(* Title: HOL/Tools/ATP/atp_proof.ML
Author: Lawrence C. Paulson, Cambridge University Computer Laboratory
Author: Claire Quigley, Cambridge University Computer Laboratory
Author: Jasmin Blanchette, TU Muenchen
Abstract representation of ATP proofs and TSTP/SPASS syntax.
*)
signature ATP_PROOF =
sig
type 'a atp_type = 'a ATP_Problem.atp_type
type ('a, 'b) atp_term = ('a, 'b) ATP_Problem.atp_term
type atp_formula_role = ATP_Problem.atp_formula_role
type ('a, 'b, 'c, 'd) atp_formula = ('a, 'b, 'c, 'd) ATP_Problem.atp_formula
type 'a atp_problem = 'a ATP_Problem.atp_problem
exception UNRECOGNIZED_ATP_PROOF of unit
datatype atp_failure =
Unprovable |
GaveUp |
ProofMissing |
ProofIncomplete |
UnsoundProof of bool * string list |
CantConnect |
TimedOut |
Inappropriate |
OutOfResources |
NoPerl |
NoLibwwwPerl |
MalformedInput |
MalformedOutput |
Interrupted |
Crashed |
InternalError |
UnknownError of string
type atp_step_name = string * string list
type ('a, 'b) atp_step =
atp_step_name * atp_formula_role * 'a * 'b * atp_step_name list
type 'a atp_proof = (('a, 'a, ('a, 'a atp_type) atp_term, 'a) atp_formula, string) atp_step list
val agsyhol_core_rule : string
val satallax_core_rule : string
val spass_input_rule : string
val spass_pre_skolemize_rule : string
val spass_skolemize_rule : string
val z3_tptp_core_rule : string
val short_output : bool -> string -> string
val string_of_atp_failure : atp_failure -> string
val extract_important_message : string -> string
val extract_known_atp_failure : (atp_failure * string) list -> string -> atp_failure option
val extract_tstplike_proof_and_outcome :
bool -> (string * string) list -> (atp_failure * string) list -> string
-> string * atp_failure option
val is_same_atp_step : atp_step_name -> atp_step_name -> bool
val scan_general_id : string list -> string * string list
val parse_formula : string list ->
(string, string atp_type, (string, string atp_type) atp_term, string) atp_formula * string list
val atp_proof_of_tstplike_proof : string atp_problem -> string -> string atp_proof
val clean_up_atp_proof_dependencies : string atp_proof -> string atp_proof
val map_term_names_in_atp_proof : (string -> string) -> string atp_proof -> string atp_proof
val nasty_atp_proof : string Symtab.table -> string atp_proof -> string atp_proof
end;
structure ATP_Proof : ATP_PROOF =
struct
open ATP_Util
open ATP_Problem
val agsyhol_core_rule = "__agsyhol_core" (* arbitrary *)
val satallax_core_rule = "__satallax_core" (* arbitrary *)
val spass_input_rule = "Inp"
val spass_pre_skolemize_rule = "__Sko0" (* arbitrary *)
val spass_skolemize_rule = "__Sko" (* arbitrary *)
val z3_tptp_core_rule = "__z3_tptp_core" (* arbitrary *)
exception UNRECOGNIZED_ATP_PROOF of unit
datatype atp_failure =
Unprovable |
GaveUp |
ProofMissing |
ProofIncomplete |
UnsoundProof of bool * string list |
CantConnect |
TimedOut |
Inappropriate |
OutOfResources |
NoPerl |
NoLibwwwPerl |
MalformedInput |
MalformedOutput |
Interrupted |
Crashed |
InternalError |
UnknownError of string
fun short_output verbose output =
if verbose then
if output = "" then "No details available" else elide_string 1000 output
else
""
val missing_message_tail =
" appears to be missing. You will need to install it if you want to invoke \
\remote provers."
fun from_lemmas [] = ""
| from_lemmas ss = " from " ^ space_implode " " (Try.serial_commas "and" (map quote ss))
fun string_of_atp_failure Unprovable = "The generated problem is unprovable."
| string_of_atp_failure GaveUp = "The prover gave up."
| string_of_atp_failure ProofMissing =
"The prover claims the conjecture is a theorem but did not provide a proof."
| string_of_atp_failure ProofIncomplete =
"The prover claims the conjecture is a theorem but provided an incomplete \
\(or unparsable) proof."
