(*  Title:      HOL/Tools/Metis/metis_reconstruct.ML

    Author:     Kong W. Susanto, Cambridge University Computer Laboratory

    Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory

    Author:     Jasmin Blanchette, TU Muenchen

    Copyright   Cambridge University 2007

Proof reconstruction for Metis.

*)

signature METIS_RECONSTRUCT =

sig

  exception METIS of string * string

  val hol_clause_from_metis :

    Proof.context -> int Symtab.table -> (string * term) list -> Metis_Thm.thm

    -> term

  val lookth : (Metis_Thm.thm * 'a) list -> Metis_Thm.thm -> 'a

  val metis_aconv_untyped : term * term -> bool

  val replay_one_inference :

    Proof.context -> (string * term) list -> int Symtab.table

    -> Metis_Thm.thm * Metis_Proof.inference -> (Metis_Thm.thm * thm) list

    -> (Metis_Thm.thm * thm) list

  val discharge_skolem_premises :

    Proof.context -> (thm * term) option list -> thm -> thm

end;

structure Metis_Reconstruct : METIS_RECONSTRUCT =

struct

open ATP_Problem

open ATP_Translate

open ATP_Reconstruct

open Metis_Translate

exception METIS of string * string

datatype term_or_type = SomeTerm of term | SomeType of typ

fun terms_of [] = []

  | terms_of (SomeTerm t :: tts) = t :: terms_of tts

  | terms_of (SomeType _ :: tts) = terms_of tts;

fun types_of [] = []

  | types_of (SomeTerm (Var ((a, idx), _)) :: tts) =

    types_of tts

    |> (if String.isPrefix metis_generated_var_prefix a then

          (* Variable generated by Metis, which might have been a type

             variable. *)

          cons (TVar (("'" ^ a, idx), HOLogic.typeS))

        else

          I)

  | types_of (SomeTerm _ :: tts) = types_of tts

  | types_of (SomeType T :: tts) = T :: types_of tts

fun atp_name_from_metis s =

  case find_first (fn (_, (s', _)) => s' = s) metis_name_table of

    SOME ((s, _), (_, swap)) => (s, swap)

  | _ => (s, false)

fun atp_term_from_metis (Metis_Term.Fn (s, tms)) =

    let val (s, swap) = atp_name_from_metis (Metis_Name.toString s) in

      ATerm (s, tms |> map atp_term_from_metis |> swap ? rev)

    end

  | atp_term_from_metis (Metis_Term.Var s) = ATerm (Metis_Name.toString s, [])

fun hol_term_from_metis ctxt sym_tab =

  atp_term_from_metis #> term_from_atp ctxt false sym_tab NONE

fun atp_literal_from_metis (pos, atom) =

  atom |> Metis_Term.Fn |> atp_term_from_metis |> AAtom |> not pos ? mk_anot

fun atp_clause_from_metis [] = AAtom (ATerm (tptp_false, []))

  | atp_clause_from_metis lits =

    lits |> map atp_literal_from_metis |> mk_aconns AOr

fun reveal_old_skolems_and_infer_types ctxt old_skolems =

  map (reveal_old_skolem_terms old_skolems)

  #> Syntax.check_terms (Proof_Context.set_mode Proof_Context.mode_pattern ctxt)

fun hol_clause_from_metis ctxt sym_tab old_skolems =

  Metis_Thm.clause

  #> Metis_LiteralSet.toList

  #> atp_clause_from_metis

  #> prop_from_atp ctxt false sym_tab

  #> singleton (reveal_old_skolems_and_infer_types ctxt old_skolems)

fun hol_terms_from_metis ctxt old_skolems sym_tab fol_tms =

  let val ts = map (hol_term_from_metis ctxt sym_tab) fol_tms

      val _ = trace_msg ctxt (fn () => "  calling type inference:")

      val _ = app (fn t => trace_msg ctxt

                                     (fn () => Syntax.string_of_term ctxt t)) ts

      val ts' =

        ts |> reveal_old_skolems_and_infer_types ctxt old_skolems

      val _ = app (fn t => trace_msg ctxt

                    (fn () => "  final term: " ^ Syntax.string_of_term ctxt t ^

                              " of type " ^ Syntax.string_of_typ ctxt (type_of t)))

                  ts'

  in  ts'  end;

(* ------------------------------------------------------------------------- *)

(* FOL step Inference Rules                                                  *)

(* ------------------------------------------------------------------------- *)

(*for debugging only*)

(*

fun print_thpair ctxt (fth,th) =

  (trace_msg ctxt (fn () => "=============================================");

   trace_msg ctxt (fn () => "Metis: " ^ Metis_Thm.toString fth);

   trace_msg ctxt (fn () => "Isabelle: " ^ Display.string_of_thm_without_context th));

