src/HOL/Tools/ATP/atp_proof.ML

simplified unsound proof detection by removing impossible case

(*  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, 'b) ho_term = ('a, 'b) ATP_Problem.ho_term
  type ('a, 'b, 'c) formula = ('a, 'b, 'c) ATP_Problem.formula
  type 'a problem = 'a ATP_Problem.problem

  exception UNRECOGNIZED_ATP_PROOF of unit

  datatype failure =
    Unprovable |
    GaveUp |
    ProofMissing |
    ProofIncomplete |
    UnsoundProof of bool * string list |
    CantConnect |
    TimedOut |
    Inappropriate |
    OutOfResources |
    SpassTooOld |
    VampireTooOld |
    NoPerl |
    NoLibwwwPerl |
    MalformedInput |
    MalformedOutput |
    Interrupted |
    Crashed |
    InternalError |
    UnknownError of string

  type step_name = string * string list option

  datatype 'a step =
    Definition of step_name * 'a * 'a |
    Inference of step_name * 'a * step_name list

  type 'a proof = ('a, 'a, ('a, 'a) ho_term) formula step list

  val short_output : bool -> string -> string
  val string_for_failure : failure -> string
  val extract_important_message : string -> string
  val extract_known_failure :
    (failure * string) list -> string -> failure option
  val extract_tstplike_proof_and_outcome :
    bool -> bool -> (string * string) list -> (failure * string) list -> string
    -> string * failure option
  val is_same_atp_step : step_name -> step_name -> bool
  val scan_general_id : string list -> string * string list
  val parse_formula :
    string list -> (string, 'a, (string, 'a) ho_term) formula * string list
  val atp_proof_from_tstplike_proof :
    string problem -> string -> string -> string proof
  val clean_up_atp_proof_dependencies : string proof -> string proof
  val map_term_names_in_atp_proof :
    (string -> string) -> string proof -> string proof
  val nasty_atp_proof : string Symtab.table -> string proof -> string proof
end;

structure ATP_Proof : ATP_PROOF =
struct

open ATP_Util
open ATP_Problem

exception UNRECOGNIZED_ATP_PROOF of unit

datatype failure =
  Unprovable |
  GaveUp |
  ProofMissing |
  ProofIncomplete |
  UnsoundProof of bool * string list |
  CantConnect |
  TimedOut |
  Inappropriate |
  OutOfResources |
  SpassTooOld |
  VampireTooOld |
  NoPerl |
  NoLibwwwPerl |
  MalformedInput |
  MalformedOutput |
  Interrupted |
  Crashed |
  InternalError |
  UnknownError of string

fun elide_string threshold s =
  if size s > threshold then
    String.extract (s, 0, SOME (threshold div 2 - 5)) ^ " ...... " ^
    String.extract (s, size s - (threshold + 1) div 2 + 6, NONE)
  else
    s
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 involving [] = ""
  | involving ss =
    "involving " ^ space_implode " " (Try.serial_commas "and" (map quote ss)) ^
    " "

fun string_for_failure Unprovable = "The problem is unprovable."
  | string_for_failure GaveUp = "The prover gave up."
  | string_for_failure ProofMissing =
    "The prover claims the conjecture is a theorem but did not provide a proof."
  | string_for_failure ProofIncomplete =
    "The prover claims the conjecture is a theorem but provided an incomplete \
    \proof."
  | string_for_failure (UnsoundProof (false, ss)) =
    "The prover found a type-unsound proof " ^ involving ss ^
    "(or, less likely, your axioms are inconsistent). Specify a sound type \
    \encoding or omit the \"type_enc\" option."
  | string_for_failure (UnsoundProof (true, ss)) =
    "The prover found a type-unsound proof " ^ involving ss ^
    "even though a supposedly type-sound encoding was used (or, less likely, \
    \your axioms are inconsistent). Please report this to the Isabelle \
    \developers."
  | string_for_failure CantConnect = "Cannot connect to remote server."
  | string_for_failure TimedOut = "Timed out."
  | string_for_failure Inappropriate =
    "The problem lies outside the prover's scope."
  | string_for_failure OutOfResources = "The prover ran out of resources."
  | string_for_failure SpassTooOld =
    "Isabelle requires a more recent version of SPASS with support for the \
    \TPTP syntax. To install it, download and extract the package \
    \\"http://isabelle.in.tum.de/dist/contrib/spass-3.7.tar.gz\" and add the \
    \\"spass-3.7\" directory's absolute path to " ^
    Path.print (Path.expand (Path.explode "$ISABELLE_HOME_USER/etc/components")) ^
    " on a line of its own."
  | string_for_failure VampireTooOld =
    "Isabelle requires a more recent version of Vampire. To install it, follow \
    \the instructions from the Sledgehammer manual (\"isabelle doc\
    \ sledgehammer\")."
  | string_for_failure NoPerl = "Perl" ^ missing_message_tail
  | string_for_failure NoLibwwwPerl =
    "The Perl module \"libwww-perl\"" ^ missing_message_tail
  | string_for_failure MalformedInput =
    "The generated problem is malformed. Please report this to the Isabelle \
    \developers."
  | string_for_failure MalformedOutput = "The prover output is malformed."
  | string_for_failure Interrupted = "The prover was interrupted."
  | string_for_failure Crashed = "The prover crashed."
  | string_for_failure InternalError = "An internal prover error occurred."
  | string_for_failure (UnknownError string) =
    "A prover error occurred" ^
    (if string = "" then ". (Pass the \"verbose\" option for details.)"
     else ":\n" ^ string)

