(*  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 option -> isar_params -> one_line_params -> string

   val proof_text :

     Proof.context -> bool option -> 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 ((_, 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 ((_, 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)

 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 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 (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

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

   (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 (role, t) (_, 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 fact_names (name as (_, ss), role, t, rule, []) lines =

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

        definitions. *)

     if is_conjecture ss then

       (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 (name, role, t, _, _)) 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

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

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

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

 val waldmeister_conjecture_num = "1.0.0.0"

 val repair_waldmeister_endgame =

   let

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

       (name, Negated_Conjecture, s_not t, rule, deps)

     fun do_body [] = []

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

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

         else line :: do_body lines

   in do_body end

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

 fun add_nontrivial_line (line as (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 fact_names frees (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

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

    else

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

 val indent_size = 2

 fun string_for_proof ctxt type_enc lam_trans i n proof =

   let

     val register_fixes = map Free #> fold Variable.auto_fixes

     fun add_suffix suffix (s, ctxt) = (s ^ suffix, ctxt)

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

     fun of_moreover ind = of_indent ind ^ "moreover\n"

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

     fun of_obtain qs nr =

       (if nr>1 orelse (nr=1 andalso member (op=) qs Then)

         then "ultimately "

       else if nr=1 orelse member (op=) qs Then

         then "then "

         else "") ^ "obtain"

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

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

     fun of_prove qs nr =

       if nr>1 orelse (nr=1 andalso member (op=) qs Then)

         then "ultimately " ^ of_show_have qs

       else if nr=1 orelse member (op=) qs Then

         then of_thus_hence qs

         else of_show_have qs

     fun add_term term (s, ctxt)=

       (s ^ (annotate_types ctxt term

             |> with_vanilla_print_mode (Syntax.string_of_term ctxt)

             |> simplify_spaces

             |> maybe_quote),

        ctxt |> Variable.auto_fixes term)

     val reconstr = Metis (type_enc, lam_trans)

     fun of_metis ind options (ls, ss) =

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

       reconstructor_command reconstr 1 1 [] 0

           (ls |> sort_distinct (prod_ord string_ord int_ord),

            ss |> sort_distinct string_ord)

     fun of_free (s, T) =

       maybe_quote s ^ " :: " ^

       maybe_quote (simplify_spaces (with_vanilla_print_mode

         (Syntax.string_of_typ ctxt) T))

     fun add_frees xs (s, ctxt) =

       (s ^ space_implode " and " (map of_free xs), ctxt |> register_fixes xs)

     fun add_fix _ [] = I

       | add_fix ind xs = add_suffix (of_indent ind ^ "fix ")

                         #> add_frees xs

                         #> add_suffix "\n"

     fun add_assm ind (l, t) =

       add_suffix (of_indent ind ^ "assume " ^ of_label l)

       #> add_term t

       #> add_suffix "\n"

     fun add_assms ind assms = fold (add_assm ind) assms

     fun add_step_post ind l t facts options =

       add_suffix (of_label l)

       #> add_term t

       #> add_suffix (of_metis ind options facts ^ "\n")

     fun of_subproof ind ctxt proof =

       let

         val ind = ind + 1

         val s = of_proof ind ctxt proof

         val prefix = "{ "

         val suffix = " }"

       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 of_subproofs _ _ _ [] = ""

       | of_subproofs ind ctxt qs subproofs =

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

         space_implode (of_moreover ind) (map (of_subproof ind ctxt) subproofs)

     and add_step_pre ind qs subproofs (s, ctxt) =

       (s ^ of_subproofs ind ctxt qs subproofs ^ of_indent ind, ctxt)

     and add_step ind (Let (t1, t2)) =

         add_suffix (of_indent ind ^ "let ")

         #> add_term t1

         #> add_suffix " = "

         #> add_term t2

         #> add_suffix "\n"

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

         add_step_pre ind qs subproofs

         #> add_suffix (of_prove qs (length subproofs) ^ " ")

         #> add_step_post ind l t facts ""

       | add_step ind (Obtain (qs, Fix xs, l, t, By_Metis (subproofs, facts))) =

         add_step_pre ind qs subproofs

         #> add_suffix (of_obtain qs (length subproofs) ^ " ")

         #> add_frees xs

         #> add_suffix " where "

         (* 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). *)

         #> add_step_post ind l t facts

                (if exists (fn (_, T) => length (binder_types T) > 1) xs then

                   "using [[metis_new_skolem]] "

                 else

                   "")

     and add_steps ind = fold (add_step ind)

     and of_proof ind ctxt (Proof (Fix xs, Assume assms, steps)) =

       ("", ctxt)

       |> add_fix ind xs

       |> add_assms ind assms

       |> add_steps ind steps

       |> fst

   in

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

        directly. *)

     case proof of

       Proof (Fix [], Assume [], [Prove (_, _, _, By_Metis ([], _))]) => ""

