src/HOL/Tools/ATP/atp_proof_reconstruct.ML

parse applie lambdas correctly plus deal gracefully with lambda-lifting in Zipperposition

(*  Title:      HOL/Tools/ATP/atp_proof_reconstruct.ML
    Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
    Author:     Claire Quigley, Cambridge University Computer Laboratory
    Author:     Jasmin Blanchette, TU Muenchen

Basic proof reconstruction from ATP proofs.
*)

signature ATP_PROOF_RECONSTRUCT =
sig
  type 'a atp_type = 'a ATP_Problem.atp_type
  type ('a, 'b) atp_term = ('a, 'b) ATP_Problem.atp_term
  type ('a, 'b, 'c, 'd) atp_formula = ('a, 'b, 'c, 'd) ATP_Problem.atp_formula
  type stature = ATP_Problem_Generate.stature
  type atp_step_name = ATP_Proof.atp_step_name
  type ('a, 'b) atp_step = ('a, 'b) ATP_Proof.atp_step
  type 'a atp_proof = 'a ATP_Proof.atp_proof

  val metisN : string
  val full_typesN : string
  val partial_typesN : string
  val no_typesN : string
  val really_full_type_enc : string
  val full_type_enc : string
  val partial_type_enc : string
  val no_type_enc : string
  val full_type_encs : string list
  val partial_type_encs : string list
  val default_metis_lam_trans : string

  val leo2_extcnf_equal_neg_rule : string
  val satallax_tab_rule_prefix : string

  val forall_of : term -> term -> term
  val exists_of : term -> term -> term
  val simplify_bool : term -> term
  val var_name_of_typ : typ -> string
  val rename_bound_vars : term -> term
  val type_enc_aliases : (string * string list) list
  val unalias_type_enc : string -> string list
  val term_of_atp : Proof.context -> ATP_Problem.atp_format -> ATP_Problem_Generate.type_enc ->
    bool -> int Symtab.table -> typ option -> (string, string atp_type) atp_term -> term
  val prop_of_atp : Proof.context -> ATP_Problem.atp_format -> ATP_Problem_Generate.type_enc ->
    bool -> int Symtab.table ->
    (string, string, (string, string atp_type) atp_term, string) atp_formula -> term

  val is_conjecture_used_in_proof : string atp_proof -> bool
  val used_facts_in_atp_proof : Proof.context -> (string * stature) list -> string atp_proof ->
    (string * stature) list
  val atp_proof_prefers_lifting : string atp_proof -> bool
  val is_typed_helper_used_in_atp_proof : string atp_proof -> bool
  val replace_dependencies_in_line : atp_step_name * atp_step_name list -> ('a, 'b) atp_step ->
    ('a, 'b) atp_step
  val termify_atp_proof : Proof.context -> string -> ATP_Problem.atp_format ->
    ATP_Problem_Generate.type_enc -> string Symtab.table -> (string * term) list ->
    int Symtab.table -> string atp_proof -> (term, string) atp_step list
  val repair_waldmeister_endgame : (term, 'a) atp_step list -> (term, 'a) atp_step list
  val infer_formulas_types : Proof.context -> term list -> term list
  val introduce_spass_skolems : (term, string) atp_step list -> (term, string) atp_step list
  val factify_atp_proof : (string * 'a) list -> term list -> term -> (term, string) atp_step list ->
    (term, string) atp_step list
  val termify_atp_abduce_candidate : Proof.context -> string -> ATP_Problem.atp_format ->
    ATP_Problem_Generate.type_enc -> string Symtab.table -> (string * term) list ->
    int Symtab.table -> (string, string, (string, string atp_type) atp_term, string) atp_formula ->
    term
  val top_abduce_candidates : int -> term list -> term list
  val sort_propositions_by_provability : Proof.context -> term list -> term list
end;

structure ATP_Proof_Reconstruct : ATP_PROOF_RECONSTRUCT =
struct

open ATP_Util
open ATP_Problem
open ATP_Proof
open ATP_Problem_Generate

val metisN = "metis"

val full_typesN = "full_types"
val partial_typesN = "partial_types"
val no_typesN = "no_types"

val really_full_type_enc = "mono_tags"
val full_type_enc = "poly_guards_query"
val partial_type_enc = "poly_args"
val no_type_enc = "erased"

val full_type_encs = [full_type_enc, really_full_type_enc]
val partial_type_encs = partial_type_enc :: full_type_encs

val type_enc_aliases =
  [(full_typesN, full_type_encs),
   (partial_typesN, partial_type_encs),
   (no_typesN, [no_type_enc])]