| string_of_atp_failure (UnsoundProof (false, ss)) =
"The prover derived \"False\"" ^ from_lemmas ss ^
". Specify a sound type encoding or omit the \"type_enc\" option."
| string_of_atp_failure (UnsoundProof (true, ss)) =
"The prover derived \"False\"" ^ from_lemmas ss ^
". This could be due to inconsistent axioms (including \"sorry\"s) or to \
\a bug in Sledgehammer. If the problem persists, please contact the \
\Isabelle developers."
| string_of_atp_failure CantConnect = "Cannot connect to remote server."
| string_of_atp_failure TimedOut = "Timed out."
| string_of_atp_failure Inappropriate =
"The generated problem lies outside the prover's scope."
| string_of_atp_failure OutOfResources = "The prover ran out of resources."
| string_of_atp_failure NoPerl = "Perl" ^ missing_message_tail
| string_of_atp_failure NoLibwwwPerl =
"The Perl module \"libwww-perl\"" ^ missing_message_tail
| string_of_atp_failure MalformedInput =
"The generated problem is malformed. Please report this to the Isabelle \
\developers."
| string_of_atp_failure MalformedOutput = "The prover output is malformed."
| string_of_atp_failure Interrupted = "The prover was interrupted."
| string_of_atp_failure Crashed = "The prover crashed."
| string_of_atp_failure InternalError = "An internal prover error occurred."
| string_of_atp_failure (UnknownError s) =
"A prover error occurred" ^
(if s = "" then ". (Pass the \"verbose\" option for details.)"
else ":\n" ^ s)
fun extract_delimited (begin_delim, end_delim) output =
(case first_field begin_delim output of
SOME (_, tail) =>
(case first_field "\n" tail of
SOME (_, tail') =>
if end_delim = "" then
tail'
else
(case first_field end_delim tail' of
SOME (body, _) => body
| NONE => "")
| NONE => "")
| NONE => "")
val tstp_important_message_delims =
("% SZS start RequiredInformation", "% SZS end RequiredInformation")
fun extract_important_message output =
(case extract_delimited tstp_important_message_delims output of
"" => ""
| s => s |> space_explode "\n" |> filter_out (curry (op =) "")
|> map (perhaps (try (unprefix "%")))
|> map (perhaps (try (unprefix " ")))
|> space_implode "\n " |> quote)
(* Splits by the first possible of a list of delimiters. *)
fun extract_tstplike_proof delims output =
(case pairself (find_first (fn s => String.isSubstring s output)) (ListPair.unzip delims) of
(SOME begin_delim, SOME end_delim) => extract_delimited (begin_delim, end_delim) output
| _ => "")
fun extract_known_atp_failure known_failures output =
known_failures
|> find_first (fn (_, pattern) => String.isSubstring pattern output)
|> Option.map fst
fun extract_tstplike_proof_and_outcome verbose proof_delims known_failures output =
(case (extract_tstplike_proof proof_delims output,
extract_known_atp_failure known_failures output) of
(_, SOME ProofIncomplete) => ("", NONE)
| ("", SOME ProofMissing) => ("", NONE)
| ("", NONE) => ("", SOME (UnknownError (short_output verbose output)))
| res as ("", _) => res
| (tstplike_proof, _) => (tstplike_proof, NONE))
type atp_step_name = string * string list
fun is_same_atp_step (s1, _) (s2, _) = s1 = s2
val vampire_fact_prefix = "f"
fun vampire_step_name_ord p =
let val q = pairself fst p in
(* The "unprefix" part is to cope with Vampire's output. *)
(case pairself (Int.fromString o perhaps (try (unprefix vampire_fact_prefix))) q of
(SOME i, SOME j) => int_ord (i, j)
| _ => raise Fail "not Vampire")
end
type ('a, 'b) atp_step = atp_step_name * atp_formula_role * 'a * 'b * atp_step_name list
