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

    Author:     Jasmin Blanchette, TU Muenchen

    Author:     Steffen Juilf Smolka, TU Muenchen

Isar proof reconstruction from ATP proofs.

*)

signature SLEDGEHAMMER_PROOF_RECONSTRUCT =

sig

  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 * bool * Time.time option * real * string Symtab.table

    * (string * stature) list vector * int Symtab.table * string proof * thm

  val smtN : string

  val string_for_reconstructor : reconstructor -> string

  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_atp_proof :

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

    (string * stature) list

  val used_facts_in_unsound_atp_proof :

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

    string list option

  val one_line_proof_text : int -> one_line_params -> string

  val isar_proof_text :

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

  val proof_text :

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

end;

structure Sledgehammer_Reconstruct : SLEDGEHAMMER_PROOF_RECONSTRUCT =

struct

open ATP_Util

open ATP_Problem

open ATP_Proof

open ATP_Problem_Generate

open ATP_Proof_Reconstruct

open Sledgehammer_Util

open Sledgehammer_Proof

open Sledgehammer_Annotate

open Sledgehammer_Compress

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

(** reconstructors **)

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

val smtN = "smt"

fun string_for_reconstructor (Metis (type_enc, lam_trans)) =

    metis_call type_enc lam_trans

  | string_for_reconstructor SMT = smtN

(** fact extraction from ATP proofs **)

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

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

val is_combinator_def = String.isPrefix (helper_prefix ^ combinator_prefix)

fun is_axiom_used_in_proof pred =

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

           | _ => false)

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

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

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

val leo2_extcnf_equal_neg_rule = "extcnf_equal_neg"

val leo2_unfold_def_rule = "unfold_def"

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

    (if rule = leo2_extcnf_equal_neg_rule then

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

     else if rule = leo2_unfold_def_rule 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_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

(** one-liner reconstructor proofs **)

fun show_time NONE = ""

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

(* FIXME: Various bugs, esp. with "unfolding"

fun unusing_chained_facts _ 0 = ""

  | unusing_chained_facts used_chaineds num_chained =

    if length used_chaineds = num_chained then ""

    else if null used_chaineds then "(* using no facts *) "

    else "(* using only " ^ space_implode " " used_chaineds ^ " *) "

*)

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 using_labels [] = ""

  | using_labels ls =

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

fun command_call name [] =

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

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

fun reconstructor_command reconstr i n used_chaineds num_chained (ls, ss) =

  (* unusing_chained_facts used_chaineds num_chained ^ *)

  using_labels ls ^ apply_on_subgoal i n ^

  command_call (string_for_reconstructor reconstr) ss

fun try_command_line banner time command =

  banner ^ ": " ^ Active.sendback_markup command ^ show_time time ^ "."

fun minimize_line _ [] = ""

  | minimize_line minimize_command ss =

    case minimize_command ss of

      "" => ""

    | command =>

      "\nTo minimize: " ^ Active.sendback_markup command ^ "."

fun split_used_facts facts =

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

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

type minimize_command = string list -> string

type one_line_params =

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

fun one_line_proof_text num_chained

        (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 (map fst chained)

                               num_chained

      |> (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

(** Isar proof construction and manipulation **)

val assume_prefix = "a"

val have_prefix = "f"

val raw_prefix = "x"

fun raw_label_for_name (num, ss) =

  case resolve_conjecture ss of

    [j] => (conjecture_prefix, j)

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

fun label_of_clause [name] = raw_label_for_name name

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

fun add_fact_from_dependencies fact_names (names as [(_, ss)]) =

    if is_fact fact_names ss then

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

    else

      apfst (insert (op =) (label_of_clause names))

  | add_fact_from_dependencies fact_names names =

    apfst (insert (op =) (label_of_clause names))

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

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

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

fun infer_formula_types ctxt =

  Type.constraint HOLogic.boolT

  #> Syntax.check_term

         (Proof_Context.set_mode Proof_Context.mode_schematic ctxt)

