(* Title: HOL/Tools/Sledgehammer/sledgehammer_mepo.ML
Author: Jia Meng, Cambridge University Computer Laboratory and NICTA
Author: Jasmin Blanchette, TU Muenchen
Sledgehammer's iterative relevance filter (MePo = Meng-Paulson).
*)
signature SLEDGEHAMMER_MEPO =
sig
type stature = ATP_Problem_Generate.stature
type raw_fact = Sledgehammer_Fact.raw_fact
type fact = Sledgehammer_Fact.fact
type params = Sledgehammer_Provers.params
type relevance_fudge = Sledgehammer_Provers.relevance_fudge
val trace : bool Config.T
val pseudo_abs_name : string
val pseudo_skolem_prefix : string
val mepo_suggested_facts :
Proof.context -> params -> string -> int -> relevance_fudge option
-> term list -> term -> raw_fact list -> fact list
end;
structure Sledgehammer_MePo : SLEDGEHAMMER_MEPO =
struct
open ATP_Problem_Generate
open Sledgehammer_Util
open Sledgehammer_Fact
open Sledgehammer_Provers
val trace =
Attrib.setup_config_bool @{binding sledgehammer_filter_iter_trace} (K false)
fun trace_msg ctxt msg = if Config.get ctxt trace then tracing (msg ()) else ()
val sledgehammer_prefix = "Sledgehammer" ^ Long_Name.separator
val pseudo_abs_name = sledgehammer_prefix ^ "abs"
val pseudo_skolem_prefix = sledgehammer_prefix ^ "sko"
val theory_const_suffix = Long_Name.separator ^ " 1"
fun order_of_type (Type (@{type_name fun}, [T1, T2])) =
Int.max (order_of_type T1 + 1, order_of_type T2)
| order_of_type (Type (_, Ts)) = fold (Integer.max o order_of_type) Ts 0
| order_of_type _ = 0
(* An abstraction of Isabelle types and first-order terms *)
datatype pattern = PVar | PApp of string * pattern list
datatype ptype = PType of int * pattern list
fun string_of_pattern PVar = "_"
| string_of_pattern (PApp (s, ps)) =
if null ps then s else s ^ string_of_patterns ps
and string_of_patterns ps = "(" ^ commas (map string_of_pattern ps) ^ ")"
fun string_of_ptype (PType (_, ps)) = string_of_patterns ps
(*Is the second type an instance of the first one?*)
fun match_pattern (PVar, _) = true
| match_pattern (PApp _, PVar) = false
| match_pattern (PApp (s, ps), PApp (t, qs)) =
s = t andalso match_patterns (ps, qs)
and match_patterns (_, []) = true
| match_patterns ([], _) = false
| match_patterns (p :: ps, q :: qs) =
match_pattern (p, q) andalso match_patterns (ps, qs)
fun match_ptype (PType (_, ps), PType (_, qs)) = match_patterns (ps, qs)
(* Is there a unifiable constant? *)
fun pconst_mem f consts (s, ps) =
exists (curry (match_ptype o f) ps)
(map snd (filter (curry (op =) s o fst) consts))
fun pconst_hyper_mem f const_tab (s, ps) =
exists (curry (match_ptype o f) ps) (these (Symtab.lookup const_tab s))
fun pattern_of_type (Type (s, Ts)) = PApp (s, map pattern_of_type Ts)
| pattern_of_type (TFree (s, _)) = PApp (s, [])
| pattern_of_type (TVar _) = PVar
(* Pairs a constant with the list of its type instantiations. *)
fun ptype thy const x =
(if const then map pattern_of_type (these (try (Sign.const_typargs thy) x))
else [])
fun rich_ptype thy const (s, T) =
PType (order_of_type T, ptype thy const (s, T))
fun rich_pconst thy const (s, T) = (s, rich_ptype thy const (s, T))
fun string_of_hyper_pconst (s, ps) =
s ^ "{" ^ commas (map string_of_ptype ps) ^ "}"
