(*  Title:      HOL/Tools/Sledgehammer/sledgehammer_proof_reconstruct.ML

    Author:     Lawrence C Paulson and Claire Quigley, Cambridge University Computer Laboratory

Transfer of proofs from external provers.

*)

signature SLEDGEHAMMER_PROOF_RECONSTRUCT =

sig

  val chained_hint: string

  val invert_const: string -> string

  val invert_type_const: string -> string

  val num_typargs: theory -> string -> int

  val make_tvar: string -> typ

  val strip_prefix: string -> string -> string option

  val setup: theory -> theory

  val is_proof_well_formed: string -> bool

  val metis_lemma_list: bool -> string ->

    string * string vector * (int * int) * Proof.context * thm * int -> string * string list

  val structured_isar_proof: string ->

    string * string vector * (int * int) * Proof.context * thm * int -> string * string list

end;

structure Sledgehammer_Proof_Reconstruct : SLEDGEHAMMER_PROOF_RECONSTRUCT =

struct

open Sledgehammer_FOL_Clause

open Sledgehammer_Fact_Preprocessor

val trace_proof_path = Path.basic "atp_trace";

fun trace_proof_msg f =

  if !trace then File.append (File.tmp_path trace_proof_path) (f ()) else ();

fun string_of_thm ctxt = PrintMode.setmp [] (Display.string_of_thm ctxt);

(*For generating structured proofs: keep every nth proof line*)

val (modulus, modulus_setup) = Attrib.config_int "sledgehammer_modulus" 1;

(*Indicates whether to include sort information in generated proofs*)

val (recon_sorts, recon_sorts_setup) = Attrib.config_bool "sledgehammer_sorts" true;

val setup = modulus_setup #> recon_sorts_setup;

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

(*Syntax trees, either termlist or formulae*)

datatype stree = Int of int | Br of string * stree list;

fun atom x = Br(x,[]);

fun scons (x,y) = Br("cons", [x,y]);

val listof = List.foldl scons (atom "nil");

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

val quoted = $$"'" |-- Scan.repeat (~$$"'") --| $$"'" >> implode;

(*Intended for $true and $false*)

fun tf s = "c_" ^ str (Char.toUpper (String.sub(s,0))) ^ String.extract(s,1,NONE);

val truefalse = $$"$" |-- Symbol.scan_id >> (atom o tf);

(*Integer constants, typically proof line numbers*)

fun is_digit s = Char.isDigit (String.sub(s,0));

val integer = Scan.many1 is_digit >> (the o Int.fromString o implode);

(*Generalized FO terms, which include filenames, numbers, etc.*)

fun termlist x = (term ::: Scan.repeat ($$"," |-- term)) x

and term x = (quoted >> atom || integer>>Int || truefalse ||

              Symbol.scan_id -- Scan.optional ($$"(" |-- termlist --| $$")") [] >> Br ||

              $$"(" |-- term --| $$")" ||

              $$"[" |-- Scan.optional termlist [] --| $$"]" >> listof) x;

fun negate t = Br("c_Not", [t]);

fun equate (t1,t2) = Br("c_equal", [t1,t2]);

(*Apply equal or not-equal to a term*)

fun syn_equal (t, NONE) = t

  | syn_equal (t1, SOME (NONE, t2)) = equate (t1,t2)

  | syn_equal (t1, SOME (SOME _, t2)) = negate (equate (t1,t2));

(*Literals can involve negation, = and !=.*)

fun literal x = ($$"~" |-- literal >> negate ||

                 (term -- Scan.option (Scan.option ($$"!") --| $$"=" -- term) >> syn_equal)) x;

val literals = literal ::: Scan.repeat ($$"|" |-- literal);

(*Clause: a list of literals separated by the disjunction sign*)

val clause = $$"(" |-- literals --| $$")" || Scan.single literal;

val annotations = $$"," |-- term -- Scan.option ($$"," |-- termlist);

(*<cnf_annotated> ::= cnf(<name>,<formula_role>,<cnf_formula><annotations>).

