monotonic type inference in ATP Sledgehammer problems -- based on Claessen & al.'s CADE 2011 paper, Sect. 2.3.
(* Title: HOL/Tools/Sledgehammer/sledgehammer_atp_translate.ML
Author: Fabian Immler, TU Muenchen
Author: Makarius
Author: Jasmin Blanchette, TU Muenchen
Translation of HOL to FOL for Sledgehammer.
*)
signature SLEDGEHAMMER_ATP_TRANSLATE =
sig
type 'a fo_term = 'a ATP_Problem.fo_term
type 'a problem = 'a ATP_Problem.problem
type locality = Sledgehammer_Filter.locality
datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic
datatype type_level =
All_Types | Nonmonotonic_Types | Finite_Types | Const_Arg_Types | No_Types
datatype type_system =
Many_Typed |
Preds of polymorphism * type_level |
Tags of polymorphism * type_level
type translated_formula
val readable_names : bool Config.T
val fact_prefix : string
val conjecture_prefix : string
val predicator_base : string
val explicit_app_base : string
val type_pred_base : string
val tff_type_prefix : string
val type_sys_from_string : string -> type_system
val polymorphism_of_type_sys : type_system -> polymorphism
val level_of_type_sys : type_system -> type_level
val is_type_sys_virtually_sound : type_system -> bool
val is_type_sys_fairly_sound : type_system -> bool
val num_atp_type_args : theory -> type_system -> string -> int
val unmangled_const : string -> string * string fo_term list
val translate_atp_fact :
Proof.context -> bool -> (string * locality) * thm
-> translated_formula option * ((string * locality) * thm)
val prepare_atp_problem :
Proof.context -> type_system -> bool -> term list -> term
-> (translated_formula option * ((string * 'a) * thm)) list
-> string problem * string Symtab.table * int * int
* (string * 'a) list vector
val atp_problem_weights : string problem -> (string * real) list
end;
structure Sledgehammer_ATP_Translate : SLEDGEHAMMER_ATP_TRANSLATE =
struct
open ATP_Problem
open Metis_Translate
open Sledgehammer_Util
open Sledgehammer_Filter
(* experimental *)
val generate_useful_info = false
(* Readable names are often much shorter, especially if types are mangled in
names. Also, the logic for generating legal SNARK sort names is only
implemented for readable names. Finally, readable names are, well, more
readable. For these reason, they are enabled by default. *)
val readable_names =
Attrib.setup_config_bool @{binding sledgehammer_atp_readable_names} (K true)
val type_decl_prefix = "type_"
val sym_decl_prefix = "sym_"
val fact_prefix = "fact_"
val conjecture_prefix = "conj_"
val helper_prefix = "help_"
val class_rel_clause_prefix = "crel_";
val arity_clause_prefix = "arity_"
val tfree_prefix = "tfree_"
val predicator_base = "hBOOL"
val explicit_app_base = "hAPP"
val type_pred_base = "is"
val tff_type_prefix = "ty_"
fun make_tff_type s = tff_type_prefix ^ ascii_of s
(* official TPTP syntax *)
val tptp_tff_type_of_types = "$tType"
val tptp_tff_bool_type = "$o"
val tptp_false = "$false"
val tptp_true = "$true"
(* Freshness almost guaranteed! *)
val sledgehammer_weak_prefix = "Sledgehammer:"
datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic
datatype type_level =
All_Types | Nonmonotonic_Types | Finite_Types | Const_Arg_Types | No_Types
datatype type_system =
Many_Typed |
Preds of polymorphism * type_level |
Tags of polymorphism * type_level
fun type_sys_from_string s =
(case try (unprefix "mangled_") s of
SOME s => (Mangled_Monomorphic, s)
| NONE =>
case try (unprefix "mono_") s of
SOME s => (Monomorphic, s)
| NONE => (Polymorphic, s))
||> (fn s =>
case try (unsuffix " ?") s of
SOME s => (Nonmonotonic_Types, s)
| NONE =>
case try (unsuffix " !") s of
SOME s => (Finite_Types, s)
| NONE => (All_Types, s))
|> (fn (polymorphism, (level, core)) =>
case (core, (polymorphism, level)) of
("many_typed", (Polymorphic (* naja *), All_Types)) =>
Many_Typed
| ("preds", extra) => Preds extra
| ("tags", extra) => Tags extra
| ("const_args", (_, All_Types (* naja *))) =>
Preds (polymorphism, Const_Arg_Types)
| ("erased", (Polymorphic, All_Types (* naja *))) =>
Preds (polymorphism, No_Types)
| _ => error ("Unknown type system: " ^ quote s ^ "."))
