src/HOL/Tools/ATP/atp_reconstruct.ML

added caching for (in)finiteness checks

(*  Title:      HOL/Tools/ATP/atp_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_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 locality = ATP_Translate.locality

  datatype reconstructor =
    Metis |
    Metis_Full_Types |
    Metis_No_Types |
    SMT of string

  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 * locality) list * minimize_command * int * int
  type isar_params =
    bool * bool * int * string Symtab.table * int list list * int
    * (string * locality) list vector * int Symtab.table * string proof * thm
  val used_facts_in_atp_proof :
    Proof.context -> int -> (string * locality) list vector -> string proof
    -> (string * locality) list
  val used_facts_in_unsound_atp_proof :
    Proof.context -> int list list -> int -> (string * locality) list vector
    -> 'a proof -> string list option
  val uses_typed_helpers : int list -> 'a proof -> bool
  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_Reconstruct : ATP_RECONSTRUCT =
struct

open ATP_Util
open ATP_Problem
open ATP_Proof
open ATP_Translate

datatype reconstructor =
  Metis |
  Metis_Full_Types |
  Metis_No_Types |
  SMT of string

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 * locality) list * minimize_command * int * int
type isar_params =
  bool * bool * int * string Symtab.table * int list list * int
  * (string * locality) list vector * int Symtab.table * string proof * thm

val is_typed_helper_name =
  String.isPrefix helper_prefix andf String.isSuffix typed_helper_suffix

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 "_"

val vampire_step_prefix = "f" (* grrr... *)

val extract_step_number =
  Int.fromString o perhaps (try (unprefix vampire_step_prefix))

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 (_, SOME ss) =
    map_filter (resolve_one_named_fact fact_names) ss
  | resolve_fact facts_offset fact_names (num, NONE) =
    (case extract_step_number num of
       SOME j =>
       let val j = j - facts_offset in
         if j > 0 andalso j <= Vector.length fact_names then
           Vector.sub (fact_names, j - 1)
         else
           []
       end
     | NONE => [])

fun is_fact conjecture_shape = not o null o resolve_fact 0 conjecture_shape

fun resolve_one_named_conjecture s =
  case try (unprefix conjecture_prefix) s of
    SOME s' => Int.fromString s'
  | NONE => NONE

fun resolve_conjecture _ (_, SOME ss) =
    map_filter resolve_one_named_conjecture ss
  | resolve_conjecture conjecture_shape (num, NONE) =
    case extract_step_number num of
      SOME i => (case find_index (exists (curry (op =) i)) conjecture_shape of
                   ~1 => []
                 | j => [j])
    | NONE => []

fun is_conjecture conjecture_shape =
  not o null o resolve_conjecture conjecture_shape

fun is_typed_helper _ (_, SOME ss) = exists is_typed_helper_name ss
  | is_typed_helper typed_helpers (num, NONE) =
    (case extract_step_number num of
       SOME i => member (op =) typed_helpers i
     | NONE => false)

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 _ facts_offset fact_names (Inference (name, _, [])) =
    union (op =) (resolve_fact facts_offset fact_names name)
  | add_fact ctxt _ _ (Inference (_, _, deps)) =
    if AList.defined (op =) deps leo2_ext then
      insert (op =) (ext_name ctxt, Extensionality)
    else
      I
  | add_fact _ _ _ _ = I

fun used_facts_in_atp_proof ctxt facts_offset fact_names atp_proof =
  if null atp_proof then Vector.foldl (uncurry (union (op =))) [] fact_names
  else fold (add_fact ctxt facts_offset fact_names) atp_proof []

fun is_conjecture_used_in_proof conjecture_shape =
  exists (fn Inference (name, _, []) => is_conjecture conjecture_shape name
           | _ => false)

