anyway *)
(assoc, ([], 0))
| off'::offs' =>
(* go to next sibling node *)
(assoc, (offs', off' + 1))
end
end) elements ([], 0)
(* we also need the reverse association (from length/offset to *)
(* index) *)
val lenoff'_lists = map Library.swap lenoff_lists
(* returns the interpretation for "(list no. m) @ (list no. n)" *)
(* nat -> nat -> interpretation *)
fun append m n =
let
val (len_m, off_m) = the (AList.lookup (op =) lenoff_lists m)
val (len_n, off_n) = the (AList.lookup (op =) lenoff_lists n)
val len_elem = len_m + len_n
val off_elem = off_m * Integer.pow len_n size_elem + off_n
in
case AList.lookup op= lenoff'_lists (len_elem, off_elem) of
NONE =>
(* undefined *)
Leaf (replicate size_list False)
| SOME element =>
Leaf ((replicate element False) @ True ::
(replicate (size_list - element - 1) False))
end
in
SOME (Node (map (fn m => Node (map (append m) elements)) elements),
model, args)
end
| _ => NONE;
(* UNSOUND
(* only an optimization: 'lfp' could in principle be interpreted with *)
(* interpreters available already (using its definition), but the code *)
(* below is more efficient *)
fun lfp_interpreter ctxt model args t =
case t of
Const (@{const_name lfp}, Type ("fun", [Type ("fun",
[Type ("fun", [T, @{typ bool}]),
Type ("fun", [_, @{typ bool}])]),
Type ("fun", [_, @{typ bool}])])) =>
let
val size_elem = size_of_type ctxt model T
(* the universe (i.e. the set that contains every element) *)
val i_univ = Node (replicate size_elem TT)
(* all sets with elements from type 'T' *)
val i_sets =
make_constants ctxt model (Type ("fun", [T, HOLogic.boolT]))
(* all functions that map sets to sets *)
val i_funs = make_constants ctxt model (Type ("fun",
[Type ("fun", [T, @{typ bool}]),
Type ("fun", [T, @{typ bool}])]))
(* "lfp(f) == Inter({u. f(u) <= u})" *)
(* interpretation * interpretation -> bool *)
fun is_subset (Node subs, Node sups) =
forall (fn (sub, sup) => (sub = FF) orelse (sup = TT)) (subs ~~ sups)
| is_subset (_, _) =
raise REFUTE ("lfp_interpreter",
"is_subset: interpretation for set is not a node")
(* interpretation * interpretation -> interpretation *)
fun intersection (Node xs, Node ys) =
Node (map (fn (x, y) => if x=TT andalso y=TT then TT else FF)
(xs ~~ ys))
| intersection (_, _) =
raise REFUTE ("lfp_interpreter",
"intersection: interpretation for set is not a node")
(* interpretation -> interpretaion *)
fun lfp (Node resultsets) =
fold (fn (set, resultset) => fn acc =>
if is_subset (resultset, set) then
intersection (acc, set)
else
acc) (i_sets ~~ resultsets) i_univ
| lfp _ =
raise REFUTE ("lfp_interpreter",
"lfp: interpretation for function is not a node")
in
SOME (Node (map lfp i_funs), model, args)
end
| _ => NONE;
(* only an optimization: 'gfp' could in principle be interpreted with *)
(* interpreters available already (using its definition), but the code *)
(* below is more efficient *)
fun gfp_interpreter ctxt model args t =
case t of
Const (@{const_name gfp}, Type ("fun", [Type ("fun",
[Type ("fun", [T, @{typ bool}]),
Type ("fun", [_, @{typ bool}])]),
Type ("fun", [_, @{typ bool}])])) =>
let
val size_elem = size_of_type ctxt model T
(* the universe (i.e. the set that contains every element) *)
val i_univ = Node (replicate size_elem TT)
(* all sets with elements from type 'T' *)
val i_sets =