*)

fun lookth th_pairs fth =

  the (AList.lookup (uncurry Metis_Thm.equal) th_pairs fth)

  handle Option.Option =>

         raise Fail ("Failed to find Metis theorem " ^ Metis_Thm.toString fth)

fun cterm_incr_types thy idx = cterm_of thy o (map_types (Logic.incr_tvar idx));

(* INFERENCE RULE: AXIOM *)

(* This causes variables to have an index of 1 by default. See also

   "term_from_atp" in "ATP_Reconstruct". *)

val axiom_inference = Thm.incr_indexes 1 oo lookth

(* INFERENCE RULE: ASSUME *)

val EXCLUDED_MIDDLE = @{lemma "P ==> ~ P ==> False" by (rule notE)}

fun inst_excluded_middle thy i_atom =

  let val th = EXCLUDED_MIDDLE

      val [vx] = Term.add_vars (prop_of th) []

      val substs = [(cterm_of thy (Var vx), cterm_of thy i_atom)]

  in  cterm_instantiate substs th  end;

fun assume_inference ctxt old_skolems sym_tab atom =

  inst_excluded_middle

      (Proof_Context.theory_of ctxt)

      (singleton (hol_terms_from_metis ctxt old_skolems sym_tab)

                 (Metis_Term.Fn atom))

(* INFERENCE RULE: INSTANTIATE (SUBST). Type instantiations are ignored. Trying

   to reconstruct them admits new possibilities of errors, e.g. concerning

   sorts. Instead we try to arrange that new TVars are distinct and that types

   can be inferred from terms. *)

fun inst_inference ctxt old_skolems sym_tab th_pairs fsubst th =

  let val thy = Proof_Context.theory_of ctxt

      val i_th = lookth th_pairs th

      val i_th_vars = Term.add_vars (prop_of i_th) []

      fun find_var x = the (List.find (fn ((a,_),_) => a=x) i_th_vars)

      fun subst_translation (x,y) =

        let val v = find_var x

            (* We call "reveal_old_skolems_and_infer_types" below. *)

            val t = hol_term_from_metis ctxt sym_tab y

        in  SOME (cterm_of thy (Var v), t)  end

        handle Option.Option =>

               (trace_msg ctxt (fn () => "\"find_var\" failed for " ^ x ^

                                         " in " ^ Display.string_of_thm ctxt i_th);

                NONE)

             | TYPE _ =>

               (trace_msg ctxt (fn () => "\"hol_term_from_metis\" failed for " ^ x ^

                                         " in " ^ Display.string_of_thm ctxt i_th);

                NONE)

      fun remove_typeinst (a, t) =

        let val a = Metis_Name.toString a in

          case strip_prefix_and_unascii schematic_var_prefix a of

            SOME b => SOME (b, t)

          | NONE =>

            case strip_prefix_and_unascii tvar_prefix a of

              SOME _ => NONE (* type instantiations are forbidden *)

            | NONE => SOME (a, t) (* internal Metis var? *)

        end

      val _ = trace_msg ctxt (fn () => "  isa th: " ^ Display.string_of_thm ctxt i_th)

      val substs = map_filter remove_typeinst (Metis_Subst.toList fsubst)

      val (vars, tms) =

        ListPair.unzip (map_filter subst_translation substs)

        ||> reveal_old_skolems_and_infer_types ctxt old_skolems

      val ctm_of = cterm_incr_types thy (1 + Thm.maxidx_of i_th)

      val substs' = ListPair.zip (vars, map ctm_of tms)

      val _ = trace_msg ctxt (fn () =>

        cat_lines ("subst_translations:" ::

          (substs' |> map (fn (x, y) =>

            Syntax.string_of_term ctxt (term_of x) ^ " |-> " ^

            Syntax.string_of_term ctxt (term_of y)))));

  in cterm_instantiate substs' i_th end

  handle THM (msg, _, _) => raise METIS ("inst_inference", msg)

       | ERROR msg => raise METIS ("inst_inference", msg)

(* INFERENCE RULE: RESOLVE *)

(* Like RSN, but we rename apart only the type variables. Vars here typically

   have an index of 1, and the use of RSN would increase this typically to 3.