(* Add a pconstant to the table, but a [] entry means a standard
connective, which we ignore.*)
fun add_pconst_to_table also_skolem (s, p) =
if (not also_skolem andalso String.isPrefix pseudo_skolem_prefix s) then I
else Symtab.map_default (s, [p]) (insert (op =) p)
(* Set constants tend to pull in too many irrelevant facts. We limit the damage
by treating them more or less as if they were built-in but add their
axiomatization at the end. *)
val set_consts = [@{const_name Collect}, @{const_name Set.member}]
val set_thms = @{thms Collect_mem_eq mem_Collect_eq Collect_cong}
fun add_pconsts_in_term thy is_built_in_const also_skolems pos =
let
val flip = Option.map not
(* We include free variables, as well as constants, to handle locales. For
each quantifiers that must necessarily be skolemized by the automatic
prover, we introduce a fresh constant to simulate the effect of
Skolemization. *)
fun do_const const ext_arg (x as (s, _)) ts =
let val (built_in, ts) = is_built_in_const x ts in
if member (op =) set_consts s then
fold (do_term ext_arg) ts
else
(not built_in
? add_pconst_to_table also_skolems (rich_pconst thy const x))
#> fold (do_term false) ts
end
and do_term ext_arg t =
case strip_comb t of
(Const x, ts) => do_const true ext_arg x ts
| (Free x, ts) => do_const false ext_arg x ts
| (Abs (_, T, t'), ts) =>
((null ts andalso not ext_arg)
(* Since lambdas on the right-hand side of equalities are usually
extensionalized later by "abs_extensionalize_term", we don't
penalize them here. *)
? add_pconst_to_table true (pseudo_abs_name,
PType (order_of_type T + 1, [])))
#> fold (do_term false) (t' :: ts)
| (_, ts) => fold (do_term false) ts
fun do_quantifier will_surely_be_skolemized abs_T body_t =
do_formula pos body_t
#> (if also_skolems andalso will_surely_be_skolemized then
add_pconst_to_table true (pseudo_skolem_prefix ^ serial_string (),
PType (order_of_type abs_T, []))
else
I)
and do_term_or_formula ext_arg T =
if T = HOLogic.boolT then do_formula NONE else do_term ext_arg
and do_formula pos t =
case t of
Const (@{const_name all}, _) $ Abs (_, T, t') =>
do_quantifier (pos = SOME false) T t'
| @{const "==>"} $ t1 $ t2 =>
do_formula (flip pos) t1 #> do_formula pos t2
| Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
do_term_or_formula false T t1 #> do_term_or_formula true T t2
| @{const Trueprop} $ t1 => do_formula pos t1
| @{const False} => I
| @{const True} => I
| @{const Not} $ t1 => do_formula (flip pos) t1
| Const (@{const_name All}, _) $ Abs (_, T, t') =>
do_quantifier (pos = SOME false) T t'
| Const (@{const_name Ex}, _) $ Abs (_, T, t') =>
do_quantifier (pos = SOME true) T t'
| @{const HOL.conj} $ t1 $ t2 => fold (do_formula pos) [t1, t2]
| @{const HOL.disj} $ t1 $ t2 => fold (do_formula pos) [t1, t2]
| @{const HOL.implies} $ t1 $ t2 =>
do_formula (flip pos) t1 #> do_formula pos t2
| Const (@{const_name HOL.eq}, Type (_, [T, _])) $ t1 $ t2 =>
do_term_or_formula false T t1 #> do_term_or_formula true T t2
| Const (@{const_name If}, Type (_, [_, Type (_, [T, _])]))
$ t1 $ t2 $ t3 =>
do_formula NONE t1 #> fold (do_term_or_formula false T) [t2, t3]
| Const (@{const_name Ex1}, _) $ Abs (_, T, t') =>
do_quantifier (is_some pos) T t'
