(*  Title:      HOL/Tools/ATP/atp_proof_reconstruct.ML

     Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory

     Author:     Claire Quigley, Cambridge University Computer Laboratory

     Author:     Jasmin Blanchette, TU Muenchen

 Proof reconstruction from ATP proofs.

 *)

 signature ATP_PROOF_RECONSTRUCT =

 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 proof = 'a ATP_Proof.proof

   type stature = ATP_Problem_Generate.stature

   datatype reconstructor =

     Metis of string * string |

     SMT

   datatype play =

     Played of reconstructor * Time.time |

     Trust_Playable of reconstructor * Time.time option |

     Failed_to_Play of reconstructor

   type minimize_command = string list -> string

   type one_line_params =

     play * string * (string * stature) list * minimize_command * int * int

   type isar_params =

     bool * int * string Symtab.table * (string * stature) list vector

     * int Symtab.table * string proof * thm

   val metisN : string

   val smtN : string

   val full_typesN : string

   val partial_typesN : string

   val no_typesN : string

   val really_full_type_enc : string

   val full_type_enc : string

   val partial_type_enc : string

   val no_type_enc : string

   val full_type_encs : string list

   val partial_type_encs : string list

   val metis_default_lam_trans : string

   val metis_call : string -> string -> string

   val string_for_reconstructor : reconstructor -> string

   val used_facts_in_atp_proof :

     Proof.context -> (string * stature) list vector -> string proof

     -> (string * stature) list

   val lam_trans_from_atp_proof : string proof -> string -> string

   val is_typed_helper_used_in_atp_proof : string proof -> bool

   val used_facts_in_unsound_atp_proof :

     Proof.context -> (string * stature) list vector -> 'a proof

     -> string list option

   val unalias_type_enc : string -> string list

   val one_line_proof_text : one_line_params -> string

   val make_tvar : string -> typ

   val make_tfree : Proof.context -> string -> typ

   val term_from_atp :

     Proof.context -> bool -> int Symtab.table -> typ option

     -> (string, string) ho_term -> term

   val prop_from_atp :

     Proof.context -> bool -> int Symtab.table

     -> (string, string, (string, string) ho_term, string) formula -> term

   val isar_proof_text :

     Proof.context -> bool -> isar_params -> one_line_params -> string

   val proof_text :

     Proof.context -> bool -> isar_params -> one_line_params -> string

 end;

 structure ATP_Proof_Reconstruct : ATP_PROOF_RECONSTRUCT =

 struct

 open ATP_Util

 open ATP_Problem

 open ATP_Proof

 open ATP_Problem_Generate

 structure String_Redirect = ATP_Proof_Redirect(

     type key = step_name

     val ord = fn ((s, _ : string list), (s', _)) => fast_string_ord (s, s')

     val string_of = fst)

 open String_Redirect

 datatype reconstructor =

   Metis of string * string |

   SMT

 datatype play =

   Played of reconstructor * Time.time |

   Trust_Playable of reconstructor * Time.time option |

   Failed_to_Play of reconstructor

 type minimize_command = string list -> string

 type one_line_params =

   play * string * (string * stature) list * minimize_command * int * int

 type isar_params =

   bool * int * string Symtab.table * (string * stature) list vector

   * int Symtab.table * string proof * thm

 val metisN = "metis"

 val smtN = "smt"

 val full_typesN = "full_types"

 val partial_typesN = "partial_types"

 val no_typesN = "no_types"

 val really_full_type_enc = "mono_tags"

 val full_type_enc = "poly_guards_query"

 val partial_type_enc = "poly_args"

 val no_type_enc = "erased"

 val full_type_encs = [full_type_enc, really_full_type_enc]

 val partial_type_encs = partial_type_enc :: full_type_encs

 val type_enc_aliases =

   [(full_typesN, full_type_encs),

    (partial_typesN, partial_type_encs),

    (no_typesN, [no_type_enc])]

 fun unalias_type_enc s =

   AList.lookup (op =) type_enc_aliases s |> the_default [s]