fun polymorphism_of_type_sys Many_Typed = Mangled_Monomorphic
| polymorphism_of_type_sys (Preds (poly, _)) = poly
| polymorphism_of_type_sys (Tags (poly, _)) = poly
fun level_of_type_sys Many_Typed = All_Types
| level_of_type_sys (Preds (_, level)) = level
| level_of_type_sys (Tags (_, level)) = level
fun is_type_level_virtually_sound s =
s = All_Types orelse s = Nonmonotonic_Types
val is_type_sys_virtually_sound =
is_type_level_virtually_sound o level_of_type_sys
fun is_type_level_fairly_sound level =
is_type_level_virtually_sound level orelse level = Finite_Types
val is_type_sys_fairly_sound = is_type_level_fairly_sound o level_of_type_sys
fun formula_map f (AQuant (q, xs, phi)) = AQuant (q, xs, formula_map f phi)
| formula_map f (AConn (c, phis)) = AConn (c, map (formula_map f) phis)
| formula_map f (AAtom tm) = AAtom (f tm)
fun formula_fold f =
let
fun aux pos (AQuant (_, _, phi)) = aux pos phi
| aux pos (AConn (ANot, [phi])) = aux (not pos) phi
| aux pos (AConn (AImplies, [phi1, phi2])) =
aux (not pos) phi1 #> aux pos phi2
| aux pos (AConn (c, phis)) =
if member (op =) [AAnd, AOr] c then fold (aux pos) phis
else raise Fail "unexpected connective with unknown polarities"
| aux pos (AAtom tm) = f pos tm
in aux true end
type translated_formula =
{name: string,
locality: locality,
kind: formula_kind,
combformula: (name, typ, combterm) formula,
atomic_types: typ list}
fun update_combformula f ({name, locality, kind, combformula, atomic_types}
: translated_formula) =
{name = name, locality = locality, kind = kind, combformula = f combformula,
atomic_types = atomic_types} : translated_formula
fun fact_lift f ({combformula, ...} : translated_formula) = f combformula
val boring_consts = [explicit_app_base, @{const_name Metis.fequal}]
fun should_omit_type_args type_sys s =
s <> type_pred_base andalso s <> type_tag_name andalso
(s = @{const_name HOL.eq} orelse level_of_type_sys type_sys = No_Types orelse
(case type_sys of
Tags (_, All_Types) => true
| _ => polymorphism_of_type_sys type_sys <> Mangled_Monomorphic andalso
member (op =) boring_consts s))
datatype type_arg_policy = No_Type_Args | Explicit_Type_Args | Mangled_Type_Args
fun general_type_arg_policy type_sys =
if level_of_type_sys type_sys = No_Types then
No_Type_Args
else if polymorphism_of_type_sys type_sys = Mangled_Monomorphic then
Mangled_Type_Args
else
Explicit_Type_Args
fun type_arg_policy type_sys s =
if should_omit_type_args type_sys s then No_Type_Args
else general_type_arg_policy type_sys
fun num_atp_type_args thy type_sys s =
if type_arg_policy type_sys s = Explicit_Type_Args then num_type_args thy s
else 0
fun atp_type_literals_for_types type_sys kind Ts =
if level_of_type_sys type_sys = No_Types then
[]
else
Ts |> type_literals_for_types
|> filter (fn TyLitVar _ => kind <> Conjecture
| TyLitFree _ => kind = Conjecture)
fun mk_aconn c phi1 phi2 = AConn (c, [phi1, phi2])
fun mk_aconns c phis =
let val (phis', phi') = split_last phis in
fold_rev (mk_aconn c) phis' phi'
end
fun mk_ahorn [] phi = phi
| mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])
fun mk_aquant _ [] phi = phi
| mk_aquant q xs (phi as AQuant (q', xs', phi')) =
if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)
| mk_aquant q xs phi = AQuant (q, xs, phi)
fun close_universally atom_vars phi =
let
fun formula_vars bounds (AQuant (_, xs, phi)) =
formula_vars (map fst xs @ bounds) phi
| formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis
| formula_vars bounds (AAtom tm) =
union (op =) (atom_vars tm []
|> filter_out (member (op =) bounds o fst))
in mk_aquant AForall (formula_vars [] phi []) phi end
fun combterm_vars (CombApp (tm1, tm2)) = fold combterm_vars [tm1, tm2]
| combterm_vars (CombConst _) = I
| combterm_vars (CombVar (name, T)) = insert (op =) (name, SOME T)
fun close_combformula_universally phi = close_universally combterm_vars phi
fun term_vars (ATerm (name as (s, _), tms)) =
is_atp_variable s ? insert (op =) (name, NONE)
#> fold term_vars tms
fun close_formula_universally phi = close_universally term_vars phi
fun fo_term_from_typ (Type (s, Ts)) =
ATerm (`make_fixed_type_const s, map fo_term_from_typ Ts)
| fo_term_from_typ (TFree (s, _)) =
ATerm (`make_fixed_type_var s, [])
| fo_term_from_typ (TVar ((x as (s, _)), _)) =
ATerm ((make_schematic_type_var x, s), [])
(* This shouldn't clash with anything else. *)
val mangled_type_sep = "\000"
fun generic_mangled_type_name f (ATerm (name, [])) = f name
| generic_mangled_type_name f (ATerm (name, tys)) =
f name ^ "(" ^ commas (map (generic_mangled_type_name f) tys) ^ ")"
val mangled_type_name =
fo_term_from_typ
#> (fn ty => (make_tff_type (generic_mangled_type_name fst ty),
generic_mangled_type_name snd ty))
