(* 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_Prover.params
type relevance_fudge =
{local_const_multiplier : real,
worse_irrel_freq : real,
higher_order_irrel_weight : real,
abs_rel_weight : real,
abs_irrel_weight : real,
theory_const_rel_weight : real,
theory_const_irrel_weight : real,
chained_const_irrel_weight : real,
intro_bonus : real,
elim_bonus : real,
simp_bonus : real,
local_bonus : real,
assum_bonus : real,
chained_bonus : real,
max_imperfect : real,
max_imperfect_exp : real,
threshold_divisor : real,
ridiculous_threshold : real}
val trace : bool Config.T
val pseudo_abs_name : string
val default_relevance_fudge : relevance_fudge
val mepo_suggested_facts :
Proof.context -> params -> 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_Prover
val trace =
Attrib.setup_config_bool @{binding sledgehammer_mepo_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 theory_const_suffix = Long_Name.separator ^ " 1"
type relevance_fudge =
{local_const_multiplier : real,
worse_irrel_freq : real,
higher_order_irrel_weight : real,
abs_rel_weight : real,
abs_irrel_weight : real,
theory_const_rel_weight : real,
theory_const_irrel_weight : real,
chained_const_irrel_weight : real,
intro_bonus : real,
elim_bonus : real,
simp_bonus : real,
local_bonus : real,
assum_bonus : real,
chained_bonus : real,
max_imperfect : real,
max_imperfect_exp : real,
threshold_divisor : real,
ridiculous_threshold : real}
(* FUDGE *)
val default_relevance_fudge =
{local_const_multiplier = 1.5,
worse_irrel_freq = 100.0,
higher_order_irrel_weight = 1.05,
abs_rel_weight = 0.5,
abs_irrel_weight = 2.0,
theory_const_rel_weight = 0.5,
theory_const_irrel_weight = 0.25,
chained_const_irrel_weight = 0.25,
intro_bonus = 0.15,
elim_bonus = 0.15,
simp_bonus = 0.15,
local_bonus = 0.55,
assum_bonus = 1.05,
chained_bonus = 1.5,
max_imperfect = 11.5,
max_imperfect_exp = 1.0,
threshold_divisor = 2.0,
ridiculous_threshold = 0.1}
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 * typ list
fun string_of_patternT (TVar _) = "_"
| string_of_patternT (Type (s, ps)) =
if null ps then s else s ^ string_of_patternsT ps
| string_of_patternT (TFree (s, _)) = s
and string_of_patternsT ps = "(" ^ commas (map string_of_patternT ps) ^ ")"
fun string_of_ptype (PType (_, ps)) = string_of_patternsT ps
(*Is the second type an instance of the first one?*)
fun match_patternT (TVar _, _) = true
| match_patternT (Type (s, ps), Type (t, qs)) =
s = t andalso match_patternsT (ps, qs)
| match_patternT (TFree (s, _), TFree (t, _)) = s = t
| match_patternT (_, _) = false
and match_patternsT (_, []) = true
| match_patternsT ([], _) = false
| match_patternsT (p :: ps, q :: qs) =
match_patternT (p, q) andalso match_patternsT (ps, qs)
fun match_ptype (PType (_, ps), PType (_, qs)) = match_patternsT (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))
(* Pairs a constant with the list of its type instantiations. *)
fun ptype thy const x =
(if const then 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) ^ "}"
fun patternT_eq (TVar _, TVar _) = true
| patternT_eq (Type (s, Ts), Type (t, Us)) = s = t andalso patternsT_eq (Ts, Us)
| patternT_eq (TFree (s, _), TFree (t, _)) = (s = t)
| patternT_eq _ = false
and patternsT_eq ([], []) = true
| patternsT_eq ([], _) = false
| patternsT_eq (_, []) = false
| patternsT_eq (T :: Ts, U :: Us) = patternT_eq (T, U) andalso patternsT_eq (Ts, Us)
fun ptype_eq (PType (m, Ts), PType (n, Us)) = m = n andalso patternsT_eq (Ts, Us)
(* Add a pconstant to the table, but a [] entry means a standard connective,
which we ignore. *)
fun add_pconst_to_table (s, p) = Symtab.map_default (s, [p]) (insert ptype_eq 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 =
let
fun do_const const (x as (s, _)) ts =
if member (op =) set_consts s then
fold (do_term false) ts
else
(not (is_irrelevant_const s) ? add_pconst_to_table (rich_pconst thy const x))
#> fold (do_term false) ts
and do_term ext_arg t =
(case strip_comb t of
(Const x, ts) => do_const true x ts
| (Free x, ts) => do_const false 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 (pseudo_abs_name,
PType (order_of_type T + 1, [])))
#> fold (do_term false) (t' :: ts)
| (_, ts) => fold (do_term false) ts)
and do_term_or_formula ext_arg T =
if T = HOLogic.boolT then do_formula else do_term ext_arg
and do_formula t =
(case t of
Const (@{const_name all}, _) $ Abs (_, _, t') => do_formula t'
| @{const "==>"} $ t1 $ t2 => do_formula t1 #> do_formula 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 t1
| @{const False} => I
| @{const True} => I
| @{const Not} $ t1 => do_formula t1
