(* Title: HOL/Tools/Sledgehammer/sledgehammer_fact_filter.ML
Author: Jia Meng, Cambridge University Computer Laboratory and NICTA
Author: Jasmin Blanchette, TU Muenchen
*)
signature SLEDGEHAMMER_FACT_FILTER =
sig
type relevance_override =
{add: Facts.ref list,
del: Facts.ref list,
only: bool}
val trace : bool Unsynchronized.ref
val chained_prefix : string
val name_thms_pair_from_ref :
Proof.context -> thm list -> Facts.ref -> string * thm list
val relevant_facts :
bool -> real -> real -> bool -> int -> bool -> relevance_override
-> Proof.context * (thm list * 'a) -> term list -> term
-> (string * thm) list
end;
structure Sledgehammer_Fact_Filter : SLEDGEHAMMER_FACT_FILTER =
struct
val trace = Unsynchronized.ref false
fun trace_msg msg = if !trace then tracing (msg ()) else ()
val respect_no_atp = true
type relevance_override =
{add: Facts.ref list,
del: Facts.ref list,
only: bool}
val sledgehammer_prefix = "Sledgehammer" ^ Long_Name.separator
(* Used to label theorems chained into the goal. *)
val chained_prefix = sledgehammer_prefix ^ "chained_"
fun name_thms_pair_from_ref ctxt chained_ths xref =
let
val ths = ProofContext.get_fact ctxt xref
val name = Facts.string_of_ref xref
|> forall (member Thm.eq_thm chained_ths) ths
? prefix chained_prefix
in (name, ths) end
(***************************************************************)
(* Relevance Filtering *)
(***************************************************************)
(*** constants with types ***)
(*An abstraction of Isabelle types*)
datatype const_typ = CTVar | CType of string * const_typ list
(*Is the second type an instance of the first one?*)
fun match_type (CType(con1,args1)) (CType(con2,args2)) =
con1=con2 andalso match_types args1 args2
| match_type CTVar _ = true
| match_type _ CTVar = false
and match_types [] [] = true
| match_types (a1::as1) (a2::as2) = match_type a1 a2 andalso match_types as1 as2;
(*Is there a unifiable constant?*)
fun uni_mem goal_const_tab (c, c_typ) =
exists (match_types c_typ) (these (Symtab.lookup goal_const_tab c))
(*Maps a "real" type to a const_typ*)
fun const_typ_of (Type (c,typs)) = CType (c, map const_typ_of typs)
| const_typ_of (TFree _) = CTVar
| const_typ_of (TVar _) = CTVar
(*Pairs a constant with the list of its type instantiations (using const_typ)*)
fun const_with_typ thy (c,typ) =
let val tvars = Sign.const_typargs thy (c,typ) in
(c, map const_typ_of tvars) end
handle TYPE _ => (c, []) (*Variable (locale constant): monomorphic*)
(*Add a const/type pair to the table, but a [] entry means a standard connective,
which we ignore.*)
fun add_const_type_to_table (c, ctyps) =
Symtab.map_default (c, [ctyps])
(fn [] => [] | ctypss => insert (op =) ctyps ctypss)
val fresh_prefix = "Sledgehammer.Fresh."