   Instantiations of those Vars could then fail. See comment on "make_var". *)

fun resolve_inc_tyvars thy tha i thb =

  let

    val tha = Drule.incr_type_indexes (1 + Thm.maxidx_of thb) tha

    fun aux tha thb =

      case Thm.bicompose false (false, tha, nprems_of tha) i thb

           |> Seq.list_of |> distinct Thm.eq_thm of

        [th] => th

      | _ => raise THM ("resolve_inc_tyvars: unique result expected", i,

                        [tha, thb])

  in

    aux tha thb

    handle TERM z =>

           (* The unifier, which is invoked from "Thm.bicompose", will sometimes

              refuse to unify "?a::?'a" with "?a::?'b" or "?a::nat" and throw a

              "TERM" exception (with "add_ffpair" as first argument). We then

              perform unification of the types of variables by hand and try

              again. We could do this the first time around but this error

              occurs seldom and we don't want to break existing proofs in subtle

              ways or slow them down needlessly. *)

           case [] |> fold (Term.add_vars o prop_of) [tha, thb]

                   |> AList.group (op =)

                   |> maps (fn ((s, _), T :: Ts) =>

                               map (fn T' => (Free (s, T), Free (s, T'))) Ts)

                   |> rpair (Envir.empty ~1)

                   |-> fold (Pattern.unify thy)

                   |> Envir.type_env |> Vartab.dest

                   |> map (fn (x, (S, T)) =>

                              pairself (ctyp_of thy) (TVar (x, S), T)) of

             [] => raise TERM z

           | ps => aux (instantiate (ps, []) tha) (instantiate (ps, []) thb)

  end

fun s_not (@{const Not} $ t) = t

  | s_not t = HOLogic.mk_not t

fun simp_not_not (@{const Trueprop} $ t) = @{const Trueprop} $ simp_not_not t

  | simp_not_not (@{const Not} $ t) = s_not (simp_not_not t)

  | simp_not_not t = t

(* Match untyped terms. *)

fun metis_aconv_untyped (Const (a, _), Const(b, _)) = (a = b)

  | metis_aconv_untyped (Free (a, _), Free (b, _)) = (a = b)

  | metis_aconv_untyped (Var ((a, _), _), Var ((b, _), _)) =

    a = b (* ignore variable numbers *)

  | metis_aconv_untyped (Bound i, Bound j) = (i = j)

  | metis_aconv_untyped (Abs (_, _, t), Abs (_, _, u)) =

    metis_aconv_untyped (t, u)

  | metis_aconv_untyped (t1 $ t2, u1 $ u2) =

    metis_aconv_untyped (t1, u1) andalso metis_aconv_untyped (t2, u2)

  | metis_aconv_untyped _ = false

val normalize_literal = simp_not_not o Envir.eta_contract

(* Find the relative location of an untyped term within a list of terms as a

   1-based index. Returns 0 in case of failure. *)

fun index_of_literal lit haystack =

  let

    fun match_lit normalize =

      HOLogic.dest_Trueprop #> normalize

      #> curry metis_aconv_untyped (lit |> normalize)

  in

    (case find_index (match_lit I) haystack of

       ~1 => find_index (match_lit (simp_not_not o Envir.eta_contract)) haystack

     | j => j) + 1

  end

(* Permute a rule's premises to move the i-th premise to the last position. *)

fun make_last i th =

  let val n = nprems_of th

  in  if 1 <= i andalso i <= n

      then Thm.permute_prems (i-1) 1 th

      else raise THM("select_literal", i, [th])

  end;

(* Maps a rule that ends "... ==> P ==> False" to "... ==> ~ P" while avoiding

   to create double negations. The "select" wrapper is a trick to ensure that

   "P ==> ~ False ==> False" is rewritten to "P ==> False", not to "~ P". We

   don't use this trick in general because it makes the proof object uglier than

   necessary. FIXME. *)

fun negate_head th =

  if exists (fn t => t aconv @{prop "~ False"}) (prems_of th) then

    (th RS @{thm select_FalseI})

    |> fold (rewrite_rule o single)

            @{thms not_atomize_select atomize_not_select}

  else

    th |> fold (rewrite_rule o single) @{thms not_atomize atomize_not}

(* Maps the clause  [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P *)

val select_literal = negate_head oo make_last

fun resolve_inference ctxt old_skolems sym_tab th_pairs atom th1 th2 =

  let

    val (i_th1, i_th2) = pairself (lookth th_pairs) (th1, th2)

    val _ = trace_msg ctxt (fn () =>

        "  isa th1 (pos): " ^ Display.string_of_thm ctxt i_th1 ^ "\n\

        \  isa th2 (neg): " ^ Display.string_of_thm ctxt i_th2)

  in

    (* Trivial cases where one operand is type info *)

    if Thm.eq_thm (TrueI, i_th1) then

      i_th2

    else if Thm.eq_thm (TrueI, i_th2) then

      i_th1

    else

      let

        val thy = Proof_Context.theory_of ctxt

        val i_atom =

          singleton (hol_terms_from_metis ctxt old_skolems sym_tab)