(* (quasi-)underapproximation of the truth *)
fun is_locality_global Local = false
  | is_locality_global Assum = false
  | is_locality_global Chained = false
  | is_locality_global _ = true

fun used_facts_in_unsound_atp_proof _ _ _ _ [] = NONE
  | used_facts_in_unsound_atp_proof ctxt conjecture_shape facts_offset
                                    fact_names atp_proof =
    let
      val used_facts =
        used_facts_in_atp_proof ctxt facts_offset fact_names atp_proof
    in
      if forall (is_locality_global o snd) used_facts andalso
         not (is_conjecture_used_in_proof conjecture_shape atp_proof) then
        SOME (map fst used_facts)
      else
        NONE
    end

fun uses_typed_helpers typed_helpers =
  exists (fn Inference (name, _, []) => is_typed_helper typed_helpers name
           | _ => false)


(** Soft-core proof reconstruction: Metis one-liner **)

(* unfortunate leaking abstraction *)
fun name_of_reconstructor Metis = "metis"
  | name_of_reconstructor Metis_Full_Types = "metis (full_types)"
  | name_of_reconstructor Metis_No_Types = "metis (no_types)"
  | name_of_reconstructor (SMT _) = "smt"

fun reconstructor_settings (SMT settings) = settings
  | reconstructor_settings _ = ""

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 set_settings "" = ""
  | set_settings settings = "using [[" ^ settings ^ "]] "
fun apply_on_subgoal settings _ 1 = set_settings settings ^ "by "
  | apply_on_subgoal settings 1 _ = set_settings settings ^ "apply "
  | apply_on_subgoal settings i n =
    "prefer " ^ string_of_int i ^ " " ^ apply_on_subgoal settings 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 Markup.sendback command ^ show_time time ^ "."
fun using_labels [] = ""
  | using_labels ls =
    "using " ^ space_implode " " (map string_for_label ls) ^ " "
fun reconstructor_command reconstructor i n (ls, ss) =
  using_labels ls ^
  apply_on_subgoal (reconstructor_settings reconstructor) i n ^
  command_call (name_of_reconstructor reconstructor) ss
fun minimize_line _ [] = ""
  | minimize_line minimize_command ss =
    case minimize_command ss of
      "" => ""
    | command => "\nTo minimize: " ^ Markup.markup Markup.sendback command ^ "."

val split_used_facts =
  List.partition (curry (op =) Chained o snd)
  #> 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, reconstructor, ext_time) =
      case preplay of
        Played (reconstructor, time) =>
        (false, reconstructor, (SOME (false, time)))
      | Trust_Playable (reconstructor, time) =>
        (false, reconstructor,
         case time of
           NONE => NONE
         | SOME time =>
           if time = Time.zeroTime then NONE else SOME (true, time))
      | Failed_to_Play reconstructor => (true, reconstructor, NONE)
    val try_line =
      ([], map fst extra)
      |> reconstructor_command reconstructor 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 conjecture_shape name =
  case resolve_conjecture conjecture_shape name of
    [j] => (conjecture_prefix, j)
  | _ => case Int.fromString (fst name) of
           SOME j => (raw_prefix, j)
         | NONE => (raw_prefix ^ fst name, 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 strip_prefix_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 strip_prefix_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 (strip_prefix_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 (strip_prefix_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 |> space_explode Long_Name.separator |> List.last |> Int.fromString |> the
  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 (a, us) =>
        if String.isPrefix simple_type_prefix a then
          @{const True} (* ignore TPTP type information *)
        else if a = 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 strip_prefix_and_unascii const_prefix a 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 ts = map (do_term [] NONE) us @ extra_ts
            val T = map slack_fastype_of ts ---> HOLogic.typeT
            val t =
              case strip_prefix_and_unascii fixed_var_prefix a of
                SOME b =>
                (* FIXME: reconstruction of lambda-lifting *)
                Free (b, T)
              | NONE =>
                case strip_prefix_and_unascii schematic_var_prefix a of
                  SOME b => Var ((b, var_index), T)
                | NONE =>
                  Var ((a |> 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_raw}),
   (@{const_name Meson.COMBK}, @{thm Meson.COMBK_def_raw}),
   (@{const_name Meson.COMBB}, @{thm Meson.COMBB_def_raw}),
   (@{const_name Meson.COMBC}, @{thm Meson.COMBC_def_raw}),
   (@{const_name Meson.COMBS}, @{thm Meson.COMBS_def_raw})]