val flip = Option.map not
(* These are typically simplified away by "Meson.presimplify". *)
val boring_consts =
[@{const_name If}, @{const_name Let} (* TODO: , @{const_name Set.member}, @{const_name Collect} *)]
fun get_consts_typs thy pos ts =
let
(* We include free variables, as well as constants, to handle locales. For
each quantifiers that must necessarily be skolemized by the ATP, we
introduce a fresh constant to simulate the effect of Skolemization. *)
fun do_term t =
case t of
Const x => add_const_type_to_table (const_with_typ thy x)
| Free (s, _) => add_const_type_to_table (s, [])
| t1 $ t2 => do_term t1 #> do_term t2
| Abs (_, _, t) =>
(* Abstractions lead to combinators, so we add a penalty for them. *)
add_const_type_to_table (gensym fresh_prefix, [])
#> do_term t
| _ => I
fun do_quantifier will_surely_be_skolemized body_t =
do_formula pos body_t
#> (if will_surely_be_skolemized then
add_const_type_to_table (gensym fresh_prefix, [])
else
I)
and do_term_or_formula T =
if T = @{typ bool} orelse T = @{typ prop} then do_formula NONE
else do_term
and do_formula pos t =
case t of
Const (@{const_name all}, _) $ Abs (_, _, body_t) =>
do_quantifier (pos = SOME false) body_t
| @{const "==>"} $ t1 $ t2 =>
do_formula (flip pos) t1 #> do_formula pos t2
| Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
fold (do_term_or_formula T) [t1, t2]
| @{const Trueprop} $ t1 => do_formula pos t1
| @{const Not} $ t1 => do_formula (flip pos) t1
| Const (@{const_name All}, _) $ Abs (_, _, body_t) =>
do_quantifier (pos = SOME false) body_t
| Const (@{const_name Ex}, _) $ Abs (_, _, body_t) =>
do_quantifier (pos = SOME true) body_t
| @{const "op &"} $ t1 $ t2 => fold (do_formula pos) [t1, t2]
| @{const "op |"} $ t1 $ t2 => fold (do_formula pos) [t1, t2]
| @{const "op -->"} $ t1 $ t2 =>
do_formula (flip pos) t1 #> do_formula pos t2
| Const (@{const_name "op ="}, Type (_, [T, _])) $ t1 $ t2 =>
fold (do_term_or_formula T) [t1, t2]
| Const (@{const_name If}, Type (_, [_, Type (_, [T, _])]))
$ t1 $ t2 $ t3 =>
do_formula NONE t1 #> fold (do_term_or_formula T) [t2, t3]
| Const (@{const_name Ex1}, _) $ Abs (_, _, body_t) =>
do_quantifier (is_some pos) body_t
| Const (@{const_name Ball}, _) $ t1 $ Abs (_, _, body_t) =>
do_quantifier (pos = SOME false)
(HOLogic.mk_imp (incr_boundvars 1 t1 $ Bound 0, body_t))
| Const (@{const_name Bex}, _) $ t1 $ Abs (_, _, body_t) =>
do_quantifier (pos = SOME true)
(HOLogic.mk_conj (incr_boundvars 1 t1 $ Bound 0, body_t))
| (t0 as Const (_, @{typ bool})) $ t1 =>
do_term t0 #> do_formula pos t1 (* theory constant *)
| _ => do_term t
in
Symtab.empty |> fold (Symtab.update o rpair []) boring_consts
|> fold (do_formula pos) ts
end
(*Inserts a dummy "constant" referring to the theory name, so that relevance
takes the given theory into account.*)
fun theory_const_prop_of theory_relevant th =
if theory_relevant then
let
val name = Context.theory_name (theory_of_thm th)
val t = Const (name ^ ". 1", @{typ bool})
in t $ prop_of th end
else
prop_of th
(**** Constant / Type Frequencies ****)
(*A two-dimensional symbol table counts frequencies of constants. It's keyed first by
constant name and second by its list of type instantiations. For the latter, we need
a linear ordering on type const_typ list.*)
local
fun cons_nr CTVar = 0
| cons_nr (CType _) = 1;
in
fun const_typ_ord TU =
case TU of
(CType (a, Ts), CType (b, Us)) =>
(case fast_string_ord(a,b) of EQUAL => dict_ord const_typ_ord (Ts,Us) | ord => ord)
| (T, U) => int_ord (cons_nr T, cons_nr U);
end;
structure CTtab = Table(type key = const_typ list val ord = dict_ord const_typ_ord);
fun count_axiom_consts theory_relevant thy (_, th) =
let
fun do_const (a, T) =
let val (c, cts) = const_with_typ thy (a, T) in
(* Two-dimensional table update. Constant maps to types maps to
count. *)
CTtab.map_default (cts, 0) (Integer.add 1)
|> Symtab.map_default (c, CTtab.empty)
end
fun do_term (Const x) = do_const x
| do_term (Free x) = do_const x
| do_term (t $ u) = do_term t #> do_term u
| do_term (Abs (_, _, t)) = do_term t
| do_term _ = I
in th |> theory_const_prop_of theory_relevant |> do_term end
(**** Actual Filtering Code ****)
(*The frequency of a constant is the sum of those of all instances of its type.*)
fun const_frequency const_tab (c, cts) =
CTtab.fold (fn (cts', m) => match_types cts cts' ? Integer.add m)
(the (Symtab.lookup const_tab c)
handle Option.Option => raise Fail ("Const: " ^ c)) 0
(* A surprising number of theorems contain only a few significant constants.