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

             of_indent 0 ^ "proof -\n" ^ of_proof 1 ctxt proof ^

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

   end

 fun add_labels_of_step step =

   case byline_of_step step of

     NONE => I

   | SOME (By_Metis (subproofs, (ls, _))) =>

     union (op =) ls #> fold add_labels_of_proof subproofs

 and add_labels_of_proof proof = fold add_labels_of_step (steps_of_proof proof)

 fun kill_useless_labels_in_proof proof =

   let

     val used_ls = add_labels_of_proof proof []

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

     fun do_assms (Assume assms) = Assume (map (apfst do_label) assms)

     fun do_step (Obtain (qs, xs, l, t, By_Metis (subproofs, facts))) =

           Obtain (qs, xs, do_label l, t, By_Metis (map do_proof subproofs, facts))

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

           Prove (qs, do_label l, t, By_Metis (map do_proof subproofs, facts))

       | do_step step = step

     and do_proof (Proof (fix, assms, steps)) =

           Proof (fix, do_assms assms, map do_step steps)

   in do_proof proof end

 fun prefix_for_depth n = replicate_string (n + 1)

 val relabel_proof =

   let

     fun fresh_label depth prefix (old as (l, subst, next)) =

       if l = no_label then

         old

       else

         let val l' = (prefix_for_depth depth prefix, next) in

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

         end

     fun do_facts subst =

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

     fun do_assm depth (l, t) (subst, next) =

       let

         val (l, subst, next) =

           (l, subst, next) |> fresh_label depth assume_prefix

       in

         ((l, t), (subst, next))

       end

     fun do_assms subst depth (Assume assms) =

       fold_map (do_assm depth) assms (subst, 1)

       |> apfst Assume

       |> apsnd fst

     fun do_steps _ _ _ [] = []

       | do_steps subst depth next (Obtain (qs, xs, l, t, by) :: steps) =

         let

           val (l, subst, next) =

             (l, subst, next) |> fresh_label depth have_prefix

           val by = by |> do_byline subst depth

         in Obtain (qs, xs, l, t, by) :: do_steps subst depth next steps end

       | do_steps subst depth next (Prove (qs, l, t, by) :: steps) =

         let

           val (l, subst, next) =

             (l, subst, next) |> fresh_label depth have_prefix

           val by = by |> do_byline subst depth

         in Prove (qs, l, t, by) :: do_steps subst depth next steps end

       | do_steps subst depth next (step :: steps) =

         step :: do_steps subst depth next steps

     and do_proof subst depth (Proof (fix, assms, steps)) =

       let val (assms, subst) = do_assms subst depth assms in

         Proof (fix, assms, do_steps subst depth 1 steps)

       end

     and do_byline subst depth (By_Metis (subproofs, facts)) =

       By_Metis (do_proofs subst depth subproofs, do_facts subst facts)

     and do_proofs subst depth = map (do_proof subst (depth + 1))

   in do_proof [] 0 end

 val chain_direct_proof =

   let

     fun do_qs_lfs NONE lfs = ([], lfs)

       | do_qs_lfs (SOME l0) lfs =

         if member (op =) lfs l0 then ([Then], lfs |> remove (op =) l0)

         else ([], lfs)

     fun chain_step lbl (Obtain (qs, xs, l, t,

                                 By_Metis (subproofs, (lfs, gfs)))) =

         let val (qs', lfs) = do_qs_lfs lbl lfs in

           Obtain (qs' @ qs, xs, l, t,

             By_Metis (chain_proofs subproofs, (lfs, gfs)))

         end

       | chain_step lbl (Prove (qs, l, t, By_Metis (subproofs, (lfs, gfs)))) =

         let val (qs', lfs) = do_qs_lfs lbl lfs in

           Prove (qs' @ qs, l, t, By_Metis (chain_proofs subproofs, (lfs, gfs)))

         end

       | chain_step _ step = step

     and chain_steps _ [] = []

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

         chain_step prev i :: chain_steps (label_of_step i) is

       | chain_steps _ (i :: is) = i :: chain_steps (label_of_step i) is

     and chain_proof (Proof (fix, Assume assms, steps)) =

       Proof (fix, Assume assms,

              chain_steps (try (List.last #> fst) assms) steps)

     and chain_proofs proofs = map (chain_proof) proofs

   in chain_proof 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 (name as (_, ss), _, _, _, []) =>

