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

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

     Author:     Jasmin Blanchette, TU Muenchen

 Transfer of proofs from external provers.

 *)

 signature SLEDGEHAMMER_PROOF_RECONSTRUCT =

 sig

   type minimize_command = string list -> string

   type name_pool = Sledgehammer_FOL_Clause.name_pool

   val chained_hint: string

   val invert_const: string -> string

   val invert_type_const: string -> string

   val num_type_args: theory -> string -> int

   val strip_prefix: string -> string -> string option

   val metis_line: int -> int -> string list -> string

   val metis_proof_text:

     minimize_command * string * string vector * thm * int

     -> string * string list

   val isar_proof_text:

     name_pool option * bool * bool * int * Proof.context * int list list

     -> minimize_command * string * string vector * thm * int

     -> string * string list

   val proof_text:

     bool -> name_pool option * bool * bool * int * Proof.context * int list list

     -> minimize_command * string * string vector * thm * int

     -> string * string list

 end;

 structure Sledgehammer_Proof_Reconstruct : SLEDGEHAMMER_PROOF_RECONSTRUCT =

 struct

 open Sledgehammer_Util

 open Sledgehammer_FOL_Clause

 open Sledgehammer_HOL_Clause

 open Sledgehammer_Fact_Preprocessor

 type minimize_command = string list -> string

 fun is_ident_char c = Char.isAlphaNum c orelse c = #"_"

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

 (* Hack: Could return false positives (e.g., a user happens to declare a

    constant called "SomeTheory.sko_means_shoe_in_$wedish". *)

 val is_skolem_const_name =

   Long_Name.base_name

   #> String.isPrefix skolem_prefix andf String.isSubstring skolem_infix

 val index_in_shape : int -> int list list -> int =

   find_index o exists o curry (op =)

 fun is_axiom_clause_number thm_names num = num <= Vector.length thm_names

 fun is_conjecture_clause_number conjecture_shape num =

   index_in_shape num conjecture_shape >= 0

 fun ugly_name NONE s = s

   | ugly_name (SOME the_pool) s =

     case Symtab.lookup (snd the_pool) s of

       SOME s' => s'

     | NONE => s

 fun smart_lambda v t =

   Abs (case v of

          Const (s, _) =>

          List.last (space_explode skolem_infix (Long_Name.base_name s))

        | Var ((s, _), _) => s

        | Free (s, _) => s

        | _ => "", fastype_of v, abstract_over (v, t))

 fun forall_of v t = HOLogic.all_const (fastype_of v) $ smart_lambda v t

 datatype ('a, 'b, 'c, 'd, 'e) raw_step =

   Definition of 'a * 'b * 'c |

   Inference of 'a * 'd * 'e list

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

 fun strip_spaces_in_list [] = ""

   | strip_spaces_in_list [c1] = if Char.isSpace c1 then "" else str c1

   | strip_spaces_in_list [c1, c2] =

     strip_spaces_in_list [c1] ^ strip_spaces_in_list [c2]

   | strip_spaces_in_list (c1 :: c2 :: c3 :: cs) =

     if Char.isSpace c1 then

       strip_spaces_in_list (c2 :: c3 :: cs)

     else if Char.isSpace c2 then

       if Char.isSpace c3 then

         strip_spaces_in_list (c1 :: c3 :: cs)

       else

         str c1 ^ (if forall is_ident_char [c1, c3] then " " else "") ^

         strip_spaces_in_list (c3 :: cs)

     else

       str c1 ^ strip_spaces_in_list (c2 :: c3 :: cs)

 val strip_spaces = strip_spaces_in_list o String.explode

 (* Syntax trees, either term list or formulae *)

 datatype node = IntLeaf of int | StrNode of string * node list

 fun str_leaf s = StrNode (s, [])

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

 val slist_of = List.foldl scons (str_leaf "nil")

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

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

 (*Integer constants, typically proof line numbers*)

 val parse_integer = Scan.many1 is_head_digit >> (the o Int.fromString o implode)

 val parse_dollar_name =

   Scan.repeat ($$ "$") -- Symbol.scan_id >> (fn (ss, s) => implode ss ^ s)

 (* needed for SPASS's output format *)

 fun repair_name _ "$true" = "c_True"

   | repair_name _ "$false" = "c_False"

   | repair_name _ "$$e" = "c_equal" (* seen in Vampire 11 proofs *)

   | repair_name _ "equal" = "c_equal" (* probably not needed *)

   | repair_name pool s = ugly_name pool s

 (* Generalized first-order terms, which include file names, numbers, etc. *)

 (* The "x" argument is not strictly necessary, but without it Poly/ML loops

    forever at compile time. *)

 fun parse_term pool x =

      (parse_quoted >> str_leaf

    || parse_integer >> IntLeaf

    || (parse_dollar_name >> repair_name pool)

       -- Scan.optional ($$ "(" |-- parse_terms pool --| $$ ")") [] >> StrNode

    || $$ "(" |-- parse_term pool --| $$ ")"

    || $$ "[" |-- Scan.optional (parse_terms pool) [] --| $$ "]" >> slist_of) x

 and parse_terms pool x =

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

 fun negate_node u = StrNode ("c_Not", [u])

 fun equate_nodes u1 u2 = StrNode ("c_equal", [u1, u2])

