src/HOL/Tools/Sledgehammer/sledgehammer_filter.ML

fiddled with fudge factor (based on Mirabelle)

(*  Title:      HOL/Tools/Sledgehammer/sledgehammer_filter.ML
    Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA
    Author:     Jasmin Blanchette, TU Muenchen
*)

signature SLEDGEHAMMER_FILTER =
sig
  datatype locality = General | Intro | Elim | Simp | Local | Assum | Chained

  type relevance_override =
    {add: Facts.ref list,
     del: Facts.ref list,
     only: bool}

  val trace : bool Unsynchronized.ref
  val worse_irrel_freq : real Unsynchronized.ref
  val higher_order_irrel_weight : real Unsynchronized.ref
  val abs_rel_weight : real Unsynchronized.ref
  val abs_irrel_weight : real Unsynchronized.ref
  val skolem_irrel_weight : real Unsynchronized.ref
  val theory_const_rel_weight : real Unsynchronized.ref
  val theory_const_irrel_weight : real Unsynchronized.ref
  val intro_bonus : real Unsynchronized.ref
  val elim_bonus : real Unsynchronized.ref
  val simp_bonus : real Unsynchronized.ref
  val local_bonus : real Unsynchronized.ref
  val assum_bonus : real Unsynchronized.ref
  val chained_bonus : real Unsynchronized.ref
  val max_imperfect : real Unsynchronized.ref
  val max_imperfect_exp : real Unsynchronized.ref
  val threshold_divisor : real Unsynchronized.ref
  val ridiculous_threshold : real Unsynchronized.ref
  val name_thm_pairs_from_ref :
    Proof.context -> unit Symtab.table -> thm list -> Facts.ref
    -> ((string * locality) * thm) list
  val relevant_facts :
    Proof.context -> bool -> real * real -> int -> bool -> relevance_override
    -> thm list -> term list -> term -> ((string * locality) * thm) list
end;

structure Sledgehammer_Filter : SLEDGEHAMMER_FILTER =
struct

open Sledgehammer_Util

val trace = Unsynchronized.ref false
fun trace_msg msg = if !trace then tracing (msg ()) else ()

(* experimental feature *)
val term_patterns = false

val respect_no_atp = true

datatype locality = General | Intro | Elim | Simp | Local | Assum | Chained

type relevance_override =
  {add: Facts.ref list,
   del: Facts.ref list,
   only: bool}

val sledgehammer_prefix = "Sledgehammer" ^ Long_Name.separator
val abs_name = "Sledgehammer.abs"
val skolem_prefix = "Sledgehammer.sko"
val theory_const_suffix = Long_Name.separator ^ " 1"

fun repair_name reserved multi j name =
  (name |> Symtab.defined reserved name ? quote) ^
  (if multi then "(" ^ Int.toString j ^ ")" else "")

fun name_thm_pairs_from_ref ctxt reserved chained_ths xref =
  let
    val ths = ProofContext.get_fact ctxt xref
    val name = Facts.string_of_ref xref
    val multi = length ths > 1
  in
    (ths, (1, []))
    |-> fold (fn th => fn (j, rest) =>
                 (j + 1, ((repair_name reserved multi j name,
                          if member Thm.eq_thm chained_ths th then Chained
                          else General), th) :: rest))
    |> snd
  end

(***************************************************************)
(* Relevance Filtering                                         *)
(***************************************************************)

(*** constants with types ***)

fun order_of_type (Type (@{type_name fun}, [T1, @{typ bool}])) =
    order_of_type T1 (* cheat: pretend sets are first-order *)
  | order_of_type (Type (@{type_name fun}, [T1, T2])) =
    Int.max (order_of_type T1 + 1, order_of_type T2)
  | order_of_type (Type (_, Ts)) = fold (Integer.max o order_of_type) Ts 0
  | order_of_type _ = 0

(* An abstraction of Isabelle types and first-order terms *)
datatype pattern = PVar | PApp of string * pattern list
datatype ptype = PType of int * pattern list

fun string_for_pattern PVar = "_"
  | string_for_pattern (PApp (s, ps)) =
    if null ps then s else s ^ string_for_patterns ps
and string_for_patterns ps = "(" ^ commas (map string_for_pattern ps) ^ ")"
fun string_for_ptype (PType (_, ps)) = string_for_patterns ps

(*Is the second type an instance of the first one?*)
fun match_pattern (PVar, _) = true
  | match_pattern (PApp _, PVar) = false
  | match_pattern (PApp (s, ps), PApp (t, qs)) =
    s = t andalso match_patterns (ps, qs)
and match_patterns (_, []) = true
  | match_patterns ([], _) = false
  | match_patterns (p :: ps, q :: qs) =
    match_pattern (p, q) andalso match_patterns (ps, qs)
fun match_ptype (PType (_, ps), PType (_, qs)) = match_patterns (ps, qs)

