val no_lamsN : string val opaque_liftingN : string val liftingN : string val opaque_combsN : string val combsN : string val combs_and_liftingN : string val combs_or_liftingN : string val keep_lamsN : string val schematic_var_prefix : string val fixed_var_prefix : string val tvar_prefix : string val tfree_prefix : string val const_prefix : string val type_const_prefix : string val class_prefix : string val lam_lifted_prefix : string val lam_lifted_mono_prefix : string val lam_lifted_poly_prefix : string val skolem_const_prefix : string val old_skolem_const_prefix : string val new_skolem_const_prefix : string val combinator_prefix : string val class_decl_prefix : string val type_decl_prefix : string val sym_decl_prefix : string val datatype_decl_prefix : string val class_memb_prefix : string val guards_sym_formula_prefix : string val tags_sym_formula_prefix : string val fact_prefix : string val conjecture_prefix : string val helper_prefix : string val subclass_prefix : string val tcon_clause_prefix : string val tfree_clause_prefix : string val lam_fact_prefix : string val typed_helper_suffix : string val untyped_helper_suffix : string val predicator_name : string val app_op_name : string val type_guard_name : string val type_tag_name : string val native_type_prefix : string val prefixed_predicator_name : string val prefixed_app_op_name : string val prefixed_type_tag_name : string val ascii_of : string -> string val unascii_of : string -> string val unprefix_and_unascii : string -> string -> stringoption val proxy_table : (string * (string * (thm * (string * string)))) list val proxify_const : string -> (string * string) option val invert_const : string -> string val unproxify_const : string -> string val new_skolem_var_name_of_const : string -> string val atp_logical_consts : stringlist val atp_irrelevant_consts : stringlist val atp_widely_irrelevant_consts : stringlist val is_irrelevant_const : string -> bool val is_widely_irrelevant_const : string -> bool val atp_schematic_consts_of : term -> typ list Symtab.table val is_type_enc_higher_order : type_enc -> bool val is_type_enc_polymorphic : type_enc -> bool val level_of_type_enc : type_enc -> type_level val is_type_enc_sound : type_enc -> bool val type_enc_of_string : strictness -> string -> type_enc val adjust_type_enc : atp_format -> type_enc -> type_enc val is_first_order_lambda_free : term -> bool val do_cheaply_conceal_lambdas : typ list -> term -> term val mk_aconns :
atp_connective -> ('a, 'b, 'c, 'd) atp_formula list
-> ('a, 'b, 'c, 'd) atp_formula val unmangled_type : string -> (string, 'a) ATP_Problem.atp_term val unmangled_const : string -> string * (string, 'b) atp_term list val unmangled_const_name : string -> stringlist val helper_table : bool -> ((string * bool) * (status * thm) list) list val trans_lams_of_string :
Proof.context -> type_enc -> string -> term list -> term list * term list val string_of_status : status -> string val factsN : string val generate_atp_problem : Proof.context -> bool -> atp_format -> atp_formula_role -> type_enc ->
mode -> string -> bool -> bool -> bool -> term list -> term ->
((string * stature) * term) list -> string atp_problem * string Symtab.table
* (string * term) list * int Symtab.table val atp_problem_term_order_info : string atp_problem -> (string * int) list end;
val no_lamsN = "no_lams"(* used internally; undocumented *) val opaque_liftingN = "opaque_lifting" val liftingN = "lifting" val opaque_combsN = "opaque_combs" val combsN = "combs" val combs_and_liftingN = "combs_and_lifting" val combs_or_liftingN = "combs_or_lifting" val keep_lamsN = "keep_lams"
(* The capitalization of the TPTP output respects the capitalzation of the prefix. *) val bound_var_prefix = "B_" val let_bound_var_prefix = "l_" val all_bound_var_prefix = "A_" val exist_bound_var_prefix = "E_" val schematic_var_prefix = "V_" val fixed_var_prefix = "v_" val tvar_prefix = "T_" val tfree_prefix = "tf_" val const_prefix = "c_" val type_const_prefix = "t_" val native_type_prefix = "n_" val class_prefix = "cl_"
(* Freshness almost guaranteed! *) val atp_prefix = "ATP" ^ Long_Name.separator val atp_weak_prefix = "ATP:" val atp_weak_suffix = ":ATP"
val lam_lifted_prefix = atp_weak_prefix ^ "Lam" val lam_lifted_mono_prefix = lam_lifted_prefix ^ "m" val lam_lifted_poly_prefix = lam_lifted_prefix ^ "p"
val skolem_const_prefix = atp_prefix ^ "Sko" val old_skolem_const_prefix = skolem_const_prefix ^ "o" val new_skolem_const_prefix = skolem_const_prefix ^ "n"
val combinator_prefix = "COMB"
val class_decl_prefix = "cl_" val type_decl_prefix = "ty_" val sym_decl_prefix = "sy_" val datatype_decl_prefix = "dt_" val class_memb_prefix = "cm_" val guards_sym_formula_prefix = "gsy_" val tags_sym_formula_prefix = "tsy_" val uncurried_alias_eq_prefix = "unc_" val fact_prefix = "fact_" val conjecture_prefix = "conj_" val helper_prefix = "help_" val subclass_prefix = "subcl_" val tcon_clause_prefix = "tcon_" val tfree_clause_prefix = "tfree_"
val lam_fact_prefix = "ATP.lambda_" val typed_helper_suffix = "_T" val untyped_helper_suffix = "_U"
val predicator_name = "pp" val app_op_name = "aa" val type_guard_name = "gg" val type_tag_name = "tt"
val prefixed_predicator_name = const_prefix ^ predicator_name val prefixed_app_op_name = const_prefix ^ app_op_name val prefixed_type_tag_name = const_prefix ^ type_tag_name
(*Escaping of special characters. Alphanumeric characters are left unchanged. The character _ goes to __. Characters in the range ASCII space to / go to _A to _P, respectively.
Other characters go to _nnn where nnn is the decimal ASCII code. *) val upper_a_minus_space = Char.ord #"A" - Char.ord #" "
fun ascii_of_char c = if Char.isAlphaNum c then String.str c elseif c = #"_"then "__" elseif #" " <= c andalso c <= #"/"then "_" ^ String.str (Char.chr (Char.ord c + upper_a_minus_space)) else (* fixed width, in case more digits follow *) "_" ^ stringN_of_int 3 (Char.ord c)
val ascii_of = String.translate ascii_of_char
(** Remove ASCII armoring from names in proof files **)
(* We don't raise error exceptions because this code can run inside a worker
thread. Also, the errors are impossible. *) val unascii_of = let fun un rcs [] = String.implode (rev rcs)
| un rcs [#"_"] = un (#"_" :: rcs) [] (* ERROR *) (* Three types of _ escapes: __, _A to _P, _nnn *)
| un rcs (#"_" :: #"_" :: cs) = un (#"_" :: rcs) cs
| un rcs (#"_" :: c :: cs) = if #"A" <= c andalso c<= #"P"then (* translation of #" " to #"/" *)
un (Char.chr (Char.ord c - upper_a_minus_space) :: rcs) cs else letval digits = List.take (c :: cs, 3) handle General.Subscript => [] in
(case Int.fromString (String.implode digits) of
SOME n => un (Char.chr n :: rcs) (List.drop (cs, 2))
| NONE => un (c :: #"_" :: rcs) cs (* ERROR *)) end
| un rcs (c :: cs) = un (c :: rcs) cs in un [] o String.explode end
(* If string s has the prefix s1, return the result of deleting it,
un-ASCII'd. *) fun unprefix_and_unascii s1 s = ifString.isPrefix s1 s then
SOME (unascii_of (String.extract (s, size s1, NONE))) else
NONE
val proxify_const = AList.lookup (op =) proxy_table #> Option.map (snd o snd)
(* Readable names for the more common symbolic functions. Do not mess with the
table unless you know what you are doing. *) val const_trans_table =
[(\<^const_name>\<open>False\<close>, "False"),
(\<^const_name>\<open>True\<close>, "True"),
(\<^const_name>\<open>Not\<close>, "Not"),
(\<^const_name>\<open>conj\<close>, "conj"),
(\<^const_name>\<open>disj\<close>, "disj"),
(\<^const_name>\<open>implies\<close>, "implies"),
(\<^const_name>\<open>HOL.eq\<close>, "equal"),
(\<^const_name>\<open>All\<close>, "All"),
(\<^const_name>\<open>Ex\<close>, "Ex"),
(\<^const_name>\<open>If\<close>, "If"),
(\<^const_name>\<open>Set.member\<close>, "member"),
(\<^const_name>\<open>HOL.Let\<close>, "Let"),
(\<^const_name>\<open>Hilbert_Choice.Eps\<close>, "Choice"),
(\<^const_name>\<open>Meson.COMBI\<close>, combinator_prefix ^ "I"),
(\<^const_name>\<open>Meson.COMBK\<close>, combinator_prefix ^ "K"),
(\<^const_name>\<open>Meson.COMBB\<close>, combinator_prefix ^ "B"),
(\<^const_name>\<open>Meson.COMBC\<close>, combinator_prefix ^ "C"),
(\<^const_name>\<open>Meson.COMBS\<close>, combinator_prefix ^ "S")]
|> Symtab.make
|> fold (Symtab.update o swap o snd o snd o snd) proxy_table
(* Invert the table of translations between Isabelle and ATPs. *) val const_trans_table_inv =
const_trans_table |> Symtab.dest |> map swap |> Symtab.make val const_trans_table_unprox =
Symtab.empty
|> fold (fn (_, (isa, (_, (_, atp)))) => Symtab.update (atp, isa)) proxy_table
val invert_const = perhaps (Symtab.lookup const_trans_table_inv) val unproxify_const = perhaps (Symtab.lookup const_trans_table_unprox)
fun lookup_const c =
(case Symtab.lookup const_trans_table c of
SOME c' => c'
| NONE => ascii_of c)
fun ascii_of_indexname (v, 0) = ascii_of v
| ascii_of_indexname (v, i) = ascii_of v ^ "_" ^ string_of_int i
fun make_bound_var x = bound_var_prefix ^ ascii_of x fun make_all_bound_var x = all_bound_var_prefix ^ ascii_of x fun make_exist_bound_var x = exist_bound_var_prefix ^ ascii_of x fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v fun make_fixed_var x = fixed_var_prefix ^ ascii_of x
fun make_tvar (s, i) = tvar_prefix ^ ascii_of_indexname (unquote_tvar s, i) fun make_tfree s = tfree_prefix ^ ascii_of (unquote_tvar s) fun tvar_name ((x as (s, _)), _) = (make_tvar x, s)
(* "HOL.eq" and choice are mapped to the ATP's equivalents *) fun make_fixed_const \<^const_name>\<open>HOL.eq\<close> = tptp_old_equal
| make_fixed_const c = const_prefix ^ lookup_const c
fun make_fixed_type_const c = type_const_prefix ^ lookup_const c
fun make_class clas = class_prefix ^ ascii_of clas
fun new_skolem_var_name_of_const s = letval ss = Long_Name.explode s in nth ss (length ss - 2) end
(* These are ignored anyway by the relevance filter (unless they appear in
higher-order places) but not by the monomorphizer. *) val atp_logical_consts =
[\<^const_name>\<open>Pure.prop\<close>, \<^const_name>\<open>Pure.conjunction\<close>,
\<^const_name>\<open>Pure.all\<close>, \<^const_name>\<open>Pure.imp\<close>, \<^const_name>\<open>Pure.eq\<close>,
\<^const_name>\<open>Trueprop\<close>, \<^const_name>\<open>All\<close>, \<^const_name>\<open>Ex\<close>,