type 'a atp_proof = (('a, 'a, ('a, 'a atp_type) atp_term, 'a) atp_formula, string) atp_step list
(**** PARSING OF TSTP FORMAT ****)
(* Strings enclosed in single quotes (e.g., file names), identifiers possibly starting
with "$" and possibly with "!" in them (for "z3_tptp"). *)
val scan_general_id =
$$ "'" |-- Scan.repeat (~$$ "'") --| $$ "'" >> implode
|| (Scan.repeat ($$ "$") -- Scan.many1 Symbol.is_letdig >> (op ^ o pairself implode))
-- Scan.optional (Scan.repeat ($$ "!") -- Scan.many1 Symbol.is_letdig >> (op ^ o pairself implode)) ""
>> op ^
val skip_term =
let
fun skip _ accum [] = (accum, [])
| skip n accum (ss as s :: ss') =
if s = "," andalso n = 0 then
(accum, ss)
else if member (op =) [")", "]"] s then
if n = 0 then (accum, ss) else skip (n - 1) (s :: accum) ss'
else if member (op =) ["(", "["] s then
skip (n + 1) (s :: accum) ss'
else
skip n (s :: accum) ss'
in
skip 0 [] #>> (rev #> implode)
end
datatype source =
File_Source of string * string option |
Inference_Source of string * string list
val dummy_phi = AAtom (ATerm (("", []), []))
val dummy_inference = Inference_Source ("", [])
(* "skip_term" is there to cope with Waldmeister nonsense such as "theory(equality)". *)
fun parse_dependency x =
(parse_inference_source >> snd
|| scan_general_id --| skip_term >> single) x
and parse_dependencies x =
(Scan.repeat (Scan.option ($$ ",") |-- parse_dependency) >> flat) x
and parse_file_source x =
(Scan.this_string "file" |-- $$ "(" |-- scan_general_id
-- Scan.option ($$ "," |-- scan_general_id) --| $$ ")") x
and parse_inference_source x =
(Scan.this_string "inference" |-- $$ "(" |-- scan_general_id
--| skip_term --| $$ "," --| skip_term --| $$ "," --| $$ "["
-- parse_dependencies --| $$ "]" --| $$ ")") x
and skip_introduced x =
(Scan.this_string "introduced" |-- $$ "(" |-- skip_term
-- Scan.repeat ($$ "," |-- skip_term) --| $$ ")") x
and parse_source x =
(parse_file_source >> File_Source
|| parse_inference_source >> Inference_Source
|| skip_introduced >> K dummy_inference (* for Vampire *)
|| scan_general_id >> (fn s => Inference_Source ("", [s])) (* for E *)
|| skip_term >> K dummy_inference) x
fun list_app (f, args) = fold (fn arg => fn f => ATerm ((tptp_app, []), [f, arg])) args f
fun parse_class x = scan_general_id x
and parse_classes x = (parse_class ::: Scan.repeat ($$ "&" |-- parse_class)) x
fun parse_type x =
(scan_general_id -- Scan.optional ($$ "{" |-- parse_classes --| $$ "}") []
-- Scan.optional ($$ "(" |-- parse_types --| $$ ")") []
>> AType) x
and parse_types x = (parse_type ::: Scan.repeat ($$ "," |-- parse_type)) x
(* We currently ignore TFF and THF types. *)
fun parse_type_signature x = Scan.repeat (($$ tptp_has_type || $$ tptp_fun_type) |-- parse_arg) x
and parse_arg x =
($$ "(" |-- parse_term --| $$ ")" --| parse_type_signature
|| scan_general_id --| parse_type_signature
-- Scan.optional ($$ "<" |-- parse_types --| $$ ">") []
-- Scan.optional ($$ "(" |-- parse_terms --| $$ ")") []
>> ATerm) x
and parse_term x =
(parse_arg -- Scan.repeat ($$ tptp_app |-- parse_arg)
--| Scan.option parse_type_signature >> list_app) x
and parse_terms x = (parse_term ::: Scan.repeat ($$ "," |-- parse_term)) x
fun parse_atom x =
(parse_term -- Scan.option (Scan.option ($$ tptp_not_infix) --| $$ tptp_equal -- parse_term)
>> (fn (u1, NONE) => AAtom u1
| (u1, SOME (neg, u2)) =>
AAtom (ATerm (("equal", []), [u1, u2])) |> is_some neg ? mk_anot)) x
(* TPTP formulas are fully parenthesized, so we don't need to worry about operator precedence. *)