  The <name> could be an identifier, but we assume integers.*)

val tstp_line = (Scan.this_string "cnf" -- $$"(") |--

                integer --| $$"," -- Symbol.scan_id --| $$"," --

                clause -- Scan.option annotations --| $$ ")";

(**** INTERPRETATION OF TSTP SYNTAX TREES ****)

exception STREE of stree;

(*If string s has the prefix s1, return the result of deleting it.*)

fun strip_prefix s1 s =

  if String.isPrefix s1 s

  then SOME (undo_ascii_of (String.extract (s, size s1, NONE)))

  else NONE;

(*Invert the table of translations between Isabelle and ATPs*)

val type_const_trans_table_inv =

      Symtab.make (map swap (Symtab.dest type_const_trans_table));

fun invert_type_const c =

    case Symtab.lookup type_const_trans_table_inv c of

        SOME c' => c'

      | NONE => c;

fun make_tvar b = TVar(("'" ^ b, 0), HOLogic.typeS);

fun make_var (b,T) = Var((b,0),T);

(*Type variables are given the basic sort, HOL.type. Some will later be constrained

  by information from type literals, or by type inference.*)

fun type_of_stree t =

  case t of

      Int _ => raise STREE t

    | Br (a,ts) =>

        let val Ts = map type_of_stree ts

        in

          case strip_prefix tconst_prefix a of

              SOME b => Type(invert_type_const b, Ts)

            | NONE =>

                if not (null ts) then raise STREE t  (*only tconsts have type arguments*)

                else

                case strip_prefix tfree_prefix a of

                    SOME b => TFree("'" ^ b, HOLogic.typeS)

                  | NONE =>

                case strip_prefix tvar_prefix a of

                    SOME b => make_tvar b

                  | NONE => make_tvar a   (*Variable from the ATP, say X1*)

        end;

(*Invert the table of translations between Isabelle and ATPs*)

val const_trans_table_inv =

      Symtab.update ("fequal", "op =")

        (Symtab.make (map swap (Symtab.dest const_trans_table)));

fun invert_const c =

    case Symtab.lookup const_trans_table_inv c of

        SOME c' => c'

      | NONE => c;

(*The number of type arguments of a constant, zero if it's monomorphic*)

fun num_typargs thy s = length (Sign.const_typargs thy (s, Sign.the_const_type thy s));

(*Generates a constant, given its type arguments*)

fun const_of thy (a,Ts) = Const(a, Sign.const_instance thy (a,Ts));

(*First-order translation. No types are known for variables. HOLogic.typeT should allow

  them to be inferred.*)

fun term_of_stree args thy t =

  case t of

      Int _ => raise STREE t

    | Br ("hBOOL",[t]) => term_of_stree [] thy t  (*ignore hBOOL*)

    | Br ("hAPP",[t,u]) => term_of_stree (u::args) thy t

    | Br (a,ts) =>

        case strip_prefix const_prefix a of

            SOME "equal" =>

              list_comb(Const (@{const_name "op ="}, HOLogic.typeT), List.map (term_of_stree [] thy) ts)

          | SOME b =>

              let val c = invert_const b

                  val nterms = length ts - num_typargs thy c

                  val us = List.map (term_of_stree [] thy) (List.take(ts,nterms) @ args)

                  (*Extra args from hAPP come AFTER any arguments given directly to the

                    constant.*)

                  val Ts = List.map type_of_stree (List.drop(ts,nterms))

              in  list_comb(const_of thy (c, Ts), us)  end

          | NONE => (*a variable, not a constant*)

              let val T = HOLogic.typeT

                  val opr = (*a Free variable is typically a Skolem function*)

                    case strip_prefix fixed_var_prefix a of

                        SOME b => Free(b,T)

                      | NONE =>

                    case strip_prefix schematic_var_prefix a of

                        SOME b => make_var (b,T)

                      | NONE => make_var (a,T)    (*Variable from the ATP, say X1*)

              in  list_comb (opr, List.map (term_of_stree [] thy) (ts@args))  end;