These include all induction rules, and other general theorems. *)
(* "log" seems best in practice. A constant function of one ignores the constant
frequencies. *)
fun log_weight2 (x:real) = 1.0 + 2.0 / Math.ln (x + 1.0)
(* Computes a constant's weight, as determined by its frequency. *)
val ct_weight = log_weight2 o real oo const_frequency
(*Relevant constants are weighted according to frequency,
but irrelevant constants are simply counted. Otherwise, Skolem functions,
which are rare, would harm a formula's chances of being picked.*)
fun formula_weight const_tab gctyps consts_typs =
let
val rel = filter (uni_mem gctyps) consts_typs
val rel_weight = fold (curry Real.+ o ct_weight const_tab) rel 0.0
val res = rel_weight / (rel_weight + real (length consts_typs - length rel))
in if Real.isFinite res then res else 0.0 end
(*Multiplies out to a list of pairs: 'a * 'b list -> ('a * 'b) list -> ('a * 'b) list*)
fun add_expand_pairs (x,ys) xys = List.foldl (fn (y,acc) => (x,y)::acc) xys ys;
fun consts_typs_of_term thy t =
Symtab.fold add_expand_pairs (get_consts_typs thy (SOME true) [t]) []
fun pair_consts_typs_axiom theory_relevant thy axiom =
(axiom, axiom |> snd |> theory_const_prop_of theory_relevant
|> consts_typs_of_term thy)
exception CONST_OR_FREE of unit
fun dest_Const_or_Free (Const x) = x
| dest_Const_or_Free (Free x) = x
| dest_Const_or_Free _ = raise CONST_OR_FREE ()
(*Look for definitions of the form f ?x1 ... ?xn = t, but not reversed.*)
fun defines thy thm gctypes =
let val tm = prop_of thm
fun defs lhs rhs =
let val (rator,args) = strip_comb lhs
val ct = const_with_typ thy (dest_Const_or_Free rator)
in
forall is_Var args andalso uni_mem gctypes ct andalso
subset (op =) (Term.add_vars rhs [], Term.add_vars lhs [])
end
handle CONST_OR_FREE () => false
in
case tm of
@{const Trueprop} $ (Const (@{const_name "op ="}, _) $ lhs $ rhs) =>
defs lhs rhs
| _ => false
end;
type annotated_thm = (string * thm) * (string * const_typ list) list
(*For a reverse sort, putting the largest values first.*)
fun compare_pairs ((_, w1), (_, w2)) = Real.compare (w2, w1)
(* Limit the number of new facts, to prevent runaway acceptance. *)
fun take_best max_new (newpairs : (annotated_thm * real) list) =
let val nnew = length newpairs in
if nnew <= max_new then
(map #1 newpairs, [])
else
let
val newpairs = sort compare_pairs newpairs
val accepted = List.take (newpairs, max_new)
in
trace_msg (fn () => ("Number of candidates, " ^ Int.toString nnew ^
", exceeds the limit of " ^ Int.toString max_new));
trace_msg (fn () => ("Effective pass mark: " ^ Real.toString (#2 (List.last accepted))));
trace_msg (fn () => "Actually passed: " ^
space_implode ", " (map (fst o fst o fst) accepted));
(map #1 accepted, map #1 (List.drop (newpairs, max_new)))
end
end;
fun relevance_filter ctxt relevance_threshold relevance_convergence
defs_relevant max_new theory_relevant
({add, del, ...} : relevance_override) axioms goal_ts =
if relevance_threshold > 1.0 then
[]
else if relevance_threshold < 0.0 then
axioms
else
let
val thy = ProofContext.theory_of ctxt
val const_tab = fold (count_axiom_consts theory_relevant thy) axioms
Symtab.empty
val goal_const_tab = get_consts_typs thy (SOME false) goal_ts
val relevance_threshold = 0.8 * relevance_threshold (* FIXME *)
val _ =
trace_msg (fn () => "Initial constants: " ^