fun parse_literal x =
((Scan.repeat ($$ tptp_not) >> length)
-- ($$ "(" |-- parse_formula --| $$ ")"
|| parse_quantified_formula
|| parse_atom)
>> (fn (n, phi) => phi |> n mod 2 = 1 ? mk_anot)) x
and parse_formula x =
(parse_literal
-- Scan.option ((Scan.this_string tptp_implies
|| Scan.this_string tptp_iff
|| Scan.this_string tptp_not_iff
|| Scan.this_string tptp_if
|| $$ tptp_or
|| $$ tptp_and) -- parse_formula)
>> (fn (phi1, NONE) => phi1
| (phi1, SOME (c, phi2)) =>
if c = tptp_implies then mk_aconn AImplies phi1 phi2
else if c = tptp_iff then mk_aconn AIff phi1 phi2
else if c = tptp_not_iff then mk_anot (mk_aconn AIff phi1 phi2)
else if c = tptp_if then mk_aconn AImplies phi2 phi1
else if c = tptp_or then mk_aconn AOr phi1 phi2
else if c = tptp_and then mk_aconn AAnd phi1 phi2
else raise Fail ("impossible connective " ^ quote c))) x
and parse_quantified_formula x =
(($$ tptp_forall >> K AForall || $$ tptp_exists >> K AExists)
--| $$ "[" -- parse_terms --| $$ "]" --| $$ ":" -- parse_literal
>> (fn ((q, ts), phi) => AQuant (q, map (fn ATerm ((s, []), _) => (s, NONE)) ts, phi))) x
val parse_tstp_extra_arguments =
Scan.optional ($$ "," |-- parse_source --| Scan.option ($$ "," |-- skip_term)) dummy_inference
val waldmeister_conjecture_name = "conjecture_1"
val tofof_fact_prefix = "fof_"
fun is_same_term subst tm1 tm2 =
let
fun do_term_pair _ NONE = NONE
| do_term_pair (ATerm ((s1, _), tm1), ATerm ((s2, _), tm2)) (SOME subst) =
(case pairself is_tptp_variable (s1, s2) of
(true, true) =>
(case AList.lookup (op =) subst s1 of
SOME s2' => if s2' = s2 then SOME subst else NONE
| NONE =>
if null (AList.find (op =) subst s2) then SOME ((s1, s2) :: subst)
else NONE)
| (false, false) =>
if s1 = s2 andalso length tm1 = length tm2 then
SOME subst |> fold do_term_pair (tm1 ~~ tm2)
else
NONE
| _ => NONE)
in
SOME subst |> do_term_pair (tm1, tm2) |> is_some
end
fun is_same_formula comm subst (AQuant (q1, xs1, phi1)) (AQuant (q2, xs2, phi2)) =
q1 = q2 andalso length xs1 = length xs2 andalso
is_same_formula comm ((map fst xs1 ~~ map fst xs2) @ subst) phi1 phi2
| is_same_formula comm subst (AConn (c1, phis1)) (AConn (c2, phis2)) =
c1 = c2 andalso length phis1 = length phis2 andalso
forall (uncurry (is_same_formula comm subst)) (phis1 ~~ phis2)
| is_same_formula comm subst
(AAtom (tm1 as ATerm (("equal", tys), [tm11, tm12]))) (AAtom tm2) =
is_same_term subst tm1 tm2 orelse
(comm andalso is_same_term subst (ATerm (("equal", tys), [tm12, tm11])) tm2)
| is_same_formula _ subst (AAtom tm1) (AAtom tm2) = is_same_term subst tm1 tm2
| is_same_formula _ _ _ _ = false
fun matching_formula_line_identifier phi (Formula ((ident, _), _, phi', _, _)) =
if is_same_formula true [] phi phi' then SOME (ident, phi') else NONE
| matching_formula_line_identifier _ _ = NONE
fun find_formula_in_problem phi =
maps snd
#> map_filter (matching_formula_line_identifier phi)
#> try (single o hd)
#> the_default []
fun commute_eq (AAtom (ATerm ((s, tys), tms))) = AAtom (ATerm ((s, tys), rev tms))
| commute_eq _ = raise Fail "expected equation"
fun role_of_tptp_string "axiom" = Axiom
| role_of_tptp_string "definition" = Definition
| role_of_tptp_string "lemma" = Lemma
| role_of_tptp_string "hypothesis" = Hypothesis
| role_of_tptp_string "conjecture" = Conjecture
| role_of_tptp_string "negated_conjecture" = Negated_Conjecture
| role_of_tptp_string "plain" = Plain
| role_of_tptp_string _ = Unknown
(* Syntax: (cnf|fof|tff|thf)\(<num>, <formula_role>, <formula> <extra_arguments>\).
