(*  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)

val dummy_phi = AAtom (ATerm ("", []))

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 (dummy_phi, skip 0 ss) end

datatype source =

  File_Source of string * string option |

  Inference_Source of string list

fun parse_dependencies x =

  (scan_general_id ::: Scan.repeat ($$ "," |-- scan_general_id)) x

fun parse_source x =

  (Scan.this_string "file" |-- $$ "(" |-- scan_general_id --

     Scan.option ($$ "," |-- scan_general_id) --| $$ ")"

     >> File_Source

   || (Scan.this_string "inference" |-- $$ "(" |-- scan_general_id

         --| skip_formula --| $$ ",")

       ::: (skip_formula |-- $$ "," |-- $$ "[" |-- parse_dependencies --| $$ "]"

            --| $$ ")")

       >> Inference_Source) 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

val parse_tstp_extra_arguments =

  Scan.optional ($$ "," |-- parse_source

                 --| Scan.option ($$ "," |-- skip_formula))

                (Inference_Source [])

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_Source (_, SOME 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_Source _ =>

                ((num, find_formula_in_problem problem phi), [])

              | Inference_Source deps => ((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 =

  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))

(* Syntax: <name> *)

fun parse_satallax_line x =

  (scan_general_id --| Scan.option ($$ " ")

   >> (fn s => Inference ((s, SOME [s]), dummy_phi, []))) x

fun parse_line problem spass_names =

  parse_tstp_line problem || parse_spass_line spass_names || parse_satallax_line

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;