commas (goal_const_tab
|> Symtab.dest
|> filter (curry (op <>) [] o snd)
|> map fst))
val add_thms = maps (ProofContext.get_fact ctxt) add
val del_thms = maps (ProofContext.get_fact ctxt) del
fun iter threshold rel_const_tab =
let
fun relevant ([], rejects) [] =
(* Nothing was added this iteration: Add "add:" facts. *)
if null add_thms then
[]
else
map_filter (fn (p as (name, th), _) =>
if member Thm.eq_thm add_thms th then SOME p
else NONE) rejects
| relevant (newpairs, rejects) [] =
let
val (newrels, more_rejects) = take_best max_new newpairs
val new_consts = maps #2 newrels
val rel_const_tab =
rel_const_tab |> fold add_const_type_to_table new_consts
val threshold =
threshold + (1.0 - threshold) / relevance_convergence
in
trace_msg (fn () => "relevant this iteration: " ^
Int.toString (length newrels));
map #1 newrels @ iter threshold rel_const_tab
(more_rejects @ rejects)
end
| relevant (newrels, rejects)
((ax as ((name, th), consts_typs)) :: axs) =
let
val weight =
if member Thm.eq_thm del_thms th then 0.0
else formula_weight const_tab rel_const_tab consts_typs
in
if weight >= threshold orelse
(defs_relevant andalso defines thy th rel_const_tab) then
(trace_msg (fn () =>
name ^ " passes: " ^ Real.toString weight
(* ^ " consts: " ^ commas (map fst consts_typs) *));
relevant ((ax, weight) :: newrels, rejects) axs)
else
relevant (newrels, ax :: rejects) axs
end
in
trace_msg (fn () => "relevant_facts, current threshold: " ^
Real.toString threshold);
relevant ([], [])
end
val relevant = iter relevance_threshold goal_const_tab
(map (pair_consts_typs_axiom theory_relevant thy)
axioms)
in
trace_msg (fn () => "Total relevant: " ^ Int.toString (length relevant));
relevant
end
(***************************************************************)
(* Retrieving and filtering lemmas *)
(***************************************************************)
(*** retrieve lemmas and filter them ***)
(*Reject theorems with names like "List.filter.filter_list_def" or
"Accessible_Part.acc.defs", as these are definitions arising from packages.*)
fun is_package_def a =
let val names = Long_Name.explode a
in
length names > 2 andalso
not (hd names = "local") andalso
String.isSuffix "_def" a orelse String.isSuffix "_defs" a
end;
fun make_fact_table xs =
fold (Termtab.update o `(prop_of o snd)) xs Termtab.empty
fun make_unique xs = Termtab.fold (cons o snd) (make_fact_table xs) []
(* FIXME: put other record thms here, or declare as "no_atp" *)
val multi_base_blacklist =
["defs", "select_defs", "update_defs", "induct", "inducts", "split", "splits",
"split_asm", "cases", "ext_cases"]
val max_lambda_nesting = 3
fun term_has_too_many_lambdas max (t1 $ t2) =
exists (term_has_too_many_lambdas max) [t1, t2]
| term_has_too_many_lambdas max (Abs (_, _, t)) =
max = 0 orelse term_has_too_many_lambdas (max - 1) t
| term_has_too_many_lambdas _ _ = false
fun is_formula_type T = (T = HOLogic.boolT orelse T = propT)
(* Don't count nested lambdas at the level of formulas, since they are
quantifiers. *)
fun formula_has_too_many_lambdas Ts (Abs (_, T, t)) =
formula_has_too_many_lambdas (T :: Ts) t
| formula_has_too_many_lambdas Ts t =
if is_formula_type (fastype_of1 (Ts, t)) then
exists (formula_has_too_many_lambdas Ts) (#2 (strip_comb t))
else
term_has_too_many_lambdas max_lambda_nesting t