| Const (@{const_name Ball}, _) $ t1 $ Abs (_, T, t') =>
do_quantifier (pos = SOME false) T
(HOLogic.mk_imp (incr_boundvars 1 t1 $ Bound 0, t'))
| Const (@{const_name Bex}, _) $ t1 $ Abs (_, T, t') =>
do_quantifier (pos = SOME true) T
(HOLogic.mk_conj (incr_boundvars 1 t1 $ Bound 0, t'))
| (t0 as Const (_, @{typ bool})) $ t1 =>
do_term false t0 #> do_formula pos t1 (* theory constant *)
| _ => do_term false t
in do_formula pos end
fun pconsts_in_fact thy is_built_in_const t =
Symtab.fold (fn (s, pss) => fold (cons o pair s) pss)
(Symtab.empty |> add_pconsts_in_term thy is_built_in_const true
(SOME true) t) []
(* Inserts a dummy "constant" referring to the theory name, so that relevance
takes the given theory into account. *)
fun theory_constify ({theory_const_rel_weight, theory_const_irrel_weight, ...}
: relevance_fudge) thy_name t =
if exists (curry (op <) 0.0) [theory_const_rel_weight,
theory_const_irrel_weight] then
Const (thy_name ^ theory_const_suffix, @{typ bool}) $ t
else
t
fun theory_const_prop_of fudge th =
theory_constify fudge (Context.theory_name (theory_of_thm th)) (prop_of th)
fun pair_consts_fact thy is_built_in_const fudge fact =
case fact |> snd |> theory_const_prop_of fudge
|> pconsts_in_fact thy is_built_in_const of
[] => NONE
| consts => SOME ((fact, consts), NONE)
(* 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 "pattern list". *)
fun pattern_ord p =
case p of
(PVar, PVar) => EQUAL
| (PVar, PApp _) => LESS
| (PApp _, PVar) => GREATER
| (PApp q1, PApp q2) =>
prod_ord fast_string_ord (dict_ord pattern_ord) (q1, q2)
fun ptype_ord (PType p, PType q) =
prod_ord (dict_ord pattern_ord) int_ord (swap p, swap q)
structure PType_Tab = Table(type key = ptype val ord = ptype_ord)
fun count_fact_consts thy fudge =
let
fun do_const const (s, T) ts =
(* Two-dimensional table update. Constant maps to types maps to count. *)
PType_Tab.map_default (rich_ptype thy const (s, T), 0) (Integer.add 1)
|> Symtab.map_default (s, PType_Tab.empty)
#> fold do_term ts
and do_term t =
case strip_comb t of
(Const x, ts) => do_const true x ts
| (Free x, ts) => do_const false x ts
| (Abs (_, _, t'), ts) => fold do_term (t' :: ts)
| (_, ts) => fold do_term ts
in do_term o theory_const_prop_of fudge o snd end
fun pow_int _ 0 = 1.0
| pow_int x 1 = x
| pow_int x n = if n > 0 then x * pow_int x (n - 1) else pow_int x (n + 1) / x
(*The frequency of a constant is the sum of those of all instances of its type.*)
fun pconst_freq match const_tab (c, ps) =
PType_Tab.fold (fn (qs, m) => match (ps, qs) ? Integer.add m)
(the (Symtab.lookup const_tab 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. Rare constants give more points if they are relevant than less
rare ones. *)
fun rel_weight_for _ freq = 1.0 + 2.0 / Math.ln (Real.fromInt freq + 1.0)
(* Irrelevant constants are treated differently. We associate lower penalties to
very rare constants and very common ones -- the former because they can't
lead to the inclusion of too many new facts, and the latter because they are
so common as to be of little interest. *)
fun irrel_weight_for ({worse_irrel_freq, higher_order_irrel_weight, ...}
: relevance_fudge) order freq =
let val (k, x) = worse_irrel_freq |> `Real.ceil in