 val metis_default_lam_trans = combsN

 fun metis_call type_enc lam_trans =

   let

     val type_enc =

       case AList.find (fn (enc, encs) => enc = hd encs) type_enc_aliases

                       type_enc of

         [alias] => alias

       | _ => type_enc

     val opts = [] |> type_enc <> partial_typesN ? cons type_enc

                   |> lam_trans <> metis_default_lam_trans ? cons lam_trans

   in metisN ^ (if null opts then "" else " (" ^ commas opts ^ ")") end

 fun string_for_reconstructor (Metis (type_enc, lam_trans)) =

     metis_call type_enc lam_trans

   | string_for_reconstructor SMT = smtN

 fun find_first_in_list_vector vec key =

   Vector.foldl (fn (ps, NONE) => AList.lookup (op =) ps key

                  | (_, value) => value) NONE vec

 val unprefix_fact_number = space_implode "_" o tl o space_explode "_"

 fun resolve_one_named_fact fact_names s =

   case try (unprefix fact_prefix) s of

     SOME s' =>

     let val s' = s' |> unprefix_fact_number |> unascii_of in

       s' |> find_first_in_list_vector fact_names |> Option.map (pair s')

     end

   | NONE => NONE

 fun resolve_fact fact_names = map_filter (resolve_one_named_fact fact_names)

 fun is_fact fact_names = not o null o resolve_fact fact_names

 fun resolve_one_named_conjecture s =

   case try (unprefix conjecture_prefix) s of

     SOME s' => Int.fromString s'

   | NONE => NONE

 val resolve_conjecture = map_filter resolve_one_named_conjecture

 val is_conjecture = not o null o resolve_conjecture

 fun is_axiom_used_in_proof pred =

   exists (fn Inference_Step ((_, ss), _, _, []) => exists pred ss | _ => false)

 val is_combinator_def = String.isPrefix (helper_prefix ^ combinator_prefix)

 val ascii_of_lam_fact_prefix = ascii_of lam_fact_prefix

 (* overapproximation (good enough) *)

 fun is_lam_lifted s =

   String.isPrefix fact_prefix s andalso

   String.isSubstring ascii_of_lam_fact_prefix s

 fun lam_trans_from_atp_proof atp_proof default =

   case (is_axiom_used_in_proof is_combinator_def atp_proof,

         is_axiom_used_in_proof is_lam_lifted atp_proof) of

     (false, false) => default

   | (false, true) => liftingN

 (*  | (true, true) => combs_and_liftingN -- not supported by "metis" *)

   | (true, _) => combsN

 val is_typed_helper_name =

   String.isPrefix helper_prefix andf String.isSuffix typed_helper_suffix

 fun is_typed_helper_used_in_atp_proof atp_proof =

   is_axiom_used_in_proof is_typed_helper_name atp_proof

 val leo2_ext = "extcnf_equal_neg"

 val leo2_unfold_def = "unfold_def"

 val isa_ext = Thm.get_name_hint @{thm ext}

 val isa_short_ext = Long_Name.base_name isa_ext

 fun ext_name ctxt =

   if Thm.eq_thm_prop (@{thm ext},

          singleton (Attrib.eval_thms ctxt) (Facts.named isa_short_ext, [])) then

     isa_short_ext

   else

     isa_ext

 fun add_non_rec_defs fact_names accum =

   Vector.foldl

       (fn (facts, facts') =>

           union (op =)

                 (filter (fn (_, (_, status)) => status = Non_Rec_Def) facts)

                 facts')

       accum fact_names

 fun add_fact ctxt fact_names (Inference_Step ((_, ss), _, rule, deps)) =

     (if rule = leo2_ext then

        insert (op =) (ext_name ctxt, (Global, General))

      else if rule = leo2_unfold_def then

        (* LEO 1.3.3 does not record definitions properly, leading to missing

          dependencies in the TSTP proof. Remove the next line once this is

          fixed. *)

        add_non_rec_defs fact_names

      else if rule = satallax_unsat_coreN then

        (fn [] =>

            (* Satallax doesn't include definitions in its unsatisfiable cores,

               so we assume the worst and include them all here. *)

            [(ext_name ctxt, (Global, General))] |> add_non_rec_defs fact_names

          | facts => facts)

      else

        I)