                    (Metis_Term.Fn atom)

        val _ = trace_msg ctxt (fn () =>

                    "  atom: " ^ Syntax.string_of_term ctxt i_atom)

      in

        case index_of_literal (s_not i_atom) (prems_of i_th1) of

          0 =>

          (trace_msg ctxt (fn () => "Failed to find literal in \"th1\"");

           i_th1)

        | j1 =>

          (trace_msg ctxt (fn () => "  index th1: " ^ string_of_int j1);

           case index_of_literal i_atom (prems_of i_th2) of

             0 =>

             (trace_msg ctxt (fn () => "Failed to find literal in \"th2\"");

              i_th2)

           | j2 =>

             (trace_msg ctxt (fn () => "  index th2: " ^ string_of_int j2);

              resolve_inc_tyvars thy (select_literal j1 i_th1) j2 i_th2

              handle TERM (s, _) => raise METIS ("resolve_inference", s)))

      end

  end

(* INFERENCE RULE: REFL *)

val REFL_THM = Thm.incr_indexes 2 @{lemma "t ~= t ==> False" by simp}

val refl_x = cterm_of @{theory} (Var (hd (Term.add_vars (prop_of REFL_THM) [])));

val refl_idx = 1 + Thm.maxidx_of REFL_THM;

fun refl_inference ctxt old_skolems sym_tab t =

  let

    val thy = Proof_Context.theory_of ctxt

    val i_t = singleton (hol_terms_from_metis ctxt old_skolems sym_tab) t

    val _ = trace_msg ctxt (fn () => "  term: " ^ Syntax.string_of_term ctxt i_t)

    val c_t = cterm_incr_types thy refl_idx i_t

  in cterm_instantiate [(refl_x, c_t)] REFL_THM end

(* INFERENCE RULE: EQUALITY *)

val subst_em = @{lemma "s = t ==> P s ==> ~ P t ==> False" by simp}

val ssubst_em = @{lemma "s = t ==> P t ==> ~ P s ==> False" by simp}

fun equality_inference ctxt old_skolems sym_tab (pos, atom) fp fr =

  let val thy = Proof_Context.theory_of ctxt

      val m_tm = Metis_Term.Fn atom

      val [i_atom, i_tm] =

        hol_terms_from_metis ctxt old_skolems sym_tab [m_tm, fr]

      val _ = trace_msg ctxt (fn () => "sign of the literal: " ^ Bool.toString pos)

      fun replace_item_list lx 0 (_::ls) = lx::ls

        | replace_item_list lx i (l::ls) = l :: replace_item_list lx (i-1) ls

      fun path_finder_fail tm ps t =

        raise METIS ("equality_inference (path_finder)",

                     "path = " ^ space_implode " " (map string_of_int ps) ^

                     " isa-term: " ^ Syntax.string_of_term ctxt tm ^

                     (case t of

                        SOME t => " fol-term: " ^ Metis_Term.toString t

                      | NONE => ""))

      fun path_finder tm [] _ = (tm, Bound 0)

        | path_finder tm (p :: ps) (t as Metis_Term.Fn (s, ts)) =

          let

            val s = s |> Metis_Name.toString

                      |> perhaps (try (strip_prefix_and_unascii const_prefix

                                       #> the #> unmangled_const_name))

          in

            if s = metis_predicator orelse s = predicator_name orelse

               s = metis_type_tag orelse s = type_tag_name then

              path_finder tm ps (nth ts p)

            else if s = metis_app_op orelse s = app_op_name then

              let

                val (tm1, tm2) = dest_comb tm

                val p' = p - (length ts - 2)

              in

                if p' = 0 then

                  path_finder tm1 ps (nth ts p) ||> (fn y => y $ tm2)

                else

                  path_finder tm2 ps (nth ts p) ||> (fn y => tm1 $ y)

              end

            else

              let

                val (tm1, args) = strip_comb tm

                val adjustment = length ts - length args

                val p' = if adjustment > p then p else p - adjustment

                val tm_p =

                  nth args p'

                  handle General.Subscript => path_finder_fail tm (p :: ps) (SOME t)

                val _ = trace_msg ctxt (fn () =>

                    "path_finder: " ^ string_of_int p ^ "  " ^

                    Syntax.string_of_term ctxt tm_p)

                val (r, t) = path_finder tm_p ps (nth ts p)

              in (r, list_comb (tm1, replace_item_list t p' args)) end

          end

        | path_finder tm ps t = path_finder_fail tm ps (SOME t)

      val (tm_subst, body) = path_finder i_atom fp m_tm

      val tm_abs = Abs ("x", type_of tm_subst, body)

      val _ = trace_msg ctxt (fn () => "abstraction: " ^ Syntax.string_of_term ctxt tm_abs)

      val _ = trace_msg ctxt (fn () => "i_tm: " ^ Syntax.string_of_term ctxt i_tm)

      val _ = trace_msg ctxt (fn () => "located term: " ^ Syntax.string_of_term ctxt tm_subst)

      val imax = maxidx_of_term (i_tm $ tm_abs $ tm_subst)  (*ill typed but gives right max*)

      val subst' = Thm.incr_indexes (imax+1) (if pos then subst_em else ssubst_em)

      val _ = trace_msg ctxt (fn () => "subst' " ^ Display.string_of_thm ctxt subst')

      val eq_terms = map (pairself (cterm_of thy))