(if freq < k then Math.ln (Real.fromInt (freq + 1)) / Math.ln x
else rel_weight_for order freq / rel_weight_for order k)
* pow_int higher_order_irrel_weight (order - 1)
end
fun multiplier_of_const_name local_const_multiplier s =
if String.isSubstring "." s then 1.0 else local_const_multiplier
(* Computes a constant's weight, as determined by its frequency. *)
fun generic_pconst_weight local_const_multiplier abs_weight skolem_weight
theory_const_weight chained_const_weight weight_for f
const_tab chained_const_tab (c as (s, PType (m, _))) =
if s = pseudo_abs_name then
abs_weight
else if String.isPrefix pseudo_skolem_prefix s then
skolem_weight
else if String.isSuffix theory_const_suffix s then
theory_const_weight
else
multiplier_of_const_name local_const_multiplier s
* weight_for m (pconst_freq (match_ptype o f) const_tab c)
|> (if chained_const_weight < 1.0 andalso
pconst_hyper_mem I chained_const_tab c then
curry (op *) chained_const_weight
else
I)
fun rel_pconst_weight ({local_const_multiplier, abs_rel_weight,
theory_const_rel_weight, ...} : relevance_fudge)
const_tab =
generic_pconst_weight local_const_multiplier abs_rel_weight 0.0
theory_const_rel_weight 0.0 rel_weight_for I const_tab
Symtab.empty
fun irrel_pconst_weight (fudge as {local_const_multiplier, abs_irrel_weight,
skolem_irrel_weight,
theory_const_irrel_weight,
chained_const_irrel_weight, ...})
const_tab chained_const_tab =
generic_pconst_weight local_const_multiplier abs_irrel_weight
skolem_irrel_weight theory_const_irrel_weight
chained_const_irrel_weight (irrel_weight_for fudge) swap
const_tab chained_const_tab
fun stature_bonus ({intro_bonus, ...} : relevance_fudge) (_, Intro) =
intro_bonus
| stature_bonus {elim_bonus, ...} (_, Elim) = elim_bonus
| stature_bonus {simp_bonus, ...} (_, Simp) = simp_bonus
| stature_bonus {local_bonus, ...} (Local, _) = local_bonus
| stature_bonus {assum_bonus, ...} (Assum, _) = assum_bonus
| stature_bonus {chained_bonus, ...} (Chained, _) = chained_bonus
| stature_bonus _ _ = 0.0
fun is_odd_const_name s =
s = pseudo_abs_name orelse String.isPrefix pseudo_skolem_prefix s orelse
String.isSuffix theory_const_suffix s
fun fact_weight fudge stature const_tab relevant_consts chained_consts
fact_consts =
case fact_consts |> List.partition (pconst_hyper_mem I relevant_consts)
||> filter_out (pconst_hyper_mem swap relevant_consts) of
([], _) => 0.0
| (rel, irrel) =>
if forall (forall (is_odd_const_name o fst)) [rel, irrel] then
0.0
else
let
val irrel = irrel |> filter_out (pconst_mem swap rel)
val rel_weight =
0.0 |> fold (curry (op +) o rel_pconst_weight fudge const_tab) rel
val irrel_weight =
~ (stature_bonus fudge stature)
|> fold (curry (op +)
o irrel_pconst_weight fudge const_tab chained_consts) irrel
val res = rel_weight / (rel_weight + irrel_weight)
in if Real.isFinite res then res else 0.0 end
fun take_most_relevant ctxt max_facts remaining_max
({max_imperfect, max_imperfect_exp, ...} : relevance_fudge)
(candidates : ((raw_fact * (string * ptype) list) * real) list) =
let
val max_imperfect =
Real.ceil (Math.pow (max_imperfect,
Math.pow (Real.fromInt remaining_max
/ Real.fromInt max_facts, max_imperfect_exp)))
val (perfect, imperfect) =