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

val default_metis_lam_trans = combsN

val leo2_extcnf_equal_neg_rule = "extcnf_equal_neg"
val satallax_tab_rule_prefix = "tab_"

fun term_name' (Var ((s, _), _)) = perhaps (try Name.dest_skolem) s
  | term_name' _ = ""

fun lambda' v = Term.lambda_name (term_name' v, v)

fun If_const T = Const (\<^const_name>\<open>If\<close>, HOLogic.boolT --> T --> T --> T)

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_tfree ctxt w =
  let val ww = "'" ^ w in
    TFree (ww, the_default \<^sort>\<open>type\<close> (Variable.def_sort ctxt (ww, ~1)))
  end

fun simplify_bool ((all as Const (\<^const_name>\<open>All\<close>, _)) $ Abs (s, T, t)) =
    let val t' = simplify_bool t in
      if loose_bvar1 (t', 0) then all $ Abs (s, T, t') else t'
    end
  | simplify_bool (Const (\<^const_name>\<open>Not\<close>, _) $ t) = s_not (simplify_bool t)
  | simplify_bool (Const (\<^const_name>\<open>conj\<close>, _) $ t $ u) =
    s_conj (simplify_bool t, simplify_bool u)
  | simplify_bool (Const (\<^const_name>\<open>disj\<close>, _) $ t $ u) =
    s_disj (simplify_bool t, simplify_bool u)
  | simplify_bool (Const (\<^const_name>\<open>implies\<close>, _) $ t $ u) =
    s_imp (simplify_bool t, simplify_bool u)
  | simplify_bool ((t as Const (\<^const_name>\<open>HOL.eq\<close>, _)) $ u $ v) =
    (case (u, v) of
      (Const (\<^const_name>\<open>True\<close>, _), _) => v
    | (u, Const (\<^const_name>\<open>True\<close>, _)) => u
    | (Const (\<^const_name>\<open>False\<close>, _), v) => s_not v
    | (u, Const (\<^const_name>\<open>False\<close>, _)) => s_not u
    | _ => if u aconv v then \<^Const>\<open>True\<close> else t $ simplify_bool u $ simplify_bool v)
  | simplify_bool (t $ u) = simplify_bool t $ simplify_bool u
  | simplify_bool (Abs (s, T, t)) = Abs (s, T, simplify_bool t)
  | simplify_bool t = t

fun single_letter upper s =
  let val s' = if String.isPrefix "o" s orelse String.isPrefix "O" s then "z" else s in
    String.extract (Name.desymbolize (SOME upper) (Long_Name.base_name s'), 0, SOME 1)
  end

fun var_name_of_typ (Type (\<^type_name>\<open>fun\<close>, [_, T])) =
    if body_type T = HOLogic.boolT then "p" else "f"
  | var_name_of_typ (Type (\<^type_name>\<open>set\<close>, [T])) =
    let fun default () = single_letter true (var_name_of_typ T) in
      (case T of
        Type (s, [T1, T2]) => if String.isSuffix "prod" s andalso T1 = T2 then "r" else default ()
      | _ => default ())
    end
  | var_name_of_typ (Type (s, Ts)) =
    if String.isSuffix "list" s then var_name_of_typ (the_single Ts) ^ "s"
    else single_letter false (Long_Name.base_name s)
  | var_name_of_typ (TFree (s, _)) = single_letter false (perhaps (try (unprefix "'")) s)
  | var_name_of_typ (TVar ((s, _), T)) = var_name_of_typ (TFree (s, T))

fun rename_bound_vars (t $ u) = rename_bound_vars t $ rename_bound_vars u
  | rename_bound_vars (Abs (_, T, t)) = Abs (var_name_of_typ T, T, rename_bound_vars t)
  | rename_bound_vars t = t

exception ATP_CLASS of string list
exception ATP_TYPE of string atp_type list
exception ATP_TERM of (string, string atp_type) atp_term list
exception ATP_FORMULA of
  (string, string, (string, string atp_type) atp_term, string) atp_formula list
exception SAME of unit

fun class_of_atp_class cls =
  (case unprefix_and_unascii class_prefix cls of
    SOME s => s
  | NONE => raise ATP_CLASS [cls])

fun atp_type_of_atp_term (ATerm ((s, _), us)) =
  let val tys = map atp_type_of_atp_term us in
    if s = tptp_fun_type then
      (case tys of
        [ty1, ty2] => AFun (ty1, ty2)
      | _ => raise ATP_TYPE tys)
    else
      AType ((s, []), tys)
  end