\<^const_name>\<open>Ex1\<close>, \<^const_name>\<open>Ball\<close>, \<^const_name>\<open>Bex\<close>]
(* These are either simplified away by "Meson.presimplify" (most of the time) or
handled specially via "fFalse", "fTrue", ..., "fequal". *) val atp_irrelevant_consts =
[\<^const_name>\<open>False\<close>, \<^const_name>\<open>True\<close>, \<^const_name>\<open>Not\<close>, \<^const_name>\<open>conj\<close>,
\<^const_name>\<open>disj\<close>, \<^const_name>\<open>implies\<close>, \<^const_name>\<open>HOL.eq\<close>, \<^const_name>\<open>If\<close>,
\<^const_name>\<open>Let\<close>]
val atp_widely_irrelevant_consts = atp_logical_consts @ atp_irrelevant_consts
val atp_irrelevant_const_tab = Symtab.make (map (rpair ()) atp_irrelevant_consts) val atp_widely_irrelevant_const_tab = Symtab.make (map (rpair ()) atp_widely_irrelevant_consts)
val is_irrelevant_const = Symtab.defined atp_irrelevant_const_tab val is_widely_irrelevant_const = Symtab.defined atp_widely_irrelevant_const_tab
fun add_schematic_const (x as (_, T)) =
Monomorph.typ_has_tvars T ? Symtab.insert_list (op =) x val add_schematic_consts_of =
Term.fold_aterms (fn Const (x as (s, _)) => not (is_widely_irrelevant_const s) ? add_schematic_const x
| _ => I) fun atp_schematic_consts_of t = add_schematic_consts_of t Symtab.empty
val tvar_a_str = "'a" val tvar_a_z = ((tvar_a_str, 0), \<^sort>\<open>type\<close>) val tvar_a = TVar tvar_a_z val tvar_a_name = tvar_name tvar_a_z val itself_name = `make_fixed_type_const \<^type_name>\<open>itself\<close> val TYPE_name = `make_fixed_const \<^const_name>\<open>Pure.type\<close> val tvar_a_atype = AType ((tvar_a_name, []), []) val a_itself_atype = AType ((itself_name, []), [tvar_a_atype])
(** Definitions and functions for FOL clauses and formulas for TPTP **)
(** Type class membership **)
(* In our data structures, [] exceptionally refers to the top class, not to
the empty class. *)
val class_of_types = the_single \<^sort>\<open>type\<close>
fun normalize_classes cls = if member (op =) cls class_of_types then [] else cls
(* Arity of type constructor "s :: (arg1, ..., argN) res" *) fun make_axiom_tcon_clause (s, name, (cl, args)) = let val args = args |> map normalize_classes val tvars = args |> map_index (fn (j, _) => TVar ((tvar_a_str, j + 1), \<^sort>\<open>type\<close>)) in (name, args ~~ tvars, (cl, Type (s, tvars))) end
(* Generate all pairs (tycon, class, sorts) such that tycon belongs to class in
theory thy provided its arguments have the corresponding sorts. *) fun class_pairs thy tycons cls = let val alg = Sign.classes_of thy fun domain_sorts tycon = Sorts.mg_domain alg tycon o single fun add_class tycon cl =
cons (cl, domain_sorts tycon cl) handle Sorts.CLASS_ERROR _ => I fun try_classes tycon = (tycon, fold (add_class tycon) cls []) inmap try_classes tycons end
(* Proving one (tycon, class) membership may require proving others, so
iterate. *) fun all_class_pairs _ _ _ [] = ([], [])
| all_class_pairs thy tycons old_cls cls = let val old_cls' = cls @ old_cls fun maybe_insert_class s = not (member (op =) old_cls' s) ? insert (op =) s
val pairs = class_pairs thy tycons cls val new_cls = fold (fold (fold (fold maybe_insert_class) o snd) o snd) pairs [] val (cls', pairs') = all_class_pairs thy tycons old_cls' new_cls in (cls' @ cls, union (op =) pairs' pairs) end
fun tcon_clause _ _ [] = []
| tcon_clause seen n ((_, []) :: rest) = tcon_clause seen n rest
| tcon_clause seen n ((tcons, (ar as (cl, _)) :: ars) :: rest) = if cl = class_of_types then
tcon_clause seen n ((tcons, ars) :: rest) elseif member (op =) seen cl then (* multiple clauses for the same (tycon, cl) pair *)
make_axiom_tcon_clause (tcons,
lookup_const tcons ^ "___" ^ ascii_of cl ^ "_" ^ string_of_int n, ar) ::
tcon_clause seen (n + 1) ((tcons, ars) :: rest) else
make_axiom_tcon_clause (tcons, lookup_const tcons ^ "___" ^ ascii_of cl, ar) ::
tcon_clause (cl :: seen) n ((tcons, ars) :: rest)
(* Generate a list ("sub", "supers") such that "sub" is a proper subclass of all
"supers". *) fun make_subclass_pairs thy subs supers = let val class_less = curry (Sorts.class_less (Sign.classes_of thy)) fun supers_of sub = (sub, filter (class_less sub) supers) inmap supers_of subs |> filter_out (null o snd) end
(* intermediate terms *) (* TODO: Merge IConst and IVar *) datatype iterm =
IConst of (string * string) * typ * typ list |
IVar of (string * string) * typ |
IApp of iterm * iterm |
IAbs of ((string * string) * typ) * iterm
fun alpha_rename from to = let fun traverse (tm as IConst (name, T, Ts)) = if name = from then IConst (to, T, Ts) else tm
| traverse (tm as IVar (name, T)) = if name = from then IVar (to, T) else tm
| traverse (tm as IApp (tm1, tm2)) = let val tm1' = traverse tm1 val tm2' = traverse tm2 in if pointer_eq (tm1, tm1') andalso pointer_eq (tm2, tm2') then tm else IApp (tm1', tm2') end
| traverse (tm as IAbs (binding as (name, _), tm1)) = if name = from then
tm else letval tm1' = traverse tm1 in if pointer_eq (tm1, tm1') then tm else IAbs (binding, tm1') end in
traverse end
fun ityp_of (IConst (_, T, _)) = T
| ityp_of (IVar (_, T)) = T
| ityp_of (IApp (t1, _)) = snd (dest_funT (ityp_of t1))
| ityp_of (IAbs ((_, T), tm)) = T --> ityp_of tm
(*gets the head of a combinator application, along with the list of arguments*) fun strip_iterm_comb u = let fun stripc (IApp (t, u), ts) = stripc (t, u :: ts)
| stripc x = x in stripc (u, []) end
fun atomic_types_of T = fold_atyps (insert (op =)) T []
fun new_skolem_const_name s num_T_args =
[new_skolem_const_prefix, s, string_of_int num_T_args]
|> Long_Name.implode
val alpha_to_beta = Logic.varifyT_global \<^typ>\<open>'a => 'b\<close> val alpha_to_beta_to_alpha_to_beta = alpha_to_beta --> alpha_to_beta
fun robust_const_type thy s = if s = app_op_name then
alpha_to_beta_to_alpha_to_beta elseifString.isPrefix lam_lifted_prefix s then
alpha_to_beta else (* Old Skolems throw a "TYPE" exception here, which will be caught. *)
s |> Sign.the_const_type thy
fun ary_of (Type (\<^type_name>\<open>fun\<close>, [_, T])) = 1 + ary_of T
| ary_of _ = 0
(* This function only makes sense if "T" is as general as possible. *) fun robust_const_type_args thy (s, T) = if s = app_op_name then letval (T1, T2) = T |> domain_type |> dest_funT in [T1, T2] end elseifString.isPrefix old_skolem_const_prefix s then
[] |> Term.add_tvarsT T |> rev |> map TVar elseifString.isPrefix lam_lifted_prefix s then ifString.isPrefix lam_lifted_poly_prefix s then letval (T1, T2) = T |> dest_funT in [T1, T2] end else
[] else
(s, T) |> Sign.const_typargs thy
(* Converts an Isabelle/HOL term (with combinators) into an intermediate term. Also accumulates sort
infomation. *) fun iterm_of_term thy type_enc = let fun iot true bs ((t0 as Const (\<^const_name>\<open>Let\<close>, _)) $ t1 $ (t2 as Abs (s, T, t'))) = let val (t0', t0_atomics_Ts) = iot true bs t0 val (t1', t1_atomics_Ts) = iot true bs t1 val (t2', t2_atomics_Ts) = iot true bs t2 in
(IApp (IApp (t0', t1'), t2'),
fold (union (op =)) [t0_atomics_Ts, t1_atomics_Ts, t2_atomics_Ts] []) end
| iot true bs ((t0 as Const (\<^const_name>\<open>Let\<close>, _)) $ t1 $ t2) =
iot true bs (t0 $ t1 $ eta_expand (map (snd o snd) bs) t2 1)
| iot fool bs (P $ Q) = let val (P', P_atomics_Ts) = iot fool bs P val (Q', Q_atomics_Ts) = iot fool bs Q in (IApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end
| iot _ _ (Const (c, T)) =
(IConst (`make_fixed_const c, T, robust_const_type_args thy (c, T)),
atomic_types_of T)
| iot _ _ (Free (s, T)) = (IConst (`make_fixed_var s, T, []), atomic_types_of T)
| iot _ _ (Var (v as (s, _), T)) =
(ifString.isPrefix Meson_Clausify.new_skolem_var_prefix s then let val Ts = T |> strip_type |> swap |> op :: val s' = new_skolem_const_name s (length Ts) in IConst (`make_fixed_const s', T, Ts) end else
IVar ((make_schematic_var v, s), T), atomic_types_of T)
| iot _ bs (Bound j) =
nth bs j |> (fn (_, (name, T)) => (IConst (name, T, []), atomic_types_of T))
| iot fool bs (Abs (s, T, t)) = let fun vary s = s |> AList.defined (op =) bs s ? vary o Symbol.bump_string val s = vary s val name = `make_bound_var s val (tm, atomic_Ts) = iot fool ((s, (name, T)) :: bs) t in
(IAbs ((name, T), tm), union (op =) atomic_Ts (atomic_types_of T)) end in iot (is_type_enc_fool type_enc) end
(* "_query" and "_at" are for the ASCII-challenged Metis and Mirabelle. *) val queries = ["?", "_query"] val ats = ["@", "_at"]
fun try_unsuffixes ss s =
fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE
fun type_enc_of_string strictness s = let val (poly, s) =
(casetry (unprefix "tc_") s of
SOME s => (SOME Type_Class_Polymorphic, s)
| NONE =>
(casetry (unprefix "poly_") s of
SOME s => (SOME (Raw_Polymorphic With_Phantom_Type_Vars), s)
| NONE =>
(casetry (unprefix "ml_poly_") s of
SOME s => (SOME (Raw_Polymorphic Without_Phantom_Type_Vars), s)
| NONE =>
(casetry (unprefix "raw_mono_") s of
SOME s => (SOME Raw_Monomorphic, s)
| NONE =>
(casetry (unprefix "mono_") s of
SOME s => (SOME Mangled_Monomorphic, s)
| NONE => (NONE, s))))))
val (level, s) = case try_unsuffixes queries s of
SOME s =>
(case try_unsuffixes queries s of
SOME s => (Nonmono_Types (strictness, Non_Uniform), s)
| NONE => (Nonmono_Types (strictness, Uniform), s))
| NONE =>
(case try_unsuffixes ats s of
SOME s => (Undercover_Types, s)
| NONE => (All_Types, s))
fun native_of_string s = let val (_, s) =
(casetry (unsuffix "_arith") s of
SOME s => (true, s)
| NONE => (false, s)) val syntax = {with_ite = true, with_let = true} val (fool, core) =
(casetry (unsuffix "_fool") s of
SOME s => (With_FOOL syntax, s)
| NONE => (Without_FOOL, s)) in
(case (core, poly) of
("native", SOME poly) =>
(case (poly, level) of
(Mangled_Monomorphic, _) => if is_type_level_uniform level then
Native (First_Order, fool, Mangled_Monomorphic, level) else raise Same.SAME
| (Raw_Monomorphic, _) => raise Same.SAME
| (poly, All_Types) => Native (First_Order, fool, poly, All_Types))
| ("native_higher", SOME poly) =>
(case (poly, level) of
(_, Nonmono_Types _) => raise Same.SAME
| (_, Undercover_Types) => raise Same.SAME
| (Mangled_Monomorphic, _) => if is_type_level_uniform level then
Native (Higher_Order THF_With_Choice, With_FOOL syntax, Mangled_Monomorphic, level) else raise Same.SAME
| (poly as Raw_Polymorphic _, All_Types) =>
Native (Higher_Order THF_With_Choice, With_FOOL syntax, poly, All_Types)