The <num> could be an identifier, but we assume integers. *)
fun parse_tstp_line problem =
((Scan.this_string tptp_cnf || Scan.this_string tptp_fof
|| Scan.this_string tptp_tff || Scan.this_string tptp_thf) -- $$ "(")
|-- scan_general_id --| $$ "," -- Symbol.scan_ascii_id --| $$ ","
-- (parse_formula || skip_term >> K dummy_phi) -- parse_tstp_extra_arguments
--| $$ ")" --| $$ "."
>> (fn (((num, role), phi), deps) =>
let
val ((name, phi), rule, deps) =
(* Waldmeister isn't exactly helping. *)
(case deps of
File_Source (_, SOME s) =>
(if s = waldmeister_conjecture_name then
(case find_formula_in_problem (mk_anot phi) problem of
(* Waldmeister hack: Get the original orientation of the
equation to avoid confusing Isar. *)
[(s, phi')] =>
((num, [s]),
phi |> not (is_same_formula false [] (mk_anot phi) phi') ? commute_eq)
| _ => ((num, []), phi))
else
((num, [s |> perhaps (try (unprefix tofof_fact_prefix))]),
phi),
"", [])
| File_Source _ =>
(((num, map fst (find_formula_in_problem phi problem)), phi), "", [])
| Inference_Source (rule, deps) => (((num, []), phi), rule, deps))
fun mk_step () = (name, role_of_tptp_string role, phi, rule, map (rpair []) deps)
in
[(case role_of_tptp_string role of
Definition =>
(case phi of
AAtom (ATerm (("equal", _), _)) =>
(* Vampire's equality proxy axiom *)
(name, Definition, phi, rule, map (rpair []) deps)
| _ => mk_step ())
| _ => mk_step ())]
end)
(**** PARSING OF SPASS OUTPUT ****)
(* SPASS returns clause references of the form "x.y". We ignore "y", whose role
is not clear anyway. *)
val parse_dot_name = scan_general_id --| $$ "." --| scan_general_id
val parse_spass_annotations =
Scan.optional ($$ ":" |-- Scan.repeat (parse_dot_name --| Scan.option ($$ ","))) []
(* It is not clear why some literals are followed by sequences of stars and/or
pluses. We ignore them. *)
fun parse_decorated_atom x =
(parse_atom --| Scan.repeat ($$ "*" || $$ "+" || $$ " ")) x
fun mk_horn ([], []) = AAtom (ATerm (("c_False", []), []))
| mk_horn ([], pos_lits) = foldr1 (uncurry (mk_aconn AOr)) pos_lits
| mk_horn (neg_lits, []) = mk_anot (foldr1 (uncurry (mk_aconn AAnd)) neg_lits)
| mk_horn (neg_lits, pos_lits) =
mk_aconn AImplies (foldr1 (uncurry (mk_aconn AAnd)) neg_lits)
(foldr1 (uncurry (mk_aconn AOr)) pos_lits)
fun parse_horn_clause x =
(Scan.repeat parse_decorated_atom --| $$ "|" --| $$ "|"
-- Scan.repeat parse_decorated_atom --| $$ "-" --| $$ ">"
-- Scan.repeat parse_decorated_atom
>> (mk_horn o apfst (op @))) x
val parse_spass_debug =
Scan.option ($$ "(" |-- Scan.repeat (scan_general_id --| Scan.option ($$ ",")) --| $$ ")")
(* Syntax: <num>[0:<inference><annotations>] <atoms> || <atoms> -> <atoms> .
derived from formulae <ident>* *)
fun parse_spass_line x =
(parse_spass_debug |-- scan_general_id --| $$ "[" --| Scan.many1 Symbol.is_digit --| $$ ":"
-- Symbol.scan_ascii_id -- parse_spass_annotations --| $$ "]" -- parse_horn_clause --| $$ "."