| Const (@{const_name All}, _) $ Abs (_, _, t') => do_formula t'
| Const (@{const_name Ex}, _) $ Abs (_, _, t') => do_formula t'
| @{const HOL.conj} $ t1 $ t2 => do_formula t1 #> do_formula t2
| @{const HOL.disj} $ t1 $ t2 => do_formula t1 #> do_formula t2
| @{const HOL.implies} $ t1 $ t2 => do_formula t1 #> do_formula 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 t1 #> fold (do_term_or_formula false T) [t2, t3]
| Const (@{const_name Ex1}, _) $ Abs (_, _, t') => do_formula t'
| Const (@{const_name Ball}, _) $ t1 $ Abs (_, _, t') =>
do_formula (t1 $ Bound ~1) #> do_formula t'
| Const (@{const_name Bex}, _) $ t1 $ Abs (_, _, t') =>
do_formula (t1 $ Bound ~1) #> do_formula t'
| (t0 as Const (_, @{typ bool})) $ t1 =>
do_term false t0 #> do_formula t1 (* theory constant *)
| _ => do_term false t)
in do_formula end
fun pconsts_in_fact thy t =
Symtab.fold (fn (s, pss) => fold (cons o pair s) pss)
(Symtab.empty |> add_pconsts_in_term thy 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 fudge fact =
(case fact |> snd |> theory_const_prop_of fudge |> pconsts_in_fact thy 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 patternT_ord p =
(case p of
(Type (s, ps), Type (t, qs)) =>
(case fast_string_ord (s, t) of
EQUAL => dict_ord patternT_ord (ps, qs)
| ord => ord)
| (TVar _, TVar _) => EQUAL
| (TVar _, _) => LESS
| (Type _, TVar _) => GREATER
| (Type _, TFree _) => LESS
| (TFree (s, _), TFree (t, _)) => fast_string_ord (s, t)
| (TFree _, _) => GREATER)
fun ptype_ord (PType (m, ps), PType (n, qs)) =
(case dict_ord patternT_ord (ps, qs) of
EQUAL => int_ord (m, n)
| ord => ord)
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 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.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
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,
theory_const_irrel_weight,
chained_const_irrel_weight, ...})
const_tab chained_const_tab =
generic_pconst_weight local_const_multiplier abs_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.isSuffix theory_const_suffix s
fun fact_weight fudge stature const_tab rel_const_tab chained_const_tab
fact_consts =
(case fact_consts |> List.partition (pconst_hyper_mem I rel_const_tab)
||> filter_out (pconst_hyper_mem swap rel_const_tab) 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_const_tab)
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 facts sc tab =
if Symtab.is_empty tab then
Symtab.empty
|> fold (add_pconsts_in_term thy)
(map_filter (fn ((_, (sc', _)), th) =>
if sc' = sc then SOME (prop_of th) else NONE) facts)
else
tab
fun consider_arities 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 (s, _) =>
(if is_widely_irrelevant_const s 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 facts =
fold (consider_arities 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 = 45 (* FUDGE *)
fun eq_prod eqx eqy ((x1, y1), (x2, y2)) = eqx (x1, x2) andalso eqy (y1, y2)
val really_hopeless_get_kicked_out_iter = 5 (* FUDGE *)
fun relevance_filter ctxt thres0 decay max_facts
(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
val chained_ts =
facts |> map_filter (fn ((_, (Chained, _)), th) => SOME (prop_of th)
| _ => NONE)
val chained_const_tab = Symtab.empty |> fold add_pconsts chained_ts
val goal_const_tab =
Symtab.empty
|> fold add_pconsts hyp_ts
|> add_pconsts concl_t
|> (fn tab => if Symtab.is_empty tab then chained_const_tab else tab)
|> fold (if_empty_replace_with_scope thy facts) [Chained, Assum, Local]
fun iter j remaining_max thres rel_const_tab hopeless hopeful =
let
val hopeless =
hopeless |> j = really_hopeless_get_kicked_out_iter
? filter_out (fn (_, w) => w < 0.001)
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 sps = maps (snd o fst) accepts
val rel_const_tab' =
rel_const_tab |> fold add_pconst_to_table sps
fun is_dirty (s, _) =
Symtab.lookup rel_const_tab' s <> Symtab.lookup rel_const_tab s
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 (eq_prod (op =) (eq_list ptype_eq))
(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 accepts ? cons @{thm ext}
|> add_set_const_thms accepts
|> insert_into_facts accepts
in
facts |> map_filter (pair_consts_fact thy 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) max_facts fudge
hyp_ts concl_t facts =
let
val thy = Proof_Context.theory_of ctxt
val fudge = fudge |> the_default default_relevance_fudge
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 orelse max_facts <= 0 then
[]
else
relevance_filter ctxt thres0 decay max_facts fudge facts hyp_ts
(concl_t |> theory_constify fudge (Context.theory_name thy)))
|> map fact_of_raw_fact
end
end;