| _ => raise Same.SAME)) end
fun nonnative_of_string core =
(case (core, poly, level) of
("guards", SOME poly, _) => if (poly = Mangled_Monomorphic andalso level = Undercover_Types) orelse
poly = Type_Class_Polymorphic then raise Same.SAME else
Guards (poly, level)
| ("tags", SOME poly, _) => if (poly = Mangled_Monomorphic andalso level = Undercover_Types) orelse
poly = Type_Class_Polymorphic then raise Same.SAME else
Tags (poly, level)
| ("args", SOME poly, All_Types (* naja *)) => if poly = Type_Class_Polymorphic thenraise Same.SAME else Guards (poly, Const_Types Without_Ctr_Optim)
| ("args", SOME poly, Nonmono_Types (_, Uniform) (* naja *)) => if poly = Mangled_Monomorphic orelse poly = Type_Class_Polymorphic then raise Same.SAME else
Guards (poly, Const_Types With_Ctr_Optim)
| ("erased", NONE, All_Types (* naja *)) =>
Guards (Raw_Polymorphic With_Phantom_Type_Vars, No_Types)
| _ => raise Same.SAME) in ifString.isPrefix "native" s then
native_of_string s else
nonnative_of_string s end handle Same.SAME => error ("Unknown type encoding: " ^ quote s)
fun lift_lams_part_2 ctxt type_enc (facts, lifted) =
(facts, lifted) (* Lambda-lifting sometimes leaves some lambdas around; we need some way to get rid of them *)
|> apply2 (map (introduce_combinators ctxt type_enc))
|> apply2 (map constify_lifted) (* Requires bound variables not to clash with any schematic variables (as should be the case right
after lambda-lifting). *)
|>> map (hol_open_form (unprefix hol_close_form_prefix))
||> map (hol_open_form I)
fun intentionalize_def (Const (\<^const_name>\<open>All\<close>, _) $ Abs (_, _, t)) =
intentionalize_def t
| intentionalize_def (Const (\<^const_name>\<open>HOL.eq\<close>, _) $ t $ u) = let fun lam T t = Abs (Name.uu, T, t) val (head, args) = strip_comb t ||> rev val head_T = fastype_of head val n = length args val arg_Ts = head_T |> binder_types |> take n |> rev val u = u |> subst_atomic (map_index (swap o apfst Bound) args) in HOLogic.eq_const head_T $ head $ fold lam arg_Ts u end
| intentionalize_def t = t
type ifact =
{name : string,
stature : stature,
role : atp_formula_role,
iformula : (string * string, typ, iterm, string * string) atp_formula,
atomic_types : typ list}
fun update_iformula f ({name, stature, role, iformula, atomic_types} : ifact) =
{name = name, stature = stature, role = role, iformula = f iformula, atomic_types = atomic_types}
: ifact
fun ifact_lift f ({iformula, ...} : ifact) = f iformula
fun insert_type thy get_T x xs = letval T = get_T x in ifexists (type_instance thy T o get_T) xs then xs else x :: filter_out (type_generalization thy T o get_T) xs end
fun chop_fun 0 T = ([], T)
| chop_fun n (Type (\<^type_name>\<open>fun\<close>, [dom_T, ran_T])) =
chop_fun (n - 1) ran_T |>> cons dom_T
| chop_fun _ T = ([], T)
fun filter_type_args thy ctrss type_enc s ary T_args = letval poly = polymorphism_of_type_enc type_enc in if s = type_tag_name then(* FIXME: why not "type_guard_name" as well? *)
T_args else
(case type_enc of
Native (_, _, Raw_Polymorphic _, _) => T_args
| Native (_, _, Type_Class_Polymorphic, _) => T_args
| _ => let fun gen_type_args _ _ [] = []
| gen_type_args keep strip_ty T_args = let val U = robust_const_type thy s val (binder_Us, body_U) = strip_ty U val in_U_vars = fold Term.add_tvarsT binder_Us [] val out_U_vars = Term.add_tvarsT body_U [] fun filt U_var T = if keep (member (op =) in_U_vars U_var,
member (op =) out_U_vars U_var) then
T else
dummyT val U_args = (s, U) |> robust_const_type_args thy in map2 (fn U_arg => filt (dest_TVar U_arg)) U_args T_args end handleTYPE _ => T_args fun is_always_ctr (s', T') =
s' = s andalso type_equiv thy (T', robust_const_type thy s') val noninfer_type_args = gen_type_args (not o fst) (chop_fun ary) val ctr_infer_type_args = gen_type_args fst strip_type val level = level_of_type_enc type_enc in if level = No_Types orelse s = \<^const_name>\<open>HOL.eq\<close> orelse
(case level of Const_Types _ => s = app_op_name | _ => false) then
[] elseif poly = Mangled_Monomorphic then
T_args elseif level = All_Types then
(case type_enc of
Guards _ => noninfer_type_args T_args
| Tags _ => []) elseif level = Undercover_Types then
noninfer_type_args T_args elseif level <> Const_Types Without_Ctr_Optim andalso exists (exists is_always_ctr) ctrss then
ctr_infer_type_args T_args else
T_args end) end
fun raw_atp_type_of_typ type_enc = let fun term (Type (s, Ts)) =
AType
((if s = \<^type_name>\<open>fun\<close> andalso is_type_enc_higher_order type_enc then
`I tptp_fun_type elseif s = \<^type_name>\<open>bool\<close> andalso
(is_type_enc_full_higher_order type_enc orelse is_type_enc_fool type_enc) then
`I tptp_bool_type else
`make_fixed_type_const s, []), map term Ts)
| term (TFree (s, _)) = AType ((`make_tfree s, []), [])
| term (TVar z) = AType ((tvar_name z, []), []) in term end
fun atp_type_of_type_arg type_enc T = if T = dummyT then NONE else SOME (raw_atp_type_of_typ type_enc T)
(* This shouldn't clash with anything else. *) val uncurried_alias_sep = "\000" val mangled_type_sep = "\001"
val ascii_of_uncurried_alias_sep = ascii_of uncurried_alias_sep
fun generic_mangled_type_name f (AType ((name, _), [])) = f name
| generic_mangled_type_name f (AType ((name, _), tys)) =
f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys) ^ ")"
| generic_mangled_type_name _ _ = raise Fail "unexpected type"
fun mangled_type type_enc = generic_mangled_type_name fst o raw_atp_type_of_typ type_enc
fun make_native_type s = if s = tptp_bool_type orelse s = tptp_fun_type orelse s = tptp_individual_type then s else native_type_prefix ^ ascii_of s
fun native_atp_type_of_raw_atp_type type_enc pred_sym ary = let fun to_mangled_atype ty =
AType (((make_native_type (generic_mangled_type_name fst ty),
generic_mangled_type_name snd ty), []), []) fun to_poly_atype (AType ((name, clss), tys)) = AType ((name, clss), map to_poly_atype tys)
| to_poly_atype _ = raise Fail "unexpected type" val to_atype = if is_type_enc_polymorphic type_enc then to_poly_atype else to_mangled_atype fun to_afun f1 f2 tys = AFun (f1 (hd tys), f2 (nth tys 1)) fun to_ho (ty as AType (((s, _), _), tys)) = if s = tptp_fun_type then to_afun to_ho to_ho tys else to_atype ty
| to_ho _ = raise Fail "unexpected type" fun to_lfho (ty as AType (((s, _), _), tys)) = if s = tptp_fun_type then to_afun to_ho to_lfho tys elseif pred_sym then bool_atype else to_atype ty
| to_lfho _ = raise Fail "unexpected type" fun to_fo 0 ty = if pred_sym then bool_atype else to_atype ty
| to_fo ary (AType (_, tys)) = to_afun to_atype (to_fo (ary - 1)) tys
| to_fo _ _ = raise Fail "unexpected type" in if is_type_enc_full_higher_order type_enc then to_ho elseif is_type_enc_higher_order type_enc then to_lfho else to_fo ary end
fun native_atp_type_of_typ type_enc pred_sym ary =
native_atp_type_of_raw_atp_type type_enc pred_sym ary o raw_atp_type_of_typ type_enc
(* Make atoms for sorted type variables. *) fun generic_add_sorts_on_type _ [] = I
| generic_add_sorts_on_type T (s :: ss) =
generic_add_sorts_on_type T ss
#> (if s = the_single \<^sort>\<open>type\<close> then I else insert (op =) (s, T)) fun add_sorts_on_tfree (T as TFree (_, S)) = generic_add_sorts_on_type T S
| add_sorts_on_tfree _ = I fun add_sorts_on_tvar (T as TVar (_, S)) = generic_add_sorts_on_type T S
| add_sorts_on_tvar _ = I
fun process_type_args type_enc T_args = if is_type_enc_native type_enc then
(map (native_atp_type_of_typ type_enc false 0) T_args, []) else
([], map_filter (Option.map atp_term_of_atp_type o atp_type_of_type_arg type_enc) T_args)
fun class_atom type_enc (cl, T) = let val cl = `make_class cl val (ty_args, tm_args) = process_type_args type_enc [T] val tm_args =
tm_args @
(case type_enc of
Native (_, _, Raw_Polymorphic Without_Phantom_Type_Vars, _) =>
[ATerm ((TYPE_name, ty_args), [])]
| _ => []) in AAtom (ATerm ((cl, ty_args), tm_args)) end
fun aliased_uncurried ary (s, s') =
(s ^ ascii_of_uncurried_alias_sep ^ string_of_int ary, s' ^ string_of_int ary) fun unaliased_uncurried (s, s') =
(case space_explode uncurried_alias_sep s of
[_] => (s, s')
| [s1, s2] => (s1, unsuffix s2 s')
| _ => raise Fail "ill-formed explicit application alias")
fun raw_mangled_const_name type_name ty_args (s, s') = let fun type_suffix f g =
fold_rev (prefix o g o prefix mangled_type_sep o type_name f) ty_args "" in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end fun mangled_const_name type_enc =
map_filter (atp_type_of_type_arg type_enc)
#> raw_mangled_const_name generic_mangled_type_name
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 o apfst (rpair []))) 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
fun unmangled_const_name s =
(s, s) |> unaliased_uncurried |> fst |> space_explode mangled_type_sep
fun unmangled_const s = letval ss = unmangled_const_name s in
(hd ss, map unmangled_type (tl ss)) end
val unmangled_invert_const = invert_const o hd o unmangled_const_name
fun generate_unique_name gen unique n = letval x = gen n in if unique x then x else generate_unique_name gen unique (n + 1) end
fun eta_expand_quantifier_body (tm as IAbs _) = tm
| eta_expand_quantifier_body tm = let (* We accumulate all variables because E 2.5 does not support variable shadowing. *) val vars = vars_of_iterm tm val x = generate_unique_name
(fn n => "X" ^ (if n = 0 then""else string_of_int n))
(fn name => not (exists (equal name) vars)) 0
|> `(prefix bound_var_prefix) val T = domain_type (ityp_of tm) in
IAbs ((x, T), IApp (tm, IConst (x, T, []))) end
fun introduce_builtins_and_proxies_in_iterm type_enc = let val is_fool = is_type_enc_fool type_enc val has_ite = has_type_enc_ite type_enc val has_let = has_type_enc_let type_enc val has_choice = has_type_enc_choice type_enc fun tweak_ho_quant ho_quant (T as Type (_, [p_T as Type (_, [x_T, _]), _])) [] = (* Eta-expand "!!" and "??", to work around LEO-II, Leo-III, and Satallax parser limitations. This works in conjunction with special code in "ATP_Problem" that uses the
syntactic sugar "!" and "?" whenever possible. *)
IAbs ((`I "P", p_T),
IApp (IConst (`I ho_quant, T, []),
IAbs ((`I "X", x_T),
IApp (IConst (`I "P", p_T, []),
IConst (`I "X", x_T, [])))))
| tweak_ho_quant ho_quant T _ = IConst (`I ho_quant, T, []) fun tweak_ho_equal T argc = if argc = 2 then