        (ListPair.zip (OldTerm.term_vars (prop_of subst'), [tm_abs, tm_subst, i_tm]))

  in  cterm_instantiate eq_terms subst'  end;

val factor = Seq.hd o distinct_subgoals_tac

fun one_step ctxt old_skolems sym_tab th_pairs p =

  case p of

    (fol_th, Metis_Proof.Axiom _) => axiom_inference th_pairs fol_th |> factor

  | (_, Metis_Proof.Assume f_atom) =>

    assume_inference ctxt old_skolems sym_tab f_atom

  | (_, Metis_Proof.Metis_Subst (f_subst, f_th1)) =>

    inst_inference ctxt old_skolems sym_tab th_pairs f_subst f_th1

    |> factor

  | (_, Metis_Proof.Resolve(f_atom, f_th1, f_th2)) =>

    resolve_inference ctxt old_skolems sym_tab th_pairs f_atom f_th1 f_th2

    |> factor

  | (_, Metis_Proof.Refl f_tm) =>

    refl_inference ctxt old_skolems sym_tab f_tm

  | (_, Metis_Proof.Equality (f_lit, f_p, f_r)) =>

    equality_inference ctxt old_skolems sym_tab f_lit f_p f_r

fun flexflex_first_order th =

  case Thm.tpairs_of th of

      [] => th

    | pairs =>

        let val thy = theory_of_thm th

            val (_, tenv) =

              fold (Pattern.first_order_match thy) pairs (Vartab.empty, Vartab.empty)

            val t_pairs = map Meson.term_pair_of (Vartab.dest tenv)

            val th' = Thm.instantiate ([], map (pairself (cterm_of thy)) t_pairs) th

        in  th'  end

        handle THM _ => th;

fun is_metis_literal_genuine (_, (s, _)) =

  not (String.isPrefix class_prefix (Metis_Name.toString s))

fun is_isabelle_literal_genuine t =

  case t of _ $ (Const (@{const_name Meson.skolem}, _) $ _) => false | _ => true

fun count p xs = fold (fn x => if p x then Integer.add 1 else I) xs 0

(* Seldomly needed hack. A Metis clause is represented as a set, so duplicate

   disjuncts are impossible. In the Isabelle proof, in spite of efforts to

   eliminate them, duplicates sometimes appear with slightly different (but

   unifiable) types. *)

fun resynchronize ctxt fol_th th =

  let

    val num_metis_lits =

      count is_metis_literal_genuine

            (Metis_LiteralSet.toList (Metis_Thm.clause fol_th))

    val num_isabelle_lits = count is_isabelle_literal_genuine (prems_of th)

  in

    if num_metis_lits >= num_isabelle_lits then

      th

    else

      let

        val (prems0, concl) = th |> prop_of |> Logic.strip_horn

        val prems = prems0 |> map normalize_literal

                           |> distinct metis_aconv_untyped

        val goal = Logic.list_implies (prems, concl)

        val tac = cut_rules_tac [th] 1

                  THEN rewrite_goals_tac @{thms not_not [THEN eq_reflection]}

                  THEN ALLGOALS assume_tac

      in

        if length prems = length prems0 then

          raise METIS ("resynchronize", "Out of sync")

        else

          Goal.prove ctxt [] [] goal (K tac)

          |> resynchronize ctxt fol_th

      end

  end

fun replay_one_inference ctxt old_skolems sym_tab (fol_th, inf) th_pairs =

  if not (null th_pairs) andalso

     prop_of (snd (hd th_pairs)) aconv @{prop False} then

    (* Isabelle sometimes identifies literals (premises) that are distinct in

       Metis (e.g., because of type variables). We give the Isabelle proof the

       benefice of the doubt. *)

    th_pairs

  else

    let

      val _ = trace_msg ctxt

                  (fn () => "=============================================")

      val _ = trace_msg ctxt

                  (fn () => "METIS THM: " ^ Metis_Thm.toString fol_th)

      val _ = trace_msg ctxt

                  (fn () => "INFERENCE: " ^ Metis_Proof.inferenceToString inf)

      val th = one_step ctxt old_skolems sym_tab th_pairs (fol_th, inf)