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, deps)) ctxt =
    let val t = u |> uncombined_etc_prop_from_atp ctxt true sym_tab in
      (Inference (name, t, 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) HOLogic.true_const

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, deps)) =
    Inference (name, t, 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 conjecture_shape fact_names (Inference (name, t, [])) lines =
    (* No dependencies: fact, conjecture, or (for Vampire) internal facts or
       definitions. *)
    if is_fact fact_names name 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 conjecture_shape name then
      Inference (name, s_not t, []) :: lines
    else
      map (replace_dependencies_in_line (name, [])) lines
  | add_line _ _ (Inference (name, t, deps)) lines =
    (* Type information will be deleted later; skip repetition test. *)
    if is_only_type_information t then
      Inference (name, t, 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, deps) :: lines
     | (pre, Inference (name', t', _) :: post) =>
       Inference (name, t', 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 (Inference (name, t, [])) lines =
    if is_only_type_information t then delete_dependency name lines
    else Inference (name, t, []) :: 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 conjecture_shape fact_names frees
                     (Inference (name, t, deps)) (j, lines) =
    (j + 1,
     if is_fact fact_names name orelse is_conjecture conjecture_shape name 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, deps) :: lines  (* keep line *)
     else
       map (replace_dependencies_in_line (name, deps)) lines)  (* drop line *)

(** Isar proof construction and manipulation **)

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

type label = string * int
type facts = label list * string list

datatype isar_qualifier = Show | Then | Moreover | Ultimately

datatype isar_step =
  Fix of (string * typ) list |
  Let of term * term |
  Assume of label * term |
  Have of isar_qualifier list * label * term * byline
and byline =
  ByMetis of facts |
  CaseSplit of isar_step list list * facts

fun smart_case_split [] facts = ByMetis facts
  | smart_case_split proofs facts = CaseSplit (proofs, facts)

fun add_fact_from_dependency conjecture_shape facts_offset fact_names name =
  if is_fact fact_names name then
    apsnd (union (op =) (map fst (resolve_fact facts_offset fact_names name)))
  else
    apfst (insert (op =) (raw_label_for_name conjecture_shape name))

fun step_for_line _ _ _ _ (Definition (_, t1, t2)) = Let (t1, t2)
  | step_for_line conjecture_shape _ _ _ (Inference (name, t, [])) =
    Assume (raw_label_for_name conjecture_shape name, t)
  | step_for_line conjecture_shape facts_offset fact_names j
                  (Inference (name, t, deps)) =
    Have (if j = 1 then [Show] else [],
          raw_label_for_name conjecture_shape name,
          fold_rev forall_of (map Var (Term.add_vars t [])) t,
          ByMetis (fold (add_fact_from_dependency conjecture_shape facts_offset
                                                  fact_names)
                        deps ([], [])))

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 isar_proof_from_atp_proof pool ctxt isar_shrink_factor conjecture_shape
        facts_offset fact_names sym_tab params frees atp_proof =
  let
    val lines =
      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 conjecture_shape fact_names)
      |> rpair [] |-> fold_rev add_nontrivial_line
      |> rpair (0, [])
      |-> fold_rev (add_desired_line isar_shrink_factor conjecture_shape
                                     fact_names frees)
      |> snd
  in
    (if null params then [] else [Fix params]) @
    map2 (step_for_line conjecture_shape facts_offset fact_names)
         (length lines downto 1) lines
  end