IConst (`I tptp_equal, T, []) else (* Eta-expand partially applied THF equality, because the LEO-II and Satallax parsers
complain about not being able to infer the type of "=". *) letval i_T = domain_type T in
IAbs ((`I "Y", i_T),
IAbs ((`I "Z", i_T),
IApp (IApp (IConst (`I tptp_equal, T, []),
IConst (`I "Y", i_T, [])),
IConst (`I "Z", i_T, [])))) end fun intro top_level args (IApp (tm1, tm2)) = let val tm1' = intro top_level (tm2 :: args) tm1 val tm2' = intro false [] tm2 val tm2'' =
(case tm1' of
IApp (IConst ((s, _), _, _), _) => if s = tptp_let then
(case tm2' of
IAbs ((name, T), tm) => let val name' =
map_prod (prefix let_bound_var_prefix o unprefix bound_var_prefix) I name in
IAbs ((name', T), alpha_rename name name' tm) end
| _ => error "Function abstraction expected") else
tm2'
| IConst ((s, _), _, _) => if s = tptp_ho_forall orelse s = tptp_ho_exists orelse s = tptp_choice then
eta_expand_quantifier_body tm2' else
tm2'
| _ => tm2') in
IApp (tm1', tm2'') end
| intro top_level args (IConst (name as (s, _), T, T_args)) = letval argc = length args in if has_ite andalso s = "c_If" andalso argc >= 3 then
IConst (`I tptp_ite, T, []) elseif has_let andalso s = "c_Let" andalso argc >= 2 then
IConst (`I tptp_let, T, []) else
(case proxify_const s of
SOME proxy_base => let fun plain_const () = IConst (name, T, []) fun proxy_const () = IConst (proxy_base |>> prefix const_prefix, T, T_args) fun handle_fool card x = if card = argc then x else proxy_const () fun handle_min_card card x = if argc < card then proxy_const () else x in if top_level then
(case s of "c_False" => IConst (`I tptp_false, T, [])
| "c_True" => IConst (`I tptp_true, T, [])
| _ => plain_const ()) elseif is_type_enc_full_higher_order type_enc then
(case s of "c_False" => IConst (`I tptp_false, T, [])
| "c_True" => IConst (`I tptp_true, T, [])
| "c_Not" => IConst (`I tptp_not, T, [])
| "c_conj" => IConst (`I tptp_and, T, [])
| "c_disj" => IConst (`I tptp_or, T, [])
| "c_implies" => IConst (`I tptp_implies, T, [])
| "c_All" => tweak_ho_quant tptp_ho_forall T args
| "c_Ex" => tweak_ho_quant tptp_ho_exists T args
| "c_Choice" => if has_choice then
handle_min_card 1 (IConst (`I tptp_choice, T, [])) else
proxy_const ()
| s => if is_tptp_equal s then
tweak_ho_equal T argc else
plain_const ()) elseif is_fool then
(case s of "c_False" => IConst (`I tptp_false, T, [])
| "c_True" => IConst (`I tptp_true, T, [])
| "c_Not" => handle_fool 1 (IConst (`I tptp_not, T, []))
| "c_conj" => handle_fool 2 (IConst (`I tptp_and, T, []))
| "c_disj" => handle_fool 2 (IConst (`I tptp_or, T, []))
| "c_implies" => handle_fool 2 (IConst (`I tptp_implies, T, []))
| "c_All" => handle_fool 1 (tweak_ho_quant tptp_ho_forall T args)
| "c_Ex" => handle_fool 1 (tweak_ho_quant tptp_ho_exists T args)
| "c_Choice" => proxy_const ()
| s => if is_tptp_equal s then
handle_fool 2 (IConst (`I tptp_equal, T, [])) else
plain_const ()) else
proxy_const () end
| NONE => if s = tptp_choice then
tweak_ho_quant tptp_choice T args else
IConst (name, T, T_args)) end
| intro _ _ (IAbs (bound, tm)) = IAbs (bound, intro false [] tm)
| intro _ _ tm = tm in intro true [] end
fun mangle_type_args_in_const type_enc (name as (s, _)) T_args = ifString.isPrefix const_prefix s andalso is_type_enc_mangling type_enc then
(mangled_const_name type_enc T_args name, []) else
(name, T_args) fun mangle_type_args_in_iterm type_enc = if is_type_enc_mangling type_enc then let fun mangle (IApp (tm1, tm2)) = IApp (mangle tm1, mangle tm2)
| mangle (tm as IConst (_, _, [])) = tm
| mangle (IConst (name, T, T_args)) =
mangle_type_args_in_const type_enc name T_args
|> (fn (name, T_args) => IConst (name, T, T_args))
| mangle (IAbs (bound, tm)) = IAbs (bound, mangle tm)
| mangle tm = tm in mangle end else
I
fun filter_type_args_in_const _ _ _ _ _ [] = []
| filter_type_args_in_const thy ctrss type_enc ary s T_args =
(case unprefix_and_unascii const_prefix s of
NONE => if level_of_type_enc type_enc = No_Types orelse s = tptp_choice then [] else T_args
| SOME s'' => filter_type_args thy ctrss type_enc (unmangled_invert_const s'') ary T_args)
fun iformula_of_prop ctxt type_enc iff_for_eq = let val thy = Proof_Context.theory_of ctxt fun do_term bs t atomic_Ts =
iterm_of_term thy type_enc bs (Envir.eta_contract t)
|>> (introduce_builtins_and_proxies_in_iterm type_enc
#> mangle_type_args_in_iterm type_enc #> AAtom)
||> union (op =) atomic_Ts fun do_quant bs q pos s T t' = let val s = singleton (Name.variant_list (map fst bs)) s val universal = Option.map (q = AExists ? not) pos val name =
s |> `(case universal of
SOME true => make_all_bound_var
| SOME false => make_exist_bound_var
| NONE => make_bound_var) in
do_formula ((s, (name, T)) :: bs) pos t'
#>> mk_aquant q [(name, SOME T)]
##> union (op =) (atomic_types_of T) end and do_conn bs c pos1 t1 pos2 t2 =
do_formula bs pos1 t1 ##>> do_formula bs pos2 t2 #>> uncurry (mk_aconn c) and do_formula bs pos t =
(case t of
\<^Const_>\<open>Trueprop for t1\<close> => do_formula bs pos t1
| \<^Const_>\<open>Not for t1\<close> => do_formula bs (Option.mapnot pos) t1 #>> mk_anot
| \<^Const_>\<open>All _ for \<open>Abs (s, T, t')\<close>\<close> => do_quant bs AForall pos s T t'
| (t0 as \<^Const_>\<open>All _\<close>) $ t1 =>
do_formula bs pos (t0 $ eta_expand (map (snd o snd) bs) t1 1)
| \<^Const_>\<open>Ex _ for \<open>Abs (s, T, t')\<close>\<close> => do_quant bs AExists pos s T t'
| (t0 as \<^Const_>\<open>Ex _\<close>) $ t1 =>
do_formula bs pos (t0 $ eta_expand (map (snd o snd) bs) t1 1)
| \<^Const_>\<open>conj for t1 t2\<close> => do_conn bs AAnd pos t1 pos t2
| \<^Const_>\<open>disj for t1 t2\<close> => do_conn bs AOr pos t1 pos t2
| \<^Const_>\<open>implies for t1 t2\<close> => do_conn bs AImplies (Option.mapnot pos) t1 pos t2
| \<^Const_>\<open>HOL.eq \<^Type>\<open>bool\<close> for t1 t2\<close> => if iff_for_eq then do_conn bs AIff NONE t1 NONE t2 else do_term bs t
| _ => do_term bs t) in do_formula [] end
fun presimplify_term simp_options ctxt t = if exists_Const (member (op =) Meson.presimplified_consts o fst) t then
t |> Skip_Proof.make_thm (Proof_Context.theory_of ctxt)
|> Meson.presimplify simp_options ctxt
|> Thm.prop_of else
t
fun preprocess_abstractions_in_terms trans_lams facts = let val (facts, lambda_ts) =
facts |> map (snd o snd) |> trans_lams
|>> map2 (fn (name, (role, _)) => fn t => (name, (role, t))) facts val lam_facts = lambda_ts
|> map_index (fn (j, t) =>
((lam_fact_prefix ^ Int.toString (j + 1), (Global, Non_Rec_Def)), (Axiom, t))) in (facts, lam_facts) end
fun presimp_prop simp_options ctxt type_enc t = let val t =
t |> Envir.beta_eta_contract
|> transform_elim_prop
|> Object_Logic.atomize_term ctxt val need_trueprop = (fastype_of t = \<^typ>\<open>bool\<close>) val is_ho = is_type_enc_full_higher_order type_enc in
t |> need_trueprop ? HOLogic.mk_Trueprop
|> (if is_ho then unextensionalize_def else cong_extensionalize_term ctxt #> abs_extensionalize_term ctxt)
|> presimplify_term simp_options ctxt
|> HOLogic.dest_Trueprop end handle TERM _ => \<^Const>\<open>True\<close>
(* Satallax prefers "=" to "<=>" (for definitions) and Metis (CNF) requires "=" for technical
reasons. *) fun should_use_iff_for_eq CNF _ = false
| should_use_iff_for_eq (THF _) format = not (is_type_enc_full_higher_order format)
| should_use_iff_for_eq _ _ = true
fun make_formula ctxt format type_enc iff_for_eq name stature role t = let val iff_for_eq = iff_for_eq andalso should_use_iff_for_eq format type_enc val (iformula, atomic_Ts) =
iformula_of_prop ctxt type_enc iff_for_eq (SOME (role <> Conjecture)) t []
|>> close_universally add_iterm_vars in
{name = name, stature = stature, role = role, iformula = iformula, atomic_types = atomic_Ts} end
fun make_fact ctxt format type_enc iff_for_eq ((name, stature), t) =
(case make_formula ctxt format type_enc iff_for_eq name stature Axiom t of
formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} => if s = tptp_true then NONE else SOME formula
| formula => SOME formula)
fun make_conjecture ctxt format type_enc = map (fn ((name, stature), (role, t)) => letval t' = t |> role = Conjecture ? s_not in
make_formula ctxt format type_enc true name stature role t' end)
(** Finite and infinite type inference **)
fun tvar_footprint thy s ary =
(case unprefix_and_unascii const_prefix s of
SOME s => letfun tvars_of T = [] |> Term.add_tvarsT T |> map fst in
s |> unmangled_invert_const |> robust_const_type thy |> chop_fun ary |> fst |> map tvars_of end
| NONE => []) handleTYPE _ => []
fun type_arg_cover thy pos s ary = if is_tptp_equal s then if pos = SOME falsethen [] else 0 upto ary - 1 else let val footprint = tvar_footprint thy s ary val eq = (s = \<^const_name>\<open>HOL.eq\<close>) fun cover _ [] = []
| cover seen ((i, tvars) :: args) =
cover (union (op =) seen tvars) args
|> (eq orelse exists (fn tvar => not (member (op =) seen tvar)) tvars)
? cons i in if forall null footprint then
[] else
map_index I footprint
|> sort (rev_order o list_ord Term_Ord.indexname_ord o apply2 snd)
|> cover [] end
type monotonicity_info =
{maybe_finite_Ts : typ list,
surely_infinite_Ts : typ list,
maybe_nonmono_Ts : typ list}
(* These types witness that the type classes they belong to allow infinite
models and hence that any types with these type classes is monotonic. *) val known_infinite_types = [\<^typ>\<open>nat\<close>, HOLogic.intT, HOLogic.realT, \<^typ>\<open>nat => bool\<close>]
fun is_type_kind_of_surely_infinite ctxt strictness cached_Ts T =
strictness <> Strict andalso is_type_surely_infinite ctxt true cached_Ts T
(* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are dangerous because their "exhaust" properties can easily lead to unsound ATP proofs. On the other hand, all HOL infinite types can be given the same
models in first-order logic (via Loewenheim-Skolem). *)
fun add_iterm_syms_to_sym_table ctxt app_op_level conj_fact = let val thy = Proof_Context.theory_of ctxt fun consider_var_ary const_T var_T max_ary = let fun iter ary T = if ary = max_ary orelse type_instance thy var_T T orelse type_instance thy T var_T orelse not (can dest_funT T) then
ary else
iter (ary + 1) (range_type T) in iter 0 const_T end fun add_universal_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) = if (app_op_level = Sufficient_App_Op andalso can dest_funT T) orelse
(app_op_level = Sufficient_App_Op_And_Predicator andalso