fun generic_mangled_type_suffix f g Ts =
fold_rev (curry (op ^) o g o prefix mangled_type_sep
o generic_mangled_type_name f) Ts ""
fun mangled_const_name T_args (s, s') =
let val ty_args = map fo_term_from_typ T_args in
(s ^ generic_mangled_type_suffix fst ascii_of ty_args,
s' ^ generic_mangled_type_suffix snd I ty_args)
end
val parse_mangled_ident =
Scan.many1 (not o member (op =) ["(", ")", ","]) >> implode
fun parse_mangled_type x =
(parse_mangled_ident
-- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")
[] >> ATerm) x
and parse_mangled_types x =
(parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x
fun unmangled_type s =
s |> suffix ")" |> raw_explode
|> Scan.finite Symbol.stopper
(Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^
quote s)) parse_mangled_type))
|> fst
val unmangled_const_name = space_explode mangled_type_sep #> hd
fun unmangled_const s =
let val ss = space_explode mangled_type_sep s in
(hd ss, map unmangled_type (tl ss))
end
fun introduce_proxies tm =
let
fun aux top_level (CombApp (tm1, tm2)) =
CombApp (aux top_level tm1, aux false tm2)
| aux top_level (CombConst (name as (s, s'), T, T_args)) =
(case proxify_const s of
SOME proxy_base =>
if top_level then
(case s of
"c_False" => (tptp_false, s')
| "c_True" => (tptp_true, s')
| _ => name, [])
else
(proxy_base |>> prefix const_prefix, T_args)
| NONE => (name, T_args))
|> (fn (name, T_args) => CombConst (name, T, T_args))
| aux _ tm = tm
in aux true tm end
fun combformula_from_prop thy eq_as_iff =
let
fun do_term bs t atomic_types =
combterm_from_term thy bs (Envir.eta_contract t)
|>> (introduce_proxies #> AAtom)
||> union (op =) atomic_types
fun do_quant bs q s T t' =
let val s = Name.variant (map fst bs) s in
do_formula ((s, T) :: bs) t'
#>> mk_aquant q [(`make_bound_var s, SOME T)]
end
and do_conn bs c t1 t2 =
do_formula bs t1 ##>> do_formula bs t2
#>> uncurry (mk_aconn c)
and do_formula bs t =
case t of
@{const Not} $ t1 => do_formula bs t1 #>> mk_anot
| Const (@{const_name All}, _) $ Abs (s, T, t') =>
do_quant bs AForall s T t'
| Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
do_quant bs AExists s T t'
| @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd t1 t2
| @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr t1 t2
| @{const HOL.implies} $ t1 $ t2 => do_conn bs AImplies t1 t2
| Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
if eq_as_iff then do_conn bs AIff t1 t2 else do_term bs t
| _ => do_term bs t
in do_formula [] end
val presimplify_term = prop_of o Meson.presimplify oo Skip_Proof.make_thm
fun concealed_bound_name j = sledgehammer_weak_prefix ^ string_of_int j
fun conceal_bounds Ts t =
subst_bounds (map (Free o apfst concealed_bound_name)
(0 upto length Ts - 1 ~~ Ts), t)
fun reveal_bounds Ts =
subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
(0 upto length Ts - 1 ~~ Ts))
(* Removes the lambdas from an equation of the form "t = (%x. u)".
(Cf. "extensionalize_theorem" in "Meson_Clausify".) *)
fun extensionalize_term t =
let
fun aux j (@{const Trueprop} $ t') = @{const Trueprop} $ aux j t'
| aux j (t as Const (s, Type (_, [Type (_, [_, T']),
Type (_, [_, res_T])]))
$ t2 $ Abs (var_s, var_T, t')) =
if s = @{const_name HOL.eq} orelse s = @{const_name "=="} then
let val var_t = Var ((var_s, j), var_T) in
Const (s, T' --> T' --> res_T)
$ betapply (t2, var_t) $ subst_bound (var_t, t')
|> aux (j + 1)
end
else
t
| aux _ t = t
in aux (maxidx_of_term t + 1) t end
fun introduce_combinators_in_term ctxt kind t =
let val thy = Proof_Context.theory_of ctxt in
if Meson.is_fol_term thy t then
t
else
let
fun aux Ts t =
case t of
@{const Not} $ t1 => @{const Not} $ aux Ts t1
| (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
t0 $ Abs (s, T, aux (T :: Ts) t')
| (t0 as Const (@{const_name All}, _)) $ t1 =>
aux Ts (t0 $ eta_expand Ts t1 1)
| (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
t0 $ Abs (s, T, aux (T :: Ts) t')
| (t0 as Const (@{const_name Ex}, _)) $ t1 =>
aux Ts (t0 $ eta_expand Ts t1 1)
| (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
| (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
| (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
| (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
$ t1 $ t2 =>
t0 $ aux Ts t1 $ aux Ts t2
| _ => if not (exists_subterm (fn Abs _ => true | _ => false) t) then
t
else
t |> conceal_bounds Ts
|> Envir.eta_contract
|> cterm_of thy
|> Meson_Clausify.introduce_combinators_in_cterm
|> prop_of |> Logic.dest_equals |> snd
|> reveal_bounds Ts
val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end
handle THM _ =>
(* A type variable of sort "{}" will make abstraction fail. *)
if kind = Conjecture then HOLogic.false_const