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

               | _ => NONE)

         val assms =

           atp_proof |> map_filter

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

                 (case resolve_conjecture ss of

                    [j] =>

                    if j = length hyp_ts then NONE

                    else SOME (raw_label_for_name name, nth hyp_ts j)

                  | _ => NONE)

               | _ => NONE)

         val bot = atp_proof |> List.last |> #1

         val refute_graph =

           atp_proof

           |> map (fn (name, _, _, _, from) => (from, name))

           |> make_refute_graph bot

           |> fold (Atom_Graph.default_node o rpair ()) conjs

         val axioms = axioms_of_refute_graph refute_graph conjs

         val tainted = tainted_atoms_of_refute_graph refute_graph conjs

         val is_clause_tainted = exists (member (op =) tainted)

         val steps =

           Symtab.empty

           |> fold (fn (name as (s, _), role, t, rule, _) =>

                       Symtab.update_new (s, (rule,

                         t |> (if is_clause_tainted [name] then

                                 s_maybe_not role

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

                               else

                                 I))))

                   atp_proof

         fun is_clause_skolemize_rule [(s, _)] =

             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 [(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 _ :: _) =>

                 s_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 isar_proof_of_direct_proof infs =

           let

             fun maybe_show outer c =

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

               ? cons Show

             val is_fixed = Variable.is_declared ctxt orf can Name.dest_skolem

             fun skolems_of t =

               Term.add_frees t [] |> filter_out (is_fixed o fst) |> rev

             fun do_steps _ _ accum [] = rev accum

               | do_steps outer _ accum (Have (gamma, c) :: infs) =

                 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 no_facts))

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

                   fun do_rest lbl step =

                     do_steps outer (SOME lbl) (step :: accum) infs

                 in

                   if is_clause_tainted c then

                     case gamma of

                       [g] =>

                       if is_clause_skolemize_rule g andalso

                          is_clause_tainted g then

                         let

                           val subproof =

                             Proof (Fix (skolems_of (prop_of_clause g)),

                                    Assume [], rev accum)

                         in

                           do_steps outer (SOME l)

                               [prove (By_Metis ([subproof], no_facts))] []

                         end

                       else

                         do_rest l (prove by)

                     | _ => do_rest l (prove by)

                   else

                     if is_clause_skolemize_rule c then

                       do_rest l (Obtain ([], Fix (skolems_of t), l, t, by))

                     else

                       do_rest l (prove by)

                 end

               | do_steps outer predecessor accum (Cases cases :: infs) =

                 let

                   fun do_case (c, infs) =

                     do_proof false [] [(label_of_clause c, prop_of_clause c)] infs

                   val c = succedent_of_cases cases

                   val l = label_of_clause c

                   val t = prop_of_clause c

                   val step =

                     (Prove (maybe_show outer c [], l, t, By_Metis

                       (map do_case cases, (the_list predecessor, []))))

                 in

                   do_steps outer (SOME l) (step :: accum) infs

                 end

             and do_proof outer fix assms infs =

               Proof (Fix fix, Assume assms, do_steps outer NONE [] infs)

           in

             do_proof true params assms infs

           end

         val cleanup_labels_in_proof =

           chain_direct_proof

           #> kill_useless_labels_in_proof

           #> relabel_proof

         val (isar_proof, (preplay_fail, preplay_time)) =

           refute_graph

           |> redirect_graph axioms tainted bot

           |> isar_proof_of_direct_proof

           |> (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 = SOME true then isar_compress else 1000.0))

           |>> cleanup_labels_in_proof

         val isar_text =

           string_for_proof ctxt type_enc lam_trans subgoal subgoal_count

                            isar_proof

       in

         case isar_text of

           "" =>

           if isar_proofs = SOME true then

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

           else

             ""

         | _ =>

           let

             val msg =

               (if verbose then

                 let

                   val num_steps = add_metis_steps (steps_of_proof isar_proof) 0

                 in [string_of_int num_steps ^ " step" ^ plural_s num_steps] end

                else

                  []) @

               (if preplay then

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

                    Sledgehammer_Preplay.string_of_preplay_time preplay_time]

                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 = SOME true then

                   "\nWarning: The Isar proof construction failed."

                 else

                   ""

   in one_line_proof ^ isar_proof end

 fun isar_proof_would_be_a_good_idea preplay =

   case preplay of

     Played (reconstr, _) => reconstr = SMT

   | Trust_Playable _ => true

   | Failed_to_Play _ => true

 fun proof_text ctxt isar_proofs isar_params num_chained

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

   (if isar_proofs = SOME true orelse

       (isar_proofs = NONE andalso isar_proof_would_be_a_good_idea preplay) then

      isar_proof_text ctxt isar_proofs isar_params

    else

      one_line_proof_text num_chained) one_line_params

 end;