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

 fun repair_predicate_term (u, NONE) = u

   | repair_predicate_term (u1, SOME (NONE, u2)) = equate_nodes u1 u2

   | repair_predicate_term (u1, SOME (SOME _, u2)) =

     negate_node (equate_nodes u1 u2)

 fun parse_predicate_term pool =

   parse_term pool -- Scan.option (Scan.option ($$ "!") --| $$ "="

                                   -- parse_term pool)

   >> repair_predicate_term

 fun parse_literal pool x =

   ($$ "~" |-- parse_literal pool >> negate_node || parse_predicate_term pool) x

 fun parse_literals pool =

   parse_literal pool ::: Scan.repeat ($$ "|" |-- parse_literal pool)

 fun parse_parenthesized_literals pool =

   $$ "(" |-- parse_literals pool --| $$ ")" || parse_literals pool

 fun parse_clause pool =

   parse_parenthesized_literals pool

     ::: Scan.repeat ($$ "|" |-- parse_parenthesized_literals pool)

   >> List.concat

 fun ints_of_node (IntLeaf n) = cons n

   | ints_of_node (StrNode (_, us)) = fold ints_of_node us

 val parse_tstp_annotations =

   Scan.optional ($$ "," |-- parse_term NONE

                    --| Scan.option ($$ "," |-- parse_terms NONE)

                  >> (fn source => ints_of_node source [])) []

 fun parse_definition pool =

   $$ "(" |-- parse_literal NONE --| Scan.this_string "<=>"

   -- parse_clause pool --| $$ ")"

 (* Syntax: cnf(<num>, <formula_role>, <cnf_formula> <annotations>).

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

 fun finish_tstp_definition_line (num, (u, us)) = Definition (num, u, us)

 fun finish_tstp_inference_line ((num, us), deps) = Inference (num, us, deps)

 fun parse_tstp_line pool =

      ((Scan.this_string "fof" -- $$ "(") |-- parse_integer --| $$ ","

        --| Scan.this_string "definition" --| $$ "," -- parse_definition pool

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

       >> finish_tstp_definition_line)

   || ((Scan.this_string "cnf" -- $$ "(") |-- parse_integer --| $$ ","

        --| Symbol.scan_id --| $$ "," -- parse_clause pool

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

       >> finish_tstp_inference_line)

 (**** 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 = parse_integer --| $$ "." --| parse_integer

 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_predicate_term pool =

   parse_predicate_term pool --| Scan.repeat ($$ "*" || $$ "+" || $$ " ")

 fun parse_horn_clause pool =

   Scan.repeat (parse_decorated_predicate_term pool) --| $$ "|" --| $$ "|"

     -- Scan.repeat (parse_decorated_predicate_term pool) --| $$ "-" --| $$ ">"

     -- Scan.repeat (parse_decorated_predicate_term pool)

   >> (fn (([], []), []) => [str_leaf "c_False"]

        | ((clauses1, clauses2), clauses3) =>

          map negate_node (clauses1 @ clauses2) @ clauses3)

 (* Syntax: <num>[0:<inference><annotations>]

    <cnf_formulas> || <cnf_formulas> -> <cnf_formulas>. *)

 fun finish_spass_line ((num, deps), us) = Inference (num, us, deps)

 fun parse_spass_line pool =

   parse_integer --| $$ "[" --| $$ "0" --| $$ ":" --| Symbol.scan_id

   -- parse_spass_annotations --| $$ "]" -- parse_horn_clause pool --| $$ "."

   >> finish_spass_line

 fun parse_line pool = parse_tstp_line pool || parse_spass_line pool

 fun parse_lines pool = Scan.repeat1 (parse_line pool)

 fun parse_proof pool =

   fst o Scan.finite Symbol.stopper

             (Scan.error (!! (fn _ => raise Fail "unrecognized ATP output")

                             (parse_lines pool)))

   o explode o strip_spaces

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

 exception NODE of node list

 (*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;

 (* 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_from_node _ (u as IntLeaf _) = raise NODE [u]

   | type_from_node tfrees (u as StrNode (a, us)) =

     let val Ts = map (type_from_node tfrees) us in

       case strip_prefix tconst_prefix a of

         SOME b => Type (invert_type_const b, Ts)

       | NONE =>

         if not (null us) then

           raise NODE [u]  (* only "tconst"s have type arguments *)

         else case strip_prefix tfree_prefix a of

           SOME b =>

           let val s = "'" ^ b in

             TFree (s, AList.lookup (op =) tfrees s |> the_default HOLogic.typeS)

           end

         | NONE =>

           case strip_prefix tvar_prefix a of

             SOME b => TVar (("'" ^ b, 0), HOLogic.typeS)

           | NONE =>

             (* Variable from the ATP, say "X1" *)

             Type_Infer.param 0 (a, HOLogic.typeS)

     end

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

 val const_trans_table_inv =

   Symtab.update ("fequal", @{const_name "op ="})

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

 fun invert_const c = c |> Symtab.lookup const_trans_table_inv |> the_default c

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

 fun num_type_args thy s =

   length (Sign.const_typargs thy (s, Sign.the_const_type thy s))

 fun fix_atp_variable_name s =

   let

     fun subscript_name s n = s ^ nat_subscript n

     val s = String.map Char.toLower s

   in

     case space_explode "_" s of

       [_] => (case take_suffix Char.isDigit (String.explode s) of

                 (cs1 as _ :: _, cs2 as _ :: _) =>

                 subscript_name (String.implode cs1)