make_constants ctxt model (Type ("fun", [T, HOLogic.boolT]))
(* all functions that map sets to sets *)
val i_funs = make_constants ctxt model (Type ("fun",
[Type ("fun", [T, HOLogic.boolT]),
Type ("fun", [T, HOLogic.boolT])]))
(* "gfp(f) == Union({u. u <= f(u)})" *)
(* interpretation * interpretation -> bool *)
fun is_subset (Node subs, Node sups) =
forall (fn (sub, sup) => (sub = FF) orelse (sup = TT))
(subs ~~ sups)
| is_subset (_, _) =
raise REFUTE ("gfp_interpreter",
"is_subset: interpretation for set is not a node")
(* interpretation * interpretation -> interpretation *)
fun union (Node xs, Node ys) =
Node (map (fn (x,y) => if x=TT orelse y=TT then TT else FF)
(xs ~~ ys))
| union (_, _) =
raise REFUTE ("gfp_interpreter",
"union: interpretation for set is not a node")
(* interpretation -> interpretaion *)
fun gfp (Node resultsets) =
fold (fn (set, resultset) => fn acc =>
if is_subset (set, resultset) then
union (acc, set)
else
acc) (i_sets ~~ resultsets) i_univ
| gfp _ =
raise REFUTE ("gfp_interpreter",
"gfp: interpretation for function is not a node")
in
SOME (Node (map gfp i_funs), model, args)
end
| _ => NONE;
*)
(* only an optimization: 'fst' could in principle be interpreted with *)
(* interpreters available already (using its definition), but the code *)
(* below is more efficient *)
fun Product_Type_fst_interpreter ctxt model args t =
case t of
Const (@{const_name fst}, Type ("fun", [Type (@{type_name Product_Type.prod}, [T, U]), _])) =>
let
val constants_T = make_constants ctxt model T
val size_U = size_of_type ctxt model U
in
SOME (Node (maps (replicate size_U) constants_T), model, args)
end
| _ => NONE;
(* only an optimization: 'snd' could in principle be interpreted with *)
(* interpreters available already (using its definition), but the code *)
(* below is more efficient *)
fun Product_Type_snd_interpreter ctxt model args t =
case t of
Const (@{const_name snd}, Type ("fun", [Type (@{type_name Product_Type.prod}, [T, U]), _])) =>
let
val size_T = size_of_type ctxt model T
val constants_U = make_constants ctxt model U
in
SOME (Node (flat (replicate size_T constants_U)), model, args)
end
| _ => NONE;
(* ------------------------------------------------------------------------- *)
(* PRINTERS *)
(* ------------------------------------------------------------------------- *)
fun stlc_printer ctxt model T intr assignment =
let
(* string -> string *)
val strip_leading_quote = perhaps (try (unprefix "'"))
(* Term.typ -> string *)
fun string_of_typ (Type (s, _)) = s
| string_of_typ (TFree (x, _)) = strip_leading_quote x
| string_of_typ (TVar ((x,i), _)) =
strip_leading_quote x ^ string_of_int i
(* interpretation -> int *)
fun index_from_interpretation (Leaf xs) =
find_index (PropLogic.eval assignment) xs
| index_from_interpretation _ =
raise REFUTE ("stlc_printer",
"interpretation for ground type is not a leaf")
in
case T of
Type ("fun", [T1, T2]) =>
let
(* create all constants of type 'T1' *)
val constants = make_constants ctxt model T1
(* interpretation list *)
val results =
(case intr of
Node xs => xs
| _ => raise REFUTE ("stlc_printer",
"interpretation for function type is a leaf"))
(* Term.term list *)
val pairs = map (fn (arg, result) =>
HOLogic.mk_prod
(print ctxt model T1 arg assignment,
print ctxt model T2 result assignment))
(constants ~~ results)