     #> (if null deps then union (op =) (resolve_fact fact_names ss)

         else I)

   | add_fact _ _ _ = I

 fun used_facts_in_atp_proof ctxt fact_names atp_proof =

   if null atp_proof then Vector.foldl (uncurry (union (op =))) [] fact_names

   else fold (add_fact ctxt fact_names) atp_proof []

 fun used_facts_in_unsound_atp_proof _ _ [] = NONE

   | used_facts_in_unsound_atp_proof ctxt fact_names atp_proof =

     let val used_facts = used_facts_in_atp_proof ctxt fact_names atp_proof in

       if forall (fn (_, (sc, _)) => sc = Global) used_facts andalso

          not (is_axiom_used_in_proof (is_conjecture o single) atp_proof) then

         SOME (map fst used_facts)

       else

         NONE

     end

 (** Soft-core proof reconstruction: one-liners **)

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

 fun show_time NONE = ""

   | show_time (SOME ext_time) = " (" ^ string_from_ext_time ext_time ^ ")"

 fun apply_on_subgoal _ 1 = "by "

   | apply_on_subgoal 1 _ = "apply "

   | apply_on_subgoal i n =

     "prefer " ^ string_of_int i ^ " " ^ apply_on_subgoal 1 n

 fun command_call name [] =

     name |> not (Lexicon.is_identifier name) ? enclose "(" ")"

   | command_call name args = "(" ^ name ^ " " ^ space_implode " " args ^ ")"

 fun try_command_line banner time command =

   banner ^ ": " ^ Markup.markup Isabelle_Markup.sendback command ^ show_time time ^ "."

 fun using_labels [] = ""

   | using_labels ls =

     "using " ^ space_implode " " (map string_for_label ls) ^ " "

 fun reconstructor_command reconstr i n (ls, ss) =

   using_labels ls ^ apply_on_subgoal i n ^

   command_call (string_for_reconstructor reconstr) ss

 fun minimize_line _ [] = ""

   | minimize_line minimize_command ss =

     case minimize_command ss of

       "" => ""

     | command =>

       "\nTo minimize: " ^ Markup.markup Isabelle_Markup.sendback command ^ "."

 fun split_used_facts facts =

   facts |> List.partition (fn (_, (sc, _)) => sc = Chained)

         |> pairself (sort_distinct (string_ord o pairself fst))

 fun one_line_proof_text (preplay, banner, used_facts, minimize_command,

                          subgoal, subgoal_count) =

   let

     val (chained, extra) = split_used_facts used_facts

     val (failed, reconstr, ext_time) =

       case preplay of

         Played (reconstr, time) => (false, reconstr, (SOME (false, time)))

       | Trust_Playable (reconstr, time) =>

         (false, reconstr,

          case time of

            NONE => NONE

          | SOME time =>

            if time = Time.zeroTime then NONE else SOME (true, time))

       | Failed_to_Play reconstr => (true, reconstr, NONE)

     val try_line =

       ([], map fst extra)

       |> reconstructor_command reconstr subgoal subgoal_count

       |> (if failed then

             enclose "One-line proof reconstruction failed: "

                      ".\n(Invoking \"sledgehammer\" with \"[strict]\" might \

                      \solve this.)"

           else

             try_command_line banner ext_time)

   in try_line ^ minimize_line minimize_command (map fst (extra @ chained)) end

 (** Hard-core proof reconstruction: structured Isar proofs **)

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

 fun exists_of v t = HOLogic.exists_const (fastype_of v) $ lambda v t

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

 fun make_tfree ctxt w =

   let val ww = "'" ^ w in

     TFree (ww, the_default HOLogic.typeS (Variable.def_sort ctxt (ww, ~1)))

   end

 val indent_size = 2

 val no_label = ("", ~1)

 val raw_prefix = "x"

 val assum_prefix = "a"

 val have_prefix = "f"

 fun raw_label_for_name (num, ss) =

   case resolve_conjecture ss of

     [j] => (conjecture_prefix, j)

   | _ => (raw_prefix ^ ascii_of num, 0)

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

 exception HO_TERM of (string, string) ho_term list

 exception FORMULA of

     (string, string, (string, string) ho_term, string) formula list

 exception SAME of unit

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

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

 fun typ_from_atp ctxt (u as ATerm ((a, _), us)) =

   let val Ts = map (typ_from_atp ctxt) us in

     case unprefix_and_unascii type_const_prefix a of

       SOME b => Type (invert_const b, Ts)

     | NONE =>

       if not (null us) then

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

       else case unprefix_and_unascii tfree_prefix a of

         SOME b => make_tfree ctxt b

       | NONE =>

         (* Could be an Isabelle variable or a variable from the ATP, say "X1"

            or "_5018". Sometimes variables from the ATP are indistinguishable

            from Isabelle variables, which forces us to use a type parameter in

            all cases. *)