                                (the (Int.fromString (String.implode cs2)))

               | (_, _) => s)

     | [s1, s2] => (case Int.fromString s2 of

                      SOME n => subscript_name s1 n

                    | NONE => s)

     | _ => s

   end

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

    should allow them to be inferred.*)

 fun term_from_node thy full_types tfrees =

   let

     fun aux opt_T args u =

       case u of

         IntLeaf _ => raise NODE [u]

       | StrNode ("hBOOL", [u1]) => aux (SOME @{typ bool}) [] u1

       | StrNode ("hAPP", [u1, u2]) => aux opt_T (u2 :: args) u1

       | StrNode ("c_Not", [u1]) => @{const Not} $ aux (SOME @{typ bool}) [] u1

       | StrNode (a, us) =>

         if a = type_wrapper_name then

           case us of

             [term_u, typ_u] =>

             aux (SOME (type_from_node tfrees typ_u)) args term_u

           | _ => raise NODE us

         else case strip_prefix const_prefix a of

           SOME "equal" =>

           list_comb (Const (@{const_name "op ="}, HOLogic.typeT),

                      map (aux NONE []) us)

         | SOME b =>

           let

             val c = invert_const b

             val num_type_args = num_type_args thy c

             val actual_num_type_args = if full_types then 0 else num_type_args

             val num_term_args = length us - actual_num_type_args

             val ts = map (aux NONE []) (take num_term_args us @ args)

             val t =

               Const (c, if full_types then

                           case opt_T of

                             SOME T => map fastype_of ts ---> T

                           | NONE =>

                             if num_type_args = 0 then

                               Sign.const_instance thy (c, [])

                             else

                               raise Fail ("no type information for " ^ quote c)

                         else

                           (* Extra args from "hAPP" come after any arguments

                              given directly to the constant. *)

                           Sign.const_instance thy (c,

                                     map (type_from_node tfrees)

                                         (drop num_term_args us)))

           in list_comb (t, ts) end

         | NONE => (* a free or schematic variable *)

           let

             val ts = map (aux NONE []) (us @ args)

             val T = map fastype_of ts ---> HOLogic.typeT

             val t =

               case strip_prefix fixed_var_prefix a of

                 SOME b => Free (b, T)

               | NONE =>

                 case strip_prefix schematic_var_prefix a of

                   SOME b => Var ((b, 0), T)

                 | NONE =>

                   (* Variable from the ATP, say "X1" *)

                   Var ((fix_atp_variable_name a, 0), T)

           in list_comb (t, ts) end

   in aux end

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

    type. *)

 fun type_constraint_from_node pos tfrees (StrNode ("c_Not", [u])) =

     type_constraint_from_node (not pos) tfrees u

   | type_constraint_from_node pos tfrees u = case u of

         IntLeaf _ => raise NODE [u]

       | StrNode (a, us) =>

             (case (strip_prefix class_prefix a,

                    map (type_from_node tfrees) us) of

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

                | _ => raise NODE [u])

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

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

 fun add_type_constraint (false, cl, TFree (a ,_)) = add_var ((a, ~1), cl)

   | add_type_constraint (false, cl, TVar (ix, _)) = add_var (ix, cl)

   | add_type_constraint _ = I

 fun is_positive_literal (@{const Not} $ _) = false

   | is_positive_literal t = true

 fun negate_term thy (Const (@{const_name All}, T) $ Abs (s, T', t')) =

     Const (@{const_name Ex}, T) $ Abs (s, T', negate_term thy t')

   | negate_term thy (Const (@{const_name Ex}, T) $ Abs (s, T', t')) =

     Const (@{const_name All}, T) $ Abs (s, T', negate_term thy t')

   | negate_term thy (@{const "op -->"} $ t1 $ t2) =

     @{const "op &"} $ t1 $ negate_term thy t2

   | negate_term thy (@{const "op &"} $ t1 $ t2) =

     @{const "op |"} $ negate_term thy t1 $ negate_term thy t2

   | negate_term thy (@{const "op |"} $ t1 $ t2) =

     @{const "op &"} $ negate_term thy t1 $ negate_term thy t2

   | negate_term _ (@{const Not} $ t) = t

   | negate_term _ t = @{const Not} $ t

 fun clause_for_literals _ [] = HOLogic.false_const

   | clause_for_literals _ [lit] = lit

   | clause_for_literals thy lits =

     case List.partition is_positive_literal lits of

       (pos_lits as _ :: _, neg_lits as _ :: _) =>

       @{const "op -->"}

           $ foldr1 HOLogic.mk_conj (map (negate_term thy) neg_lits)

           $ foldr1 HOLogic.mk_disj pos_lits

     | _ => foldr1 HOLogic.mk_disj lits

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

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

 fun finish_clause _ [] = HOLogic.true_const

   | finish_clause thy lits =

     lits |> filter_out (curry (op =) HOLogic.false_const) |> rev

          |> clause_for_literals thy

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

 fun lits_of_nodes thy full_types tfrees =

   let

     fun aux (vt, lits) [] = (vt, finish_clause thy lits)

       | aux (vt, lits) (u :: us) =

         aux (add_type_constraint

                  (type_constraint_from_node true tfrees u) vt, lits) us

         handle NODE _ =>

                aux (vt, term_from_node thy full_types tfrees (SOME @{typ bool})