               |> flexflex_first_order

               |> resynchronize ctxt fol_th

      val _ = trace_msg ctxt

                  (fn () => "ISABELLE THM: " ^ Display.string_of_thm ctxt th)

      val _ = trace_msg ctxt

                  (fn () => "=============================================")

    in (fol_th, th) :: th_pairs end

(* It is normally sufficient to apply "assume_tac" to unify the conclusion with

   one of the premises. Unfortunately, this sometimes yields "Variable

   ?SK_a_b_c_x has two distinct types" errors. To avoid this, we instantiate the

   variables before applying "assume_tac". Typical constraints are of the form

     ?SK_a_b_c_x SK_d_e_f_y ... SK_a_b_c_x ... SK_g_h_i_z =?= SK_a_b_c_x,

   where the nonvariables are goal parameters. *)

fun unify_first_prem_with_concl thy i th =

  let

    val goal = Logic.get_goal (prop_of th) i |> Envir.beta_eta_contract

    val prem = goal |> Logic.strip_assums_hyp |> hd

    val concl = goal |> Logic.strip_assums_concl

    fun pair_untyped_aconv (t1, t2) (u1, u2) =

      metis_aconv_untyped (t1, u1) andalso metis_aconv_untyped (t2, u2)

    fun add_terms tp inst =

      if exists (pair_untyped_aconv tp) inst then inst

      else tp :: map (apsnd (subst_atomic [tp])) inst

    fun is_flex t =

      case strip_comb t of

        (Var _, args) => forall is_Bound args

      | _ => false

    fun unify_flex flex rigid =

      case strip_comb flex of

        (Var (z as (_, T)), args) =>

        add_terms (Var z,

          fold_rev (curry absdummy) (take (length args) (binder_types T)) rigid)

      | _ => I

    fun unify_potential_flex comb atom =

      if is_flex comb then unify_flex comb atom

      else if is_Var atom then add_terms (atom, comb)

      else I

    fun unify_terms (t, u) =

      case (t, u) of

        (t1 $ t2, u1 $ u2) =>

        if is_flex t then unify_flex t u

        else if is_flex u then unify_flex u t

        else fold unify_terms [(t1, u1), (t2, u2)]

      | (_ $ _, _) => unify_potential_flex t u

      | (_, _ $ _) => unify_potential_flex u t

      | (Var _, _) => add_terms (t, u)

      | (_, Var _) => add_terms (u, t)

      | _ => I

    val t_inst =

      [] |> try (unify_terms (prem, concl) #> map (pairself (cterm_of thy)))

         |> the_default [] (* FIXME *)

  in th |> cterm_instantiate t_inst end

val copy_prem = @{lemma "P ==> (P ==> P ==> Q) ==> Q" by fast}

fun copy_prems_tac [] ns i =

    if forall (curry (op =) 1) ns then all_tac else copy_prems_tac (rev ns) [] i

  | copy_prems_tac (1 :: ms) ns i =

    rotate_tac 1 i THEN copy_prems_tac ms (1 :: ns) i

  | copy_prems_tac (m :: ms) ns i =

    etac copy_prem i THEN copy_prems_tac ms (m div 2 :: (m + 1) div 2 :: ns) i

(* Metis generates variables of the form _nnn. *)

val is_metis_fresh_variable = String.isPrefix "_"

fun instantiate_forall_tac thy t i st =

  let

    val params = Logic.strip_params (Logic.get_goal (prop_of st) i) |> rev

    fun repair (t as (Var ((s, _), _))) =

        (case find_index (fn (s', _) => s' = s) params of

           ~1 => t

         | j => Bound j)

      | repair (t $ u) =

        (case (repair t, repair u) of

           (t as Bound j, u as Bound k) =>

           (* This is a rather subtle trick to repair the discrepancy between

              the fully skolemized term that MESON gives us (where existentials

              were pulled out) and the reality. *)

           if k > j then t else t $ u

         | (t, u) => t $ u)

      | repair t = t

    val t' = t |> repair |> fold (curry absdummy) (map snd params)

    fun do_instantiate th =

      case Term.add_vars (prop_of th) []

           |> filter_out ((Meson_Clausify.is_zapped_var_name orf

                           is_metis_fresh_variable) o fst o fst) of

        [] => th

      | [var as (_, T)] =>

        let

          val var_binder_Ts = T |> binder_types |> take (length params) |> rev

          val var_body_T = T |> funpow (length params) range_type

          val tyenv =

            Vartab.empty |> Type.raw_unifys (fastype_of t :: map snd params,

                                             var_body_T :: var_binder_Ts)

          val env =

            Envir.Envir {maxidx = Vartab.fold (Integer.max o snd o fst) tyenv 0,

                         tenv = Vartab.empty, tyenv = tyenv}

          val ty_inst =

            Vartab.fold (fn (x, (S, T)) =>

                            cons (pairself (ctyp_of thy) (TVar (x, S), T)))