(* The max apply depth of any "metis" call in "Metis_Examples" (on 31-10-2007)
was 11. *)
val max_apply_depth = 15
fun apply_depth (f $ t) = Int.max (apply_depth f, apply_depth t + 1)
| apply_depth (Abs (_, _, t)) = apply_depth t
| apply_depth _ = 0
fun is_formula_too_complex t =
apply_depth t > max_apply_depth orelse formula_has_too_many_lambdas [] t
val exists_sledgehammer_const =
exists_Const (fn (s, _) => String.isPrefix sledgehammer_prefix s)
fun is_strange_thm th =
case head_of (concl_of th) of
Const (a, _) => (a <> @{const_name Trueprop} andalso
a <> @{const_name "=="})
| _ => false
val type_has_top_sort =
exists_subtype (fn TFree (_, []) => true | TVar (_, []) => true | _ => false)
fun is_theorem_bad_for_atps thm =
let val t = prop_of thm in
is_formula_too_complex t orelse exists_type type_has_top_sort t orelse
exists_sledgehammer_const t orelse is_strange_thm thm
end
fun all_name_thms_pairs ctxt add_thms chained_ths =
let
val global_facts = PureThy.facts_of (ProofContext.theory_of ctxt);
val local_facts = ProofContext.facts_of ctxt;
val full_space =
Name_Space.merge (Facts.space_of global_facts, Facts.space_of local_facts);
fun valid_facts facts =
(facts, []) |-> Facts.fold_static (fn (name, ths0) =>
if (Facts.is_concealed facts name orelse
(respect_no_atp andalso is_package_def name) orelse
member (op =) multi_base_blacklist (Long_Name.base_name name) orelse
String.isSuffix "_def_raw" (* FIXME: crude hack *) name) andalso
forall (not o member Thm.eq_thm add_thms) ths0 then
I
else
let
fun check_thms a =
(case try (ProofContext.get_thms ctxt) a of
NONE => false
| SOME ths1 => Thm.eq_thms (ths0, ths1))
val name1 = Facts.extern facts name;
val name2 = Name_Space.extern full_space name;
val ths = filter (fn th => not (is_theorem_bad_for_atps th) orelse
member Thm.eq_thm add_thms th) ths0
in
case find_first check_thms [name1, name2, name] of
NONE => I
| SOME name' =>
cons (name' |> forall (member Thm.eq_thm chained_ths) ths
? prefix chained_prefix, ths)
end)
in valid_facts global_facts @ valid_facts local_facts end;
fun multi_name a th (n, pairs) =
(n + 1, (a ^ "(" ^ Int.toString n ^ ")", th) :: pairs);
fun add_names (_, []) pairs = pairs
| add_names (a, [th]) pairs = (a, th) :: pairs
| add_names (a, ths) pairs = #2 (fold (multi_name a) ths (1, pairs))
fun is_multi (a, ths) = length ths > 1 orelse String.isSuffix ".axioms" a;
(* The single-name theorems go after the multiple-name ones, so that single
names are preferred when both are available. *)
fun name_thm_pairs ctxt respect_no_atp name_thms_pairs =
let
val (mults, singles) = List.partition is_multi name_thms_pairs
val ps = [] |> fold add_names singles |> fold add_names mults
in ps |> respect_no_atp ? filter_out (No_ATPs.member ctxt o snd) end;
fun is_named ("", th) =
(warning ("No name for theorem " ^
Display.string_of_thm_without_context th); false)
| is_named _ = true
fun checked_name_thm_pairs respect_no_atp ctxt =
name_thm_pairs ctxt respect_no_atp
#> tap (fn ps => trace_msg
(fn () => ("Considering " ^ Int.toString (length ps) ^
" theorems")))
#> filter is_named
(***************************************************************)
(* ATP invocation methods setup *)
(***************************************************************)
(**** Predicates to detect unwanted facts (prolific or likely to cause