         (a |> perhaps (unprefix_and_unascii tvar_prefix), HOLogic.typeS)

         |> Type_Infer.param 0

   end

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

    type. *)

 fun type_constraint_from_term ctxt (u as ATerm ((a, _), us)) =

   case (unprefix_and_unascii class_prefix a, map (typ_from_atp ctxt) us) of

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

   | _ => raise HO_TERM [u]

 (* Accumulate type constraints in a formula: 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 repair_variable_name f s =

   let

     fun subscript_name s n = s ^ nat_subscript n

     val s = String.map f 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

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

    (instances of) Skolem pseudoconstants, this information is encoded in the

    constant name. *)

 fun num_type_args thy s =

   if String.isPrefix skolem_const_prefix s then

     s |> Long_Name.explode |> List.last |> Int.fromString |> the

   else if String.isPrefix lam_lifted_prefix s then

     if String.isPrefix lam_lifted_poly_prefix s then 2 else 0

   else

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

 fun slack_fastype_of t = fastype_of t handle TERM _ => HOLogic.typeT

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

    should allow them to be inferred. *)

 fun term_from_atp ctxt textual sym_tab =

   let

     val thy = Proof_Context.theory_of ctxt

     (* For Metis, we use 1 rather than 0 because variable references in clauses

        may otherwise conflict with variable constraints in the goal. At least,

        type inference often fails otherwise. See also "axiom_inference" in

        "Metis_Reconstruct". *)

     val var_index = if textual then 0 else 1

     fun do_term extra_ts opt_T u =

       case u of

         ATerm ((s, _), us) =>

         if String.isPrefix native_type_prefix s then

           @{const True} (* ignore TPTP type information *)

         else if s = tptp_equal then

           let val ts = map (do_term [] NONE) us in

             if textual andalso length ts = 2 andalso

               hd ts aconv List.last ts then

               (* Vampire is keen on producing these. *)

               @{const True}

             else

               list_comb (Const (@{const_name HOL.eq}, HOLogic.typeT), ts)

           end

         else case unprefix_and_unascii const_prefix s of

           SOME s' =>

           let

             val ((s', s''), mangled_us) =

               s' |> unmangled_const |>> `invert_const

           in

             if s' = type_tag_name then

               case mangled_us @ us of

                 [typ_u, term_u] =>

                 do_term extra_ts (SOME (typ_from_atp ctxt typ_u)) term_u

               | _ => raise HO_TERM us

             else if s' = predicator_name then

               do_term [] (SOME @{typ bool}) (hd us)

             else if s' = app_op_name then

               let val extra_t = do_term [] NONE (List.last us) in

                 do_term (extra_t :: extra_ts)

                         (case opt_T of

                            SOME T => SOME (slack_fastype_of extra_t --> T)

                          | NONE => NONE)

                         (nth us (length us - 2))

               end

             else if s' = type_guard_name then

               @{const True} (* ignore type predicates *)

             else

               let

                 val new_skolem = String.isPrefix new_skolem_const_prefix s''

                 val num_ty_args =

                   length us - the_default 0 (Symtab.lookup sym_tab s)

                 val (type_us, term_us) =

                   chop num_ty_args us |>> append mangled_us

                 val term_ts = map (do_term [] NONE) term_us

                 val T =

                   (if not (null type_us) andalso

                       num_type_args thy s' = length type_us then

                      let val Ts = type_us |> map (typ_from_atp ctxt) in

                        if new_skolem then

                          SOME (Type_Infer.paramify_vars (tl Ts ---> hd Ts))

                        else if textual then

                          try (Sign.const_instance thy) (s', Ts)