                                        [] u :: lits) us

   in aux end

 (* Update TVars with detected sort constraints. It's not totally clear when

    this code is necessary. *)

 fun repair_tvar_sorts vt =

   let

     fun do_type (Type (a, Ts)) = Type (a, map do_type Ts)

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

       | do_type (TFree z) = TFree z

     fun do_term (Const (a, T)) = Const (a, do_type T)

       | do_term (Free (a, T)) = Free (a, do_type T)

       | do_term (Var (xi, T)) = Var (xi, do_type T)

       | do_term (t as Bound _) = t

       | do_term (Abs (a, T, t)) = Abs (a, do_type T, do_term t)

       | do_term (t1 $ t2) = do_term t1 $ do_term t2

   in not (Vartab.is_empty vt) ? do_term end

 fun unskolemize_term t =

   Term.add_consts t []

   |> filter (is_skolem_const_name o fst) |> map Const

   |> rpair t |-> fold forall_of

 val combinator_table =

   [(@{const_name COMBI}, @{thm COMBI_def_raw}),

    (@{const_name COMBK}, @{thm COMBK_def_raw}),

    (@{const_name COMBB}, @{thm COMBB_def_raw}),

    (@{const_name COMBC}, @{thm COMBC_def_raw}),

    (@{const_name COMBS}, @{thm COMBS_def_raw})]

 fun uncombine_term (t1 $ t2) = betapply (pairself uncombine_term (t1, t2))

   | uncombine_term (Abs (s, T, t')) = Abs (s, T, uncombine_term t')

   | uncombine_term (t as Const (x as (s, _))) =

     (case AList.lookup (op =) combinator_table s of

        SOME thm => thm |> prop_of |> specialize_type @{theory} x |> Logic.dest_equals |> snd

      | NONE => t)

   | uncombine_term t = t

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

    sort constraints. "vt" holds the initial sort constraints, from the

    conjecture clauses. *)

 fun clause_of_nodes ctxt full_types tfrees us =

   let

     val thy = ProofContext.theory_of ctxt

     val (vt, t) = lits_of_nodes thy full_types tfrees (Vartab.empty, []) us

   in repair_tvar_sorts vt t end

 fun check_formula ctxt =

   Type_Infer.constrain @{typ bool}

   #> Syntax.check_term (ProofContext.set_mode ProofContext.mode_schematic ctxt)

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

 fun unvarify_term (Var ((s, 0), T)) = Free (s, T)

   | unvarify_term t = raise TERM ("unvarify_term: non-Var", [t])

 fun decode_line full_types tfrees (Definition (num, u, us)) ctxt =

     let

       val t1 = clause_of_nodes ctxt full_types tfrees [u]

       val vars = snd (strip_comb t1)

       val frees = map unvarify_term vars

       val unvarify_args = subst_atomic (vars ~~ frees)

       val t2 = clause_of_nodes ctxt full_types tfrees us

       val (t1, t2) =

         HOLogic.eq_const HOLogic.typeT $ t1 $ t2

         |> unvarify_args |> uncombine_term |> check_formula ctxt

         |> HOLogic.dest_eq

     in

       (Definition (num, t1, t2),

        fold Variable.declare_term (maps OldTerm.term_frees [t1, t2]) ctxt)

     end

   | decode_line full_types tfrees (Inference (num, us, deps)) ctxt =

     let

       val t = us |> clause_of_nodes ctxt full_types tfrees

                  |> unskolemize_term |> uncombine_term |> check_formula ctxt

     in

       (Inference (num, t, deps),

        fold Variable.declare_term (OldTerm.term_frees t) ctxt)

     end

 fun decode_lines ctxt full_types tfrees lines =

   fst (fold_map (decode_line full_types tfrees) lines ctxt)

 fun aint_inference _ (Definition _) = true

   | aint_inference t (Inference (_, t', _)) = not (t aconv t')

 (* No "real" literals means only type information (tfree_tcs, clsrel, or

    clsarity). *)

 val is_only_type_information = curry (op aconv) HOLogic.true_const

 fun replace_one_dep (old, new) dep = if dep = old then new else [dep]

 fun replace_deps_in_line _ (line as Definition _) = line

   | replace_deps_in_line p (Inference (num, t, deps)) =

     Inference (num, t, fold (union (op =) o replace_one_dep p) deps [])

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

   only in type information.*)

 fun add_line _ _ (line as Definition _) lines = line :: lines

   | add_line conjecture_shape thm_names (Inference (num, t, [])) lines =

     (* No dependencies: axiom, conjecture clause, or internal axioms or

        definitions (Vampire). *)

     if is_axiom_clause_number thm_names num then

       (* Axioms are not proof lines. *)

       if is_only_type_information t then

         map (replace_deps_in_line (num, [])) lines

       (* Is there a repetition? If so, replace later line by earlier one. *)

       else case take_prefix (aint_inference t) lines of

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

       | (pre, Inference (num', _, _) :: post) =>

         pre @ map (replace_deps_in_line (num', [num])) post

     else if is_conjecture_clause_number conjecture_shape num then

       Inference (num, t, []) :: lines

     else

       map (replace_deps_in_line (num, [])) lines

   | add_line _ _ (Inference (num, t, deps)) lines =

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

     if is_only_type_information t then

       Inference (num, t, deps) :: lines

     (* Is there a repetition? If so, replace later line by earlier one. *)

     else case take_prefix (aint_inference t) lines of

       (* FIXME: Doesn't this code risk conflating proofs involving different

          types?? *)