(* 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_of_atp_type ctxt (ty as AType ((a, clss), tys)) =
    let val Ts = map (typ_of_atp_type ctxt) tys in
      (case unprefix_and_unascii native_type_prefix a of
        SOME b => typ_of_atp_type ctxt (atp_type_of_atp_term (unmangled_type b))
      | NONE =>
        (case unprefix_and_unascii type_const_prefix a of
          SOME b => Type (invert_const b, Ts)
        | NONE =>
          if not (null tys) then
            raise ATP_TYPE [ty] (* only "tconst"s have type arguments *)
          else
            (case unprefix_and_unascii tfree_prefix a of
              SOME b => make_tfree ctxt b
            | NONE =>
              (* The term 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. *)
              Type_Infer.param 0 ("'" ^ perhaps (unprefix_and_unascii tvar_prefix) a,
                (if null clss then \<^sort>\<open>type\<close> else map class_of_atp_class clss)))))
    end
  | typ_of_atp_type ctxt (AFun (ty1, ty2)) = typ_of_atp_type ctxt ty1 --> typ_of_atp_type ctxt ty2

fun typ_of_atp_term ctxt = typ_of_atp_type ctxt o atp_type_of_atp_term

(* Type class literal applied to a type. Returns triple of polarity, class, type. *)
fun type_constraint_of_term ctxt (u as ATerm ((a, _), us)) =
  (case (unprefix_and_unascii class_prefix a, map (typ_of_atp_term ctxt) us) of
    (SOME b, [T]) => (b, T)
  | _ => raise ATP_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_var_name s =
  (case unprefix_and_unascii schematic_var_prefix s of
    SOME s' => s'
  | NONE => s)

(* 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 robust_const_num_type_args thy s =
  if String.isPrefix skolem_const_prefix s then
    s |> Long_Name.explode |> List.last |> Int.fromString |> the
  else if String.isPrefix lam_lifted_prefix s then
    if String.isPrefix lam_lifted_poly_prefix s then 2 else 0
  else
    (s, Sign.the_const_type thy s) |> Sign.const_typargs thy |> length

fun slack_fastype_of t = fastype_of t handle TERM _ => Type_Infer.anyT \<^sort>\<open>type\<close>

val spass_skolem_prefix = "sk" (* "skc" or "skf" *)
val vampire_skolem_prefix = "sK"

val zip_internal_ite_prefix = "zip_internal_ite_"

fun var_index_of_textual textual = if textual then 0 else 1

fun quantify_over_var textual quant_of var_s var_T t =
  let
    val vars =
      ((var_s, var_index_of_textual textual), var_T) ::
      filter (fn ((s, _), _) => s = var_s) (Term.add_vars t [])
    val normTs = vars |> AList.group (op =) |> map (apsnd hd)
    fun norm_var_types (Var (x, T)) =
        Var (x, the_default T (AList.lookup (op =) normTs x))
      | norm_var_types t = t
  in t |> map_aterms norm_var_types |> fold_rev quant_of (map Var normTs) end

(* This assumes that distinct names are mapped to distinct names by "Variable.variant_frees". This
   does not hold in general but should hold for ATP-generated Skolem function names, since these end
   with a digit and "variant_frees" appends letters. *)
fun fresh_up ctxt s = fst (hd (Variable.variant_frees ctxt [] [(s, ())]))

(* Higher-order translation. Variables are typed (although we don't use that information). Lambdas
   are typed. The code is similar to "term_of_atp_fo". *)
fun term_of_atp_ho ctxt sym_tab =
  let
    val thy = Proof_Context.theory_of ctxt
    val var_index = var_index_of_textual true