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

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, role, u, rule, deps)) ctxt =

    let

      val thy = Proof_Context.theory_of ctxt

      val t = u |> prop_from_atp ctxt true sym_tab

                |> uncombine_term thy |> infer_formula_types ctxt

    in

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

    Inference_Step (name, role, t, rule,

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

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

   clsarity). *)

fun is_only_type_information t = t aconv @{term True}

fun s_maybe_not role = role <> Conjecture ? s_not

fun is_same_inference _ (Definition_Step _) = false

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

    s_maybe_not role t aconv s_maybe_not role' t'

(* 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), role, t, rule, []))

             lines =

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

       definitions. *)

    if is_conjecture ss then

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

    else 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

      else

        lines

    else

      map (replace_dependencies_in_line (name, [])) lines

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

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

    if is_only_type_information t then

      line :: lines

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

    else case take_prefix (not o is_same_inference (role, t)) lines of

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

         types? *)

      (_, []) => line :: lines

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

      line :: 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, Negated_Conjecture, 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

val e_skolemize_rule = "skolemize"

val vampire_skolemisation_rule = "skolemisation"

val is_skolemize_rule =

  member (op =) [e_skolemize_rule, vampire_skolemisation_rule]

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

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

  | add_desired_line fact_names frees

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

    (j + 1,

     if is_fact fact_names ss orelse

        is_conjecture ss orelse

        is_skolemize_rule rule 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

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

         j <> 1) then

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

     else

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

val indent_size = 2

val no_label = ("", ~1)

fun string_for_proof ctxt type_enc lam_trans i n =

  let

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

    fun do_free (s, T) =

      maybe_quote s ^ " :: " ^

      maybe_quote (simplify_spaces (with_vanilla_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 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")

    fun do_obtain qs xs =

      (if member (op =) qs Then then "then " else "") ^ "obtain " ^

      (space_implode " " (map fst xs)) ^ " where"

    val do_term =

      annotate_types ctxt

      #> with_vanilla_print_mode (Syntax.string_of_term ctxt)

      #> simplify_spaces

      #> maybe_quote

    val reconstr = Metis (type_enc, lam_trans)

    fun do_metis ind options (ls, ss) =

      "\n" ^ do_indent (ind + 1) ^ options ^

      reconstructor_command reconstr 1 1 [] 0

          (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 (Obtain (qs, xs, l, t, By_Metis facts)) =

        do_indent ind ^ do_obtain qs xs ^ " " ^

        do_label l ^ do_term t ^

        (* The new skolemizer puts the arguments in the same order as the ATPs

           (E and Vampire -- but see also "atp_proof_reconstruct.ML" regarding

           Vampire). *)

        do_metis ind "using [[metis_new_skolem]] " facts ^ "\n"

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

        do_prove ind qs l t facts

      | 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_prove ind qs l t facts

      | do_step ind (Prove (qs, l, t, Subblock proof)) =

        do_block ind proof ^ do_prove ind qs l t ([], [])

    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

    and do_prove ind qs l t facts =

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

      do_metis ind "" facts ^ "\n"

    (* 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 used_labels_of_step (Obtain (_, _, _, _, By_Metis (ls, _))) = ls

  | 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

     | Subblock proof => used_labels_of proof)

  | 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 (Obtain (qs, xs, l, t, by)) = Obtain (qs, xs, do_label l, t, by)

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

        Prove (qs, do_label l, t,

               case by of

                 Case_Split (proofs, facts) =>

                 Case_Split (map do_proof proofs, facts)

               | Subblock proof => Subblock (do_proof proof)

               | _ => by)

      | do_step step = step

    and do_proof proof = map do_step proof

  in do_proof proof end

fun prefix_for_depth n = replicate_string (n + 1)

val relabel_proof =

  let

    fun fresh_label depth (old as (l, subst, next_have)) =

      if l = no_label then

        old

      else

        let val l' = (prefix_for_depth depth have_prefix, next_have) in

          (l', (l, l') :: subst, next_have + 1)

        end

    fun do_facts subst =

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

    fun do_byline subst depth nexts by =

      case by of

        By_Metis facts => By_Metis (do_facts subst facts)

      | Case_Split (proofs, facts) =>

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

                    do_facts subst facts)

      | Subblock proof => Subblock (do_proof subst depth nexts proof)

    and do_proof _ _ _ [] = []

      | do_proof subst depth (next_assum, next_have) (Assume (l, t) :: proof) =

        if l = no_label then

          Assume (l, t) :: do_proof subst depth (next_assum, next_have) proof

        else

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

            Assume (l', t) ::

            do_proof ((l, l') :: subst) depth (next_assum + 1, next_have) proof

          end

      | do_proof subst depth (nexts as (next_assum, next_have))