        (_, []) => Inference (num, t, deps) :: lines

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

        Inference (num, t', deps) ::

        pre @ map (replace_deps_in_line (num', [num])) post

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

 fun add_nontrivial_line (Inference (num, t, [])) lines =

     if is_only_type_information t then delete_dep num lines

     else Inference (num, t, []) :: lines

   | add_nontrivial_line line lines = line :: lines

 and delete_dep num lines =

   fold_rev add_nontrivial_line (map (replace_deps_in_line (num, [])) lines) []

 (* ATPs sometimes reuse free variable names in the strangest ways. Surprisingly,

    removing the offending lines often does the trick. *)

 fun is_bad_free frees (Free x) = not (member (op =) frees x)

   | is_bad_free _ _ = false

 (* Vampire is keen on producing these. *)

 fun is_trivial_formula (@{const Not} $ (Const (@{const_name "op ="}, _)

                                         $ t1 $ t2)) = (t1 aconv t2)

   | is_trivial_formula t = false

 fun add_desired_line _ _ _ _ _ (line as Definition _) (j, lines) =

     (j, line :: lines)

   | add_desired_line ctxt isar_shrink_factor conjecture_shape thm_names frees

                      (Inference (num, t, deps)) (j, lines) =

     (j + 1,

      if is_axiom_clause_number thm_names num orelse

         is_conjecture_clause_number conjecture_shape num orelse

         (not (is_only_type_information t) andalso

          null (Term.add_tvars t []) andalso

          not (exists_subterm (is_bad_free frees) t) andalso

          not (is_trivial_formula t) andalso

          (null lines orelse (* last line must be kept *)

           (length deps >= 2 andalso j mod isar_shrink_factor = 0))) then

        Inference (num, t, deps) :: lines  (* keep line *)

      else

        map (replace_deps_in_line (num, deps)) lines)  (* drop line *)

 (** EXTRACTING LEMMAS **)

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

    TSTP: Interesting lines have the form "cnf(108, axiom, ...)", where the

    number (108) is extracted.

    SPASS: Lines have the form "108[0:Inp] ...", where the first number (108) is

    extracted. *)

 fun extract_clause_numbers_in_atp_proof atp_proof =

   let

     val tokens_of = String.tokens (not o is_ident_char)

     fun extract_num ("cnf" :: num :: "axiom" :: _) = Int.fromString num

       | extract_num (num :: "0" :: "Inp" :: _) = Int.fromString num

       | extract_num _ = NONE

   in atp_proof |> split_lines |> map_filter (extract_num o tokens_of) end

 (* Used to label theorems chained into the goal. *)

 val chained_hint = "sledgehammer_chained"

 fun apply_command _ 1 = "by "

   | apply_command 1 _ = "apply "

   | apply_command i _ = "prefer " ^ string_of_int i ^ " apply "

 fun metis_command i n [] = apply_command i n ^ "metis"

   | metis_command i n ss =

     apply_command i n ^ "(metis " ^ space_implode " " ss ^ ")"

 fun metis_line i n xs =

   "Try this command: " ^

   Markup.markup Markup.sendback (metis_command i n xs) ^ ".\n" 

 fun minimize_line _ [] = ""

   | minimize_line minimize_command facts =

     case minimize_command facts of

       "" => ""

     | command =>

       "To minimize the number of lemmas, try this command: " ^

       Markup.markup Markup.sendback command ^ ".\n"

 fun metis_proof_text (minimize_command, atp_proof, thm_names, goal, i) =

   let

     val lemmas =

       atp_proof |> extract_clause_numbers_in_atp_proof

                 |> filter (is_axiom_clause_number thm_names)

                 |> map (fn i => Vector.sub (thm_names, i - 1))

                 |> filter_out (fn s => s = "??.unknown" orelse s = chained_hint)

                 |> sort_distinct string_ord

     val n = Logic.count_prems (prop_of goal)

   in (metis_line i n lemmas ^ minimize_line minimize_command lemmas, lemmas) end

 (** Isar proof construction and manipulation **)

 fun merge_fact_sets (ls1, ss1) (ls2, ss2) =

   (union (op =) ls1 ls2, union (op =) ss1 ss2)

 type label = string * int

 type facts = label list * string list

 datatype qualifier = Show | Then | Moreover | Ultimately

 datatype step =

   Fix of (string * typ) list |

   Let of term * term |

   Assume of label * term |

   Have of qualifier list * label * term * byline

 and byline =

   ByMetis of facts |

   CaseSplit of step list list * facts

 fun smart_case_split [] facts = ByMetis facts

   | smart_case_split proofs facts = CaseSplit (proofs, facts)

 val raw_prefix = "X"

 val assum_prefix = "A"

 val fact_prefix = "F"

 fun string_for_label (s, num) = s ^ string_of_int num

 fun add_fact_from_dep thm_names num =

   if is_axiom_clause_number thm_names num then

     apsnd (insert (op =) (Vector.sub (thm_names, num - 1)))