(can dest_funT T orelse T = \<^typ>\<open>bool\<close>)) then let val bool_vars' =
bool_vars orelse
(app_op_level = Sufficient_App_Op_And_Predicator andalso body_type T = \<^typ>\<open>bool\<close>) fun repair_min_ary {pred_sym, min_ary, max_ary, types, in_conj} =
{pred_sym = pred_sym andalso not bool_vars',
min_ary = fold (fn T' => consider_var_ary T' T) types min_ary,
max_ary = max_ary, types = types, in_conj = in_conj} val fun_var_Ts' = fun_var_Ts |> can dest_funT T ? insert_type thy I T in if bool_vars' = bool_vars andalso fun_var_Ts' = fun_var_Ts then accum else ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_ary) sym_tab) end else
accum fun add_iterm_syms top_level tm (accum as ((bool_vars, fun_var_Ts), sym_tab)) = letval (head, args) = strip_iterm_comb tm in
(case head of
IConst ((s, _), T, _) => if is_maybe_universal_name s orelse String.isPrefix let_bound_var_prefix s then
add_universal_var T accum elseifString.isPrefix exist_bound_var_prefix s then
accum else letval ary = length args in
((bool_vars, fun_var_Ts),
(case Symtab.lookup sym_tab s of
SOME {pred_sym, min_ary, max_ary, types, in_conj} => let val pred_sym = pred_sym andalso top_level andalso not bool_vars val types' = types |> insert_type thy I T val in_conj = in_conj orelse conj_fact val min_ary = if (app_op_level = Sufficient_App_Op orelse
app_op_level = Sufficient_App_Op_And_Predicator)
andalso types' <> types then
fold (consider_var_ary T) fun_var_Ts min_ary else
min_ary in
Symtab.update (s, {pred_sym = pred_sym, min_ary = Int.min (ary, min_ary),
max_ary = Int.max (ary, max_ary), types = types', in_conj = in_conj}) sym_tab end
| NONE => let val max_ary =
(case unprefix_and_unascii const_prefix s of
SOME s =>
(ifString.isSubstring uncurried_alias_sep s then
ary else
(casetry (ary_of o robust_const_type thy o unmangled_invert_const) s of
SOME ary0 => Int.min (ary0, ary)
| NONE => ary))
| NONE => ary) val pred_sym = top_level andalso max_ary = ary andalso not bool_vars val min_ary =
(case app_op_level of
Min_App_Op => max_ary
| Full_App_Op_And_Predicator => 0
| _ => fold (consider_var_ary T) fun_var_Ts max_ary) in
Symtab.update_new (s, {pred_sym = pred_sym, min_ary = min_ary, max_ary = max_ary,
types = [T], in_conj = conj_fact}) sym_tab end)) end
| IVar (_, T) => add_universal_var T accum
| IAbs ((_, T), tm) => accum |> add_universal_var T |> add_iterm_syms false tm
| _ => accum)
|> fold (add_iterm_syms false) args end in add_iterm_syms end
fun sym_table_of_facts ctxt type_enc app_op_level conjs facts = let fun add_iterm_syms conj_fact = add_iterm_syms_to_sym_table ctxt app_op_level conj_fact true fun add_fact_syms conj_fact = ifact_lift (formula_fold NONE (K (add_iterm_syms conj_fact))) in
((false, []), Symtab.empty)
|> fold (add_fact_syms true) conjs
|> fold (add_fact_syms false) facts
||> fold Symtab.update (default_sym_tab_entries type_enc) end
fun min_ary_of sym_tab s =
(case Symtab.lookup sym_tab s of
SOME ({min_ary, ...} : sym_info) => min_ary
| NONE =>
(case unprefix_and_unascii const_prefix s of
SOME s => letval s = s |> unmangled_invert_const in if s = predicator_name then 1 elseif s = app_op_name then 2 elseif s = type_guard_name 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 fTrue_iconst = IConst ((const_prefix ^ "fTrue", \<^const_name>\<open>fTrue\<close>), \<^typ>\<open>bool\<close>, []) val predicator_iconst = IConst (`make_fixed_const predicator_name, \<^typ>\<open>bool => bool\<close>, [])
fun predicatify completish tm = if completish > 1 then
IApp (IApp (IConst (`I tptp_equal, \<^typ>\<open>bool => bool => bool\<close>, []), tm), fTrue_iconst) else
IApp (predicator_iconst, tm)
val app_op = `make_fixed_const app_op_name
fun list_app head args = fold (curry (IApp o swap)) args head
fun mk_app_op type_enc head arg = let val head_T = ityp_of head val (arg_T, res_T) = dest_funT head_T valapp =
IConst (app_op, head_T --> head_T, [arg_T, res_T])
|> mangle_type_args_in_iterm type_enc in list_app app [head, arg] end
fun firstorderize_fact thy ctrss type_enc uncurried_aliases completish sym_tab = let fun do_app arg head = mk_app_op type_enc head arg fun list_app_ops (head, args) = fold do_app args head fun introduce_app_ops tm = letval (head, args) = tm |> strip_iterm_comb ||> map introduce_app_ops in
(case head of
IConst (name as (s, _), T, T_args) => let val min_ary = min_ary_of sym_tab s val ary = if uncurried_aliases andalso String.isPrefix const_prefix s then let val ary = length args (* In polymorphic native type encodings, it is impossible to declare a fully polymorphic symbol that takes more arguments than its signature (even though such concrete instances, where a type variable is instantiated by a
function type, are possible.) *) val official_ary = if is_type_enc_polymorphic type_enc then
(case unprefix_and_unascii const_prefix s of
SOME s' =>
(casetry (ary_of o robust_const_type thy) (invert_const s') of
SOME ary => ary
| NONE => min_ary)
| NONE => min_ary) else
1000000000 (* irrealistically big arity *) in Int.min (ary, official_ary) end else
min_ary val head = if ary = min_ary then head else IConst (aliased_uncurried ary name, T, T_args) in
args |> chop ary |>> list_app head |> list_app_ops end
| IAbs ((name, T), tm) =>
list_app_ops (IAbs ((name, T), introduce_app_ops tm), args)
| _ => list_app_ops (head, args)) end fun introduce_predicators tm =
(case strip_iterm_comb tm of
(IConst ((s, _), _, _), _) => if is_pred_sym sym_tab s then tm else predicatify completish tm
| _ => predicatify completish tm) val is_ho = is_type_enc_higher_order type_enc val is_full_ho = is_type_enc_full_higher_order type_enc val is_fool = is_type_enc_fool type_enc val do_iterm =
(not is_ho ? introduce_app_ops)
#> (not (is_full_ho orelse is_fool) ? introduce_predicators)
#> filter_type_args_in_iterm thy ctrss type_enc in update_iformula (formula_map do_iterm) end
(** Helper facts **)
val not_ffalse = @{lemma "\<not> fFalse" by (unfold fFalse_def) fast} val ftrue = @{lemma "fTrue" by (unfold fTrue_def) fast}
(* The Boolean indicates that a fairly sound type encoding is needed. *) fun helper_table with_combs =
(if with_combs then
[(("COMBI", false), [(Non_Rec_Def, @{thm Meson.COMBI_def})]),
(("COMBK", false), [(Non_Rec_Def, @{thm Meson.COMBK_def})]),
(("COMBB", false), [(Non_Rec_Def, @{thm Meson.COMBB_def})]),
(("COMBC", false), [(Non_Rec_Def, @{thm Meson.COMBC_def})]),
(("COMBS", false), [(Non_Rec_Def, @{thm Meson.COMBS_def})])] else
[]) @
[((predicator_name, false), [(General, not_ffalse), (General, ftrue)]),
(("fFalse", false), [(General, not_ffalse)]),
(("fFalse", true), [(General, @{thm True_or_False})]),
(("fTrue", false), [(General, ftrue)]),
(("fTrue", true), [(General, @{thm True_or_False})]),
(("If", true),
[(Non_Rec_Def, @{thm if_True}), (Non_Rec_Def, @{thm if_False}),
(General, @{thm True_or_False})]),
(("fNot", false),
@{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]
fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}
|> map (pair Non_Rec_Def)),
(("fconj", false),
@{lemma "\<not> P \<or> \<not> Q \<or> fconj P Q""\<not> fconj P Q \<or> P""\<not> fconj P Q \<or> Q" by (unfold fconj_def) fast+}
|> map (pair General)),
(("fdisj", false),
@{lemma "\<not> P \<or> fdisj P Q""\<not> Q \<or> fdisj P Q""\<not> fdisj P Q \<or> P \<or> Q" by (unfold fdisj_def) fast+}
|> map (pair General)),
(("fimplies", false),
@{lemma "P \<or> fimplies P Q""\<not> Q \<or> fimplies P Q""\<not> fimplies P Q \<or> \<not> P \<or> Q"
by (unfold fimplies_def) fast+}
|> map (pair General)),
(("fequal", true), (* This is a lie: Higher-order equality doesn't need a sound type encoding. However, this is done so for backward compatibility: Including the
equality helpers by default in Metis breaks a few existing proofs. *)
@{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]
fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}
|> map (pair General)), (* Partial characterization of "fAll" and "fEx". A complete characterization
would require the axiom of choice for replay with Metis. *)
(("fAll", false), [(General, @{lemma "\<not> fAll P \<or> P x" by (auto simp: fAll_def)})]),
(("fEx", false), [(General, @{lemma "\<not> P x \<or> fEx P" by (auto simp: fEx_def)})]),
(("fChoice", true), [(General, @{thm fChoice_iff_Ex})])]
|> map (apsnd (map (apsnd zero_var_indexes)))
val () = let fun is_skolemizable \<^Const_>\<open>Ex _ for \<open>Abs _\<close>\<close> = true
| is_skolemizable _ = false
fun check_no_skolemizable_thm thm = if Term.exists_subterm is_skolemizable (Thm.full_prop_of thm) then
error "Theorems of the helper table cannot contain skolemizable terms because they don't \
\get skolimized in metis." else
() in
helper_table true
|> List.app (fn (_, thms) => List.app (check_no_skolemizable_thm o snd) thms) end
val helper_rank = default_rank val min_rank = 9 * helper_rank div 10 val max_rank = 4 * min_rank
fun rank_of_fact_num n j = min_rank + (max_rank - min_rank) * j div n
val type_tag = `make_fixed_const type_tag_name
fun could_specialize_helpers type_enc = not (is_type_enc_polymorphic type_enc) andalso level_of_type_enc type_enc <> No_Types
fun should_specialize_helper type_enc t =
could_specialize_helpers type_enc andalso not (null (Term.hidden_polymorphism t))
fun add_helper_facts_of_sym ctxt format type_enc lam_trans completish (s, {types, ...} : sym_info) =
(case unprefix_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 |> hd fun dub needs_sound j k =
ascii_of unmangled_s ^ "_" ^ string_of_int j ^ "_" ^ string_of_int k ^
(if mangled_s = unmangled_s then""else"_" ^ ascii_of mangled_s) ^
(if needs_sound then typed_helper_suffix else untyped_helper_suffix) fun specialize_helper t T = if unmangled_s = app_op_name then letval tyenv = Sign.typ_match thy (alpha_to_beta, domain_type T) Vartab.empty in
Envir.subst_term_types tyenv t end else
specialize_type thy (invert_const unmangled_s, T) t fun dub_and_inst needs_sound (j, (status, t)) =
(if should_specialize_helper type_enc t then
map_filter (try (specialize_helper t)) types else
[t])
|> map_index (fn (k, t) => ((dub needs_sound j (k + 1), (Global, status)), t)) fun make_facts type_enc = map_filter (make_fact ctxt format type_enc false) val sound = is_type_enc_sound type_enc val could_specialize = could_specialize_helpers type_enc val with_combs = lam_trans <> opaque_combsN in