                        else

                          NONE

                      end

                    else

                      NONE)

                   |> (fn SOME T => T

                        | NONE => map slack_fastype_of term_ts --->

                                  (case opt_T of

                                     SOME T => T

                                   | NONE => HOLogic.typeT))

                 val t =

                   if new_skolem then

                     Var ((new_skolem_var_name_from_const s'', var_index), T)

                   else

                     Const (unproxify_const s', T)

               in list_comb (t, term_ts @ extra_ts) end

           end

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

           let

             val term_ts = map (do_term [] NONE) us

             val ts = term_ts @ extra_ts

             val T =

               case opt_T of

                 SOME T => map slack_fastype_of term_ts ---> T

               | NONE => map slack_fastype_of ts ---> HOLogic.typeT

             val t =

               case unprefix_and_unascii fixed_var_prefix s of

                 SOME s => Free (s, T)

               | NONE =>

                 case unprefix_and_unascii schematic_var_prefix s of

                   SOME s => Var ((s, var_index), T)

                 | NONE =>

                   Var ((s |> textual ? repair_variable_name Char.toLower,

                         var_index), T)

           in list_comb (t, ts) end

   in do_term [] end

 fun term_from_atom ctxt textual sym_tab pos (u as ATerm ((s, _), _)) =

   if String.isPrefix class_prefix s then

     add_type_constraint pos (type_constraint_from_term ctxt u)

     #> pair @{const True}

   else

     pair (term_from_atp ctxt textual sym_tab (SOME @{typ bool}) u)

 val combinator_table =

   [(@{const_name Meson.COMBI}, @{thm Meson.COMBI_def [abs_def]}),

    (@{const_name Meson.COMBK}, @{thm Meson.COMBK_def [abs_def]}),

    (@{const_name Meson.COMBB}, @{thm Meson.COMBB_def [abs_def]}),

    (@{const_name Meson.COMBC}, @{thm Meson.COMBC_def [abs_def]}),

    (@{const_name Meson.COMBS}, @{thm Meson.COMBS_def [abs_def]})]

 fun uncombine_term thy =

   let

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

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

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

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

            SOME thm => thm |> prop_of |> specialize_type thy x

                            |> Logic.dest_equals |> snd

          | NONE => t)

       | aux t = t

   in aux end

 (* Update schematic type variables with detected sort constraints. It's not

    totally clear whether this code is necessary. *)

 fun repair_tvar_sorts (t, tvar_tab) =

   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 tvar_tab 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 t |> not (Vartab.is_empty tvar_tab) ? do_term end

 fun quantify_over_var quant_of var_s t =

   let

     val vars = [] |> Term.add_vars t |> filter (fn ((s, _), _) => s = var_s)

                   |> map Var

   in fold_rev quant_of vars t end

 (* Interpret an ATP formula as a HOL term, extracting sort constraints as they

    appear in the formula. *)

 fun prop_from_atp ctxt textual sym_tab phi =

   let

     fun do_formula pos phi =

       case phi of

         AQuant (_, [], phi) => do_formula pos phi

       | AQuant (q, (s, _) :: xs, phi') =>

         do_formula pos (AQuant (q, xs, phi'))

         (* FIXME: TFF *)

         #>> quantify_over_var (case q of

                                  AForall => forall_of

                                | AExists => exists_of)

                               (s |> textual ? repair_variable_name Char.toLower)

       | AConn (ANot, [phi']) => do_formula (not pos) phi' #>> s_not

       | AConn (c, [phi1, phi2]) =>

         do_formula (pos |> c = AImplies ? not) phi1

         ##>> do_formula pos phi2

         #>> (case c of

                AAnd => s_conj

              | AOr => s_disj

              | AImplies => s_imp

              | AIff => s_iff

              | ANot => raise Fail "impossible connective")

       | AAtom tm => term_from_atom ctxt textual sym_tab pos tm

       | _ => raise FORMULA [phi]

   in repair_tvar_sorts (do_formula true phi Vartab.empty) end

 fun infer_formula_types ctxt =

   Type.constraint HOLogic.boolT

   #> Syntax.check_term

          (Proof_Context.set_mode Proof_Context.mode_schematic ctxt)

 fun uncombined_etc_prop_from_atp ctxt textual sym_tab =

   let val thy = Proof_Context.theory_of ctxt in

     prop_from_atp ctxt textual sym_tab

     #> textual ? uncombine_term thy #> infer_formula_types ctxt

   end

 (**** 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 sym_tab (Definition_Step (name, phi1, phi2)) ctxt =