(*Type class literal applied to a type. Returns triple of polarity, class, type.*)

fun constraint_of_stree pol (Br("c_Not",[t])) = constraint_of_stree (not pol) t

  | constraint_of_stree pol t = case t of

        Int _ => raise STREE t

      | Br (a,ts) =>

            (case (strip_prefix class_prefix a, map type_of_stree ts) of

                 (SOME b, [T]) => (pol, b, T)

               | _ => raise STREE t);

(** Accumulate type constraints in a clause: negative type literals **)

fun addix (key,z)  = Vartab.map_default (key,[]) (cons z);

fun add_constraint ((false, cl, TFree(a,_)), vt) = addix ((a,~1),cl) vt

  | add_constraint ((false, cl, TVar(ix,_)), vt) = addix (ix,cl) vt

  | add_constraint (_, vt) = vt;

(*False literals (which E includes in its proofs) are deleted*)

val nofalses = filter (not o equal HOLogic.false_const);

(*Final treatment of the list of "real" literals from a clause.*)

fun finish [] = HOLogic.true_const  (*No "real" literals means only type information*)

  | finish lits =

      case nofalses lits of

          [] => HOLogic.false_const  (*The empty clause, since we started with real literals*)

        | xs => foldr1 HOLogic.mk_disj (rev xs);

(*Accumulate sort constraints in vt, with "real" literals in lits.*)

fun lits_of_strees _ (vt, lits) [] = (vt, finish lits)

  | lits_of_strees ctxt (vt, lits) (t::ts) =

      lits_of_strees ctxt (add_constraint (constraint_of_stree true t, vt), lits) ts

      handle STREE _ =>

      lits_of_strees ctxt (vt, term_of_stree [] (ProofContext.theory_of ctxt) t :: lits) ts;

(*Update TVars/TFrees with detected sort constraints.*)

fun fix_sorts vt =

  let fun tysubst (Type (a, Ts)) = Type (a, map tysubst Ts)

        | tysubst (TVar (xi, s)) = TVar (xi, the_default s (Vartab.lookup vt xi))

        | tysubst (TFree (x, s)) = TFree (x, the_default s (Vartab.lookup vt (x, ~1)))

      fun tmsubst (Const (a, T)) = Const (a, tysubst T)

        | tmsubst (Free (a, T)) = Free (a, tysubst T)

        | tmsubst (Var (xi, T)) = Var (xi, tysubst T)

        | tmsubst (t as Bound _) = t

        | tmsubst (Abs (a, T, t)) = Abs (a, tysubst T, tmsubst t)

        | tmsubst (t $ u) = tmsubst t $ tmsubst u;

  in fn t => if Vartab.is_empty vt then t else tmsubst t end;

(*Interpret a list of syntax trees as a clause, given by "real" literals and sort constraints.

  vt0 holds the initial sort constraints, from the conjecture clauses.*)

fun clause_of_strees ctxt vt0 ts =

  let val (vt, dt) = lits_of_strees ctxt (vt0,[]) ts in

    singleton (Syntax.check_terms ctxt) (TypeInfer.constrain HOLogic.boolT (fix_sorts vt dt))

  end;

fun gen_all_vars t = fold_rev Logic.all (OldTerm.term_vars t) t;

fun ints_of_stree_aux (Int n, ns) = n::ns

  | ints_of_stree_aux (Br(_,ts), ns) = List.foldl ints_of_stree_aux ns ts;

fun ints_of_stree t = ints_of_stree_aux (t, []);

fun decode_tstp vt0 (name, role, ts, annots) ctxt =

  let val deps = case annots of NONE => [] | SOME (source,_) => ints_of_stree source

      val cl = clause_of_strees ctxt vt0 ts

  in  ((name, role, cl, deps), fold Variable.declare_term (OldTerm.term_frees cl) ctxt)  end;

fun dest_tstp ((((name, role), ts), annots), chs) =

  case chs of

          "."::_ => (name, role, ts, annots)

        | _ => error ("TSTP line not terminated by \".\": " ^ implode chs);