fold (fn ((helper_s, needs_sound), ths) => let fun map_syntax f (Native (order, With_FOOL syntax, polymorphism, type_level)) =
Native (order, With_FOOL (f syntax), polymorphism, type_level)
| map_syntax _ type_enc = type_enc val remove_ite_syntax = map_syntax
(fn {with_let, ...} => {with_ite = false, with_let = with_let}) in if (needs_sound andalso not sound) orelse
(helper_s <> unmangled_s andalso
(completish < 3 orelse could_specialize)) then
I else
ths
|> map_index (apfst (curry op+ 1))
|> maps (dub_and_inst needs_sound o apsnd (apsnd Thm.prop_of))
|> make_facts ((helper_s = "If" ? remove_ite_syntax) type_enc)
|> union (op = o apply2 #iformula) end)
((if completish >= 3 then completish_helper_table else helper_table) with_combs) end
| NONE => I) fun helper_facts_of_sym_table ctxt format type_enc lam_trans completish sym_tab =
Symtab.fold_rev (add_helper_facts_of_sym ctxt format type_enc lam_trans completish) sym_tab []
(***************************************************************) (* Type Classes Present in the Axiom or Conjecture Clauses *) (***************************************************************)
fun set_insert (x, s) = Symtab.update (x, ()) s
fun add_classes (cls, cset) = List.foldl set_insert cset (flat cls)
fun classes_of_terms get_Ts = map (map snd o get_Ts)
#> List.foldl add_classes Symtab.empty #> Symtab.delete_safe class_of_types
#> Symtab.keys
val tfree_classes_of_terms = classes_of_terms Misc_Legacy.term_tfrees val tvar_classes_of_terms = classes_of_terms Misc_Legacy.term_tvars
fun fold_type_ctrs f (Type (s, Ts)) x = fold (fold_type_ctrs f) Ts (f (s, x))
| fold_type_ctrs _ _ x = x
(* Type constructors used to instantiate overloaded constants are the only ones
needed. *) fun add_type_ctrs_in_term thy = let fun add (Const (\<^const_name>\<open>Meson.skolem\<close>, _) $ _) = I
| add (t $ u) = add t #> add u
| add (Const x) =
x |> robust_const_type_args thy |> fold (fold_type_ctrs set_insert)
| add (Abs (_, _, u)) = add u
| add _ = I in add end
fun trans_lams_of_string ctxt type_enc lam_trans = if lam_trans = no_lamsN then
rpair [] elseif lam_trans = opaque_liftingN then
lift_lams ctxt type_enc ##> K [] elseif lam_trans = liftingN then
lift_lams ctxt type_enc elseif lam_trans = opaque_combsN orelse lam_trans = combsN then map (introduce_combinators ctxt type_enc) #> rpair [] elseif lam_trans = combs_and_liftingN then
lift_lams_part_1 ctxt type_enc
##> maps (fn t => [t, introduce_combinators ctxt type_enc (intentionalize_def t)])
#> lift_lams_part_2 ctxt type_enc elseif lam_trans = combs_or_liftingN then
lift_lams_part_1 ctxt type_enc
##> map (fn t => (case head_of (strip_qnt_body \<^const_name>\<open>All\<close> t) of
\<^term>\<open>(=) ::bool => bool => bool\<close> => t
| _ => introduce_combinators ctxt type_enc (intentionalize_def t)))
#> lift_lams_part_2 ctxt type_enc elseif lam_trans = keep_lamsN then map (Envir.eta_contract) #> rpair [] else
error ("Unknown lambda translation scheme: " ^ quote lam_trans)
val pull_and_reorder_definitions = let fun add_consts (IApp (t, u)) = fold add_consts [t, u]
| add_consts (IAbs (_, t)) = add_consts t
| add_consts (IConst (name, _, _)) = insert (op =) name
| add_consts (IVar _) = I fun consts_of_hs l_or_r ({iformula, ...} : ifact) =
(case iformula of
AAtom (IApp (IApp (IConst _, t), u)) => add_consts (l_or_r (t, u)) []
| _ => []) (* Quadratic, but usually OK. *) fun reorder [] [] = []
| reorder (fact :: skipped) [] =
fact :: reorder [] skipped (* break cycle *)
| reorder skipped (fact :: facts) = letval rhs_consts = consts_of_hs snd fact in ifexists (exists (exists (member (op =) rhs_consts)
o consts_of_hs fst)) [skipped, facts] then
reorder (fact :: skipped) facts else
fact :: reorder [] (facts @ skipped) end inList.partition (curry (op =) Definition o #role) #>> reorder [] #> op @ end
fun s_not_prop \<^Const_>\<open>Trueprop for t\<close> = \<^Const>\<open>Trueprop for \<open>s_not t\<close>\<close>
| s_not_prop \<^Const_>\<open>Pure.imp for t \<^prop>\<open>False\<close>\<close> = t
| s_not_prop t = \<^Const>\<open>Pure.imp for t \<^prop>\<open>False\<close>\<close>
fun translate_formulas simp_options ctxt prem_role format type_enc lam_trans presimp hyp_ts concl_t
facts = let val thy = Proof_Context.theory_of ctxt val trans_lams = trans_lams_of_string ctxt type_enc lam_trans val fact_ts = facts |> map snd (* Remove existing facts from the conjecture, as this can dramatically boost an ATP's
performance (for some reason). *) val hyp_ts = hyp_ts |> map (fn t => if member (op aconv) fact_ts t then \<^prop>\<open>True\<close> else t)
val maybe_presimp_prop = presimp ? presimp_prop simp_options ctxt type_enc
val facts = facts |> map (apsnd (pair Axiom o maybe_presimp_prop)) val conjs = map (pair prem_role) hyp_ts @ [(Conjecture, s_not_prop concl_t)]
|> map_index (map_prod (rpair (Local, General) o string_of_int) (apsnd maybe_presimp_prop)) val ((conjs, facts), lam_facts) =
(conjs, facts)
|> (if lam_trans = no_lamsN then
rpair [] else
op @
#> preprocess_abstractions_in_terms trans_lams
#>> chop (length conjs)) val conjs =
conjs |> make_conjecture ctxt format type_enc
|> pull_and_reorder_definitions val facts =
facts |> map_filter (fn (name, (_, t)) => make_fact ctxt format type_enc true (name, t))
|> pull_and_reorder_definitions val lifted = lam_facts |> map (extract_lambda_def dest_Const o snd o snd) val lam_facts = lam_facts |> map_filter (make_fact ctxt format type_enc true o apsnd snd) val all_ts = concl_t :: hyp_ts @ fact_ts val subs = tfree_classes_of_terms all_ts val supers = tvar_classes_of_terms all_ts val tycons = type_ctrs_of_terms thy all_ts val (supers, tcon_clauses) = if level_of_type_enc type_enc = No_Types then ([], []) else make_tcon_clauses thy tycons supers val subclass_pairs = make_subclass_pairs thy subs supers in
(union (op =) subs supers, conjs, facts @ lam_facts, subclass_pairs, tcon_clauses, lifted) end
val type_guard = `make_fixed_const type_guard_name
fun type_guard_iterm type_enc T tm =
IApp (IConst (type_guard, T --> \<^typ>\<open>bool\<close>, [T])
|> mangle_type_args_in_iterm type_enc, tm)
fun is_var_undercover_in_term thy name pos tm accum =
accum orelse let val var = ATerm ((name, []), []) fun is_undercover (ATerm (_, [])) = false
| is_undercover (ATerm (((s, _), _), tms)) = let val ary = length tms val cover = type_arg_cover thy pos s ary in exists (fn (j, tm) => tm = var andalso member (op =) cover j) (map_index I tms) orelse exists is_undercover tms end
| is_undercover _ = true in is_undercover tm end
(* Each fact is given a unique fact number to avoid name clashes (e.g., because of monomorphization). The TPTP forbids name clashes, and some of the remote
provers might care. *) fun line_of_fact ctxt generate_isabelle_info prefix encode alt freshen pos mono type_enc rank
(j, {name, stature = (_, status), role, iformula, atomic_types}) =
Formula ((prefix ^ (if freshen then string_of_int j ^ "_"else"") ^
encode name, alt name),
role,
iformula
|> formula_of_iformula ctxt mono type_enc
should_guard_var_in_formula (if pos then SOME trueelse NONE)
|> close_formula_universally
|> bound_tvars type_enc true atomic_types,
NONE, isabelle_info generate_isabelle_info (string_of_status status) (rank j))
fun lines_of_subclass generate_isabelle_info type_enc sub super =
Formula ((subclass_prefix ^ ascii_of sub ^ "___" ^ ascii_of super, ""), Axiom,
AConn (AImplies,
[sub, super] |> map (fn s => class_atom type_enc (s, tvar_a)))
|> bound_tvars type_enc false [tvar_a],
NONE, isabelle_info generate_isabelle_info inductiveN helper_rank)
fun lines_of_subclass_pair generate_isabelle_info type_enc (sub, supers) = if polymorphism_of_type_enc type_enc = Type_Class_Polymorphic then
[Class_Decl (class_decl_prefix ^ ascii_of sub, `make_class sub, map (`make_class) supers)] else map (lines_of_subclass generate_isabelle_info type_enc sub) supers
fun lines_of_free_types type_enc (facts : ifact list) = if is_type_enc_polymorphic type_enc then let val type_classes = (polymorphism_of_type_enc type_enc = Type_Class_Polymorphic) fun line (j, (cl, T)) = if type_classes then
Class_Memb (class_memb_prefix ^ string_of_int j, [],
native_atp_type_of_typ type_enc false 0 T, `make_class cl) else
Formula ((tfree_clause_prefix ^ string_of_int j, ""), Hypothesis,
class_atom type_enc (cl, T), NONE, []) val membs =
fold (union (op =)) (map #atomic_types facts) []
|> class_membs_of_types type_enc add_sorts_on_tfree in
map_index line membs end else
[]
(** Symbol declarations **)
fun decl_line_of_class phantoms s = letval name as (s, _) = `make_class s in
Sym_Decl (sym_decl_prefix ^ s, name,
APi ([tvar_a_name], if phantoms = Without_Phantom_Type_Vars then
AFun (a_itself_atype, bool_atype) else
bool_atype)) end
fun decl_lines_of_classes type_enc =
(case type_enc of
Native (_, _, Raw_Polymorphic phantoms, _) => map (decl_line_of_class phantoms)
| _ => K [])
fun sym_decl_table_of_facts thy type_enc sym_tab (conjs, facts, extra_tms) = let fun add_iterm_syms tm = letval (head, args) = strip_iterm_comb tm in
(case head of
IConst ((s, s'), T, T_args) => let val (pred_sym, in_conj) =
(case Symtab.lookup sym_tab s of
SOME ({pred_sym, in_conj, ...} : sym_info) => (pred_sym, in_conj)
| NONE => (false, false)) val decl_sym =
(case type_enc of Guards _ => not pred_sym | _ => true) andalso not (String.isPrefix let_bound_var_prefix s) andalso
is_tptp_user_symbol s in if decl_sym then
Symtab.map_default (s, [])
(insert_type thy #3 (s', T_args, T, pred_sym, length args, in_conj)) else
I end
| IAbs (_, tm) => add_iterm_syms tm
| _ => I)
#> fold add_iterm_syms args end val add_fact_syms = ifact_lift (formula_fold NONE (K add_iterm_syms)) fun add_formula_var_types (ATyQuant (_, _, phi)) = add_formula_var_types phi
| add_formula_var_types (AQuant (_, xs, phi)) =
fold (fn (_, SOME T) => insert_type thy I T | _ => I) xs
#> add_formula_var_types phi
| add_formula_var_types (AConn (_, phis)) =
fold add_formula_var_types phis
| add_formula_var_types _ = I fun var_types () = if is_type_enc_polymorphic type_enc then [tvar_a] else fold (ifact_lift add_formula_var_types) (conjs @ facts) [] (* Don't add declaration for undefined_bool as bool already has fTrue and fFalse als witnesses and this declaration causes problems in FOF if undefined_bool occurs without predicator pp.