-- Scan.option (Scan.this_string "derived from formulae "
|-- Scan.repeat (scan_general_id --| Scan.option ($$ " ")))
>> (fn ((((num, rule), deps), u), names) =>
[((num, these names), Unknown, u, rule, map (rpair []) deps)])) x
fun parse_spass_pirate_dependency x = (Scan.option ($$ "-") |-- scan_general_id) x
fun parse_spass_pirate_dependencies x =
Scan.repeat (parse_spass_pirate_dependency --| Scan.option ($$ "," || $$ " ")) x
fun parse_spass_pirate_file_source x =
((Scan.this_string "Input" || Scan.this_string "Conj") |-- $$ "(" |-- scan_general_id
--| $$ ")") x
fun parse_spass_pirate_inference_source x =
(scan_general_id -- ($$ "(" |-- parse_spass_pirate_dependencies --| $$ ")")) x
fun parse_spass_pirate_source x =
(parse_spass_pirate_file_source >> (fn s => File_Source ("", SOME s))
|| parse_spass_pirate_inference_source >> Inference_Source) x
(* Syntax: <num> <stuff> || <atoms> -> <atoms> . origin\(<origin>\) *)
fun parse_spass_pirate_line x =
(scan_general_id --| Scan.repeat (~$$ "|") -- parse_horn_clause --| $$ "."
--| Scan.this_string "origin" --| $$ "(" -- parse_spass_pirate_source --| $$ ")"
>> (fn ((((num, u), source))) =>
let
val (names, rule, deps) =
(case source of
File_Source (_, SOME s) => ([s], spass_input_rule, [])
| Inference_Source (rule, deps) => ([], rule, deps))
in
[((num, names), Unknown, u, rule, map (rpair []) deps)]
end)) x
fun core_inference inf fact = ((fact, [fact]), Unknown, dummy_phi, inf, [])
(* Syntax: <name> *)
fun parse_satallax_core_line x =
(scan_general_id --| Scan.option ($$ " ") >> (single o core_inference satallax_core_rule)) x
(* Syntax: SZS core <name> ... <name> *)
fun parse_z3_tptp_core_line x =
(Scan.this_string "SZS core" |-- Scan.repeat ($$ " " |-- scan_general_id)
>> map (core_inference z3_tptp_core_rule)) x
fun parse_line problem =
parse_tstp_line problem || parse_z3_tptp_core_line || parse_spass_line || parse_spass_pirate_line
|| parse_satallax_core_line
fun parse_proof problem =
strip_spaces_except_between_idents
#> raw_explode
#> Scan.error (!! (fn _ => raise UNRECOGNIZED_ATP_PROOF ())
(Scan.finite Symbol.stopper (Scan.repeat1 (parse_line problem) >> flat)))
#> fst
fun core_of_agsyhol_proof s =
(case split_lines s of
"The transformed problem consists of the following conjectures:" :: conj ::
_ :: proof_term :: _ => SOME (unprefix " " conj :: find_enclosed "<<" ">>" proof_term)
| _ => NONE)
fun atp_proof_of_tstplike_proof _ "" = []
| atp_proof_of_tstplike_proof problem tstp =
(case core_of_agsyhol_proof tstp of
SOME facts => facts |> map (core_inference agsyhol_core_rule)
| NONE =>
tstp ^ "$" (* the $ sign acts as a sentinel (FIXME: needed?) *)
|> parse_proof problem
|> perhaps (try (sort (vampire_step_name_ord o pairself #1))))
fun clean_up_dependencies _ [] = []
| clean_up_dependencies seen ((name, role, u, rule, deps) :: steps) =
(name, role, u, rule, map_filter (fn dep => find_first (is_same_atp_step dep) seen) deps) ::
clean_up_dependencies (name :: seen) steps
fun clean_up_atp_proof_dependencies proof = clean_up_dependencies [] proof
fun map_term_names_in_atp_proof f =
let
fun map_type (AType ((s, clss), tys)) = AType ((f s, map f clss), map map_type tys)
| map_type (AFun (ty, ty')) = AFun (map_type ty, map_type ty')
| map_type (APi (ss, ty)) = APi (map f ss, map_type ty)
fun map_term (ATerm ((s, tys), ts)) = ATerm ((f s, map map_type tys), map map_term ts)
| map_term (AAbs (((s, ty), tm), args)) =
AAbs (((f s, map_type ty), map_term tm), map map_term args)
fun map_formula (AQuant (q, xs, phi)) = AQuant (q, map (apfst f) xs, map_formula phi)
| map_formula (AConn (c, phis)) = AConn (c, map map_formula phis)
| map_formula (AAtom t) = AAtom (map_term t)
fun map_step (name, role, phi, rule, deps) = (name, role, map_formula phi, rule, deps)
in
map map_step
end
fun nasty_name pool s = s |> Symtab.lookup pool |> the_default s
fun nasty_atp_proof pool =
not (Symtab.is_empty pool) ? map_term_names_in_atp_proof (nasty_name pool)
end;