(** Global sort constraints on TFrees (from tfree_tcs) are positive unit clauses. **)

fun add_tfree_constraint ((true, cl, TFree(a,_)), vt) = addix ((a,~1),cl) vt

  | add_tfree_constraint (_, vt) = vt;

fun tfree_constraints_of_clauses vt [] = vt

  | tfree_constraints_of_clauses vt ([lit]::tss) =

      (tfree_constraints_of_clauses (add_tfree_constraint (constraint_of_stree true lit, vt)) tss

       handle STREE _ => (*not a positive type constraint: ignore*)

       tfree_constraints_of_clauses vt tss)

  | tfree_constraints_of_clauses vt (_::tss) = tfree_constraints_of_clauses vt tss;

(**** Translation of TSTP files to Isar Proofs ****)

fun decode_tstp_list ctxt tuples =

  let val vt0 = tfree_constraints_of_clauses Vartab.empty (map #3 tuples)

  in  #1 (fold_map (decode_tstp vt0) tuples ctxt) end;

(** Finding a matching assumption. The literals may be permuted, and variable names

    may disagree. We have to try all combinations of literals (quadratic!) and

    match up the variable names consistently. **)

fun strip_alls_aux n (Const(@{const_name all}, _)$Abs(a,T,t))  =

      strip_alls_aux (n+1) (subst_bound (Var ((a,n), T), t))

  | strip_alls_aux _ t  =  t;

val strip_alls = strip_alls_aux 0;

exception MATCH_LITERAL;

(*Ignore types: they are not to be trusted...*)

fun match_literal (t1$u1) (t2$u2) env =

      match_literal t1 t2 (match_literal u1 u2 env)

  | match_literal (Abs (_,_,t1)) (Abs (_,_,t2)) env =

      match_literal t1 t2 env

  | match_literal (Bound i1) (Bound i2) env =

      if i1=i2 then env else raise MATCH_LITERAL

  | match_literal (Const(a1,_)) (Const(a2,_)) env =

      if a1=a2 then env else raise MATCH_LITERAL

  | match_literal (Free(a1,_)) (Free(a2,_)) env =

      if a1=a2 then env else raise MATCH_LITERAL

  | match_literal (Var(ix1,_)) (Var(ix2,_)) env = insert (op =) (ix1,ix2) env

  | match_literal _ _ _ = raise MATCH_LITERAL;

(*Checking that all variable associations are unique. The list env contains no

  repetitions, but does it contain say (x,y) and (y,y)? *)

fun good env =

  let val (xs,ys) = ListPair.unzip env

  in  not (has_duplicates (op=) xs orelse has_duplicates (op=) ys)  end;

(*Match one list of literals against another, ignoring types and the order of

  literals. Sorting is unreliable because we don't have types or variable names.*)

fun matches_aux _ [] [] = true

  | matches_aux env (lit::lits) ts =

      let fun match1 us [] = false

            | match1 us (t::ts) =

                let val env' = match_literal lit t env

                in  (good env' andalso matches_aux env' lits (us@ts)) orelse

                    match1 (t::us) ts

                end

                handle MATCH_LITERAL => match1 (t::us) ts

      in  match1 [] ts  end;

(*Is this length test useful?*)

fun matches (lits1,lits2) =

  length lits1 = length lits2  andalso

  matches_aux [] (map Envir.eta_contract lits1) (map Envir.eta_contract lits2);

fun permuted_clause t =

  let val lits = HOLogic.disjuncts t

      fun perm [] = NONE

        | perm (ctm::ctms) =

            if matches (lits, HOLogic.disjuncts (HOLogic.dest_Trueprop (strip_alls ctm)))

            then SOME ctm else perm ctms

  in perm end;

(*ctms is a list of conjecture clauses as yielded by Isabelle. Those returned by the

  ATP may have their literals reordered.*)

fun isar_proof_body ctxt ctms =

  let

    val _ = trace_proof_msg (K "\n\nisar_proof_body: start\n")

    val string_of_term = PrintMode.setmp [] (Syntax.string_of_term ctxt)

    fun have_or_show "show" _ = "show \""

      | have_or_show have lname = have ^ " " ^ lname ^ ": \""

    fun do_line _ (lname, t, []) =

       (* No deps: it's a conjecture clause, with no proof. *)