fun extract_delimited (begin_delim, end_delim) output =
  output |> first_field begin_delim |> the |> snd
         |> first_field end_delim |> the |> fst
         |> first_field "\n" |> the |> snd
  handle Option.Option => ""

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_failure known_failures output =
  known_failures
  |> find_first (fn (_, pattern) => String.isSubstring pattern output)
  |> Option.map fst

fun extract_tstplike_proof_and_outcome verbose complete proof_delims
                                       known_failures output =
  case (extract_tstplike_proof proof_delims output,
        extract_known_failure known_failures output) of
    (_, SOME ProofIncomplete) => ("", SOME ProofIncomplete)
  | ("", SOME ProofMissing) => ("", NONE)
  | ("", SOME failure) =>
    ("", SOME (if failure = GaveUp andalso complete then Unprovable
               else failure))
  | ("", NONE) => ("", SOME (UnknownError (short_output verbose output)))
  | (tstplike_proof, _) => (tstplike_proof, NONE)

type step_name = string * string list option

fun is_same_atp_step (s1, _) (s2, _) = s1 = s2

val vampire_fact_prefix = "f"

fun step_name_ord p =
  let val q = pairself fst p in
    (* The "unprefix" part is to cope with remote Vampire's output. The proper
       solution would be to perform a topological sort, e.g. using the nice
       "Graph" functor. *)
    case pairself (Int.fromString
                   o perhaps (try (unprefix vampire_fact_prefix))) q of
      (NONE, NONE) => string_ord q
    | (NONE, SOME _) => LESS
    | (SOME _, NONE) => GREATER
    | (SOME i, SOME j) => int_ord (i, j)
  end

datatype 'a step =
  Definition of step_name * 'a * 'a |
  Inference of step_name * 'a * step_name list

type 'a proof = ('a, 'a, ('a, 'a) ho_term) formula step list

fun step_name (Definition (name, _, _)) = name
  | step_name (Inference (name, _, _)) = name

(**** PARSING OF TSTP FORMAT ****)

(* FIXME: temporary hack *)
fun repair_waldmeister_step_name s =
  case space_explode "." s of
    [a, b, c, d] =>
    (case a of "0" => "X" | "1" => "Y" | _ => "Z" ^ a) ^
    (if size b = 1 then "0" else "") ^ b ^ c ^ d
  | _ => s

(* Strings enclosed in single quotes (e.g., file names) *)
val scan_general_id =
  $$ "'" |-- Scan.repeat (~$$ "'") --| $$ "'"
     >> implode >> repair_waldmeister_step_name
  || Scan.repeat ($$ "$") -- Scan.many1 Symbol.is_letdig
     >> (fn (ss1, ss2) => implode ss1 ^ implode ss2)

(* Generalized first-order terms, which include file names, numbers, etc. *)
fun parse_annotation x =
  ((scan_general_id ::: Scan.repeat ($$ " " |-- scan_general_id))
     -- Scan.optional parse_annotation [] >> op @
   || $$ "(" |-- parse_annotations --| $$ ")"
   || $$ "[" |-- parse_annotations --| $$ "]") x
and parse_annotations x =
  (Scan.optional (parse_annotation
                  ::: Scan.repeat ($$ "," |-- parse_annotation)) []
   >> flat) x

fun list_app (f, args) =
  fold (fn arg => fn f => ATerm (tptp_app, [f, arg])) args f