(* When redirecting proofs, we keep information about the labels seen so far in
   the "backpatches" data structure. The first component indicates which facts
   should be associated with forthcoming proof steps. The second component is a
   pair ("assum_ls", "drop_ls"), where "assum_ls" are the labels that should
   become assumptions and "drop_ls" are the labels that should be dropped in a
   case split. *)
type backpatches = (label * facts) list * (label list * label list)

fun used_labels_of_step (Have (_, _, _, by)) =
    (case by of
       ByMetis (ls, _) => ls
     | CaseSplit (proofs, (ls, _)) =>
       fold (union (op =) o used_labels_of) proofs ls)
  | used_labels_of_step _ = []
and used_labels_of proof = fold (union (op =) o used_labels_of_step) proof []

fun new_labels_of_step (Fix _) = []
  | new_labels_of_step (Let _) = []
  | new_labels_of_step (Assume (l, _)) = [l]
  | new_labels_of_step (Have (_, l, _, _)) = [l]
val new_labels_of = maps new_labels_of_step

val join_proofs =
  let
    fun aux _ [] = NONE
      | aux proof_tail (proofs as (proof1 :: _)) =
        if exists null proofs then
          NONE
        else if forall (curry (op =) (hd proof1) o hd) (tl proofs) then
          aux (hd proof1 :: proof_tail) (map tl proofs)
        else case hd proof1 of
          Have ([], l, t, _) => (* FIXME: should we really ignore the "by"? *)
          if forall (fn Have ([], l', t', _) :: _ => (l, t) = (l', t')
                      | _ => false) (tl proofs) andalso
             not (exists (member (op =) (maps new_labels_of proofs))
                         (used_labels_of proof_tail)) then
            SOME (l, t, map rev proofs, proof_tail)
          else
            NONE
        | _ => NONE
  in aux [] o map rev end

fun case_split_qualifiers proofs =
  case length proofs of
    0 => []
  | 1 => [Then]
  | _ => [Ultimately]

fun redirect_proof hyp_ts concl_t proof =
  let
    (* The first pass outputs those steps that are independent of the negated
       conjecture. The second pass flips the proof by contradiction to obtain a
       direct proof, introducing case splits when an inference depends on
       several facts that depend on the negated conjecture. *)
     val concl_l = (conjecture_prefix, length hyp_ts)
     fun first_pass ([], contra) = ([], contra)
       | first_pass ((step as Fix _) :: proof, contra) =
         first_pass (proof, contra) |>> cons step
       | first_pass ((step as Let _) :: proof, contra) =
         first_pass (proof, contra) |>> cons step
       | first_pass ((step as Assume (l as (_, j), _)) :: proof, contra) =
         if l = concl_l then first_pass (proof, contra ||> cons step)
         else first_pass (proof, contra) |>> cons (Assume (l, nth hyp_ts j))
       | first_pass (Have (qs, l, t, ByMetis (ls, ss)) :: proof, contra) =
         let val step = Have (qs, l, t, ByMetis (ls, ss)) in
           if exists (member (op =) (fst contra)) ls then
             first_pass (proof, contra |>> cons l ||> cons step)
           else
             first_pass (proof, contra) |>> cons step
         end
       | first_pass _ = raise Fail "malformed proof"
    val (proof_top, (contra_ls, contra_proof)) =
      first_pass (proof, ([concl_l], []))
    val backpatch_label = the_default ([], []) oo AList.lookup (op =) o fst
    fun backpatch_labels patches ls =
      fold merge_fact_sets (map (backpatch_label patches) ls) ([], [])
    fun second_pass end_qs ([], assums, patches) =
        ([Have (end_qs, no_label, concl_t,
                ByMetis (backpatch_labels patches (map snd assums)))], patches)
      | second_pass end_qs (Assume (l, t) :: proof, assums, patches) =
        second_pass end_qs (proof, (t, l) :: assums, patches)
      | second_pass end_qs (Have (qs, l, t, ByMetis (ls, ss)) :: proof, assums,
                            patches) =
        (if member (op =) (snd (snd patches)) l andalso
            not (member (op =) (fst (snd patches)) l) andalso
            not (AList.defined (op =) (fst patches) l) then
           second_pass end_qs (proof, assums, patches ||> apsnd (append ls))
         else case List.partition (member (op =) contra_ls) ls of
           ([contra_l], co_ls) =>
           if member (op =) qs Show then
             second_pass end_qs (proof, assums,
                                 patches |>> cons (contra_l, (co_ls, ss)))
           else
             second_pass end_qs
                         (proof, assums,
                          patches |>> cons (contra_l, (l :: co_ls, ss)))
             |>> cons (if member (op =) (fst (snd patches)) l then
                         Assume (l, s_not t)
                       else
                         Have (qs, l, s_not t,
                               ByMetis (backpatch_label patches l)))
         | (contra_ls as _ :: _, co_ls) =>
           let
             val proofs =
               map_filter
                   (fn l =>
                       if l = concl_l then
                         NONE
                       else
                         let
                           val drop_ls = filter (curry (op <>) l) contra_ls
                         in
                           second_pass []
                               (proof, assums,
                                patches ||> apfst (insert (op =) l)
                                        ||> apsnd (union (op =) drop_ls))
                           |> fst |> SOME
                         end) contra_ls
             val (assumes, facts) =
               if member (op =) (fst (snd patches)) l then
                 ([Assume (l, s_not t)], (l :: co_ls, ss))
               else
                 ([], (co_ls, ss))
           in
             (case join_proofs proofs of
                SOME (l, t, proofs, proof_tail) =>
                Have (case_split_qualifiers proofs @
                      (if null proof_tail then end_qs else []), l, t,
                      smart_case_split proofs facts) :: proof_tail
              | NONE =>
                [Have (case_split_qualifiers proofs @ end_qs, no_label,
                       concl_t, smart_case_split proofs facts)],
              patches)
             |>> append assumes
           end
         | _ => raise Fail "malformed proof")
       | second_pass _ _ = raise Fail "malformed proof"
    val proof_bottom =
      second_pass [Show] (contra_proof, [], ([], ([], []))) |> fst
  in proof_top @ proof_bottom end