else HOLogic.true_const
end
(* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
fun freeze_term t =
let
fun aux (t $ u) = aux t $ aux u
| aux (Abs (s, T, t)) = Abs (s, T, aux t)
| aux (Var ((s, i), T)) =
Free (sledgehammer_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
| aux t = t
in t |> exists_subterm is_Var t ? aux end
(* making fact and conjecture formulas *)
fun make_formula ctxt eq_as_iff presimp name loc kind t =
let
val thy = Proof_Context.theory_of ctxt
val t = t |> Envir.beta_eta_contract
|> transform_elim_term
|> Object_Logic.atomize_term thy
val need_trueprop = (fastype_of t = @{typ bool})
val t = t |> need_trueprop ? HOLogic.mk_Trueprop
|> extensionalize_term
|> presimp ? presimplify_term thy
|> perhaps (try (HOLogic.dest_Trueprop))
|> introduce_combinators_in_term ctxt kind
|> kind <> Axiom ? freeze_term
val (combformula, atomic_types) =
combformula_from_prop thy eq_as_iff t []
in
{name = name, locality = loc, kind = kind, combformula = combformula,
atomic_types = atomic_types}
end
fun make_fact ctxt keep_trivial eq_as_iff presimp ((name, loc), t) =
case (keep_trivial, make_formula ctxt eq_as_iff presimp name loc Axiom t) of
(false, {combformula = AAtom (CombConst (("c_True", _), _, _)), ...}) =>
NONE
| (_, formula) => SOME formula
fun make_conjecture ctxt ts =
let val last = length ts - 1 in
map2 (fn j => make_formula ctxt true true (string_of_int j) Chained
(if j = last then Conjecture else Hypothesis))
(0 upto last) ts
end
(** "hBOOL" and "hAPP" **)
type sym_info =
{pred_sym : bool, min_ary : int, max_ary : int, typ : typ option}
fun add_combterm_syms_to_table explicit_apply =
let
fun aux top_level tm =
let val (head, args) = strip_combterm_comb tm in
(case head of
CombConst ((s, _), T, _) =>
if String.isPrefix bound_var_prefix s then
I
else
let val ary = length args in
Symtab.map_default
(s, {pred_sym = true,
min_ary = if explicit_apply then 0 else ary,
max_ary = 0, typ = SOME T})
(fn {pred_sym, min_ary, max_ary, typ} =>
{pred_sym = pred_sym andalso top_level,
min_ary = Int.min (ary, min_ary),
max_ary = Int.max (ary, max_ary),
typ = if typ = SOME T then typ else NONE})
end
| _ => I)
#> fold (aux false) args
end
in aux true end
fun add_fact_syms_to_table explicit_apply =
fact_lift (formula_fold (K (add_combterm_syms_to_table explicit_apply)))
val default_sym_table_entries : (string * sym_info) list =
[("equal", {pred_sym = true, min_ary = 2, max_ary = 2, typ = NONE}),
(make_fixed_const predicator_base,
{pred_sym = true, min_ary = 1, max_ary = 1, typ = NONE})] @
([tptp_false, tptp_true]
|> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, typ = NONE}))
fun sym_table_for_facts explicit_apply facts =
Symtab.empty |> fold Symtab.default default_sym_table_entries
|> fold (add_fact_syms_to_table explicit_apply) facts
fun min_arity_of sym_tab s =
case Symtab.lookup sym_tab s of
SOME ({min_ary, ...} : sym_info) => min_ary
| NONE =>
case strip_prefix_and_unascii const_prefix s of
SOME s =>
let val s = s |> unmangled_const_name |> invert_const in
if s = predicator_base then 1
else if s = explicit_app_base then 2
else if s = type_pred_base then 1
else 0
end
| NONE => 0
(* True if the constant ever appears outside of the top-level position in
literals, or if it appears with different arities (e.g., because of different
type instantiations). If false, the constant always receives all of its
arguments and is used as a predicate. *)
fun is_pred_sym sym_tab s =
case Symtab.lookup sym_tab s of
SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>
pred_sym andalso min_ary = max_ary
| NONE => false
val predicator_combconst =
CombConst (`make_fixed_const predicator_base, @{typ "bool => bool"}, [])
fun predicator tm = CombApp (predicator_combconst, tm)
fun introduce_predicators_in_combterm sym_tab tm =
case strip_combterm_comb tm of
(CombConst ((s, _), _, _), _) =>
if is_pred_sym sym_tab s then tm else predicator tm
| _ => predicator tm
fun list_app head args = fold (curry (CombApp o swap)) args head
fun explicit_app arg head =
let
val head_T = combtyp_of head
val (arg_T, res_T) = dest_funT head_T
val explicit_app =
CombConst (`make_fixed_const explicit_app_base, head_T --> head_T,
[arg_T, res_T])
in list_app explicit_app [head, arg] end
fun list_explicit_app head args = fold explicit_app args head
fun introduce_explicit_apps_in_combterm sym_tab =
let
fun aux tm =
case strip_combterm_comb tm of
(head as CombConst ((s, _), _, _), args) =>
args |> map aux
|> chop (min_arity_of sym_tab s)
|>> list_app head
|-> list_explicit_app
| (head, args) => list_explicit_app head (map aux args)
in aux end
fun impose_type_arg_policy_in_combterm type_sys =
let
fun aux (CombApp tmp) = CombApp (pairself aux tmp)
