(*  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) formula = ('a, 'b, 'c) 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) 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 ((_, 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 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_fact _ fact_names (Inference ((_, ss), _, _, [])) =

    union (op =) (resolve_fact fact_names ss)

  | add_fact ctxt _ (Inference (_, _, rule, _)) =

    if rule = leo2_ext then insert (op =) (ext_name ctxt, (Global, General))

    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: " "."

          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)

  | _ => case Int.fromString num of

           SOME j => (raw_prefix, j)

         | NONE => (raw_prefix ^ 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) 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 (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 (name, t1, t2),

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

    end

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

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

      (Inference (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 _) = false

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

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

   clsarity). *)

val is_only_type_information = curry (op 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 _) = line

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

    Inference (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 _) lines = line :: lines

  | add_line fact_names (Inference (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 (name', _, _, _) :: post) =>

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

      | _ => raise Fail "unexpected inference"

    else if is_conjecture ss then

      Inference (name, s_not t, rule, []) :: lines

    else

      map (replace_dependencies_in_line (name, [])) lines

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

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

    if is_only_type_information t then

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

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

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

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

     | _ => raise Fail "unexpected inference"

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

fun add_nontrivial_line (line as Inference (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 (name, _, _)) (j, lines) =

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

  | add_desired_line isar_shrink_factor fact_names frees

                     (Inference (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 (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 |> Config.put show_free_types false

            |> Config.put show_types true

            |> Config.put show_sorts true

    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)

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

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

                          | _ => NONE)

        fun dep_of_step (Definition _) = NONE

          | dep_of_step (Inference (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 _ => I (* FIXME *)

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

                      Symtab.update_new (s,

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

                  atp_proof

        (* FIXME: add "fold_rev forall_of (map Var (Term.add_vars t []))"? *)

        fun prop_of_clause c =

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

               @{term False}

        fun 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;