       (case permuted_clause t ctms of

          SOME u => "assume " ^ lname ^ ": \"" ^ string_of_term u ^ "\"\n"

        | NONE => raise TERM ("Sledgehammer_Proof_Reconstruct.isar_proof_body",

                              [t]))

      | do_line have (lname, t, deps) =

        have_or_show have lname ^

        string_of_term (gen_all_vars (HOLogic.mk_Trueprop t)) ^

        "\"\n  by (metis " ^ space_implode " " deps ^ ")\n"

    fun do_lines [(lname, t, deps)] = [do_line "show" (lname, t, deps)]

      | do_lines ((lname, t, deps) :: lines) =

        do_line "have" (lname, t, deps) :: do_lines lines

  in setmp_CRITICAL show_sorts (Config.get ctxt recon_sorts) do_lines end;

fun unequal t (_, t', _) = not (t aconv t');

(*No "real" literals means only type information*)

fun eq_types t = t aconv HOLogic.true_const;

fun replace_dep (old:int, new) dep = if dep=old then new else [dep];

fun replace_deps (old:int, new) (lno, t, deps) =

      (lno, t, List.foldl (uncurry (union (op =))) [] (map (replace_dep (old, new)) deps));

(*Discard axioms; consolidate adjacent lines that prove the same clause, since they differ

  only in type information.*)

fun add_prfline ((lno, "axiom", t, []), lines) =  (*axioms are not proof lines*)

      if eq_types t (*must be clsrel/clsarity: type information, so delete refs to it*)

      then map (replace_deps (lno, [])) lines

      else

       (case take_prefix (unequal t) lines of

           (_,[]) => lines                  (*no repetition of proof line*)

         | (pre, (lno', _, _) :: post) =>   (*repetition: replace later line by earlier one*)

             pre @ map (replace_deps (lno', [lno])) post)

  | add_prfline ((lno, _, t, []), lines) =  (*no deps: conjecture clause*)

      (lno, t, []) :: lines

  | add_prfline ((lno, _, t, deps), lines) =

      if eq_types t then (lno, t, deps) :: lines

      (*Type information will be deleted later; skip repetition test.*)

      else (*FIXME: Doesn't this code risk conflating proofs involving different types??*)

      case take_prefix (unequal t) lines of

         (_,[]) => (lno, t, deps) :: lines  (*no repetition of proof line*)

       | (pre, (lno', t', _) :: post) =>

           (lno, t', deps) ::               (*repetition: replace later line by earlier one*)

           (pre @ map (replace_deps (lno', [lno])) post);

(*Recursively delete empty lines (type information) from the proof.*)

fun add_nonnull_prfline ((lno, t, []), lines) = (*no dependencies, so a conjecture clause*)

     if eq_types t (*must be type information, tfree_tcs, clsrel, clsarity: delete refs to it*)

     then delete_dep lno lines

     else (lno, t, []) :: lines

  | add_nonnull_prfline ((lno, t, deps), lines) = (lno, t, deps) :: lines

and delete_dep lno lines = List.foldr add_nonnull_prfline [] (map (replace_deps (lno, [])) lines);

fun bad_free (Free (a,_)) = String.isPrefix skolem_prefix a

  | bad_free _ = false;

(*TVars are forbidden in goals. Also, we don't want lines with <2 dependencies.

  To further compress proofs, setting modulus:=n deletes every nth line, and nlines

  counts the number of proof lines processed so far.