*) fun add_undefined_const \<^Type>\<open>bool\<close> = I
| add_undefined_const T = let val name = `make_fixed_const \<^const_name>\<open>undefined\<close> val ((s, s'), Ts) = if is_type_enc_mangling type_enc then
(mangled_const_name type_enc [T] name, []) else
(name, [T]) in
Symtab.map_default (s, []) (insert_type thy #3 (s', Ts, T, false, 0, false)) end fun add_TYPE_const () = letval (s, s') = TYPE_name in
Symtab.map_default (s, [])
(insert_type thy #3
(s', [tvar_a], \<^typ>\<open>'a itself\<close>, false, 0, false)) end in
Symtab.empty
|> is_type_enc_sound type_enc
? (fold (fold add_fact_syms) [conjs, facts]
#> fold add_iterm_syms extra_tms
#> (case type_enc of
Native (_, _, Raw_Polymorphic phantoms, _) =>
phantoms = Without_Phantom_Type_Vars ? add_TYPE_const ()
| Native _ => I
| _ => (* Add constants "undefined" as witnesses that the types are inhabited. *)
fold add_undefined_const (var_types ()))) end
(* We add "bool" in case the helper "True_or_False" is included later. *) fun default_mono level completish =
{maybe_finite_Ts = [\<^typ>\<open>bool\<close>],
surely_infinite_Ts = (case level of Nonmono_Types (Strict, _) => [] | _ => known_infinite_types),
maybe_nonmono_Ts = [if completish >= 3 then tvar_a else \<^typ>\<open>bool\<close>]}
(* This inference is described in section 4 of Blanchette et al., "Encoding
monomorphic and polymorphic types", TACAS 2013. *) fun add_iterm_mononotonicity_info ctxt level polarity tm
(mono as {maybe_finite_Ts, surely_infinite_Ts, maybe_nonmono_Ts}) = let val thy = Proof_Context.theory_of ctxt
fun update_mono T mono =
(case level of
Nonmono_Types (strictness, _) => ifexists (type_instance thy T) surely_infinite_Ts orelse
member (type_equiv thy) maybe_finite_Ts T then
mono elseif is_type_kind_of_surely_infinite ctxt strictness
surely_infinite_Ts T then
{maybe_finite_Ts = maybe_finite_Ts,
surely_infinite_Ts = surely_infinite_Ts |> insert_type thy I T,
maybe_nonmono_Ts = maybe_nonmono_Ts} else
{maybe_finite_Ts = maybe_finite_Ts |> insert (type_equiv thy) T,
surely_infinite_Ts = surely_infinite_Ts,
maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type thy I T}
| _ => mono)
fun update_mono_rec (IConst ((_, s'), Type (_, [T, _]), _)) = ifString.isPrefix \<^const_name>\<open>fequal\<close> s' then update_mono T else I
| update_mono_rec (IApp (tm1, tm2)) = fold update_mono_rec [tm1, tm2]
| update_mono_rec (IAbs (_, tm)) = update_mono_rec tm
| update_mono_rec _ = I in
mono |>
(case tm of
IApp (IApp (IConst ((s, _), Type (_, [T, _]), _), tm1), tm2) =>
((polarity <> SOME false andalso is_tptp_equal s
andalso exists is_maybe_universal_var [tm1, tm2])
? update_mono T)
#> fold update_mono_rec [tm1, tm2]
| _ => update_mono_rec tm) end fun add_fact_mononotonicity_info ctxt level ({role, iformula, ...} : ifact) =
formula_fold (SOME (role <> Conjecture)) (add_iterm_mononotonicity_info ctxt level) iformula fun mononotonicity_info_of_facts ctxt type_enc completish facts = letval level = level_of_type_enc type_enc in
default_mono level completish
|> is_type_level_monotonicity_based level
? fold (add_fact_mononotonicity_info ctxt level) facts end
fun fold_arg_types f (IApp (tm1, tm2)) =
fold_arg_types f tm1 #> fold_term_types f tm2
| fold_arg_types _ _ = I and fold_term_types f tm = f (ityp_of tm) #> fold_arg_types f tm
fun add_iformula_monotonic_types ctxt mono type_enc = let val thy = Proof_Context.theory_of ctxt val level = level_of_type_enc type_enc val should_encode = should_encode_type ctxt mono level fun add_type T = not (should_encode T) ? insert_type thy I T in formula_fold NONE (K (fold_term_types add_type)) end
fun decl_line_of_sym ctxt type_enc s (s', T_args, T, pred_sym, ary, _) = let val thy = Proof_Context.theory_of ctxt val (T, T_args) = if null T_args then
(T, []) else
(case unprefix_and_unascii const_prefix s of
SOME s' => let val s' = s' |> unmangled_invert_const val T = s' |> robust_const_type thy in (T, robust_const_type_args thy (s', T)) end
| NONE => raise Fail "unexpected type arguments") in
Sym_Decl (sym_decl_prefix ^ s, (s, s'),
T |> native_atp_type_of_typ type_enc pred_sym ary
|> not (null T_args) ? curry APi (map (tvar_name o dest_TVar) T_args)) end
fun honor_conj_sym_role in_conj = (if in_conj then Hypothesis else Axiom, I)
fun line_of_guards_sym_decl ctxt generate_isabelle_info mono type_enc n s j
(s', T_args, T, _, ary, in_conj) = let val thy = Proof_Context.theory_of ctxt val (role, maybe_negate) = honor_conj_sym_role in_conj val (arg_Ts, res_T) = chop_fun ary T val bound_names = 1 upto ary |> map (`I o make_bound_var o string_of_int) val bounds = map2 (fn name => fn T => IConst (name, T, [])) bound_names arg_Ts val bound_Ts =
(case level_of_type_enc type_enc of
All_Types => if null T_args then replicate ary NONE elsemap SOME arg_Ts
| Undercover_Types => letval cover = type_arg_cover thy NONE s ary in
map_index (fn (j, arg_T) => if member (op =) cover j then SOME arg_T else NONE) arg_Ts end
| _ => replicate ary NONE) in
Formula ((guards_sym_formula_prefix ^ s ^
(if n > 1 then"_" ^ string_of_int j else""), ""),
role,
IConst ((s, s'), T, T_args)
|> fold (curry (IApp o swap)) bounds
|> type_guard_iterm type_enc res_T
|> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
|> formula_of_iformula ctxt mono type_enc
always_guard_var_in_formula (SOME true)
|> close_formula_universally
|> bound_tvars type_enc (n > 1) (atomic_types_of T)
|> maybe_negate,
NONE, isabelle_info generate_isabelle_info inductiveN helper_rank) end
fun lines_of_tags_sym_decl ctxt generate_isabelle_info mono type_enc n s
(j, (s', T_args, T, pred_sym, ary, in_conj)) = let val thy = Proof_Context.theory_of ctxt val level = level_of_type_enc type_enc val ident =
tags_sym_formula_prefix ^ s ^
(if n > 1 then"_" ^ string_of_int j else"") val (role, maybe_negate) = honor_conj_sym_role in_conj val (arg_Ts, res_T) = chop_fun ary T val bound_names = 1 upto ary |> map (`I o make_bound_var o string_of_int) val bounds = bound_names |> map (fn name => ATerm ((name, []), [])) val cst = mk_aterm type_enc (s, s') T_args val eq = maybe_negate oo eq_formula type_enc (atomic_types_of T) [] pred_sym val tag_with = tag_with_type ctxt mono type_enc NONE fun formula c =
[Formula ((ident, ""), role, eq (tag_with res_T c) c, NONE,
isabelle_info generate_isabelle_info non_rec_defN helper_rank)] in if pred_sym orelse not (should_encode_type ctxt mono level res_T) then
[] elseif level = Undercover_Types then let val cover = type_arg_cover thy NONE s ary fun maybe_tag (j, arg_T) = member (op =) cover j ? tag_with arg_T val bounds = bounds |> map2 maybe_tag (map_index I arg_Ts) in formula (cst bounds) end else
formula (cst bounds) end
fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd
fun rationalize_decls thy (decls as decl :: (decls' as _ :: _)) = let val T = result_type_of_decl decl |> map_type_tvar (fn (z, _) => TVar (z, \<^sort>\<open>type\<close>)) in if forall (type_generalization thy T o result_type_of_decl) decls' then [decl] else decls end
| rationalize_decls _ decls = decls
fun lines_of_sym_decls ctxt generate_isabelle_info mono type_enc (s, decls) =
(case type_enc of
Native _ => [decl_line_of_sym ctxt type_enc s (hd decls)]
| Guards (_, level) => let val thy = Proof_Context.theory_of ctxt val decls = decls |> rationalize_decls thy val n = length decls val decls = decls |> filter (should_encode_type ctxt mono level o result_type_of_decl) in
map_index (uncurry (line_of_guards_sym_decl ctxt generate_isabelle_info mono type_enc n s)) decls end
| Tags (_, level) => if is_type_level_uniform level then
[] else letval n = length decls in
map_index I decls
|> maps (lines_of_tags_sym_decl ctxt generate_isabelle_info mono type_enc n s) end)
fun lines_of_sym_decl_table ctxt generate_isabelle_info mono type_enc mono_Ts sym_decl_tab = let val syms = sym_decl_tab |> Symtab.dest |> sort_by fst val mono_lines = lines_of_mono_types ctxt generate_isabelle_info mono type_enc mono_Ts val decl_lines = maps (lines_of_sym_decls ctxt generate_isabelle_info mono type_enc) syms in mono_lines @ decl_lines end
fun datatypes_of_sym_table ctxt ctrss (DFG Polymorphic) (type_enc as Native _) uncurried_aliases
sym_tab = if is_type_enc_polymorphic type_enc then let val thy = Proof_Context.theory_of ctxt
fun do_ctr (s, T) = let val s' = make_fixed_const s val ary = ary_of T fun mk name = SOME (mk_aterm type_enc name (robust_const_type_args thy (s, T)) []) in if T = HOLogic.boolT then
(case proxify_const s' of
SOME proxy_base => mk (proxy_base |>> prefix const_prefix)
| NONE => NONE) else
(case Symtab.lookup sym_tab s' of
NONE => NONE
| SOME ({min_ary, ...} : sym_info) => if ary = min_ary then mk (s', s) elseif uncurried_aliases then mk (aliased_uncurried ary (s', s)) else NONE) end
fun datatype_of_ctrs (ctrs as (_, T1) :: _) = letval ctrs' = map do_ctr ctrs in
(native_atp_type_of_typ type_enc false 0 (body_type T1), map_filter I ctrs',
forall is_some ctrs') end in
ctrss |> map datatype_of_ctrs |> filter #3 end else
[]
| datatypes_of_sym_table _ _ _ _ _ _ = []
fun decl_line_of_datatype (ty as AType (((_, s'), _), ty_args), ctrs, exhaust) = letval xs = map (fn AType ((name, _), []) => name) ty_args in
Datatype_Decl (datatype_decl_prefix ^ ascii_of s', map (rpair []) xs, ty, ctrs, exhaust) end
val implicit_declsN = "Could-be-implicit typings" val explicit_declsN = "Explicit typings" val uncurried_alias_eqsN = "Uncurried aliases" val factsN = "Relevant facts" val subclassesN = "Subclasses" val tconsN = "Type constructors" val helpersN = "Helper facts" val conjsN = "Conjectures" val free_typesN = "Free types"
(* TFF allows implicit declarations of types, function symbols, and predicate symbols (with "$i" as the type of individuals), but some provers (e.g.,
SNARK) require explicit declarations. The situation is similar for THF. *)
fun default_type pred_sym = let fun typ 0 0 = if pred_sym then bool_atype else individual_atype