    fun do_term opt_T u =
      (case u of
        AAbs (((var, ty), term), us) =>
        let
          val typ = typ_of_atp_type ctxt ty
          val var_name = repair_var_name var
          val tm = do_term NONE term
          val lam = quantify_over_var true lambda' var_name typ tm
          val args = map (do_term NONE) us
        in
          list_comb (lam, args)
        end
      | ATerm ((s, tys), us) =>
        if s = "" (* special marker generated on parse error *) then
          error "Isar proof reconstruction failed because the ATP proof contains unparsable \
            \material"
        else if s = tptp_equal then
          list_comb (Const (\<^const_name>\<open>HOL.eq\<close>, Type_Infer.anyT \<^sort>\<open>type\<close>),
            map (do_term NONE) us)
        else if s = tptp_not_equal andalso length us = 2 then
          \<^const>\<open>HOL.Not\<close> $ list_comb (Const (\<^const_name>\<open>HOL.eq\<close>, Type_Infer.anyT \<^sort>\<open>type\<close>),
            map (do_term NONE) us)
        else if not (null us) then
          let
            val args = map (do_term NONE) us
            val opt_T' = SOME (map slack_fastype_of args ---> the_default dummyT opt_T)
            val func = do_term opt_T' (ATerm ((s, tys), []))
          in foldl1 (op $) (func :: args) end
        else if s = tptp_or then HOLogic.disj
        else if s = tptp_and then HOLogic.conj
        else if s = tptp_implies then HOLogic.imp
        else if s = tptp_iff orelse s = tptp_equal then HOLogic.eq_const dummyT
        else if s = tptp_not_iff orelse s = tptp_not_equal then \<^term>\<open>\<lambda>x y. x \<noteq> y\<close>
        else if s = tptp_if then \<^term>\<open>\<lambda>P Q. Q \<longrightarrow> P\<close>
        else if s = tptp_not_and then \<^term>\<open>\<lambda>P Q. \<not> (P \<and> Q)\<close>
        else if s = tptp_not_or then \<^term>\<open>\<lambda>P Q. \<not> (P \<or> Q)\<close>
        else if s = tptp_not then HOLogic.Not
        else if s = tptp_ho_forall then HOLogic.all_const dummyT
        else if s = tptp_ho_exists then HOLogic.exists_const dummyT
        else if s = tptp_hilbert_choice then HOLogic.choice_const dummyT
        else if s = tptp_hilbert_the then \<^term>\<open>The\<close>
        else if String.isPrefix zip_internal_ite_prefix s then If_const dummyT
        else
          (case unprefix_and_unascii const_prefix s of
            SOME s' =>
            let
              val ((s', _), mangled_us) = s' |> unmangled_const |>> `invert_const
              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 Ts = map (typ_of_atp_type ctxt) tys @ map (typ_of_atp_term ctxt) type_us
              val T =
                (if not (null Ts) andalso robust_const_num_type_args thy s' = length Ts then
                   try (Sign.const_instance thy) (s', Ts)
                 else
                   NONE)
                |> (fn SOME T => T
                     | NONE =>
                       map slack_fastype_of term_ts --->
                       the_default (Type_Infer.anyT \<^sort>\<open>type\<close>) opt_T)
              val t = Const (unproxify_const s', T)
            in list_comb (t, term_ts) end
          | NONE => (* a free or schematic variable *)
            let
              val ts = map (do_term NONE) us
              val T =
                (case tys of
                  [ty] => typ_of_atp_type ctxt ty
                | _ =>
                  map slack_fastype_of ts --->
                  (case opt_T of
                    SOME T => T
                  | NONE => Type_Infer.anyT \<^sort>\<open>type\<close>))
              val t =
                (case unprefix_and_unascii fixed_var_prefix s of
                  SOME s => Free (s, T)
                | NONE =>
                  if not (is_tptp_variable s) then Free (fresh_up ctxt s, T)
                  else Var ((repair_var_name s, var_index), T))
            in list_comb (t, ts) end))
  in do_term end

(* First-order translation. No types are known for variables. "Type_Infer.anyT @{sort type}"
   should allow them to be inferred. *)
fun term_of_atp_fo 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 = var_index_of_textual textual

    fun do_term extra_ts opt_T u =
      (case u of
        ATerm ((s, tys), us) =>
        if s = "" (* special marker generated on parse error *) then
          error "Isar proof reconstruction failed because the ATP proof contains unparsable \
            \material"
        else if String.isPrefix native_type_prefix s then
          \<^Const>\<open>True\<close> (* ignore TPTP type information (needed?) *)
        else if s = tptp_equal then
          list_comb (Const (\<^const_name>\<open>HOL.eq\<close>, Type_Infer.anyT \<^sort>\<open>type\<close>),
            map (do_term [] NONE) us)
        else
          (case unprefix_and_unascii const_prefix s of
            SOME s' =>
            let val ((s', s''), mangled_us) = s' |> unmangled_const |>> `invert_const in
              if s' = type_tag_name then
                (case mangled_us @ us of
                  [typ_u, term_u] => do_term extra_ts (SOME (typ_of_atp_term ctxt typ_u)) term_u
                | _ => raise ATP_TERM us)
              else if s' = predicator_name then
                do_term [] (SOME \<^typ>\<open>bool\<close>) (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>\<open>True\<close> (* 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 Ts = map (typ_of_atp_type ctxt) tys @ map (typ_of_atp_term ctxt) type_us
                  val T =
                    (if not (null Ts) andalso robust_const_num_type_args thy s' = length Ts then
                       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
                     else
                       NONE)
                    |> (fn SOME T => T
                         | NONE =>
                           map slack_fastype_of term_ts --->
                           the_default (Type_Infer.anyT \<^sort>\<open>type\<close>) opt_T)
                  val t =
                    if new_skolem then Var ((new_skolem_var_name_of_const s'', var_index), T)
                    else Const (unproxify_const s', T)
                in
                  list_comb (t, term_ts @ extra_ts)
                end
            end
          | NONE => (* a free or schematic variable *)
            let
              val term_ts =
                map (do_term [] NONE) us
                (* SPASS (3.8ds) and Vampire (2.6) pass arguments to Skolem functions in reverse
                   order, which is incompatible with "metis"'s new skolemizer. *)
                |> exists (fn pre => String.isPrefix pre s)
                  [spass_skolem_prefix, vampire_skolem_prefix] ? rev
              val ts = term_ts @ extra_ts
              val T =
                (case tys of
                  [ty] => typ_of_atp_type ctxt ty
                | _ =>
                  (case opt_T of
                    SOME T => map slack_fastype_of term_ts ---> T
                  | NONE => map slack_fastype_of ts ---> Type_Infer.anyT \<^sort>\<open>type\<close>))
              val t =
                (case unprefix_and_unascii fixed_var_prefix s of
                  SOME s => Free (s, T)
                | NONE =>
                  if textual andalso not (is_tptp_variable s) then
                    Free (s |> textual ? fresh_up ctxt, T)
                  else
                    Var ((repair_var_name s, var_index), T))
            in list_comb (t, ts) end))
  in do_term [] end