(* We ignore TFF and THF types for now. *)
fun parse_type_stuff x =
  Scan.repeat (($$ tptp_has_type || $$ tptp_fun_type) |-- parse_arg) x
and parse_arg x =
  ($$ "(" |-- parse_term --| $$ ")" --| parse_type_stuff
   || scan_general_id --| parse_type_stuff
        -- Scan.optional ($$ "(" |-- parse_terms --| $$ ")") []
      >> ATerm) x
and parse_app x =
  (parse_arg -- Scan.repeat ($$ tptp_app |-- parse_arg) >> list_app) x
and parse_term x =
  (parse_app -- Scan.option (Scan.option ($$ tptp_not_infix) --| $$ tptp_equal
                             -- parse_app)
   >> (fn (u1, NONE) => u1
        | (u1, SOME (NONE, u2)) => ATerm ("equal", [u1, u2])
        | (u1, SOME (SOME _, u2)) =>
          ATerm (tptp_not, [ATerm ("equal", [u1, u2])]))) x
and parse_terms x =
  (parse_term ::: Scan.repeat ($$ "," |-- parse_term)) x

(* TODO: Avoid duplication with "parse_term" above. *)
fun parse_atom x =
  (parse_term -- Scan.option (Scan.option ($$ tptp_not_infix) --| $$ tptp_equal
                              -- parse_term)
   >> (fn (u1, NONE) => AAtom u1
        | (u1, SOME (NONE, u2)) => AAtom (ATerm ("equal", [u1, u2]))
        | (u1, SOME (SOME _, u2)) =>
          mk_anot (AAtom (ATerm ("equal", [u1, u2]))))) x

fun ho_term_head (ATerm (s, _)) = s

(* 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) =>
          (* We ignore TFF and THF types for now. *)
          AQuant (q, map (rpair NONE o ho_term_head) ts, phi))) x

fun skip_formula ss =
  let
    fun skip _ [] = []
      | skip 0 (ss as "," :: _) = ss
      | skip 0 (ss as ")" :: _) = ss
      | skip 0 (ss as "]" :: _) = ss
      | skip n ("(" :: ss) = skip (n + 1) ss
      | skip n ("[" :: ss) = skip (n + 1) ss
      | skip n ("]" :: ss) = skip (n - 1) ss
      | skip n (")" :: ss) = skip (n - 1) ss
      | skip n (_ :: ss) = skip n ss
  in (AAtom (ATerm ("", [])), skip 0 ss) end

val parse_tstp_extra_arguments =
  Scan.optional ($$ "," |-- parse_annotation
                 --| Scan.option ($$ "," |-- parse_annotations)) []

val waldmeister_conjecture = "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 subst (AQuant (q1, xs1, phi1)) (AQuant (q2, xs2, phi2)) =
    q1 = q2 andalso length xs1 = length xs2 andalso
    is_same_formula ((map fst xs1 ~~ map fst xs2) @ subst) phi1 phi2
  | is_same_formula subst (AConn (c1, phis1)) (AConn (c2, phis2)) =
    c1 = c2 andalso length phis1 = length phis2 andalso
    forall (uncurry (is_same_formula subst)) (phis1 ~~ phis2)
  | is_same_formula subst (AAtom (ATerm ("equal", [tm11, tm12]))) (AAtom tm2) =
    is_same_term subst (ATerm ("equal", [tm11, tm12])) tm2 orelse
    is_same_term subst (ATerm ("equal", [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 [] phi phi' then SOME ident else NONE
  | matching_formula_line_identifier _ _ = NONE

fun find_formula_in_problem problem phi =
  problem |> maps snd |> map_filter (matching_formula_line_identifier phi)
          |> try (single o hd)

(* 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_id --| $$ ","
    -- (parse_formula || skip_formula) -- parse_tstp_extra_arguments --| $$ ")"
    --| $$ "."
   >> (fn (((num, role), phi), deps) =>
          let
            val (name, deps) =
              (* Waldmeister isn't exactly helping. *)
              case deps of
                ["file", _, s] =>
                ((num,
                  if s = waldmeister_conjecture then
                    find_formula_in_problem problem (mk_anot phi)
                  else
                    SOME [s |> perhaps (try (unprefix tofof_fact_prefix))]),
                 [])
              | ["file", _] => ((num, find_formula_in_problem problem phi), [])
              | _ => ((num, NONE), deps)
          in
            case role of
              "definition" =>
              (case phi of
                 AConn (AIff, [phi1 as AAtom _, phi2]) =>
                 Definition (name, phi1, phi2)
               | AAtom (ATerm ("equal", _)) =>
                 (* Vampire's equality proxy axiom *)
                 Inference (name, phi, map (rpair NONE) deps)
               | _ => raise UNRECOGNIZED_ATP_PROOF ())
            | _ => Inference (name, phi, map (rpair NONE) deps)
          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

fun resolve_spass_num spass_names num =
  case Int.fromString num of
    SOME j => if j > 0 andalso j <= Vector.length spass_names then
                SOME (Vector.sub (spass_names, j - 1))
              else
                NONE
  | NONE => NONE