            (Obtain (qs, xs, l, t, by) :: proof) =

        let

          val (l, subst, next_have) = (l, subst, next_have) |> fresh_label depth

          val by = by |> do_byline subst depth nexts

        in

          Obtain (qs, xs, l, t, by) ::

          do_proof subst depth (next_assum, next_have) proof

        end

      | do_proof subst depth (nexts as (next_assum, next_have))

            (Prove (qs, l, t, by) :: proof) =

        let

          val (l, subst, next_have) = (l, subst, next_have) |> fresh_label depth

          val by = by |> do_byline subst depth nexts

        in

          Prove (qs, l, t, by) ::

          do_proof subst depth (next_assum, next_have) proof

        end

      | do_proof subst depth nexts (step :: proof) =

        step :: do_proof subst depth nexts proof

  in do_proof [] 0 (1, 1) end

val chain_direct_proof =

  let

    fun label_of (Assume (l, _)) = SOME l

      | label_of (Obtain (_, _, l, _, _)) = SOME l

      | label_of (Prove (_, l, _, _)) = SOME l

      | label_of _ = NONE

    fun chain_step (SOME l0)

                   (step as Obtain (qs, xs, l, t, By_Metis (lfs, gfs))) =

        if member (op =) lfs l0 then

          Obtain (Then :: qs, xs, l, t, By_Metis (lfs |> remove (op =) l0, gfs))

        else

          step

      | chain_step (SOME l0)

                   (step as Prove (qs, l, t, By_Metis (lfs, gfs))) =

        if member (op =) lfs l0 then

          Prove (Then :: qs, l, t, By_Metis (lfs |> remove (op =) l0, gfs))

        else

          step

      | chain_step _ (Prove (qs, l, t, Case_Split (proofs, facts))) =

        Prove (qs, l, t, Case_Split (map (chain_proof NONE) proofs, facts))

      | chain_step _ (Prove (qs, l, t, Subblock proof)) =

        Prove (qs, l, t, Subblock (chain_proof NONE proof))

      | chain_step _ step = step

    and chain_proof _ [] = []

      | chain_proof (prev as SOME _) (i :: is) =

        chain_step prev i :: chain_proof (label_of i) is

      | chain_proof _ (i :: is) = i :: chain_proof (label_of i) is

  in chain_proof NONE end

type isar_params =

  bool * bool * Time.time option * real * string Symtab.table

  * (string * stature) list vector * int Symtab.table * string proof * thm

fun isar_proof_text ctxt isar_proofs

    (debug, verbose, preplay_timeout, isar_compress, pool, fact_names, sym_tab,

     atp_proof, goal)

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

  let

    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 0 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

    val preplay = preplay_timeout <> SOME Time.zeroTime

    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

          |> repair_waldmeister_endgame

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

          |> rpair [] |-> fold_rev add_nontrivial_line

          |> rpair (0, [])

          |-> fold_rev (add_desired_line 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)

        val assms =

          atp_proof |> map_filter

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

                (case resolve_conjecture ss of

                   [j] =>

                   if j = length hyp_ts then NONE

                   else SOME (Assume (raw_label_for_name name, nth hyp_ts j))

                 | _ => NONE)

              | _ => NONE)

        fun dep_of_step (Definition_Step _) = NONE

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

            SOME (from, name)

        val refute_graph =

          atp_proof |> map_filter dep_of_step |> make_refute_graph

        val axioms = axioms_of_refute_graph refute_graph conjs

        val tainted = tainted_atoms_of_refute_graph refute_graph conjs

        val bot = atp_proof |> List.last |> dep_of_step |> the |> snd

        val steps =

          Symtab.empty

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

                    | Inference_Step (name as (s, ss), role, t, rule, _) =>

                      Symtab.update_new (s, (rule,

                        t |> (if member (op = o apsnd fst) tainted s then

                                s_maybe_not role

                                #> fold exists_of (map Var (Term.add_vars t []))