fun term_of_atp ctxt (ATP_Problem.THF _) type_enc =
    if ATP_Problem_Generate.is_type_enc_higher_order type_enc then K (term_of_atp_ho ctxt)
    else error "Unsupported Isar reconstruction"
  | term_of_atp ctxt _ type_enc =
    if not (ATP_Problem_Generate.is_type_enc_higher_order type_enc) then term_of_atp_fo ctxt
    else error "Unsupported Isar reconstruction"

fun term_of_atom ctxt format type_enc textual sym_tab pos (u as ATerm ((s, _), _)) =
  if String.isPrefix class_prefix s then
    add_type_constraint pos (type_constraint_of_term ctxt u)
    #> pair \<^Const>\<open>True\<close>
  else
    pair (term_of_atp ctxt format type_enc textual sym_tab (SOME \<^typ>\<open>bool\<close>) u)

(* 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

(* Interpret an ATP formula as a HOL term, extracting sort constraints as they appear in the
   formula. *)
fun prop_of_atp ctxt format type_enc 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 textual (case q of AForall => forall_of | AExists => exists_of)
          (repair_var_name s) dummyT
      | 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_of_atom ctxt format type_enc textual sym_tab pos tm
      | _ => raise ATP_FORMULA [phi])
  in
    repair_tvar_sorts (do_formula true phi Vartab.empty)
  end

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

fun resolve_fact facts s =
  (case try (unprefix fact_prefix) s of
    SOME s' =>
    let val s' = s' |> unprefix_fact_number |> unascii_of in
      AList.lookup (op =) facts s' |> Option.map (pair s')
    end
  | NONE => NONE)

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

fun resolve_facts facts = map_filter (resolve_fact facts)
val resolve_conjectures = map_filter resolve_conjecture

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

val is_conjecture_used_in_proof = is_axiom_used_in_proof (is_some o resolve_conjecture)

fun add_fact ctxt facts ((num, ss), _, _, rule, deps) =
  (if member (op =) [agsyhol_core_rule, leo2_extcnf_equal_neg_rule] rule orelse
      String.isPrefix satallax_tab_rule_prefix rule then
     insert (op =) (short_thm_name ctxt ext, (Global, General))
   else
     I)
  #> (if null deps then union (op =) (resolve_facts facts (num :: ss)) else I)

fun used_facts_in_atp_proof ctxt facts atp_proof =
  if null atp_proof then facts else fold (add_fact ctxt facts) atp_proof []

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 atp_proof_prefers_lifting atp_proof =
  (is_axiom_used_in_proof is_combinator_def atp_proof,
   is_axiom_used_in_proof is_lam_lifted atp_proof) = (false, true)

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 replace_one_dependency (old, new) dep = if is_same_atp_step dep old then new else [dep]
fun replace_dependencies_in_line old_new (name, role, t, rule, deps) =
  (name, role, t, rule, fold (union (op =) o replace_one_dependency old_new) 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 set_var_index j = map_aterms (fn Var ((s, 0), T) => Var ((s, j), T) | t => t)

fun infer_formulas_types ctxt =
  map_index (uncurry (fn j => set_var_index j #> Type.constraint HOLogic.boolT))
  #> Syntax.check_terms (Proof_Context.set_mode Proof_Context.mode_schematic ctxt)

val combinator_table =
  [(\<^const_name>\<open>Meson.COMBI\<close>, @{thm Meson.COMBI_def [abs_def]}),
   (\<^const_name>\<open>Meson.COMBK\<close>, @{thm Meson.COMBK_def [abs_def]}),
   (\<^const_name>\<open>Meson.COMBB\<close>, @{thm Meson.COMBB_def [abs_def]}),
   (\<^const_name>\<open>Meson.COMBC\<close>, @{thm Meson.COMBC_def [abs_def]}),
   (\<^const_name>\<open>Meson.COMBS\<close>, @{thm Meson.COMBS_def [abs_def]})]