(* FIXME: Still needed? Probably not. *)
val kill_duplicate_assumptions_in_proof =
  let
    fun relabel_facts subst =
      apfst (map (fn l => AList.lookup (op =) subst l |> the_default l))
    fun do_step (step as Assume (l, t)) (proof, subst, assums) =
        (case AList.lookup (op aconv) assums t of
           SOME l' => (proof, (l, l') :: subst, assums)
         | NONE => (step :: proof, subst, (t, l) :: assums))
      | do_step (Have (qs, l, t, by)) (proof, subst, assums) =
        (Have (qs, l, t,
               case by of
                 ByMetis facts => ByMetis (relabel_facts subst facts)
               | CaseSplit (proofs, facts) =>
                 CaseSplit (map do_proof proofs, relabel_facts subst facts)) ::
         proof, subst, assums)
      | do_step step (proof, subst, assums) = (step :: proof, subst, assums)
    and do_proof proof = fold do_step proof ([], [], []) |> #1 |> rev
  in do_proof end

val then_chain_proof =
  let
    fun aux _ [] = []
      | aux _ ((step as Assume (l, _)) :: proof) = step :: aux l proof
      | aux l' (Have (qs, l, t, by) :: proof) =
        (case by of
           ByMetis (ls, ss) =>
           Have (if member (op =) ls l' then
                   (Then :: qs, l, t,
                    ByMetis (filter_out (curry (op =) l') ls, ss))
                 else
                   (qs, l, t, ByMetis (ls, ss)))
         | CaseSplit (proofs, facts) =>
           Have (qs, l, t, CaseSplit (map (aux no_label) proofs, facts))) ::
        aux l proof
      | aux _ (step :: proof) = step :: aux no_label proof
  in aux no_label end

fun kill_useless_labels_in_proof proof =
  let
    val used_ls = used_labels_of proof
    fun do_label l = if member (op =) used_ls l then l else no_label
    fun do_step (Assume (l, t)) = Assume (do_label l, t)
      | do_step (Have (qs, l, t, by)) =
        Have (qs, do_label l, t,
              case by of
                CaseSplit (proofs, facts) =>
                CaseSplit (map (map do_step) proofs, facts)
              | _ => by)
      | do_step step = step
  in map do_step proof end