(* Syntax: <num>[0:<inference><annotations>]
   <atoms> || <atoms> -> <atoms>. *)
fun parse_spass_line spass_names x =
  (scan_general_id --| $$ "[" --| $$ "0" --| $$ ":" --| Symbol.scan_id
     -- parse_spass_annotations --| $$ "]" -- parse_horn_clause --| $$ "."
   >> (fn ((num, deps), u) =>
          Inference ((num, resolve_spass_num spass_names num), u,
                     map (swap o `(resolve_spass_num spass_names)) deps))) x

fun parse_line problem spass_names =
  parse_tstp_line problem || parse_spass_line spass_names
fun parse_proof problem spass_names tstp =
  tstp |> strip_spaces_except_between_idents
       |> raw_explode
       |> Scan.finite Symbol.stopper
              (Scan.error (!! (fn _ => raise UNRECOGNIZED_ATP_PROOF ())
                              (Scan.repeat1 (parse_line problem spass_names))))
       |> fst

(** SPASS's FLOTTER hack **)

(* This is a hack required for keeping track of facts after they have been
   clausified by SPASS's FLOTTER preprocessor. The "ATP/scripts/spass" script is
   also part of this hack. *)

val set_ClauseFormulaRelationN = "set_ClauseFormulaRelation"

fun extract_clause_sequence output =
  let
    val tokens_of = String.tokens (not o Char.isAlphaNum)
    fun extract_num ("clause" :: (ss as _ :: _)) = Int.fromString (List.last ss)
      | extract_num _ = NONE
  in output |> split_lines |> map_filter (extract_num o tokens_of) end

fun is_head_digit s = Char.isDigit (String.sub (s, 0))
val scan_integer = Scan.many1 is_head_digit >> (the o Int.fromString o implode)

val parse_clause_formula_pair =
  $$ "(" |-- scan_integer --| $$ ","
  -- (Symbol.scan_id ::: Scan.repeat ($$ "," |-- Symbol.scan_id)) --| $$ ")"
  --| Scan.option ($$ ",")
val parse_clause_formula_relation =
  Scan.this_string set_ClauseFormulaRelationN |-- $$ "("
  |-- Scan.repeat parse_clause_formula_pair
val extract_clause_formula_relation =
  Substring.full #> Substring.position set_ClauseFormulaRelationN
  #> snd #> Substring.position "." #> fst #> Substring.string
  #> raw_explode #> filter_out Symbol.is_blank #> parse_clause_formula_relation
  #> fst

fun extract_spass_name_vector output =
  (if String.isSubstring set_ClauseFormulaRelationN output then
     let
       val num_seq = extract_clause_sequence output
       val name_map = extract_clause_formula_relation output
       val name_seq = num_seq |> map (these o AList.lookup (op =) name_map)
     in name_seq end
   else
     [])
  |> Vector.fromList

fun atp_proof_from_tstplike_proof _ _ "" = []
  | atp_proof_from_tstplike_proof problem output tstp =
    tstp ^ "$" (* the $ sign acts as a sentinel (FIXME: needed?) *)
    |> parse_proof problem (extract_spass_name_vector output)
    |> sort (step_name_ord o pairself step_name)

fun clean_up_dependencies _ [] = []
  | clean_up_dependencies seen ((step as Definition (name, _, _)) :: steps) =
    step :: clean_up_dependencies (name :: seen) steps
  | clean_up_dependencies seen (Inference (name, u, deps) :: steps) =
    Inference (name, u,
               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_term f (ATerm (s, ts)) =
  ATerm (f s, map (map_term_names_in_term f) ts)
fun map_term_names_in_formula f (AQuant (q, xs, phi)) =
    AQuant (q, xs, map_term_names_in_formula f phi)
  | map_term_names_in_formula f (AConn (c, phis)) =
    AConn (c, map (map_term_names_in_formula f) phis)
  | map_term_names_in_formula f (AAtom t) = AAtom (map_term_names_in_term f t)
fun map_term_names_in_step f (Definition (name, phi1, phi2)) =
    Definition (name, map_term_names_in_formula f phi1,
                map_term_names_in_formula f phi2)
  | map_term_names_in_step f (Inference (name, phi, deps)) =
    Inference (name, map_term_names_in_formula f phi, deps)
fun map_term_names_in_atp_proof f = map (map_term_names_in_step f)

fun nasty_name pool s = s |> Symtab.lookup pool |> the_default s
fun nasty_atp_proof pool =
  if Symtab.is_empty pool then I
  else map_term_names_in_atp_proof (nasty_name pool)

end;