fun uncombine_term thy =
  let
    fun uncomb (t1 $ t2) = betapply (uncomb t1, uncomb t2)
      | uncomb (Abs (s, T, t)) = Abs (s, T, uncomb t)
      | uncomb (t as Const (x as (s, _))) =
        (case AList.lookup (op =) combinator_table s of
          SOME thm => thm |> Thm.prop_of |> specialize_type thy x |> Logic.dest_equals |> snd
        | NONE => t)
      | uncomb t = t
  in uncomb end

fun unlift_aterm lifted (t as Const (s, _)) =
    if String.isPrefix lam_lifted_prefix s then
      (* FIXME: do something about the types *)
      (case AList.lookup (op =) lifted s of
        SOME t' => unlift_term lifted t'
      | NONE => t)
    else
      t
  | unlift_aterm _ t = t
and unlift_term lifted =
  map_aterms (unlift_aterm lifted)

fun termify_line _ _ _ _ _ (_, Type_Role, _, _, _) = NONE
  | termify_line ctxt format type_enc lifted sym_tab (name, role, u, rule, deps) =
    let
      val thy = Proof_Context.theory_of ctxt
      val t = u
        |> prop_of_atp ctxt format type_enc true sym_tab
        |> unlift_term lifted
        |> uncombine_term thy
        |> simplify_bool
    in
      SOME (name, role, t, rule, deps)
    end

val waldmeister_conjecture_num = "1.0.0.0"

fun repair_waldmeister_endgame proof =
  let
    fun repair_tail (name, _, \<^Const>\<open>Trueprop for t\<close>, rule, deps) =
      (name, Negated_Conjecture, \<^Const>\<open>Trueprop for \<open>s_not t\<close>\<close>, rule, deps)
    fun repair_body [] = []
      | repair_body ((line as ((num, _), _, _, _, _)) :: lines) =
        if num = waldmeister_conjecture_num then map repair_tail (line :: lines)
        else line :: repair_body lines
  in
    repair_body proof
  end

fun map_proof_terms f (lines : ('a * 'b * 'c * 'd * 'e) list) =
  map2 (fn c => fn (a, b, _, d, e) => (a, b, c, d, e)) (f (map #3 lines)) lines

fun termify_atp_proof ctxt local_prover format type_enc pool lifted sym_tab =
  nasty_atp_proof pool
  #> map_term_names_in_atp_proof repair_name
  #> map_filter (termify_line ctxt format type_enc lifted sym_tab)
  #> map_proof_terms (infer_formulas_types ctxt #> map HOLogic.mk_Trueprop)
  #> local_prover = waldmeisterN ? repair_waldmeister_endgame

fun unskolemize_spass_term skos =
  let
    val is_skolem_name = member (op =) skos

    fun find_argless_skolem (Free _ $ Var _) = NONE
      | find_argless_skolem (Free (x as (s, _))) = if is_skolem_name s then SOME x else NONE
      | find_argless_skolem (t $ u) =
        (case find_argless_skolem t of NONE => find_argless_skolem u | sk => sk)
      | find_argless_skolem (Abs (_, _, t)) = find_argless_skolem t
      | find_argless_skolem _ = NONE

    fun find_skolem_arg (Free (s, _) $ Var v) = if is_skolem_name s then SOME v else NONE
      | find_skolem_arg (t $ u) = (case find_skolem_arg t of NONE => find_skolem_arg u | v => v)
      | find_skolem_arg (Abs (_, _, t)) = find_skolem_arg t
      | find_skolem_arg _ = NONE

    fun kill_skolem_arg (t as Free (s, T) $ Var _) =
        if is_skolem_name s then Free (s, range_type T) else t
      | kill_skolem_arg (t $ u) = kill_skolem_arg t $ kill_skolem_arg u
      | kill_skolem_arg (Abs (s, T, t)) = Abs (s, T, kill_skolem_arg t)
      | kill_skolem_arg t = t

    fun find_var (Var v) = SOME v
      | find_var (t $ u) = (case find_var t of NONE => find_var u | v => v)
      | find_var (Abs (_, _, t)) = find_var t
      | find_var _ = NONE

    val safe_abstract_over = abstract_over o apsnd (incr_boundvars 1)

    fun unskolem_inner t =
      (case find_argless_skolem t of
        SOME (x as (s, T)) =>
        HOLogic.exists_const T $ Abs (s, T, unskolem_inner (safe_abstract_over (Free x, t)))
      | NONE =>
        (case find_skolem_arg t of
          SOME (v as ((s, _), T)) =>
          let
            val (haves, have_nots) =
              HOLogic.disjuncts t
              |> List.partition (exists_subterm (curry (op =) (Var v)))
              |> apply2 (fn lits => fold (curry s_disj) lits \<^term>\<open>False\<close>)
          in
            s_disj (HOLogic.all_const T
                $ Abs (s, T, unskolem_inner (safe_abstract_over (Var v, kill_skolem_arg haves))),
              unskolem_inner have_nots)
          end
        | NONE =>
          (case find_var t of
            SOME (v as ((s, _), T)) =>
            HOLogic.all_const T $ Abs (s, T, unskolem_inner (safe_abstract_over (Var v, t)))
          | NONE => t)))