| aux (CombConst (name as (s, _), T, T_args)) =
(case strip_prefix_and_unascii const_prefix s of
NONE => (name, T_args)
| SOME s'' =>
let val s'' = invert_const s'' in
case type_arg_policy type_sys s'' of
No_Type_Args => (name, [])
| Explicit_Type_Args => (name, T_args)
| Mangled_Type_Args => (mangled_const_name T_args name, [])
end)
|> (fn (name, T_args) => CombConst (name, T, T_args))
| aux tm = tm
in aux end
fun repair_combterm type_sys sym_tab =
introduce_explicit_apps_in_combterm sym_tab
#> introduce_predicators_in_combterm sym_tab
#> impose_type_arg_policy_in_combterm type_sys
fun repair_fact type_sys sym_tab =
update_combformula (formula_map (repair_combterm type_sys sym_tab))
(** Helper facts **)
fun ti_ti_helper_fact () =
let
fun var s = ATerm (`I s, [])
fun tag tm = ATerm (`make_fixed_const type_tag_name, [var "X", tm])
in
Formula (helper_prefix ^ "ti_ti", Axiom,
AAtom (ATerm (`I "equal", [tag (tag (var "Y")), tag (var "Y")]))
|> close_formula_universally, NONE, NONE)
end
fun helper_facts_for_sym ctxt type_sys (s, {typ, ...} : sym_info) =
case strip_prefix_and_unascii const_prefix s of
SOME mangled_s =>
let
val thy = Proof_Context.theory_of ctxt
val unmangled_s = mangled_s |> unmangled_const_name
fun dub_and_inst c needs_some_types (th, j) =
((c ^ "_" ^ string_of_int j ^ (if needs_some_types then "T" else ""),
Chained),
let val t = th |> prop_of in
t |> (general_type_arg_policy type_sys = Mangled_Type_Args andalso
not (null (Term.hidden_polymorphism t)))
? (case typ of
SOME T => specialize_type thy (invert_const unmangled_s, T)
| NONE => I)
end)
fun make_facts eq_as_iff =
map_filter (make_fact ctxt false eq_as_iff false)
val has_some_types = is_type_sys_fairly_sound type_sys
in
metis_helpers
|> maps (fn (metis_s, (needs_some_types, ths)) =>
if metis_s <> unmangled_s orelse
(needs_some_types andalso not has_some_types) then
[]
else
ths ~~ (1 upto length ths)
|> map (dub_and_inst mangled_s needs_some_types)
|> make_facts (not needs_some_types))
end
| NONE => []
fun helper_facts_for_sym_table ctxt type_sys sym_tab =
Symtab.fold_rev (append o helper_facts_for_sym ctxt type_sys) sym_tab []
fun translate_atp_fact ctxt keep_trivial =
`(make_fact ctxt keep_trivial true true o apsnd prop_of)
fun translate_formulas ctxt type_sys hyp_ts concl_t rich_facts =
let
val thy = Proof_Context.theory_of ctxt
val fact_ts = map (prop_of o snd o snd) rich_facts
val (facts, fact_names) =
rich_facts
|> map_filter (fn (NONE, _) => NONE
| (SOME fact, (name, _)) => SOME (fact, name))
|> ListPair.unzip
(* Remove existing facts from the conjecture, as this can dramatically
boost an ATP's performance (for some reason). *)
val hyp_ts = hyp_ts |> filter_out (member (op aconv) fact_ts)
val goal_t = Logic.list_implies (hyp_ts, concl_t)
val all_ts = goal_t :: fact_ts
val subs = tfree_classes_of_terms all_ts
val supers = tvar_classes_of_terms all_ts
val tycons = type_consts_of_terms thy all_ts
val conjs = make_conjecture ctxt (hyp_ts @ [concl_t])
val (supers', arity_clauses) =
if level_of_type_sys type_sys = No_Types then ([], [])
else make_arity_clauses thy tycons supers
val class_rel_clauses = make_class_rel_clauses thy subs supers'
in
(fact_names |> map single, (conjs, facts, class_rel_clauses, arity_clauses))
end
fun fo_literal_from_type_literal (TyLitVar (class, name)) =
(true, ATerm (class, [ATerm (name, [])]))
| fo_literal_from_type_literal (TyLitFree (class, name)) =
(true, ATerm (class, [ATerm (name, [])]))
fun formula_from_fo_literal (pos, t) = AAtom t |> not pos ? mk_anot
(* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
considered dangerous because their "exhaust" properties can easily lead to
unsound ATP proofs. The checks below are an (unsound) approximation of
finiteness. *)
fun is_dtyp_finite _ (Datatype_Aux.DtTFree _) = true
| is_dtyp_finite ctxt (Datatype_Aux.DtType (s, Us)) =
is_type_constr_finite ctxt s andalso forall (is_dtyp_finite ctxt) Us
| is_dtyp_finite _ (Datatype_Aux.DtRec _) = false
and is_type_finite ctxt (Type (s, Ts)) =
is_type_constr_finite ctxt s andalso forall (is_type_finite ctxt) Ts
| is_type_finite _ _ = false
and is_type_constr_finite ctxt s =
let val thy = Proof_Context.theory_of ctxt in
case Datatype_Data.get_info thy s of
SOME {descr, ...} =>
forall (fn (_, (_, _, constrs)) =>
forall (forall (is_dtyp_finite ctxt) o snd) constrs) descr
| NONE =>
case Typedef.get_info ctxt s of
({rep_type, ...}, _) :: _ => is_type_finite ctxt rep_type
| [] => true
end
fun should_encode_type _ All_Types _ = true
| should_encode_type ctxt Finite_Types T = is_type_finite ctxt T
| should_encode_type _ Nonmonotonic_Types _ =
error "Monotonicity inference not implemented yet."