| typ 0 tm_ary = AFun (individual_atype, typ 0 (tm_ary - 1))
| typ ty_ary tm_ary = APi (replicate ty_ary tvar_a_name, typ 0 tm_ary) in typ end
fun undeclared_in_problem problem = let fun do_sym (name as (s, _)) value = if is_tptp_user_symbol s andalso not (String.isPrefix let_bound_var_prefix s) then
Symtab.default (s, (name, value)) else
I fun do_class name = apfst (apfst (do_sym name ())) val do_bound_tvars = fold do_class o snd fun do_type (AType ((name, _), tys)) =
apfst (apsnd (do_sym name (length tys))) #> fold do_type tys
| do_type (AFun (ty1, ty2)) = do_type ty1 #> do_type ty2
| do_type (APi (_, ty)) = do_type ty fun do_term pred_sym (ATerm ((name, tys), tms)) =
apsnd (do_sym name (fn _ => default_type pred_sym (length tys) (length tms)))
#> fold do_type tys #> fold (do_term false) tms
| do_term _ (AAbs (((_, ty), tm), args)) =
do_type ty #> do_term false tm #> fold (do_term false) args fun do_formula (ATyQuant (_, xs, phi)) =
fold (do_type o fst) xs #> fold (fold do_class o snd) xs #> do_formula phi
| do_formula (AQuant (_, xs, phi)) = fold do_type (map_filter snd xs) #> do_formula phi
| do_formula (AConn (_, phis)) = fold do_formula phis
| do_formula (AAtom tm) = do_term true tm fun do_line (Class_Decl (_, _, cls)) = fold do_class cls
| do_line (Type_Decl _) = I
| do_line (Sym_Decl (_, _, ty)) = do_type ty
| do_line (Datatype_Decl (_, xs, ty, tms, _)) =
fold do_bound_tvars xs #> do_type ty #> fold (do_term false) tms
| do_line (Class_Memb (_, xs, ty, cl)) = fold do_bound_tvars xs #> do_type ty #> do_class cl
| do_line (Formula (_, _, phi, _, _)) = do_formula phi val ((cls, tys), syms) = declared_in_atp_problem problem in
((Symtab.empty, Symtab.empty), Symtab.empty)
|>> apfst (fold (fn (s, _) => Symtab.default (s, (("", ""), ()))) cls)
|>> apsnd (fold (fn (s, _) => Symtab.default (s, (("", ""), 0))) tys)
||> fold (fn (s, _) => Symtab.default (s, (("", ""), K tvar_a_atype))) syms
|> fold (fold do_line o snd) problem end
fun declare_undeclared_in_problem heading problem = let val ((cls, tys), syms) = undeclared_in_problem problem val decls =
Symtab.fold (fn (_, (("", ""), _)) => I (* already declared *)
| (s, (cls, ())) => cons (Class_Decl (class_decl_prefix ^ s, cls, []))) cls [] @
Symtab.fold (fn (_, (("", ""), _)) => I (* already declared *)
| (s, (ty, ary)) => cons (Type_Decl (type_decl_prefix ^ s, ty, ary))) tys [] @
Symtab.fold (fn (_, (("", ""), _)) => I (* already declared *)
| (s, (sym, ty)) => cons (Sym_Decl (sym_decl_prefix ^ s, sym, ty ()))) syms [] in (heading, decls) :: problem end
val all_ctrss_of_datatypes = map (map_filter (try dest_Const) o #ctrs) o Ctr_Sugar.ctr_sugars_of
val app_op_and_predicator_threshold = 45
fun generate_atp_problem ctxt generate_isabelle_info format prem_role type_enc mode lam_trans
uncurried_aliases readable_names presimp hyp_ts concl_t facts = let val thy = Proof_Context.theory_of ctxt val type_enc = type_enc |> adjust_type_enc format val completish = (case mode of Sledgehammer_Completish k => k | _ => 0) (* Forcing explicit applications is expensive for polymorphic encodings, because it takes only one existential variable ranging over "'a => 'b" to ruin everything. Hence we do it only if there are few facts (which is
normally the case for "metis" and the minimizer). *) val app_op_level = if completish > 0 then
Full_App_Op_And_Predicator elseif is_greater_equal
(compare_length_with facts (app_op_and_predicator_threshold - length hyp_ts)) then if is_type_enc_polymorphic type_enc then Min_App_Op else Sufficient_App_Op else
Sufficient_App_Op_And_Predicator val lam_trans = if lam_trans = keep_lamsN andalso not (is_type_enc_full_higher_order type_enc) then liftingN else lam_trans val simp_options =
{if_simps = not (has_type_enc_ite type_enc),
let_simps = not (has_type_enc_let type_enc)} val (classes, conjs, facts, subclass_pairs, tcon_clauses, lifted) =
translate_formulas simp_options ctxt prem_role format type_enc lam_trans presimp hyp_ts
concl_t facts val (_, sym_tab0) = sym_table_of_facts ctxt type_enc app_op_level conjs facts val mono = conjs @ facts |> mononotonicity_info_of_facts ctxt type_enc completish val ctrss = all_ctrss_of_datatypes ctxt fun firstorderize in_helper =
firstorderize_fact thy ctrss type_enc (uncurried_aliases andalso not in_helper) completish
sym_tab0 val (conjs, facts) = (conjs, facts) |> apply2 (map (firstorderize false)) val (ho_stuff, sym_tab) =
sym_table_of_facts ctxt type_enc Min_App_Op conjs facts val (uncurried_alias_eq_tms, uncurried_alias_eq_lines) =
uncurried_alias_lines_of_sym_table ctxt generate_isabelle_info ctrss mono type_enc
uncurried_aliases sym_tab0 sym_tab val (_, sym_tab) =
(ho_stuff, sym_tab)
|> fold (add_iterm_syms_to_sym_table ctxt Min_App_Op falsefalse)
uncurried_alias_eq_tms val helpers =
sym_tab |> helper_facts_of_sym_table ctxt format type_enc lam_trans completish
|> map (firstorderize true) val all_facts = helpers @ conjs @ facts val mono_Ts = monotonic_types_of_facts ctxt mono type_enc all_facts val datatypes = datatypes_of_sym_table ctxt ctrss format type_enc uncurried_aliases sym_tab val class_decl_lines = decl_lines_of_classes type_enc classes val sym_decl_lines =
(conjs, helpers @ facts, uncurried_alias_eq_tms)
|> sym_decl_table_of_facts thy type_enc sym_tab
|> lines_of_sym_decl_table ctxt generate_isabelle_info mono type_enc mono_Ts val datatype_decl_lines = map decl_line_of_datatype datatypes val decl_lines = class_decl_lines @ sym_decl_lines @ datatype_decl_lines val num_facts = length facts val freshen = mode <> Exporter andalso mode <> Translator val pos = mode <> Exporter val rank_of = rank_of_fact_num num_facts val fact_lines = facts
|> map_index (line_of_fact ctxt generate_isabelle_info fact_prefix ascii_of I freshen pos mono
type_enc rank_of)
val subclass_lines = maps (lines_of_subclass_pair generate_isabelle_info type_enc) subclass_pairs val tcon_lines = map (line_of_tcon_clause generate_isabelle_info type_enc) tcon_clauses val helper_lines = helpers
|> map_index (line_of_fact ctxt generate_isabelle_info helper_prefix I (K "") falsetrue mono
type_enc (K default_rank)) val free_type_lines = lines_of_free_types type_enc (facts @ conjs) val conj_lines = map (line_of_conjecture ctxt mono type_enc) conjs (* Reordering these might confuse the proof reconstruction code. *) val problem =
[(explicit_declsN, decl_lines),
(uncurried_alias_eqsN, uncurried_alias_eq_lines),
(factsN, fact_lines),
(subclassesN, subclass_lines),
(tconsN, tcon_lines),
(helpersN, helper_lines),
(free_typesN, free_type_lines),
(conjsN, conj_lines)] val problem =
problem
|> (case format of
CNF => ensure_cnf_problem
| CNF_UEQ => filter_cnf_ueq_problem
| FOF => I
| _ => declare_undeclared_in_problem implicit_declsN) val (problem, pool) = problem |> nice_atp_problem readable_names format fun add_sym_ary (s, {min_ary, ...} : sym_info) = min_ary > 0 ? Symtab.insert (op =) (s, min_ary) in
(problem, Option.map snd pool |> the_default Symtab.empty, lifted,
Symtab.fold add_sym_ary sym_tab Symtab.empty) 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
(* Ugly hack: may make innocent victims (collateral damage) *) fun may_be_app s args = String.isPrefix app_op_name s andalso length args = 2 fun may_be_predicator s =
member (op =) [predicator_name, prefixed_predicator_name] s
fun strip_predicator (tm as ATerm ((s, _), [tm'])) = if may_be_predicator s then tm'else tm
| strip_predicator tm = tm
fun make_head_roll (ATerm ((s, _), tms)) = if may_be_app s tms then make_head_roll (hd tms) ||> append (tl tms) else (s, tms)
| make_head_roll _ = ("", [])
fun atp_problem_term_order_info problem = let fun add_edge s s' =
Graph.default_node (s, ())
#> Graph.default_node (s', ())
#> Graph.add_edge_acyclic (s, s') fun add_term_deps head (ATerm ((s, _), args)) = if is_tptp_user_symbol head then
(if is_tptp_user_symbol s then perhaps (try (add_edge s head)) else I)
#> fold (add_term_deps head) args else
I
| add_term_deps head (AAbs ((_, tm), args)) =
add_term_deps head tm #> fold (add_term_deps head) args fun add_intro_deps pred (Formula (_, role, phi, _, info)) = if pred (role, info) then letval (hyps, concl) = strip_ahorn_etc phi in
(case make_head_roll concl of
(head, args as _ :: _) => fold (add_term_deps head) (hyps @ args)
| _ => I) end else
I
| add_intro_deps _ _ = I fun add_atom_eq_deps (SOME true) (ATerm ((s, _), [lhs as _, rhs])) = if is_tptp_equal s then
(case make_head_roll lhs of
(head, args as _ :: _) => fold (add_term_deps head) (rhs :: args)
| _ => I) else
I
| add_atom_eq_deps _ _ = I fun add_eq_deps pred (Formula (_, role, phi, _, info)) = if pred (role, info) then
(case strip_iff_etc phi of
([lhs], rhs) =>
(case make_head_roll lhs of
(head, args as _ :: _) => fold (add_term_deps head) (rhs @ args)
| _ => I)
| _ => formula_fold (SOME (role <> Conjecture)) add_atom_eq_deps phi) else
I
| add_eq_deps _ _ = I fun has_status status (_, info) = extract_isabelle_status info = SOME status fun is_conj (role, _) = (role = Conjecture orelse role = Hypothesis) val graph =
Graph.empty
|> fold (fold (add_eq_deps (has_status non_rec_defN)) o snd) problem
|> fold (fold (add_eq_deps (has_status rec_defN orf has_status simpN
orf is_conj)) o snd) problem
|> fold (fold (add_intro_deps (has_status inductiveN)) o snd) problem
|> fold (fold (add_intro_deps (has_status introN)) o snd) problem fun next_weight w = if w + w <= max_term_order_weight then w + w else w + 1 fun add_weights _ [] = I
| add_weights weight syms =
fold (AList.update (op =) o rpair weight) syms
#> add_weights (next_weight weight)
(fold (union (op =) o Graph.immediate_succs graph) syms []) in (* Sorting is not just for aesthetics: It specifies the precedence order for
the term ordering (KBO or LPO), from smaller to larger values. *)
[] |> add_weights 1 (Graph.minimals graph) |> sort (int_ord o apply2 snd) end
end;
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