     let

       val thy = Proof_Context.theory_of ctxt

       val t1 = prop_from_atp ctxt true sym_tab phi1

       val vars = snd (strip_comb t1)

       val frees = map unvarify_term vars

       val unvarify_args = subst_atomic (vars ~~ frees)

       val t2 = prop_from_atp ctxt true sym_tab phi2

       val (t1, t2) =

         HOLogic.eq_const HOLogic.typeT $ t1 $ t2

         |> unvarify_args |> uncombine_term thy |> infer_formula_types ctxt

         |> HOLogic.dest_eq

     in

       (Definition_Step (name, t1, t2),

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

     end

   | decode_line sym_tab (Inference_Step (name, u, rule, deps)) ctxt =

     let val t = u |> uncombined_etc_prop_from_atp ctxt true sym_tab in

       (Inference_Step (name, t, rule, deps),

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

     end

 fun decode_lines ctxt sym_tab lines =

   fst (fold_map (decode_line sym_tab) lines ctxt)

 fun is_same_inference _ (Definition_Step _) = false

   | is_same_inference t (Inference_Step (_, t', _, _)) = t aconv t'

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

    clsarity). *)

 fun is_only_type_information t = t aconv @{term True}

 fun replace_one_dependency (old, new) dep =

   if is_same_atp_step dep old then new else [dep]

 fun replace_dependencies_in_line _ (line as Definition_Step _) = line

   | replace_dependencies_in_line p (Inference_Step (name, t, rule, deps)) =

     Inference_Step (name, t, rule,

                     fold (union (op =) o replace_one_dependency p) deps [])

 (* Discard facts; consolidate adjacent lines that prove the same formula, since

    they differ only in type information.*)

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

   | add_line fact_names (Inference_Step (name as (_, ss), t, rule, [])) lines =

     (* No dependencies: fact, conjecture, or (for Vampire) internal facts or

        definitions. *)

     if is_fact fact_names ss then

       (* Facts are not proof lines. *)

       if is_only_type_information t then

         map (replace_dependencies_in_line (name, [])) lines

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

       else case take_prefix (not o is_same_inference t) lines of

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

       | (pre, Inference_Step (name', _, _, _) :: post) =>

         pre @ map (replace_dependencies_in_line (name', [name])) post

       | _ => raise Fail "unexpected inference"

     else if is_conjecture ss then

       Inference_Step (name, t, rule, []) :: lines

     else

       map (replace_dependencies_in_line (name, [])) lines

   | add_line _ (Inference_Step (name, t, rule, deps)) lines =

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

     if is_only_type_information t then

       Inference_Step (name, t, rule, deps) :: lines

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

     else case take_prefix (not o is_same_inference t) lines of

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

          types? *)

        (_, []) => Inference_Step (name, t, rule, deps) :: lines

      | (pre, Inference_Step (name', t', rule, _) :: post) =>

        Inference_Step (name, t', rule, deps) ::

        pre @ map (replace_dependencies_in_line (name', [name])) post

      | _ => raise Fail "unexpected inference"

 val waldmeister_conjecture_num = "1.0.0.0"

 val repair_waldmeister_endgame =

   let

     fun do_tail (Inference_Step (name, t, rule, deps)) =

         Inference_Step (name, s_not t, rule, deps)

       | do_tail line = line

     fun do_body [] = []

       | do_body ((line as Inference_Step ((num, _), _, _, _)) :: lines) =

         if num = waldmeister_conjecture_num then map do_tail (line :: lines)

         else line :: do_body lines

       | do_body (line :: lines) = line :: do_body lines

   in do_body end

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

 fun add_nontrivial_line (line as Inference_Step (name, t, _, [])) lines =

     if is_only_type_information t then delete_dependency name lines

     else line :: lines

   | add_nontrivial_line line lines = line :: lines

 and delete_dependency name lines =

   fold_rev add_nontrivial_line

            (map (replace_dependencies_in_line (name, [])) lines) []

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

    offending lines often does the trick. *)