| should_encode_type _ _ _ = false
fun should_predicate_on_type ctxt (Preds (_, level)) T =
should_encode_type ctxt level T
| should_predicate_on_type _ _ _ = false
fun should_tag_with_type ctxt (Tags (_, level)) T =
should_encode_type ctxt level T
| should_tag_with_type _ _ _ = false
fun type_pred_combatom type_sys T tm =
CombApp (CombConst (`make_fixed_const type_pred_base, T --> @{typ bool}, [T]),
tm)
|> impose_type_arg_policy_in_combterm type_sys
|> AAtom
fun formula_from_combformula ctxt type_sys =
let
fun tag_with_type type_sys T tm =
CombConst (`make_fixed_const type_tag_name, T --> T, [T])
|> impose_type_arg_policy_in_combterm type_sys
|> do_term true
|> (fn ATerm (s, tms) => ATerm (s, tms @ [tm]))
and do_term top_level u =
let
val (head, args) = strip_combterm_comb u
val (x, T_args) =
case head of
CombConst (name, _, T_args) => (name, T_args)
| CombVar (name, _) => (name, [])
| CombApp _ => raise Fail "impossible \"CombApp\""
val t = ATerm (x, map fo_term_from_typ T_args @
map (do_term false) args)
val T = combtyp_of u
in
t |> (if not top_level andalso should_tag_with_type ctxt type_sys T then
tag_with_type type_sys T
else
I)
end
val do_bound_type =
if type_sys = Many_Typed then SOME o mangled_type_name else K NONE
fun do_out_of_bound_type (s, T) =
if should_predicate_on_type ctxt type_sys T then
type_pred_combatom type_sys T (CombVar (s, T))
|> do_formula |> SOME
else
NONE
and do_formula (AQuant (q, xs, phi)) =
AQuant (q, xs |> map (apsnd (fn NONE => NONE
| SOME T => do_bound_type T)),
(if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))
(map_filter
(fn (_, NONE) => NONE
| (s, SOME T) => do_out_of_bound_type (s, T)) xs)
(do_formula phi))
| do_formula (AConn (c, phis)) = AConn (c, map do_formula phis)
| do_formula (AAtom tm) = AAtom (do_term true tm)
in do_formula end
fun formula_for_fact ctxt type_sys
({combformula, atomic_types, ...} : translated_formula) =
mk_ahorn (map (formula_from_fo_literal o fo_literal_from_type_literal)
(atp_type_literals_for_types type_sys Axiom atomic_types))
(formula_from_combformula ctxt type_sys
(close_combformula_universally combformula))
|> close_formula_universally
fun useful_isabelle_info s = SOME (ATerm ("[]", [ATerm ("isabelle_" ^ s, [])]))
(* Each fact is given a unique fact number to avoid name clashes (e.g., because
of monomorphization). The TPTP explicitly forbids name clashes, and some of
the remote provers might care. *)
fun formula_line_for_fact ctxt prefix type_sys
(j, formula as {name, locality, kind, ...}) =
Formula (prefix ^
(if polymorphism_of_type_sys type_sys = Polymorphic then ""
else string_of_int j ^ "_") ^
ascii_of name,
kind, formula_for_fact ctxt type_sys formula, NONE,
if generate_useful_info then
case locality of
Intro => useful_isabelle_info "intro"
| Elim => useful_isabelle_info "elim"
| Simp => useful_isabelle_info "simp"
| _ => NONE
else
NONE)
fun formula_line_for_class_rel_clause (ClassRelClause {name, subclass,
superclass, ...}) =
let val ty_arg = ATerm (`I "T", []) in
Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,
AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),
AAtom (ATerm (superclass, [ty_arg]))])
|> close_formula_universally, NONE, NONE)
end
fun fo_literal_from_arity_literal (TConsLit (c, t, args)) =
(true, ATerm (c, [ATerm (t, map (fn arg => ATerm (arg, [])) args)]))
| fo_literal_from_arity_literal (TVarLit (c, sort)) =
(false, ATerm (c, [ATerm (sort, [])]))
fun formula_line_for_arity_clause (ArityClause {name, conclLit, premLits,
...}) =
Formula (arity_clause_prefix ^ ascii_of name, Axiom,
mk_ahorn (map (formula_from_fo_literal o apfst not
o fo_literal_from_arity_literal) premLits)
(formula_from_fo_literal
(fo_literal_from_arity_literal conclLit))
|> close_formula_universally, NONE, NONE)
fun formula_line_for_conjecture ctxt type_sys
({name, kind, combformula, ...} : translated_formula) =
Formula (conjecture_prefix ^ name, kind,
formula_from_combformula ctxt type_sys
(close_combformula_universally combformula)
|> close_formula_universally, NONE, NONE)
fun free_type_literals type_sys ({atomic_types, ...} : translated_formula) =
atomic_types |> atp_type_literals_for_types type_sys Conjecture
|> map fo_literal_from_type_literal
fun formula_line_for_free_type j lit =
Formula (tfree_prefix ^ string_of_int j, Hypothesis,
formula_from_fo_literal lit, NONE, NONE)