(* Term.typ *)
val HOLogic_prodT = HOLogic.mk_prodT (T1, T2)
val HOLogic_setT = HOLogic.mk_setT HOLogic_prodT
(* Term.term *)
val HOLogic_empty_set = Const (@{const_abbrev Set.empty}, HOLogic_setT)
val HOLogic_insert =
Const (@{const_name insert}, HOLogic_prodT --> HOLogic_setT --> HOLogic_setT)
in
SOME (fold_rev (fn pair => fn acc => HOLogic_insert $ pair $ acc) pairs HOLogic_empty_set)
end
| Type ("prop", []) =>
(case index_from_interpretation intr of
~1 => SOME (HOLogic.mk_Trueprop (Const (@{const_name undefined}, HOLogic.boolT)))
| 0 => SOME (HOLogic.mk_Trueprop HOLogic.true_const)
| 1 => SOME (HOLogic.mk_Trueprop HOLogic.false_const)
| _ => raise REFUTE ("stlc_interpreter",
"illegal interpretation for a propositional value"))
| Type _ =>
if index_from_interpretation intr = (~1) then
SOME (Const (@{const_name undefined}, T))
else
SOME (Const (string_of_typ T ^
string_of_int (index_from_interpretation intr), T))
| TFree _ =>
if index_from_interpretation intr = (~1) then
SOME (Const (@{const_name undefined}, T))
else
SOME (Const (string_of_typ T ^
string_of_int (index_from_interpretation intr), T))
| TVar _ =>
if index_from_interpretation intr = (~1) then
SOME (Const (@{const_name undefined}, T))
else
SOME (Const (string_of_typ T ^
string_of_int (index_from_interpretation intr), T))
end;
fun IDT_printer ctxt model T intr assignment =
let
val thy = ProofContext.theory_of ctxt
in
(case T of
Type (s, Ts) =>
(case Datatype.get_info thy s of
SOME info => (* inductive datatype *)
let
val (typs, _) = model
val index = #index info
val descr = #descr info
val (_, dtyps, constrs) = the (AList.lookup (op =) descr index)
val typ_assoc = dtyps ~~ Ts
(* sanity check: every element in 'dtyps' must be a 'DtTFree' *)
val _ =
if Library.exists (fn d =>
case d of Datatype_Aux.DtTFree _ => false | _ => true) dtyps
then
raise REFUTE ("IDT_printer", "datatype argument (for type " ^
Syntax.string_of_typ ctxt (Type (s, Ts)) ^ ") is not a variable")
else ()
(* the index of the element in the datatype *)
val element =
(case intr of
Leaf xs => find_index (PropLogic.eval assignment) xs
| Node _ => raise REFUTE ("IDT_printer",
"interpretation is not a leaf"))
in
if element < 0 then
SOME (Const (@{const_name undefined}, Type (s, Ts)))
else
let
(* takes a datatype constructor, and if for some arguments this *)
(* constructor generates the datatype's element that is given by *)
(* 'element', returns the constructor (as a term) as well as the *)
(* indices of the arguments *)
fun get_constr_args (cname, cargs) =
let
val cTerm = Const (cname,
map (typ_of_dtyp descr typ_assoc) cargs ---> Type (s, Ts))
val (iC, _, _) = interpret ctxt (typs, []) {maxvars=0,
def_eq=false, next_idx=1, bounds=[], wellformed=True} cTerm
(* interpretation -> int list option *)
fun get_args (Leaf xs) =
if find_index (fn x => x = True) xs = element then
SOME []
else
NONE
| get_args (Node xs) =
let
(* interpretation * int -> int list option *)
fun search ([], _) =
NONE
| search (x::xs, n) =
(case get_args x of
SOME result => SOME (n::result)
| NONE => search (xs, n+1))
in
search (xs, 0)
end
in
Option.map (fn args => (cTerm, cargs, args)) (get_args iC)
end
val (cTerm, cargs, args) =
(* we could speed things up by computing the correct *)
(* constructor directly (rather than testing all *)