   else

     apfst (insert (op =) (raw_prefix, num))

 fun forall_vars t = fold_rev forall_of (map Var (Term.add_vars t [])) t

 fun step_for_line _ _ (Definition (num, t1, t2)) = Let (t1, t2)

   | step_for_line _ _ (Inference (num, t, [])) = Assume ((raw_prefix, num), t)

   | step_for_line thm_names j (Inference (num, t, deps)) =

     Have (if j = 1 then [Show] else [], (raw_prefix, num),

           forall_vars t,

           ByMetis (fold (add_fact_from_dep thm_names) deps ([], [])))

 fun proof_from_atp_proof pool ctxt full_types tfrees isar_shrink_factor

                          atp_proof conjecture_shape thm_names params frees =

   let

     val lines =

       atp_proof ^ "$" (* the $ sign acts as a sentinel *)

       |> parse_proof pool

       |> decode_lines ctxt full_types tfrees

       |> rpair [] |-> fold_rev (add_line conjecture_shape thm_names)

       |> rpair [] |-> fold_rev add_nontrivial_line

       |> rpair (0, []) |-> fold_rev (add_desired_line ctxt isar_shrink_factor

                                                conjecture_shape thm_names frees)

       |> snd

   in

     (if null params then [] else [Fix params]) @

     map2 (step_for_line thm_names) (length lines downto 1) lines

   end

 val indent_size = 2

 val no_label = ("", ~1)

 fun no_show qs = not (member (op =) qs Show)

 (* When redirecting proofs, we keep information about the labels seen so far in

    the "backpatches" data structure. The first component indicates which facts

    should be associated with forthcoming proof steps. The second component is a

    pair ("keep_ls", "drop_ls"), where "keep_ls" are the labels to keep and

    "drop_ls" are those that should be dropped in a case split. *)

 type backpatches = (label * facts) list * (label list * label list)

 fun used_labels_of_step (Have (_, _, _, by)) =

     (case by of

        ByMetis (ls, _) => ls

      | CaseSplit (proofs, (ls, _)) =>

        fold (union (op =) o used_labels_of) proofs ls)

   | used_labels_of_step _ = []

 and used_labels_of proof = fold (union (op =) o used_labels_of_step) proof []

 fun new_labels_of_step (Fix _) = []

   | new_labels_of_step (Let _) = []

   | new_labels_of_step (Assume (l, _)) = [l]

   | new_labels_of_step (Have (_, l, _, _)) = [l]

 val new_labels_of = maps new_labels_of_step

 val join_proofs =

   let

     fun aux _ [] = NONE

       | aux proof_tail (proofs as (proof1 :: _)) =

         if exists null proofs then

           NONE

         else if forall (curry (op =) (hd proof1) o hd) (tl proofs) then

           aux (hd proof1 :: proof_tail) (map tl proofs)

         else case hd proof1 of

           Have ([], l, t, by) =>

           if forall (fn Have ([], l', t', _) :: _ => (l, t) = (l', t')

                       | _ => false) (tl proofs) andalso

              not (exists (member (op =) (maps new_labels_of proofs))

                          (used_labels_of proof_tail)) then

             SOME (l, t, map rev proofs, proof_tail)

           else

             NONE

         | _ => NONE

   in aux [] o map rev end

 fun case_split_qualifiers proofs =

   case length proofs of

     0 => []

   | 1 => [Then]

   | _ => [Ultimately]

 fun redirect_proof thy conjecture_shape hyp_ts concl_t proof =

   let

     val concl_ls = map (pair raw_prefix) (List.last conjecture_shape)

     fun find_hyp num = nth hyp_ts (index_in_shape num conjecture_shape)

     fun first_pass ([], contra) = ([], contra)

       | first_pass ((step as Fix _) :: proof, contra) =

         first_pass (proof, contra) |>> cons step

       | first_pass ((step as Let _) :: proof, contra) =

         first_pass (proof, contra) |>> cons step

       | first_pass ((step as Assume (l as (_, num), t)) :: proof, contra) =

         if member (op =) concl_ls l then

           first_pass (proof, contra ||> cons step)

         else

           first_pass (proof, contra) |>> cons (Assume (l, find_hyp num))

       | first_pass ((step as Have (qs, l, t, ByMetis (ls, ss))) :: proof,

                     contra) =

         if exists (member (op =) (fst contra)) ls then

           first_pass (proof, contra |>> cons l ||> cons step)

         else

           first_pass (proof, contra) |>> cons step

       | first_pass _ = raise Fail "malformed proof"

     val (proof_top, (contra_ls, contra_proof)) =

       first_pass (proof, (concl_ls, []))

     val backpatch_label = the_default ([], []) oo AList.lookup (op =) o fst

     fun backpatch_labels patches ls =

       fold merge_fact_sets (map (backpatch_label patches) ls) ([], [])

     fun second_pass end_qs ([], assums, patches) =

         ([Have (end_qs, no_label,

                 if length assums < length concl_ls then

                   clause_for_literals thy (map (negate_term thy o fst) assums)

                 else

                   concl_t,

                 ByMetis (backpatch_labels patches (map snd assums)))], patches)

       | second_pass end_qs (Assume (l, t) :: proof, assums, patches) =

         second_pass end_qs (proof, (t, l) :: assums, patches)

       | second_pass end_qs (Have (qs, l, t, ByMetis (ls, ss)) :: proof, assums,

                             patches) =

         if member (op =) (snd (snd patches)) l andalso

            not (AList.defined (op =) (fst patches) l) then

           second_pass end_qs (proof, assums, patches ||> apsnd (append ls))

         else

           (case List.partition (member (op =) contra_ls) ls of

              ([contra_l], co_ls) =>

              if no_show qs then

                second_pass end_qs

                            (proof, assums,

                             patches |>> cons (contra_l, (l :: co_ls, ss)))

                |>> cons (if member (op =) (fst (snd patches)) l then

                            Assume (l, negate_term thy t)