  Deleted lines are replaced by their own dependencies. Note that the "add_nonnull_prfline"

  phase may delete some dependencies, hence this phase comes later.*)

fun add_wanted_prfline ctxt ((lno, t, []), (nlines, lines)) =

      (nlines, (lno, t, []) :: lines)   (*conjecture clauses must be kept*)

  | add_wanted_prfline ctxt ((lno, t, deps), (nlines, lines)) =

      if eq_types t orelse not (null (Term.add_tvars t [])) orelse

         exists_subterm bad_free t orelse

         (not (null lines) andalso   (*final line can't be deleted for these reasons*)

          (length deps < 2 orelse nlines mod (Config.get ctxt modulus) <> 0))

      then (nlines+1, map (replace_deps (lno, deps)) lines) (*Delete line*)

      else (nlines+1, (lno, t, deps) :: lines);

(*Replace numeric proof lines by strings, either from thm_names or sequential line numbers*)

fun stringify_deps thm_names deps_map [] = []

  | stringify_deps thm_names deps_map ((lno, t, deps) :: lines) =

      if lno <= Vector.length thm_names  (*axiom*)

      then (Vector.sub(thm_names,lno-1), t, []) :: stringify_deps thm_names deps_map lines

      else let val lname = Int.toString (length deps_map)

               fun fix lno = if lno <= Vector.length thm_names

                             then SOME(Vector.sub(thm_names,lno-1))

                             else AList.lookup op= deps_map lno;

           in  (lname, t, map_filter fix (distinct (op=) deps)) ::

               stringify_deps thm_names ((lno,lname)::deps_map) lines

           end;

val proofstart = "proof (neg_clausify)\n";

fun isar_header [] = proofstart

  | isar_header ts = proofstart ^ "fix " ^ space_implode " " ts ^ "\n";

fun isar_proof_from_tstp_file cnfs ctxt th sgno thm_names =

  let

    val _ = trace_proof_msg (K "\nisar_proof_from_tstp_file: start\n")

    val tuples = map (dest_tstp o tstp_line o explode) cnfs

    val _ = trace_proof_msg (fn () =>

      Int.toString (length tuples) ^ " tuples extracted\n")

    val ctxt = ProofContext.set_mode ProofContext.mode_schematic ctxt

    val raw_lines = List.foldr add_prfline [] (decode_tstp_list ctxt tuples)

    val _ = trace_proof_msg (fn () =>

      Int.toString (length raw_lines) ^ " raw_lines extracted\n")

    val nonnull_lines = List.foldr add_nonnull_prfline [] raw_lines

    val _ = trace_proof_msg (fn () =>

      Int.toString (length nonnull_lines) ^ " nonnull_lines extracted\n")

    val (_, lines) = List.foldr (add_wanted_prfline ctxt) (0,[]) nonnull_lines

    val _ = trace_proof_msg (fn () =>

      Int.toString (length lines) ^ " lines extracted\n")

    val (ccls, fixes) = neg_conjecture_clauses ctxt th sgno

    val _ = trace_proof_msg (fn () =>

      Int.toString (length ccls) ^ " conjecture clauses\n")

    val ccls = map forall_intr_vars ccls

    val _ = app (fn th => trace_proof_msg

                              (fn () => "\nccl: " ^ string_of_thm ctxt th)) ccls

    val body = isar_proof_body ctxt (map prop_of ccls)

                               (stringify_deps thm_names [] lines)

    val _ = trace_proof_msg (K "\nisar_proof_from_tstp_file: finishing\n")

  in isar_header (map #1 fixes) ^ implode body ^ "qed\n" end

  handle STREE _ => error "Could not extract proof (ATP output malformed?)";

(*=== EXTRACTING PROOF-TEXT === *)

val begin_proof_strs = ["# SZS output start CNFRefutation.",

  "=========== Refutation ==========",

  "Here is a proof"];

val end_proof_strs = ["# SZS output end CNFRefutation",

  "======= End of refutation =======",

  "Formulae used in the proof"];

fun get_proof_extract proof =

  let

    (*splits to_split by the first possible of a list of splitters*)

    val (begin_string, end_string) =

      (find_first (fn s => String.isSubstring s proof) begin_proof_strs,

      find_first (fn s => String.isSubstring s proof) end_proof_strs)

  in

    if is_none begin_string orelse is_none end_string

    then error "Could not extract proof (no substring indicating a proof)"

    else proof |> first_field (the begin_string) |> the |> snd

               |> first_field (the end_string) |> the |> fst

  end;