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

   | is_bad_free _ _ = false

 fun add_desired_line _ _ _ (line as Definition_Step (name, _, _)) (j, lines) =

     (j, line :: map (replace_dependencies_in_line (name, [])) lines)

   | add_desired_line isar_shrink_factor fact_names frees

         (Inference_Step (name as (_, ss), t, rule, deps)) (j, lines) =

     (j + 1,

      if is_fact fact_names ss orelse

         is_conjecture ss orelse

         (* the last line must be kept *)

         j = 0 orelse

         (not (is_only_type_information t) andalso

          null (Term.add_tvars t []) andalso

          not (exists_subterm (is_bad_free frees) t) andalso

          length deps >= 2 andalso j mod isar_shrink_factor = 0 andalso

          (* kill next to last line, which usually results in a trivial step *)

          j <> 1) then

        Inference_Step (name, t, rule, deps) :: lines  (* keep line *)

      else

        map (replace_dependencies_in_line (name, deps)) lines)  (* drop line *)

 (** Isar proof construction and manipulation **)

 type label = string * int

 type facts = label list * string list

 datatype isar_qualifier = Show | Then | Moreover | Ultimately

 datatype isar_step =

   Fix of (string * typ) list |

   Let of term * term |

   Assume of label * term |

   Prove of isar_qualifier list * label * term * byline

 and byline =

   By_Metis of facts |

   Case_Split of isar_step list list * facts

 fun add_fact_from_dependency fact_names (name as (_, ss)) =

   if is_fact fact_names ss then

     apsnd (union (op =) (map fst (resolve_fact fact_names ss)))

   else

     apfst (insert (op =) (raw_label_for_name name))

 fun repair_name "$true" = "c_True"

   | repair_name "$false" = "c_False"

   | repair_name "$$e" = tptp_equal (* seen in Vampire proofs *)

   | repair_name s =

     if is_tptp_equal s orelse

        (* seen in Vampire proofs *)

        (String.isPrefix "sQ" s andalso String.isSuffix "_eqProxy" s) then

       tptp_equal

     else

       s

 (* FIXME: Still needed? Try with SPASS proofs perhaps. *)

 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 (Prove (qs, l, t, by)) (proof, subst, assums) =

         (Prove (qs, l, t,

                 case by of

                   By_Metis facts => By_Metis (relabel_facts subst facts)

                 | Case_Split (proofs, facts) =>

                   Case_Split (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

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

     (case by of

        By_Metis (ls, _) => ls

      | Case_Split (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 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 (Prove (qs, l, t, by)) =

         Prove (qs, do_label l, t,

                case by of

                  Case_Split (proofs, facts) =>

                  Case_Split (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)

             (Prove (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 have_prefix, next_fact)

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

           val relabel_facts =

             apfst (maps (the_list o AList.lookup (op =) subst))

           val by =

             case by of

               By_Metis facts => By_Metis (relabel_facts facts)

             | Case_Split (proofs, facts) =>

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

                           relabel_facts facts)

         in

           Prove (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 ctxt0 type_enc lam_trans i n =

   let

     val ctxt = ctxt0

 (* FIXME: Implement proper handling of type constraints:

       |> Config.put show_free_types false

       |> Config.put show_types false

       |> Config.put show_sorts false

 *)

     fun fix_print_mode f x =

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

                                (print_mode_value ())) f x

     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)

     val reconstr = Metis (type_enc, lam_trans)

     fun do_facts (ls, ss) =

       reconstructor_command reconstr 1 1

           (ls |> sort_distinct (prod_ord string_ord int_ord),

            ss |> sort_distinct string_ord)

     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 (Prove (qs, l, t, By_Metis facts)) =

         do_indent ind ^ do_have qs ^ " " ^

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

       | do_step ind (Prove (qs, l, t, Case_Split (proofs, facts))) =

         implode (map (prefix (do_indent ind ^ "moreover\n") o 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 [Prove (_, _, _, By_Metis _)] = ""

       | 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")

   in do_proof end

 fun isar_proof_text ctxt isar_proof_requested

         (debug, isar_shrink_factor, pool, fact_names, sym_tab, atp_proof, goal)