    fun unskolem_outer (@{const Trueprop} $ t) = @{const Trueprop} $ unskolem_outer t
      | unskolem_outer t = unskolem_inner t
  in
    unskolem_outer
  end

fun rename_skolem_args t =
  let
    fun add_skolem_args (Abs (_, _, t)) = add_skolem_args t
      | add_skolem_args t =
        (case strip_comb t of
          (Free (s, _), args as _ :: _) =>
          if String.isPrefix spass_skolem_prefix s then
            insert (op =) (s, take_prefix is_Var args)
          else
            fold add_skolem_args args
        | (u, args as _ :: _) => fold add_skolem_args (u :: args)
        | _ => I)

    fun subst_of_skolem (sk, args) =
      map_index (fn (j, Var (z, T)) => (z, Var ((sk ^ "_" ^ string_of_int j, 0), T))) args

    val subst = maps subst_of_skolem (add_skolem_args t [])
  in
    subst_vars ([], subst) t
  end

fun introduce_spass_skolems proof =
  let
    fun add_skolem (Free (s, _)) = String.isPrefix spass_skolem_prefix s ? insert (op =) s
      | add_skolem _ = I

    (* union-find would be faster *)
    fun add_cycle [] = I
      | add_cycle ss =
        fold (fn s => Graph.default_node (s, ())) ss
        #> fold Graph.add_edge (ss ~~ tl ss @ [hd ss])

    val (input_steps, other_steps) = List.partition (null o #5) proof

    (* The names of the formulas are added to the Skolem constants, to ensure that formulas giving
       rise to several clauses are skolemized together. *)
    val skoXss = map (fn ((_, ss), _, t, _, _) => Term.fold_aterms add_skolem t ss) input_steps
    val groups0 = Graph.strong_conn (fold add_cycle skoXss Graph.empty)
    val groups = filter (exists (String.isPrefix spass_skolem_prefix)) groups0

    val skoXss_input_steps = skoXss ~~ input_steps

    fun step_name_of_group skoXs = (implode skoXs, [])
    fun in_group group = member (op =) group o hd
    fun group_of skoX = find_first (fn group => in_group group skoX) groups

    fun new_steps (skoXss_steps : (string list * (term, 'a) atp_step) list) group =
      let
        val name = step_name_of_group group
        val name0 = name |>> prefix "0"
        val t =
          (case map (snd #> #3) skoXss_steps of
            [t] => t
          | ts => ts
            |> map (HOLogic.dest_Trueprop #> rename_skolem_args)
            |> Library.foldr1 s_conj
            |> HOLogic.mk_Trueprop)
        val skos =
          fold (union (op =) o filter (String.isPrefix spass_skolem_prefix) o fst) skoXss_steps []
        val deps = map (snd #> #1) skoXss_steps
      in
        [(name0, Unknown, unskolemize_spass_term skos t, spass_pre_skolemize_rule, deps),
         (name, Unknown, t, spass_skolemize_rule, [name0])]
      end

    val sko_steps =
      maps (fn group => new_steps (filter (in_group group o fst) skoXss_input_steps) group) groups

    val old_news =
      map_filter (fn (skoXs, (name, _, _, _, _)) =>
          Option.map (pair name o single o step_name_of_group) (group_of skoXs))
        skoXss_input_steps
    val repair_deps = fold replace_dependencies_in_line old_news
  in
    input_steps @ sko_steps @ map repair_deps other_steps
  end

val zf_stmt_prefix = "zf_stmt_"

fun is_if_True_or_False_axiom true_or_false t =
  (case t of
    @{const Trueprop} $
     (Const (@{const_name HOL.eq}, _) $
       (Const (@{const_name If}, _) $ cond $ Var _ $ Var _) $
       Var _) =>
     cond aconv true_or_false
  | _ => false)

fun factify_atp_proof facts hyp_ts concl_t atp_proof =
  let
    fun factify_step ((num, ss), role, t, rule, deps) =
      let
        val (ss', role', t') =
          (case resolve_conjectures ss of
            [j] =>
            if j = length hyp_ts then ([], Conjecture, concl_t)
            else ([], Hypothesis, close_form (nth hyp_ts j))
          | _ =>
            (case resolve_facts facts (num :: ss) of
              [] =>
              if role = Axiom andalso String.isPrefix zf_stmt_prefix num
                 andalso is_if_True_or_False_axiom @{const True} t then
                (["if_True"], Axiom, t)
              else if role = Axiom andalso String.isPrefix zf_stmt_prefix num
                 andalso is_if_True_or_False_axiom @{const False} t then
                (["if_False"], Axiom, t)
              else
                (ss,
                 if role = Definition then Definition
                 else if role = Lemma then Lemma
                 else Plain, t)
            | facts => (map fst facts, Axiom, t)))
      in
        ((num, ss'), role', t', rule, deps)
      end