(* ==== CHECK IF PROOF WAS SUCCESSFUL === *)

fun is_proof_well_formed proof =

  exists (fn s => String.isSubstring s proof) begin_proof_strs andalso

  exists (fn s => String.isSubstring s proof) end_proof_strs

(* === EXTRACTING LEMMAS === *)

(* lines have the form "cnf(108, axiom, ...",

the number (108) has to be extracted)*)

fun get_step_nums false extract =

  let

    val toks = String.tokens (not o Char.isAlphaNum)

    fun inputno ("cnf" :: ntok :: "axiom" :: _) = Int.fromString ntok

      | inputno ("cnf" :: ntok :: "negated" :: "conjecture" :: _) =

        Int.fromString ntok

      | inputno _ = NONE

    val lines = split_lines extract

  in map_filter (inputno o toks) lines end

(*String contains multiple lines. We want those of the form

  "253[0:Inp] et cetera..."

  A list consisting of the first number in each line is returned. *)

|  get_step_nums true proofextract =

  let val toks = String.tokens (not o Char.isAlphaNum)

  fun inputno (ntok::"0"::"Inp"::_) = Int.fromString ntok

    | inputno _ = NONE

  val lines = split_lines proofextract

  in  map_filter (inputno o toks) lines  end

(*extracting lemmas from tstp-output between the lines from above*)

fun extract_lemmas get_step_nums (proof, thm_names, conj_count, _, _, _) =

  let

    (* get the names of axioms from their numbers*)

    fun get_axiom_names thm_names step_nums =

      let

        val last_axiom = Vector.length thm_names

        fun is_axiom n = n <= last_axiom

        fun is_conj n = n >= fst conj_count andalso

                        n < fst conj_count + snd conj_count

        fun getname i = Vector.sub(thm_names, i-1)

      in

        (sort_distinct string_ord (filter (fn x => x <> "??.unknown")

          (map getname (filter is_axiom step_nums))),

        exists is_conj step_nums)

      end

  in get_axiom_names thm_names (get_step_nums (get_proof_extract proof)) end;

(*Used to label theorems chained into the sledgehammer call*)

val chained_hint = "CHAINED";

val kill_chained = filter_out (curry (op =) chained_hint)

(* metis-command *)

fun metis_line [] = "apply metis"

  | metis_line xs = "apply (metis " ^ space_implode " " xs ^ ")"

(* atp_minimize [atp=<prover>] <lemmas> *)

fun minimize_line _ [] = ""

  | minimize_line name lemmas = "For minimizing the number of lemmas try this command:\n" ^

        Markup.markup Markup.sendback ("atp_minimize [atp = " ^ name ^ "] " ^

                                       space_implode " " (kill_chained lemmas))

fun metis_lemma_list dfg name result =

  let val (lemmas, used_conj) = extract_lemmas (get_step_nums dfg) result in

    (Markup.markup Markup.sendback (metis_line (kill_chained lemmas)) ^ "\n" ^

     minimize_line name lemmas ^

     (if used_conj then

        ""

      else

        "\nWarning: The goal is provable because the context is inconsistent."),

     kill_chained lemmas)

  end;

fun structured_isar_proof name (result as (proof, thm_names, conj_count, ctxt,

                                           goal, subgoalno)) =

  let

    (* Could use "split_lines", but it can return blank lines *)

    val lines = String.tokens (equal #"\n");

    val kill_spaces =

      String.translate (fn c => if Char.isSpace c then "" else str c)

    val extract = get_proof_extract proof

    val cnfs = filter (String.isPrefix "cnf(") (map kill_spaces (lines extract))

    val (one_line_proof, lemma_names) = metis_lemma_list false name result

    val tokens = String.tokens (fn c => c = #" ") one_line_proof

    val isar_proof =

      if member (op =) tokens chained_hint then ""

      else isar_proof_from_tstp_file cnfs ctxt goal subgoalno thm_names

  in

    (one_line_proof ^ "\n\n" ^ Markup.markup Markup.sendback isar_proof,

     lemma_names)

  end

end;