         (one_line_params as (_, _, _, _, subgoal, subgoal_count)) =

   let

     val isar_shrink_factor =

       (if isar_proof_requested then 1 else 2) * isar_shrink_factor

     val (params, hyp_ts, concl_t) = strip_subgoal ctxt goal subgoal

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

     val one_line_proof = one_line_proof_text one_line_params

     val type_enc =

       if is_typed_helper_used_in_atp_proof atp_proof then full_typesN

       else partial_typesN

     val lam_trans = lam_trans_from_atp_proof atp_proof metis_default_lam_trans

     fun isar_proof_of () =

       let

         val atp_proof =

           atp_proof

           |> clean_up_atp_proof_dependencies

           |> nasty_atp_proof pool

           |> map_term_names_in_atp_proof repair_name

           |> decode_lines ctxt sym_tab

           |> rpair [] |-> fold_rev (add_line fact_names)

           |> repair_waldmeister_endgame

           |> rpair [] |-> fold_rev add_nontrivial_line

           |> rpair (0, [])

           |-> fold_rev (add_desired_line isar_shrink_factor fact_names frees)

           |> snd

         val conj_name = conjecture_prefix ^ string_of_int (length hyp_ts)

         val conjs =

           atp_proof

           |> map_filter (fn Inference_Step (name as (_, ss), _, _, []) =>

                             if member (op =) ss conj_name then SOME name else NONE

                           | _ => NONE)

         fun dep_of_step (Definition_Step _) = NONE

           | dep_of_step (Inference_Step (name, _, _, from)) = SOME (from, name)

         val ref_graph = atp_proof |> map_filter dep_of_step |> make_ref_graph

         val axioms = axioms_of_ref_graph ref_graph conjs

         val tainted = tainted_atoms_of_ref_graph ref_graph conjs

         val props =

           Symtab.empty

           |> fold (fn Definition_Step _ => I (* FIXME *)

                     | Inference_Step ((s, _), t, _, _) =>

                       Symtab.update_new (s,

                           t |> fold forall_of (map Var (Term.add_vars t []))

                             |> member (op = o apsnd fst) tainted s ? s_not))

                   atp_proof

         fun prop_of_clause c =

           fold (curry s_disj) (map_filter (Symtab.lookup props o fst) c)

                @{term False}

         fun label_of_clause [name] = raw_label_for_name name

           | label_of_clause c = (space_implode "___" (map fst c), 0)

         fun maybe_show outer c =

           (outer andalso length c = 1 andalso subset (op =) (c, conjs))

           ? cons Show

         fun do_have outer qs (gamma, c) =

           Prove (maybe_show outer c qs, label_of_clause c, prop_of_clause c,

                  By_Metis (fold (add_fact_from_dependency fact_names

                                  o the_single) gamma ([], [])))

         fun do_inf outer (Have z) = do_have outer [] z

           | do_inf outer (Hence z) = do_have outer [Then] z

           | do_inf outer (Cases cases) =

             let val c = succedent_of_cases cases in

               Prove (maybe_show outer c [Ultimately], label_of_clause c,

                      prop_of_clause c,

                      Case_Split (map (do_case false) cases, ([], [])))

             end

         and do_case outer (c, infs) =

           Assume (label_of_clause c, prop_of_clause c) ::

           map (do_inf outer) infs

         val isar_proof =

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

           (ref_graph

            |> redirect_graph axioms tainted

            |> chain_direct_proof

            |> map (do_inf true)

            |> kill_duplicate_assumptions_in_proof

            |> kill_useless_labels_in_proof

            |> relabel_proof)

           |> string_for_proof ctxt type_enc lam_trans subgoal subgoal_count

       in

         case isar_proof of

           "" =>

           if isar_proof_requested then

             "\nNo structured proof available (proof too short)."

           else

             ""

         | _ =>

           "\n\n" ^ (if isar_proof_requested then "Structured proof"

                     else "Perhaps this will work") ^

           ":\n" ^ Markup.markup Isabelle_Markup.sendback isar_proof

       end

     val isar_proof =

       if debug then

         isar_proof_of ()

       else case try isar_proof_of () of

         SOME s => s

       | NONE => if isar_proof_requested then

                   "\nWarning: The Isar proof construction failed."

                 else

                   ""

   in one_line_proof ^ isar_proof end

 fun proof_text ctxt isar_proof isar_params

                (one_line_params as (preplay, _, _, _, _, _)) =

   (if case preplay of Failed_to_Play _ => true | _ => isar_proof then

      isar_proof_text ctxt isar_proof isar_params

    else

      one_line_proof_text) one_line_params

 end;