                        tyenv []

          val t_inst =

            [pairself (cterm_of thy o Envir.norm_term env) (Var var, t')]

        in th |> instantiate (ty_inst, t_inst) end

      | _ => raise Fail "expected a single non-zapped, non-Metis Var"

  in

    (DETERM (etac @{thm allE} i THEN rotate_tac ~1 i)

     THEN PRIMITIVE do_instantiate) st

  end

fun fix_exists_tac t =

  etac @{thm exE} THEN' rename_tac [t |> dest_Var |> fst |> fst]

fun release_quantifier_tac thy (skolem, t) =

  (if skolem then fix_exists_tac else instantiate_forall_tac thy) t

fun release_clusters_tac _ _ _ [] = K all_tac

  | release_clusters_tac thy ax_counts substs

                         ((ax_no, cluster_no) :: clusters) =

    let

      val cluster_of_var =

        Meson_Clausify.cluster_of_zapped_var_name o fst o fst o dest_Var

      fun in_right_cluster ((_, (cluster_no', _)), _) = cluster_no' = cluster_no

      val cluster_substs =

        substs

        |> map_filter (fn (ax_no', (_, (_, tsubst))) =>

                          if ax_no' = ax_no then

                            tsubst |> map (apfst cluster_of_var)

                                   |> filter (in_right_cluster o fst)

                                   |> map (apfst snd)

                                   |> SOME

                          else

                            NONE)

      fun do_cluster_subst cluster_subst =

        map (release_quantifier_tac thy) cluster_subst @ [rotate_tac 1]

      val first_prem = find_index (fn (ax_no', _) => ax_no' = ax_no) substs

    in

      rotate_tac first_prem

      THEN' (EVERY' (maps do_cluster_subst cluster_substs))

      THEN' rotate_tac (~ first_prem - length cluster_substs)

      THEN' release_clusters_tac thy ax_counts substs clusters

    end

fun cluster_key ((ax_no, (cluster_no, index_no)), skolem) =

  (ax_no, (cluster_no, (skolem, index_no)))

fun cluster_ord p =

  prod_ord int_ord (prod_ord int_ord (prod_ord bool_ord int_ord))

           (pairself cluster_key p)

val tysubst_ord =

  list_ord (prod_ord Term_Ord.fast_indexname_ord

                     (prod_ord Term_Ord.sort_ord Term_Ord.typ_ord))

structure Int_Tysubst_Table =

  Table(type key = int * (indexname * (sort * typ)) list

        val ord = prod_ord int_ord tysubst_ord)

structure Int_Pair_Graph =

  Graph(type key = int * int val ord = prod_ord int_ord int_ord)

fun shuffle_key (((axiom_no, (_, index_no)), _), _) = (axiom_no, index_no)

fun shuffle_ord p = prod_ord int_ord int_ord (pairself shuffle_key p)

(* Attempts to derive the theorem "False" from a theorem of the form

   "P1 ==> ... ==> Pn ==> False", where the "Pi"s are to be discharged using the

   specified axioms. The axioms have leading "All" and "Ex" quantifiers, which

   must be eliminated first. *)

fun discharge_skolem_premises ctxt axioms prems_imp_false =

  if prop_of prems_imp_false aconv @{prop False} then

    prems_imp_false

  else

    let

      val thy = Proof_Context.theory_of ctxt

      fun match_term p =

        let

          val (tyenv, tenv) =

            Pattern.first_order_match thy p (Vartab.empty, Vartab.empty)

          val tsubst =

            tenv |> Vartab.dest

                 |> filter (Meson_Clausify.is_zapped_var_name o fst o fst)

                 |> sort (cluster_ord

                          o pairself (Meson_Clausify.cluster_of_zapped_var_name

                                      o fst o fst))

                 |> map (Meson.term_pair_of

                         #> pairself (Envir.subst_term_types tyenv))

          val tysubst = tyenv |> Vartab.dest

        in (tysubst, tsubst) end

      fun subst_info_for_prem subgoal_no prem =

        case prem of

          _ $ (Const (@{const_name Meson.skolem}, _) $ (_ $ t $ num)) =>

          let val ax_no = HOLogic.dest_nat num in

            (ax_no, (subgoal_no,

                     match_term (nth axioms ax_no |> the |> snd, t)))

          end

        | _ => raise TERM ("discharge_skolem_premises: Malformed premise",

                           [prem])

      fun cluster_of_var_name skolem s =

        case try Meson_Clausify.cluster_of_zapped_var_name s of

          NONE => NONE

        | SOME ((ax_no, (cluster_no, _)), skolem') =>

          if skolem' = skolem andalso cluster_no > 0 then

            SOME (ax_no, cluster_no)