fun formula_lines_for_free_types type_sys facts =
let
val litss = map (free_type_literals type_sys) facts
val lits = fold (union (op =)) litss []
in map2 formula_line_for_free_type (0 upto length lits - 1) lits end
(** Symbol declarations **)
fun insert_type get_T x xs =
let val T = get_T x in
if exists (curry Type.raw_instance T o get_T) xs then xs
else x :: filter_out ((fn T' => Type.raw_instance (T', T)) o get_T) xs
end
fun should_declare_sym type_sys pred_sym s =
not (String.isPrefix bound_var_prefix s) andalso s <> "equal" andalso
not (String.isPrefix "$" s) andalso
(type_sys = Many_Typed orelse not pred_sym)
fun add_combterm_syms_to_decl_table type_sys repaired_sym_tab =
let
fun declare_sym decl decls =
(* FIXME: use "insert_type" in all cases? *)
case type_sys of
Preds (Polymorphic, All_Types) => insert_type #3 decl decls
| _ => insert (op =) decl decls
fun do_term tm =
let val (head, args) = strip_combterm_comb tm in
(case head of
CombConst ((s, s'), T, T_args) =>
let val pred_sym = is_pred_sym repaired_sym_tab s in
if should_declare_sym type_sys pred_sym s then
Symtab.map_default (s, [])
(declare_sym (s', T_args, T, pred_sym, length args))
else
I
end
| _ => I)
#> fold do_term args
end
in do_term end
fun add_fact_syms_to_decl_table type_sys repaired_sym_tab =
fact_lift (formula_fold
(K (add_combterm_syms_to_decl_table type_sys repaired_sym_tab)))
fun sym_decl_table_for_facts type_sys repaired_sym_tab facts =
Symtab.empty |> is_type_sys_fairly_sound type_sys
? fold (add_fact_syms_to_decl_table type_sys repaired_sym_tab)
facts
(* FIXME: use CombVar not CombConst for bound variables? *)
fun is_var_or_bound_var (CombConst ((s, _), _, _)) =
String.isPrefix bound_var_prefix s
| is_var_or_bound_var (CombVar _) = true
| is_var_or_bound_var _ = false
fun add_combterm_nonmonotonic_types true
(CombApp (CombConst (("equal", _), Type (_, [T, _]), _),
CombApp (tm1, tm2))) =
exists is_var_or_bound_var [tm1, tm2] ? insert_type I T
| add_combterm_nonmonotonic_types _ _ = I
val add_fact_nonmonotonic_types =
fact_lift (formula_fold add_combterm_nonmonotonic_types)
fun nonmonotonic_types_for_facts type_sys facts =
[] |> level_of_type_sys type_sys = Nonmonotonic_Types
? fold add_fact_nonmonotonic_types facts
fun n_ary_strip_type 0 T = ([], T)
| n_ary_strip_type n (Type (@{type_name fun}, [dom_T, ran_T])) =
n_ary_strip_type (n - 1) ran_T |>> cons dom_T
| n_ary_strip_type _ _ = raise Fail "unexpected non-function"
fun result_type_of_decl (_, _, T, _, ary) = n_ary_strip_type ary T |> snd
fun decl_line_for_sym_decl s (s', _, T, pred_sym, ary) =
let val (arg_Ts, res_T) = n_ary_strip_type ary T in
Decl (sym_decl_prefix ^ s, (s, s'), map mangled_type_name arg_Ts,
if pred_sym then `I tptp_tff_bool_type else mangled_type_name res_T)
end
fun is_polymorphic_type T = fold_atyps (fn TVar _ => K true | _ => I) T false
fun formula_line_for_sym_decl ctxt type_sys n s j (s', T_args, T, _, ary) =
let
val (arg_Ts, res_T) = n_ary_strip_type ary T
val bound_names =
1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)
val bound_tms =
bound_names ~~ arg_Ts |> map (fn (name, T) => CombConst (name, T, []))
val bound_Ts =
arg_Ts |> map (fn T => if n > 1 orelse is_polymorphic_type T then SOME T
else NONE)
in
Formula (sym_decl_prefix ^ s ^
(if n > 1 then "_" ^ string_of_int j else ""), Axiom,
CombConst ((s, s'), T, T_args)
|> fold (curry (CombApp o swap)) bound_tms
|> type_pred_combatom type_sys res_T
|> mk_aquant AForall (bound_names ~~ bound_Ts)
|> formula_from_combformula ctxt type_sys
|> close_formula_universally,
NONE, NONE)
end
fun problem_lines_for_sym_decls ctxt type_sys (s, decls) =
if type_sys = Many_Typed then
map (decl_line_for_sym_decl s) decls
else
let
val decls =
case decls of
decl :: (decls' as _ :: _) =>
let val T = result_type_of_decl decl in
if forall ((fn T' => Type.raw_instance (T', T))
o result_type_of_decl) decls' then
[decl]
else
decls
end
| _ => decls
val n = length decls
val decls =
decls |> filter (should_predicate_on_type ctxt type_sys
o result_type_of_decl)
in
map2 (formula_line_for_sym_decl ctxt type_sys n s)
(0 upto length decls - 1) decls
end
fun problem_lines_for_sym_decl_table ctxt type_sys sym_decl_tab =
Symtab.fold_rev (append o problem_lines_for_sym_decls ctxt type_sys)
sym_decl_tab []
fun add_tff_types_in_formula (AQuant (_, xs, phi)) =
union (op =) (map_filter snd xs) #> add_tff_types_in_formula phi