    val atp_proof = map factify_step atp_proof
    val names = map #1 atp_proof

    fun repair_dep (num, ss) = (num, the_default ss (AList.lookup (op =) names num))
    fun repair_deps (name, role, t, rule, deps) = (name, role, t, rule, map repair_dep deps)
  in
    map repair_deps atp_proof
  end

fun termify_atp_abduce_candidate ctxt local_prover format type_enc pool lifted sym_tab phi =
  let
    val proof = [(("", []), Conjecture, mk_anot phi, "", [])]
    val new_proof = termify_atp_proof ctxt local_prover format type_enc pool lifted sym_tab proof
  in
    (case new_proof of
      [(_, _, phi', _, _)] => phi'
    | _ => error "Impossible case in termify_atp_abduce_candidate")
  end

fun sort_top n scored_items =
  if n <= 0 orelse null scored_items then
    []
  else
    let
      fun split_min accum [] (_, best_item) = (best_item, List.rev accum)
        | split_min accum ((score, item) :: scored_items) (best_score, best_item) =
          if score < best_score then
            split_min ((best_score, best_item) :: accum) scored_items (score, item)
          else
            split_min ((score, item) :: accum) scored_items (best_score, best_item)

      val (min_item, other_scored_items) = split_min [] (tl scored_items) (hd scored_items)
    in
      min_item :: sort_top (n - 1) (filter_out (equal min_item o snd) other_scored_items)
      |> distinct (op aconv)
    end

fun top_abduce_candidates max_candidates candidates =
  let
    (* We prefer free variables to other constructs, so that e.g. "x \<le> y" is
       prioritized over "x \<le> 5". *)
    fun score t =
      Term.fold_aterms (fn t => fn score => score + (case t of Free _ => 1 | _ => 2)) t 0

    (* Equations of the form "x = ..." or "... = x" or similar are too specific
       to be useful. Quantified formulas are also filtered out. As for "True",
       it may seem an odd choice for abduction, but it sometimes arises in
       conjunction with type class constraints, which are removed by the
       termifier. *)
    fun maybe_score t =
      (case t of
        @{prop True} => NONE
      | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ Free _ $ _) => NONE
      | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ _ $ Free _) => NONE
      | @{const Trueprop} $ (@{const less(nat)} $ _ $ @{const zero_class.zero(nat)}) => NONE
      | @{const Trueprop} $ (@{const less_eq(nat)} $ _ $ @{const zero_class.zero(nat)}) => NONE
      | @{const Trueprop} $ (@{const less(nat)} $ _ $ @{const one_class.one(nat)}) => NONE
      | @{const Trueprop} $ (@{const Not} $
          (@{const less(nat)} $ @{const zero_class.zero(nat)} $ _)) => NONE
      | @{const Trueprop} $ (@{const Not} $
          (@{const less_eq(nat)} $ @{const zero_class.zero(nat)} $ _)) => NONE
      | @{const Trueprop} $ (@{const Not} $
          (@{const less_eq(nat)} $ @{const one_class.one(nat)} $ _)) => NONE
      | @{const Trueprop} $ (Const (@{const_name All}, _) $ _) => NONE
      | @{const Trueprop} $ (Const (@{const_name Ex}, _) $ _) => NONE
      | _ =>
        (* We require the presence of at least one free variable. A "missing
           assumption" that does not talk about any free variable is likely
           spurious. *)
        if exists_subterm (fn Free _ => true | _ => false) t then SOME (score t, t)
        else NONE)
  in
    sort_top max_candidates (map_filter maybe_score candidates)
  end

fun provability_status ctxt t =
  let
    val res = Timeout.apply (seconds 0.1)
      (Thm.term_of o Thm.rhs_of o Simplifier.full_rewrite ctxt) (Thm.cterm_of ctxt t)
  in
    if res aconv @{prop True} then SOME true
    else if res aconv @{prop False} then SOME false
    else NONE
  end
  handle Timeout.TIMEOUT _ => NONE

(* Put propositions that simplify to "True" first, and filter out propositions
   that simplify to "False". *)
fun sort_propositions_by_provability ctxt ts =
  let
    val statuses_ts = map (`(provability_status ctxt)) ts
  in
    maps (fn (SOME true, t) => [t] | _ => []) statuses_ts @
    maps (fn (NONE, t) => [t] | _ => []) statuses_ts
  end

end;