candidates |> sort (Real.compare o swap o pairself snd)
|> take_prefix (fn (_, w) => w > 0.99999)
val ((accepts, more_rejects), rejects) =
chop max_imperfect imperfect |>> append perfect |>> chop remaining_max
in
trace_msg ctxt (fn () =>
"Actually passed (" ^ string_of_int (length accepts) ^ " of " ^
string_of_int (length candidates) ^ "): " ^
(accepts |> map (fn ((((name, _), _), _), weight) =>
name () ^ " [" ^ Real.toString weight ^ "]")
|> commas));
(accepts, more_rejects @ rejects)
end
fun if_empty_replace_with_scope thy is_built_in_const facts sc tab =
if Symtab.is_empty tab then
Symtab.empty
|> fold (add_pconsts_in_term thy is_built_in_const false (SOME false))
(map_filter (fn ((_, (sc', _)), th) =>
if sc' = sc then SOME (prop_of th) else NONE) facts)
else
tab
fun consider_arities is_built_in_const th =
let
fun aux _ _ NONE = NONE
| aux t args (SOME tab) =
case t of
t1 $ t2 => SOME tab |> aux t1 (t2 :: args) |> aux t2 []
| Const (x as (s, _)) =>
(if is_built_in_const x args |> fst then
SOME tab
else case Symtab.lookup tab s of
NONE => SOME (Symtab.update (s, length args) tab)
| SOME n => if n = length args then SOME tab else NONE)
| _ => SOME tab
in aux (prop_of th) [] end
(* FIXME: This is currently only useful for polymorphic type encodings. *)
fun could_benefit_from_ext is_built_in_const facts =
fold (consider_arities is_built_in_const o snd) facts (SOME Symtab.empty)
|> is_none
(* High enough so that it isn't wrongly considered as very relevant (e.g., for E
weights), but low enough so that it is unlikely to be truncated away if few
facts are included. *)
val special_fact_index = 75
fun relevance_filter ctxt thres0 decay max_facts is_built_in_const
(fudge as {threshold_divisor, ridiculous_threshold, ...}) facts hyp_ts
concl_t =
let
val thy = Proof_Context.theory_of ctxt
val const_tab = fold (count_fact_consts thy fudge) facts Symtab.empty
val add_pconsts = add_pconsts_in_term thy is_built_in_const false o SOME
val chained_ts =
facts |> map_filter (fn ((_, (Chained, _)), th) => SOME (prop_of th)
| _ => NONE)
val chained_const_tab = Symtab.empty |> fold (add_pconsts true) chained_ts
val goal_const_tab =
Symtab.empty |> fold (add_pconsts true) hyp_ts
|> add_pconsts false concl_t
|> (fn tab => if Symtab.is_empty tab then chained_const_tab else tab)
|> fold (if_empty_replace_with_scope thy is_built_in_const facts)
[Chained, Assum, Local]
fun iter j remaining_max thres rel_const_tab hopeless hopeful =
let
fun relevant [] _ [] =
(* Nothing has been added this iteration. *)
if j = 0 andalso thres >= ridiculous_threshold then
(* First iteration? Try again. *)
iter 0 max_facts (thres / threshold_divisor) rel_const_tab
hopeless hopeful
else
[]
| relevant candidates rejects [] =
let
val (accepts, more_rejects) =
take_most_relevant ctxt max_facts remaining_max fudge candidates
val rel_const_tab' =
rel_const_tab
|> fold (add_pconst_to_table false) (maps (snd o fst) accepts)
fun is_dirty (c, _) =
Symtab.lookup rel_const_tab' c <> Symtab.lookup rel_const_tab c
val (hopeful_rejects, hopeless_rejects) =
(rejects @ hopeless, ([], []))
|-> fold (fn (ax as (_, consts), old_weight) =>
if exists is_dirty consts then
apfst (cons (ax, NONE))
else