(* constructors), based on the order in which constructors *)
(* generate elements of datatypes; the current implementation *)
(* of 'IDT_printer' however is independent of the internals *)
(* of 'IDT_constructor_interpreter' *)
(case get_first get_constr_args constrs of
SOME x => x
| NONE => raise REFUTE ("IDT_printer",
"no matching constructor found for element " ^
string_of_int element))
val argsTerms = map (fn (d, n) =>
let
val dT = typ_of_dtyp descr typ_assoc d
(* we only need the n-th element of this list, so there *)
(* might be a more efficient implementation that does not *)
(* generate all constants *)
val consts = make_constants ctxt (typs, []) dT
in
print ctxt (typs, []) dT (List.nth (consts, n)) assignment
end) (cargs ~~ args)
in
SOME (list_comb (cTerm, argsTerms))
end
end
| NONE => (* not an inductive datatype *)
NONE)
| _ => (* a (free or schematic) type variable *)
NONE)
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! *)
(* ------------------------------------------------------------------------- *)
val setup =
add_interpreter "stlc" stlc_interpreter #>
add_interpreter "Pure" Pure_interpreter #>
add_interpreter "HOLogic" HOLogic_interpreter #>
add_interpreter "set" set_interpreter #>
add_interpreter "IDT" IDT_interpreter #>
add_interpreter "IDT_constructor" IDT_constructor_interpreter #>
add_interpreter "IDT_recursion" IDT_recursion_interpreter #>
add_interpreter "Finite_Set.card" Finite_Set_card_interpreter #>
add_interpreter "Finite_Set.finite" Finite_Set_finite_interpreter #>
add_interpreter "Nat_Orderings.less" Nat_less_interpreter #>
add_interpreter "Nat_HOL.plus" Nat_plus_interpreter #>
add_interpreter "Nat_HOL.minus" Nat_minus_interpreter #>
add_interpreter "Nat_HOL.times" Nat_times_interpreter #>
add_interpreter "List.append" List_append_interpreter #>
(* UNSOUND
add_interpreter "lfp" lfp_interpreter #>
add_interpreter "gfp" gfp_interpreter #>
*)
add_interpreter "Product_Type.fst" Product_Type_fst_interpreter #>
add_interpreter "Product_Type.snd" Product_Type_snd_interpreter #>
add_printer "stlc" stlc_printer #>
add_printer "IDT" IDT_printer;
(** outer syntax commands 'refute' and 'refute_params' **)
(* argument parsing *)
(*optional list of arguments of the form [name1=value1, name2=value2, ...]*)
val scan_parm = Parse.name -- (Scan.optional (Parse.$$$ "=" |-- Parse.name) "true")
val scan_parms = Scan.optional (Parse.$$$ "[" |-- Parse.list scan_parm --| Parse.$$$ "]") [];
(* 'refute' command *)
val _ =
Outer_Syntax.improper_command "refute"
"try to find a model that refutes a given subgoal" Keyword.diag
(scan_parms -- Scan.optional Parse.nat 1 >>
(fn (parms, i) =>
Toplevel.keep (fn state =>
let
val ctxt = Toplevel.context_of state;
val {goal = st, ...} = Proof.raw_goal (Toplevel.proof_of state);
in refute_goal ctxt parms st i end)));
(* 'refute_params' command *)
val _ =
Outer_Syntax.command "refute_params"
"show/store default parameters for the 'refute' command" Keyword.thy_decl
(scan_parms >> (fn parms =>
Toplevel.theory (fn thy =>
let
val thy' = fold set_default_param parms thy;
val output =
(case get_default_params (ProofContext.init_global thy') of
[] => "none"
| new_defaults => cat_lines (map (fn (x, y) => x ^ "=" ^ y) new_defaults));
val _ = writeln ("Default parameters for 'refute':\n" ^ output);
in thy' end)));
end;