                              else

                                I))))

                  atp_proof

        fun is_clause_skolemize_rule [atom as (s, _)] =

            not (member (op =) tainted atom) andalso

            Option.map (is_skolemize_rule o fst) (Symtab.lookup steps s) =

            SOME true

          | is_clause_skolemize_rule _ = false

        (* The assumptions and conjecture are "prop"s; the other formulas are

           "bool"s. *)

        fun prop_of_clause [name as (s, ss)] =

            (case resolve_conjecture ss of

               [j] => if j = length hyp_ts then concl_t else nth hyp_ts j

             | _ => the_default ("", @{term False}) (Symtab.lookup steps s)

                    |> snd |> HOLogic.mk_Trueprop |> close_form)

          | prop_of_clause names =

            let

              val lits = map snd (map_filter (Symtab.lookup steps o fst) names)

            in

              case List.partition (can HOLogic.dest_not) lits of

                (negs as _ :: _, pos as _ :: _) =>

                HOLogic.mk_imp

                  (Library.foldr1 s_conj (map HOLogic.dest_not negs),

                   Library.foldr1 s_disj pos)

              | _ => fold (curry s_disj) lits @{term False}

            end

            |> HOLogic.mk_Trueprop |> close_form

        fun maybe_show outer c =

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

          ? cons Show

        fun isar_step_of_direct_inf outer (Have (gamma, c)) =

            let

              val l = label_of_clause c

              val t = prop_of_clause c

              val by =

                By_Metis (fold (add_fact_from_dependencies fact_names) gamma

                               ([], []))

            in

              if is_clause_skolemize_rule c then

                let

                  val is_fixed =

                    Variable.is_declared ctxt orf can Name.dest_skolem

                  val xs = Term.add_frees t [] |> filter_out (is_fixed o fst)

                in Obtain ([], xs, l, t, by) end

              else

                Prove (maybe_show outer c [], l, t, by)

            end

          | isar_step_of_direct_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 (isar_step_of_direct_inf outer) infs

        val (isar_proof, (preplay_fail, preplay_time)) =

          refute_graph

          |> redirect_graph axioms tainted bot

          |> map (isar_step_of_direct_inf true)

          |> append assms

          |> (if not preplay andalso isar_compress <= 1.0 then

                rpair (false, (true, seconds 0.0))

              else

                compress_proof debug ctxt type_enc lam_trans preplay

                  preplay_timeout

                  (if isar_proofs then isar_compress else 1000.0))

       (* |>> reorder_proof_to_minimize_jumps (* ? *) *)

          |>> chain_direct_proof

          |>> kill_useless_labels_in_proof

          |>> relabel_proof

          |>> not (null params) ? cons (Fix params)

        val isar_text =

          string_for_proof ctxt type_enc lam_trans subgoal subgoal_count

                           isar_proof

      in

        case isar_text of

          "" =>

          if isar_proofs then

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

          else

            ""

        | _ =>

          let

            val msg =

              (if preplay then

                [(if preplay_fail then "may fail, " else "") ^

                   Sledgehammer_Preplay.string_of_preplay_time preplay_time]

               else

                 []) @

              (if verbose then

                 let val num_steps = metis_steps_total isar_proof in

                   [string_of_int num_steps ^ " step" ^ plural_s num_steps]

                 end

               else

                 [])

          in

            "\n\nStructured proof "

              ^ (commas msg |> not (null msg) ? enclose "(" ")")

              ^ ":\n" ^ Active.sendback_markup isar_text

          end

      end

    val isar_proof =

      if debug then

        isar_proof_of ()

      else case try isar_proof_of () of

        SOME s => s

      | NONE => if isar_proofs then

                  "\nWarning: The Isar proof construction failed."

                else

                  ""

  in one_line_proof ^ isar_proof end

fun proof_text ctxt isar_proofs isar_params num_chained

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

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

     isar_proof_text ctxt isar_proofs isar_params

   else

     one_line_proof_text num_chained) one_line_params

end;