apsnd (cons (ax, old_weight)))
|>> append (more_rejects
|> map (fn (ax as (_, consts), old_weight) =>
(ax, if exists is_dirty consts then NONE
else SOME old_weight)))
val thres =
1.0 - (1.0 - thres)
* Math.pow (decay, Real.fromInt (length accepts))
val remaining_max = remaining_max - length accepts
in
trace_msg ctxt (fn () => "New or updated constants: " ^
commas (rel_const_tab' |> Symtab.dest
|> subtract (op =) (rel_const_tab |> Symtab.dest)
|> map string_of_hyper_pconst));
map (fst o fst) accepts @
(if remaining_max = 0 then
[]
else
iter (j + 1) remaining_max thres rel_const_tab'
hopeless_rejects hopeful_rejects)
end
| relevant candidates rejects
(((ax as (((_, stature), _), fact_consts)), cached_weight)
:: hopeful) =
let
val weight =
case cached_weight of
SOME w => w
| NONE => fact_weight fudge stature const_tab rel_const_tab
chained_const_tab fact_consts
in
if weight >= thres then
relevant ((ax, weight) :: candidates) rejects hopeful
else
relevant candidates ((ax, weight) :: rejects) hopeful
end
in
trace_msg ctxt (fn () =>
"ITERATION " ^ string_of_int j ^ ": current threshold: " ^
Real.toString thres ^ ", constants: " ^
commas (rel_const_tab |> Symtab.dest
|> filter (curry (op <>) [] o snd)
|> map string_of_hyper_pconst));
relevant [] [] hopeful
end
fun uses_const s t =
fold_aterms (curry (fn (Const (s', _), false) => s' = s | (_, b) => b)) t
false
fun uses_const_anywhere accepts s =
exists (uses_const s o prop_of o snd) accepts orelse
exists (uses_const s) (concl_t :: hyp_ts)
fun add_set_const_thms accepts =
exists (uses_const_anywhere accepts) set_consts ? append set_thms
fun insert_into_facts accepts [] = accepts
| insert_into_facts accepts ths =
let
val add = facts |> filter (member Thm.eq_thm_prop ths o snd)
val (bef, after) =
accepts |> filter_out (member Thm.eq_thm_prop ths o snd)
|> take (max_facts - length add)
|> chop special_fact_index
in bef @ add @ after end
fun insert_special_facts accepts =
(* FIXME: get rid of "ext" here once it is treated as a helper *)
[] |> could_benefit_from_ext is_built_in_const accepts ? cons @{thm ext}
|> add_set_const_thms accepts
|> insert_into_facts accepts
in
facts |> map_filter (pair_consts_fact thy is_built_in_const fudge)
|> iter 0 max_facts thres0 goal_const_tab []
|> insert_special_facts
|> tap (fn accepts => trace_msg ctxt (fn () =>
"Total relevant: " ^ string_of_int (length accepts)))
end
fun mepo_suggested_facts ctxt
({fact_thresholds = (thres0, thres1), ...} : params) prover
max_facts fudge hyp_ts concl_t facts =
let
val thy = Proof_Context.theory_of ctxt
val is_built_in_const =
Sledgehammer_Provers.is_built_in_const_of_prover ctxt prover
val fudge =
case fudge of
SOME fudge => fudge
| NONE => Sledgehammer_Provers.relevance_fudge_of_prover ctxt prover
val decay = Math.pow ((1.0 - thres1) / (1.0 - thres0),
1.0 / Real.fromInt (max_facts + 1))
in
trace_msg ctxt (fn () => "Considering " ^ string_of_int (length facts) ^
" facts");
(if thres1 < 0.0 then
facts
else if thres0 > 1.0 orelse thres0 > thres1 then
[]
else
relevance_filter ctxt thres0 decay max_facts is_built_in_const fudge
facts hyp_ts
(concl_t |> theory_constify fudge (Context.theory_name thy)))
|> map fact_of_raw_fact
end
end;