          else

            NONE

      fun clusters_in_term skolem t =

        Term.add_var_names t [] |> map_filter (cluster_of_var_name skolem o fst)

      fun deps_for_term_subst (var, t) =

        case clusters_in_term false var of

          [] => NONE

        | [(ax_no, cluster_no)] =>

          SOME ((ax_no, cluster_no),

                clusters_in_term true t

                |> cluster_no > 1 ? cons (ax_no, cluster_no - 1))

        | _ => raise TERM ("discharge_skolem_premises: Expected Var", [var])

      val prems = Logic.strip_imp_prems (prop_of prems_imp_false)

      val substs = prems |> map2 subst_info_for_prem (1 upto length prems)

                         |> sort (int_ord o pairself fst)

      val depss = maps (map_filter deps_for_term_subst o snd o snd o snd) substs

      val clusters = maps (op ::) depss

      val ordered_clusters =

        Int_Pair_Graph.empty

        |> fold Int_Pair_Graph.default_node (map (rpair ()) clusters)

        |> fold Int_Pair_Graph.add_deps_acyclic depss

        |> Int_Pair_Graph.topological_order

        handle Int_Pair_Graph.CYCLES _ =>

               error "Cannot replay Metis proof in Isabelle without \

                     \\"Hilbert_Choice\"."

      val ax_counts =

        Int_Tysubst_Table.empty

        |> fold (fn (ax_no, (_, (tysubst, _))) =>

                    Int_Tysubst_Table.map_default ((ax_no, tysubst), 0)

                                                  (Integer.add 1)) substs

        |> Int_Tysubst_Table.dest

      val needed_axiom_props =

        0 upto length axioms - 1 ~~ axioms

        |> map_filter (fn (_, NONE) => NONE

                        | (ax_no, SOME (_, t)) =>

                          if exists (fn ((ax_no', _), n) =>

                                        ax_no' = ax_no andalso n > 0)

                                    ax_counts then

                            SOME t

                          else

                            NONE)

      val outer_param_names =

        [] |> fold Term.add_var_names needed_axiom_props

           |> filter (Meson_Clausify.is_zapped_var_name o fst)

           |> map (`(Meson_Clausify.cluster_of_zapped_var_name o fst))

           |> filter (fn (((_, (cluster_no, _)), skolem), _) =>

                         cluster_no = 0 andalso skolem)

           |> sort shuffle_ord |> map (fst o snd)

(* for debugging only:

      fun string_for_subst_info (ax_no, (subgoal_no, (tysubst, tsubst))) =

        "ax: " ^ string_of_int ax_no ^ "; asm: " ^ string_of_int subgoal_no ^

        "; tysubst: " ^ PolyML.makestring tysubst ^ "; tsubst: {" ^

        commas (map ((fn (s, t) => s ^ " |-> " ^ t)

                     o pairself (Syntax.string_of_term ctxt)) tsubst) ^ "}"

      val _ = tracing ("ORDERED CLUSTERS: " ^ PolyML.makestring ordered_clusters)

      val _ = tracing ("AXIOM COUNTS: " ^ PolyML.makestring ax_counts)

      val _ = tracing ("OUTER PARAMS: " ^ PolyML.makestring outer_param_names)

      val _ = tracing ("SUBSTS (" ^ string_of_int (length substs) ^ "):\n" ^

                       cat_lines (map string_for_subst_info substs))

*)

      fun cut_and_ex_tac axiom =

        cut_rules_tac [axiom] 1

        THEN TRY (REPEAT_ALL_NEW (etac @{thm exE}) 1)

      fun rotation_for_subgoal i =

        find_index (fn (_, (subgoal_no, _)) => subgoal_no = i) substs

    in

      Goal.prove ctxt [] [] @{prop False}

          (K (DETERM (EVERY (map (cut_and_ex_tac o fst o the o nth axioms o fst

                                  o fst) ax_counts)

                      THEN rename_tac outer_param_names 1

                      THEN copy_prems_tac (map snd ax_counts) [] 1)

              THEN release_clusters_tac thy ax_counts substs ordered_clusters 1

              THEN match_tac [prems_imp_false] 1

              THEN ALLGOALS (fn i =>

                       rtac @{thm Meson.skolem_COMBK_I} i

                       THEN rotate_tac (rotation_for_subgoal i) i

                       THEN PRIMITIVE (unify_first_prem_with_concl thy i)

                       THEN assume_tac i

                       THEN flexflex_tac)))

      handle ERROR _ =>

             error ("Cannot replay Metis proof in Isabelle:\n\

                    \Error when discharging Skolem assumptions.")

    end

end;