fun prefix_for_depth n = replicate_string (n + 1)

val relabel_proof =
  let
    fun aux _ _ _ [] = []
      | aux subst depth (next_assum, next_fact) (Assume (l, t) :: proof) =
        if l = no_label then
          Assume (l, t) :: aux subst depth (next_assum, next_fact) proof
        else
          let val l' = (prefix_for_depth depth assum_prefix, next_assum) in
            Assume (l', t) ::
            aux ((l, l') :: subst) depth (next_assum + 1, next_fact) proof
          end
      | aux subst depth (next_assum, next_fact) (Have (qs, l, t, by) :: proof) =
        let
          val (l', subst, next_fact) =
            if l = no_label then
              (l, subst, next_fact)
            else
              let
                val l' = (prefix_for_depth depth 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
              ByMetis facts => ByMetis (relabel_facts facts)
            | CaseSplit (proofs, facts) =>
              CaseSplit (map (aux subst (depth + 1) (1, 1)) proofs,
                         relabel_facts facts)
        in
          Have (qs, l', t, by) ::
          aux subst depth (next_assum, next_fact) proof
        end
      | aux subst depth nextp (step :: proof) =
        step :: aux subst depth nextp proof
  in aux [] 0 (1, 1) end

fun string_for_proof ctxt0 full_types 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 reconstructor = if full_types then Metis_Full_Types else Metis
    fun do_facts (ls, ss) =
      reconstructor_command reconstructor 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 (Have (qs, l, t, ByMetis facts)) =
        do_indent ind ^ do_have qs ^ " " ^
        do_label l ^ do_term t ^ " " ^ do_facts facts ^ "\n"
      | do_step ind (Have (qs, l, t, CaseSplit (proofs, facts))) =
        space_implode (do_indent ind ^ "moreover\n")
                      (map (do_block ind) proofs) ^
        do_indent ind ^ do_have qs ^ " " ^ do_label l ^ do_term t ^ " " ^
        do_facts facts ^ "\n"
    and do_steps prefix suffix ind steps =
      let val s = implode (map (do_step ind) steps) in
        replicate_string (ind * indent_size - size prefix) " " ^ prefix ^
        String.extract (s, ind * indent_size,
                        SOME (size s - ind * indent_size - 1)) ^
        suffix ^ "\n"
      end
    and do_block ind proof = do_steps "{ " " }" (ind + 1) proof
    (* One-step proofs are pointless; better use the Metis one-liner
       directly. *)
    and do_proof [Have (_, _, _, ByMetis _)] = ""
      | do_proof proof =
        (if i <> 1 then "prefer " ^ string_of_int i ^ "\n" else "") ^
        do_indent 0 ^ "proof -\n" ^ do_steps "" "" 1 proof ^ do_indent 0 ^
        (if n <> 1 then "next" else "qed")
  in do_proof end

fun isar_proof_text ctxt isar_proof_requested
        (debug, full_types, isar_shrink_factor, pool, conjecture_shape,
         facts_offset, 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
    fun isar_proof_for () =
      case atp_proof
           |> isar_proof_from_atp_proof pool ctxt isar_shrink_factor
                  conjecture_shape facts_offset fact_names sym_tab params frees
           |> redirect_proof hyp_ts concl_t
           |> kill_duplicate_assumptions_in_proof
           |> then_chain_proof
           |> kill_useless_labels_in_proof
           |> relabel_proof
           |> string_for_proof ctxt full_types subgoal subgoal_count of
        "" =>
        if isar_proof_requested then
          "\nNo structured proof available (proof too short)."
        else
          ""
      | proof =>
        "\n\n" ^ (if isar_proof_requested then "Structured proof"
                  else "Perhaps this will work") ^
        ":\n" ^ Markup.markup Markup.sendback proof
    val isar_proof =
      if debug then
        isar_proof_for ()
      else
        case try isar_proof_for () 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;