(*  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 |

     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 * string * 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 |

   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 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 * string * 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 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 ("", [])

 fun parse_dependencies x = (skip_term ::: Scan.repeat ($$ "," |-- skip_term)) 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_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 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_term))

                 dummy_inference

 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_term >> K dummy_phi) -- parse_tstp_extra_arguments

     --| $$ ")" --| $$ "."

    >> (fn (((num, role), phi), deps) =>

           let

             val (name, rule, 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 (rule, deps) => ((num, NONE), rule, 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, rule, map (rpair NONE) deps)

                | _ => raise UNRECOGNIZED_ATP_PROOF ())

             | _ => Inference (name, phi, rule, 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, rule), deps), u) =>

          Inference ((num, resolve_spass_num spass_names num), u, rule,

                     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, rule, deps) :: steps) =

     Inference (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, 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, rule, deps)) =

     Inference (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;