                          else

                            Have (qs, l, negate_term thy t,

                                  ByMetis (backpatch_label patches l)))

              else

                second_pass end_qs (proof, assums,

                                    patches |>> cons (contra_l, (co_ls, ss)))

            | (contra_ls as _ :: _, co_ls) =>

              let

                val proofs =

                  map_filter

                      (fn l =>

                          if member (op =) concl_ls l then

                            NONE

                          else

                            let

                              val drop_ls = filter (curry (op <>) l) contra_ls

                            in

                              second_pass []

                                  (proof, assums,

                                   patches ||> apfst (insert (op =) l)

                                           ||> apsnd (union (op =) drop_ls))

                              |> fst |> SOME

                            end) contra_ls

                val facts = (co_ls, [])

              in

                (case join_proofs proofs of

                   SOME (l, t, proofs, proof_tail) =>

                   Have (case_split_qualifiers proofs @

                         (if null proof_tail then end_qs else []), l, t,

                         smart_case_split proofs facts) :: proof_tail

                 | NONE =>

                   [Have (case_split_qualifiers proofs @ end_qs, no_label,

                          concl_t, smart_case_split proofs facts)],

                 patches)

              end

            | _ => raise Fail "malformed proof")

        | second_pass _ _ = raise Fail "malformed proof"

     val proof_bottom =

       second_pass [Show] (contra_proof, [], ([], ([], []))) |> fst

   in proof_top @ proof_bottom end

 val kill_duplicate_assumptions_in_proof =

   let

     fun relabel_facts subst =

       apfst (map (fn l => AList.lookup (op =) subst l |> the_default l))

     fun do_step (step as Assume (l, t)) (proof, subst, assums) =

         (case AList.lookup (op aconv) assums t of

            SOME l' => (proof, (l, l') :: subst, assums)

          | NONE => (step :: proof, subst, (t, l) :: assums))

       | do_step (Have (qs, l, t, by)) (proof, subst, assums) =

         (Have (qs, l, t,

                case by of

                  ByMetis facts => ByMetis (relabel_facts subst facts)

                | CaseSplit (proofs, facts) =>

                  CaseSplit (map do_proof proofs, relabel_facts subst facts)) ::

          proof, subst, assums)

       | do_step step (proof, subst, assums) = (step :: proof, subst, assums)

     and do_proof proof = fold do_step proof ([], [], []) |> #1 |> rev

   in do_proof end

 val then_chain_proof =

   let

     fun aux _ [] = []

       | aux _ ((step as Assume (l, _)) :: proof) = step :: aux l proof

       | aux l' (Have (qs, l, t, by) :: proof) =

         (case by of

            ByMetis (ls, ss) =>

            Have (if member (op =) ls l' then

                    (Then :: qs, l, t,

                     ByMetis (filter_out (curry (op =) l') ls, ss))

                  else

                    (qs, l, t, ByMetis (ls, ss)))

          | CaseSplit (proofs, facts) =>

            Have (qs, l, t, CaseSplit (map (aux no_label) proofs, facts))) ::

         aux l proof

       | aux _ (step :: proof) = step :: aux no_label proof

   in aux no_label end

 fun kill_useless_labels_in_proof proof =

   let

     val used_ls = used_labels_of proof

     fun do_label l = if member (op =) used_ls l then l else no_label

     fun do_step (Assume (l, t)) = Assume (do_label l, t)

       | do_step (Have (qs, l, t, by)) =

         Have (qs, do_label l, t,

               case by of

                 CaseSplit (proofs, facts) =>

                 CaseSplit (map (map do_step) proofs, facts)

               | _ => by)

       | do_step step = step

   in map do_step proof end

 fun prefix_for_depth n = replicate_string (n + 1)

 val relabel_proof =

   let

     fun aux _ _ _ [] = []

       | aux subst depth (next_assum, next_fact) (Assume (l, t) :: proof) =

         if l = no_label then

           Assume (l, t) :: aux subst depth (next_assum, next_fact) proof

         else

           let val l' = (prefix_for_depth depth assum_prefix, next_assum) in

             Assume (l', t) ::

             aux ((l, l') :: subst) depth (next_assum + 1, next_fact) proof

           end

       | aux subst depth (next_assum, next_fact) (Have (qs, l, t, by) :: proof) =

         let

           val (l', subst, next_fact) =

             if l = no_label then

               (l, subst, next_fact)

             else

               let

                 val l' = (prefix_for_depth depth fact_prefix, next_fact)

               in (l', (l, l') :: subst, next_fact + 1) end

           val relabel_facts =

             apfst (map (fn l =>

                            case AList.lookup (op =) subst l of

                              SOME l' => l'

