(*  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, 'd) formula = ('a, 'b, 'c, 'd) 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 |

     OldSPASS |

     NoPerl |

     NoLibwwwPerl |

     MalformedInput |

     MalformedOutput |

     Interrupted |

     Crashed |

     InternalError |

     UnknownError of string

   type step_name = string * string list

   datatype 'a step =

     Definition_Step of step_name * 'a * 'a |

     Inference_Step of step_name * 'a * string * step_name list

   type 'a proof = ('a, 'a, ('a, 'a) ho_term, 'a) 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 satallax_coreN : string

   val z3_tptp_coreN : string

   val parse_formula :

     string list

     -> (string, 'a, (string, 'a) ho_term, string) formula * string list

   val atp_proof_from_tstplike_proof : string problem -> 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 |

   OldSPASS |

   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 involving [] = ""

   | involving ss =

     "involving " ^ space_implode " " (Try.serial_commas "and" (map quote ss)) ^

     " "

 fun string_for_failure Unprovable = "The generated 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 \

     \(or unparsable) 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 generated problem lies outside the prover's scope."

   | string_for_failure OutOfResources = "The prover ran out of resources."

   | string_for_failure OldSPASS =

     "The version of SPASS you are using is obsolete. Please upgrade to \

     \SPASS 3.8ds. To install it, download and extract the package \

     \\"http://www21.in.tum.de/~blanchet/spass-3.8ds.tar.gz\" and add the \

     \\"spass-3.8ds\" directory's absolute path to " ^

     quote (Path.implode (Path.expand (Path.appends

                (Path.variable "ISABELLE_HOME_USER" ::

                 map Path.basic ["etc", "components"])))) ^

     " on a line of its own."

   | 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

          |> perhaps (try (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

 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_Step of step_name * 'a * 'a |

   Inference_Step of step_name * 'a * string * step_name list

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

 fun step_name (Definition_Step (name, _, _)) = name

   | step_name (Inference_Step (name, _, _, _)) = name

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

 (* Strings enclosed in single quotes (e.g., file names) *)

 val scan_general_id =

   $$ "'" |-- Scan.repeat (~$$ "'") --| $$ "'" >> implode

   || Scan.repeat ($$ "$") -- Scan.many1 Symbol.is_letdig

      >> (fn (ss1, ss2) => implode ss1 ^ implode ss2)

 val skip_term =

   let

     fun skip _ accum [] = (accum, [])

       | skip 0 accum (ss as "," :: _) = (accum, ss)

       | skip 0 accum (ss as ")" :: _) = (accum, ss)

       | skip 0 accum (ss as "]" :: _) = (accum, ss)

       | skip n accum ((s as "(") :: ss) = skip (n + 1) (s :: accum) ss

       | skip n accum ((s as "[") :: ss) = skip (n + 1) (s :: accum) ss

       | skip n accum ((s as "]") :: ss) = skip (n - 1) (s :: accum) ss

       | skip n accum ((s as ")") :: ss) = skip (n - 1) (s :: accum) ss

       | skip n accum (s :: ss) = 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)". *)

 val parse_dependency = scan_general_id --| skip_term

 val parse_dependencies =

   parse_dependency ::: Scan.repeat ($$ "," |-- parse_dependency)

 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_term --| $$ "," --| skip_term --| $$ "," --| $$ "["

         -- parse_dependencies --| $$ "]" --| $$ ")"

        >> Inference_Source

    || skip_term >> K dummy_inference) x

 fun list_app (f, args) =

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

 (* We currently ignore TFF and THF types. *)

 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 o apfst (rpair []))) x

 and parse_term x =

   (parse_arg -- Scan.repeat ($$ tptp_app |-- parse_arg) >> 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) =>

           (* We ignore TFF and THF types for now. *)

           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", []), [tm11, tm12]))) (AAtom tm2) =

     is_same_term subst tm1 tm2 orelse

     (comm andalso 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 true [] phi phi' then SOME (ident, phi') 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) |> the_default []

 fun commute_eq (AAtom (ATerm ((s, []), tms))) = AAtom (ATerm ((s, []), rev tms))

   | commute_eq _ = raise Fail "expected equation"

 (* 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_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 problem (mk_anot phi) 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, phi |> find_formula_in_problem problem |> map fst),

                   phi), "", [])

               | Inference_Source (rule, deps) => (((num, []), phi), rule, deps)

             fun mk_step () =

               Inference_Step (name, phi, rule, map (rpair []) deps)

           in

             case role of

               "definition" =>

               (case phi of

                  AConn (AIff, [phi1 as AAtom _, phi2]) =>

                  Definition_Step (name, phi1, phi2)

                | AAtom (ATerm (("equal", []), _)) =>

                  (* Vampire's equality proxy axiom *)

                  Inference_Step (name, 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 --| $$ "[" --| $$ "0" --| $$ ":"

      -- Symbol.scan_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) =>

           Inference_Step ((num, these names), u, rule, map (rpair []) deps))) x

 val satallax_coreN = "__satallax_core" (* arbitrary *)

 val z3_tptp_coreN = "__z3_tptp_core" (* arbitrary *)

 (* Syntax: core(<name>,[<name>,...,<name>]). *)

 fun parse_z3_tptp_line x =

   (scan_general_id --| $$ "," --| $$ "[" -- parse_dependencies --| $$ "]"

    >> (fn (name, names) =>

           Inference_Step (("", name :: names), dummy_phi, z3_tptp_coreN, []))) x

 (* Syntax: <name> *)

 fun parse_satallax_line x =

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

    >> (fn s => Inference_Step ((s, [s]), dummy_phi, satallax_coreN, []))) x

 fun parse_line problem =

   parse_tstp_line problem || parse_spass_line || parse_z3_tptp_line

   || parse_satallax_line

 fun parse_proof problem 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))))

        |> fst

 fun atp_proof_from_tstplike_proof _ "" = []

   | atp_proof_from_tstplike_proof problem tstp =

     tstp ^ "$" (* the $ sign acts as a sentinel (FIXME: needed?) *)

     |> parse_proof problem

     |> sort (step_name_ord o pairself step_name) (* FIXME: needed? *)

 fun clean_up_dependencies _ [] = []

   | clean_up_dependencies seen

                           ((step as Definition_Step (name, _, _)) :: steps) =

     step :: clean_up_dependencies (name :: seen) steps

   | clean_up_dependencies seen (Inference_Step (name, u, rule, deps) :: steps) =

     Inference_Step (name, 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_term f (ATerm ((s, tys), ts)) =

   ATerm ((f s, tys), 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_Step (name, phi1, phi2)) =

     Definition_Step (name, map_term_names_in_formula f phi1,

                      map_term_names_in_formula f phi2)

   | map_term_names_in_step f (Inference_Step (name, phi, rule, deps)) =

     Inference_Step (name, map_term_names_in_formula f phi, rule, 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;