| add_tff_types_in_formula (AConn (_, phis)) =
fold add_tff_types_in_formula phis
| add_tff_types_in_formula (AAtom _) = I
fun add_tff_types_in_problem_line (Decl (_, _, arg_Ts, res_T)) =
union (op =) (res_T :: arg_Ts)
| add_tff_types_in_problem_line (Formula (_, _, phi, _, _)) =
add_tff_types_in_formula phi
fun tff_types_in_problem problem =
fold (fold add_tff_types_in_problem_line o snd) problem []
fun decl_line_for_tff_type (s, s') =
Decl (type_decl_prefix ^ ascii_of s, (s, s'), [], `I tptp_tff_type_of_types)
val type_declsN = "Types"
val sym_declsN = "Symbol types"
val factsN = "Relevant facts"
val class_relsN = "Class relationships"
val aritiesN = "Arities"
val helpersN = "Helper facts"
val conjsN = "Conjectures"
val free_typesN = "Type variables"
fun offset_of_heading_in_problem _ [] j = j
| offset_of_heading_in_problem needle ((heading, lines) :: problem) j =
if heading = needle then j
else offset_of_heading_in_problem needle problem (j + length lines)
fun prepare_atp_problem ctxt type_sys explicit_apply hyp_ts concl_t facts =
let
val (fact_names, (conjs, facts, class_rel_clauses, arity_clauses)) =
translate_formulas ctxt type_sys hyp_ts concl_t facts
val sym_tab = conjs @ facts |> sym_table_for_facts explicit_apply
val (conjs, facts) =
(conjs, facts) |> pairself (map (repair_fact type_sys sym_tab))
val conjs_and_facts = conjs @ facts
val repaired_sym_tab = conjs_and_facts |> sym_table_for_facts false
val sym_decl_lines =
conjs_and_facts
|> sym_decl_table_for_facts type_sys repaired_sym_tab
|> problem_lines_for_sym_decl_table ctxt type_sys
val helpers =
helper_facts_for_sym_table ctxt type_sys repaired_sym_tab
|> map (repair_fact type_sys sym_tab)
val nonmonotonic_Ts =
nonmonotonic_types_for_facts type_sys (helpers @ conjs_and_facts)
(* Reordering these might confuse the proof reconstruction code or the SPASS
Flotter hack. *)
val problem =
[(sym_declsN, sym_decl_lines),
(factsN, map (formula_line_for_fact ctxt fact_prefix type_sys)
(0 upto length facts - 1 ~~ facts)),
(class_relsN, map formula_line_for_class_rel_clause class_rel_clauses),
(aritiesN, map formula_line_for_arity_clause arity_clauses),
(helpersN, map (formula_line_for_fact ctxt helper_prefix type_sys)
(0 upto length helpers - 1 ~~ helpers)
|> (case type_sys of
Tags (Polymorphic, level) =>
member (op =) [Finite_Types, Nonmonotonic_Types] level
? cons (ti_ti_helper_fact ())
| _ => I)),
(conjsN, map (formula_line_for_conjecture ctxt type_sys) conjs),
(free_typesN, formula_lines_for_free_types type_sys (facts @ conjs))]
val problem =
problem
|> (if type_sys = Many_Typed then
cons (type_declsN,
map decl_line_for_tff_type (tff_types_in_problem problem))
else
I)
val (problem, pool) =
problem |> nice_atp_problem (Config.get ctxt readable_names)
in
(problem,
case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
offset_of_heading_in_problem conjsN problem 0,
offset_of_heading_in_problem factsN problem 0,
fact_names |> Vector.fromList)
end
(* FUDGE *)
val conj_weight = 0.0
val hyp_weight = 0.1
val fact_min_weight = 0.2
val fact_max_weight = 1.0
val type_info_default_weight = 0.8
fun add_term_weights weight (ATerm (s, tms)) =
(not (is_atp_variable s) andalso s <> "equal") ? Symtab.default (s, weight)
#> fold (add_term_weights weight) tms
fun add_problem_line_weights weight (Formula (_, _, phi, _, _)) =
formula_fold (K (add_term_weights weight)) phi
| add_problem_line_weights _ _ = I
fun add_conjectures_weights [] = I
| add_conjectures_weights conjs =
let val (hyps, conj) = split_last conjs in
add_problem_line_weights conj_weight conj
#> fold (add_problem_line_weights hyp_weight) hyps
end
fun add_facts_weights facts =
let
val num_facts = length facts
fun weight_of j =
fact_min_weight + (fact_max_weight - fact_min_weight) * Real.fromInt j
/ Real.fromInt num_facts
in
map weight_of (0 upto num_facts - 1) ~~ facts
|> fold (uncurry add_problem_line_weights)
end
(* Weights are from 0.0 (most important) to 1.0 (least important). *)
fun atp_problem_weights problem =
let val get = these o AList.lookup (op =) problem in
Symtab.empty
|> add_conjectures_weights (get free_typesN @ get conjsN)
|> add_facts_weights (get factsN)
|> fold (fold (add_problem_line_weights type_info_default_weight) o get)
[sym_declsN, class_relsN, aritiesN]
|> Symtab.dest
|> sort (prod_ord Real.compare string_ord o pairself swap)
end
end;