                            | NONE => raise Fail ("unknown label " ^

                                                  quote (string_for_label l))))

           val by =

             case by of

               ByMetis facts => ByMetis (relabel_facts facts)

             | CaseSplit (proofs, facts) =>

               CaseSplit (map (aux subst (depth + 1) (1, 1)) proofs,

                          relabel_facts facts)

         in

           Have (qs, l', t, by) ::

           aux subst depth (next_assum, next_fact) proof

         end

       | aux subst depth nextp (step :: proof) =

         step :: aux subst depth nextp proof

   in aux [] 0 (1, 1) end

 fun string_for_proof ctxt i n =

   let

     fun fix_print_mode f =

       PrintMode.setmp (filter (curry (op =) Symbol.xsymbolsN)

                       (print_mode_value ())) f

     fun do_indent ind = replicate_string (ind * indent_size) " "

     fun do_free (s, T) =

       maybe_quote s ^ " :: " ^

       maybe_quote (fix_print_mode (Syntax.string_of_typ ctxt) T)

     fun do_label l = if l = no_label then "" else string_for_label l ^ ": "

     fun do_have qs =

       (if member (op =) qs Moreover then "moreover " else "") ^

       (if member (op =) qs Ultimately then "ultimately " else "") ^

       (if member (op =) qs Then then

          if member (op =) qs Show then "thus" else "hence"

        else

          if member (op =) qs Show then "show" else "have")

     val do_term = maybe_quote o fix_print_mode (Syntax.string_of_term ctxt)

     fun do_facts (ls, ss) =

       let

         val ls = ls |> sort_distinct (prod_ord string_ord int_ord)

         val ss = ss |> sort_distinct string_ord

       in metis_command 1 1 (map string_for_label ls @ ss) end

     and do_step ind (Fix xs) =

         do_indent ind ^ "fix " ^ space_implode " and " (map do_free xs) ^ "\n"

       | do_step ind (Let (t1, t2)) =

         do_indent ind ^ "let " ^ do_term t1 ^ " = " ^ do_term t2 ^ "\n"

       | do_step ind (Assume (l, t)) =

         do_indent ind ^ "assume " ^ do_label l ^ do_term t ^ "\n"

       | do_step ind (Have (qs, l, t, ByMetis facts)) =

         do_indent ind ^ do_have qs ^ " " ^

         do_label l ^ do_term t ^ " " ^ do_facts facts ^ "\n"

       | do_step ind (Have (qs, l, t, CaseSplit (proofs, facts))) =

         space_implode (do_indent ind ^ "moreover\n")

                       (map (do_block ind) proofs) ^

         do_indent ind ^ do_have qs ^ " " ^ do_label l ^ do_term t ^ " " ^

         do_facts facts ^ "\n"

     and do_steps prefix suffix ind steps =

       let val s = implode (map (do_step ind) steps) in

         replicate_string (ind * indent_size - size prefix) " " ^ prefix ^

         String.extract (s, ind * indent_size,

                         SOME (size s - ind * indent_size - 1)) ^

         suffix ^ "\n"

       end

     and do_block ind proof = do_steps "{ " " }" (ind + 1) proof

     (* One-step proofs are pointless; better use the Metis one-liner

        directly. *)

     and do_proof [Have (_, _, _, ByMetis _)] = ""

       | do_proof proof =

         (if i <> 1 then "prefer " ^ string_of_int i ^ "\n" else "") ^

         do_indent 0 ^ "proof -\n" ^

         do_steps "" "" 1 proof ^

         do_indent 0 ^ (if n <> 1 then "next" else "qed") ^ "\n"

   in do_proof end

 fun isar_proof_text (pool, debug, full_types, isar_shrink_factor, ctxt,

                      conjecture_shape)

                     (minimize_command, atp_proof, thm_names, goal, i) =

   let

     val thy = ProofContext.theory_of ctxt

     val (params, hyp_ts, concl_t) = strip_subgoal goal i

     val frees = fold Term.add_frees (concl_t :: hyp_ts) []

     val tfrees = fold Term.add_tfrees (concl_t :: hyp_ts) []

     val n = Logic.count_prems (prop_of goal)

     val (one_line_proof, lemma_names) =

       metis_proof_text (minimize_command, atp_proof, thm_names, goal, i)

     fun isar_proof_for () =

       case proof_from_atp_proof pool ctxt full_types tfrees isar_shrink_factor

                                 atp_proof conjecture_shape thm_names params

                                 frees

            |> redirect_proof thy conjecture_shape hyp_ts concl_t

            |> kill_duplicate_assumptions_in_proof

            |> then_chain_proof

            |> kill_useless_labels_in_proof

            |> relabel_proof

            |> string_for_proof ctxt i n of

         "" => ""

       | proof => "\nStructured proof:\n" ^ Markup.markup Markup.sendback proof

     val isar_proof =

       if debug then

         isar_proof_for ()

       else

         try isar_proof_for ()

         |> the_default "Warning: The Isar proof construction failed.\n"

   in (one_line_proof ^ isar_proof, lemma_names) end

 fun proof_text isar_proof isar_params other_params =

   (if isar_proof then isar_proof_text isar_params else metis_proof_text)

       other_params

 end;