unsoundness) ****)
(* Too general means, positive equality literal with a variable X as one
operand, when X does not occur properly in the other operand. This rules out
clearly inconsistent facts such as X = a | X = b, though it by no means
guarantees soundness. *)
(* Unwanted equalities are those between a (bound or schematic) variable that
does not properly occur in the second operand. *)
fun too_general_eqterms (Var z) t =
not (exists_subterm (fn Var z' => z = z' | _ => false) t)
| too_general_eqterms (Bound j) t = not (loose_bvar1 (t, j))
| too_general_eqterms _ _ = false
val is_exhaustive_finite =
let
fun do_equals t1 t2 =
too_general_eqterms t1 t2 orelse too_general_eqterms t2 t1
fun do_formula pos t =
case (pos, t) of
(_, @{const Trueprop} $ t1) => do_formula pos t1
| (true, Const (@{const_name all}, _) $ Abs (_, _, t')) =>
do_formula pos t'
| (true, Const (@{const_name All}, _) $ Abs (_, _, t')) =>
do_formula pos t'
| (false, Const (@{const_name Ex}, _) $ Abs (_, _, t')) =>
do_formula pos t'
| (_, @{const "==>"} $ t1 $ t2) =>
do_formula (not pos) t1 andalso do_formula pos t2
| (_, @{const "op -->"} $ t1 $ t2) =>
do_formula (not pos) t1 andalso do_formula pos t2
| (_, @{const Not} $ t1) => do_formula (not pos) t1
| (true, @{const "op |"} $ t1 $ t2) => forall (do_formula pos) [t1, t2]
| (false, @{const "op &"} $ t1 $ t2) => forall (do_formula pos) [t1, t2]
| (true, Const (@{const_name "op ="}, _) $ t1 $ t2) => do_equals t1 t2
| (true, Const (@{const_name "=="}, _) $ t1 $ t2) => do_equals t1 t2
| (_, @{const False}) => true
| (_, @{const True}) => true
| _ => false
in do_formula true end
fun has_bound_or_var_of_type tycons =
exists_subterm (fn Var (_, Type (s, _)) => member (op =) tycons s
| Abs (_, Type (s, _), _) => member (op =) tycons s
| _ => false)
(* Facts are forbidden to contain variables of these types. The typical reason
is that they lead to unsoundness. Note that "unit" satisfies numerous
equations like "?x = ()". The resulting clauses will have no type constraint,
yielding false proofs. Even "bool" leads to many unsound proofs, though only
for higher-order problems. *)
val dangerous_types = [@{type_name unit}, @{type_name bool}, @{type_name prop}];
(* Facts containing variables of type "unit" or "bool" or of the form
"ALL x. x = A | x = B | x = C" are likely to lead to unsound proofs if types
are omitted. *)
fun is_dangerous_term full_types t =
not full_types andalso
(has_bound_or_var_of_type dangerous_types t orelse is_exhaustive_finite t)
fun relevant_facts full_types relevance_threshold relevance_convergence
defs_relevant max_new theory_relevant
(relevance_override as {add, del, only})
(ctxt, (chained_ths, _)) hyp_ts concl_t =
let
val add_thms = maps (ProofContext.get_fact ctxt) add
val axioms =
checked_name_thm_pairs (respect_no_atp andalso not only) ctxt
(if only then map (name_thms_pair_from_ref ctxt chained_ths) add
else all_name_thms_pairs ctxt add_thms chained_ths)
|> make_unique
|> map (apsnd Clausifier.transform_elim_theorem)
|> filter (fn (_, th) => member Thm.eq_thm add_thms th orelse
not (is_dangerous_term full_types (prop_of th)))
in
relevance_filter ctxt relevance_threshold relevance_convergence
defs_relevant max_new theory_relevant relevance_override
axioms (concl_t :: hyp_ts)
|> sort_wrt fst
end
end;