(* Is there a unifiable constant? *)
fun pconst_mem f consts (s, ps) =
  exists (curry (match_ptype o f) ps)
         (map snd (filter (curry (op =) s o fst) consts))
fun pconst_hyper_mem f const_tab (s, ps) =
  exists (curry (match_ptype o f) ps) (these (Symtab.lookup const_tab s))

fun pattern_for_type (Type (s, Ts)) = PApp (s, map pattern_for_type Ts)
  | pattern_for_type (TFree (s, _)) = PApp (s, [])
  | pattern_for_type (TVar _) = PVar

fun pterm thy t =
  case strip_comb t of
    (Const x, ts) => PApp (pconst thy true x ts)
  | (Free x, ts) => PApp (pconst thy false x ts)
  | (Var x, []) => PVar
  | _ => PApp ("?", [])  (* equivalence class of higher-order constructs *)
(* Pairs a constant with the list of its type instantiations. *)
and ptype thy const x ts =
  (if const then map pattern_for_type (these (try (Sign.const_typargs thy) x))
   else []) @
  (if term_patterns then map (pterm thy) ts else [])
and pconst thy const (s, T) ts = (s, ptype thy const (s, T) ts)
and rich_ptype thy const (s, T) ts =
  PType (order_of_type T, ptype thy const (s, T) ts)
and rich_pconst thy const (s, T) ts = (s, rich_ptype thy const (s, T) ts)

fun string_for_hyper_pconst (s, ps) =
  s ^ "{" ^ commas (map string_for_ptype ps) ^ "}"

(* These are typically simplified away by "Meson.presimplify". Equality is
   handled specially via "fequal". *)
val boring_consts =
  [@{const_name False}, @{const_name True}, @{const_name If}, @{const_name Let},
   @{const_name HOL.eq}]

(* Add a pconstant to the table, but a [] entry means a standard
   connective, which we ignore.*)
fun add_pconst_to_table also_skolem (c, p) =
  if member (op =) boring_consts c orelse
     (not also_skolem andalso String.isPrefix skolem_prefix c) then
    I
  else
    Symtab.map_default (c, [p]) (insert (op =) p)

fun is_formula_type T = (T = HOLogic.boolT orelse T = propT)

fun pconsts_in_terms thy also_skolems pos ts =
  let
    val flip = Option.map not
    (* We include free variables, as well as constants, to handle locales. For
       each quantifiers that must necessarily be skolemized by the ATP, we
       introduce a fresh constant to simulate the effect of Skolemization. *)
    fun do_const const (s, T) ts =
      add_pconst_to_table also_skolems (rich_pconst thy const (s, T) ts)
      #> fold do_term ts
    and do_term t =
      case strip_comb t of
        (Const x, ts) => do_const true x ts
      | (Free x, ts) => do_const false x ts
      | (Abs (_, T, t'), ts) =>
        (null ts
         ? add_pconst_to_table true (abs_name, PType (order_of_type T + 1, [])))
        #> fold do_term (t' :: ts)
      | (_, ts) => fold do_term ts
    fun do_quantifier will_surely_be_skolemized abs_T body_t =
      do_formula pos body_t
      #> (if also_skolems andalso will_surely_be_skolemized then
            add_pconst_to_table true
                         (gensym skolem_prefix, PType (order_of_type abs_T, []))
          else
            I)
    and do_term_or_formula T =
      if is_formula_type T then do_formula NONE else do_term
    and do_formula pos t =
      case t of
        Const (@{const_name all}, _) $ Abs (_, T, t') =>
        do_quantifier (pos = SOME false) T t'
      | @{const "==>"} $ t1 $ t2 =>
        do_formula (flip pos) t1 #> do_formula pos t2
      | Const (@{const_name "=="}, Type (_, [T, _])) $ t1 $ t2 =>
        fold (do_term_or_formula T) [t1, t2]
      | @{const Trueprop} $ t1 => do_formula pos t1
      | @{const Not} $ t1 => do_formula (flip pos) t1
      | Const (@{const_name All}, _) $ Abs (_, T, t') =>
        do_quantifier (pos = SOME false) T t'
      | Const (@{const_name Ex}, _) $ Abs (_, T, t') =>
        do_quantifier (pos = SOME true) T t'
      | @{const HOL.conj} $ t1 $ t2 => fold (do_formula pos) [t1, t2]
      | @{const HOL.disj} $ t1 $ t2 => fold (do_formula pos) [t1, t2]
      | @{const HOL.implies} $ t1 $ t2 =>
        do_formula (flip pos) t1 #> do_formula pos t2
      | Const (@{const_name HOL.eq}, Type (_, [T, _])) $ t1 $ t2 =>
        fold (do_term_or_formula T) [t1, t2]
      | Const (@{const_name If}, Type (_, [_, Type (_, [T, _])]))
        $ t1 $ t2 $ t3 =>
        do_formula NONE t1 #> fold (do_term_or_formula T) [t2, t3]
      | Const (@{const_name Ex1}, _) $ Abs (_, T, t') =>
        do_quantifier (is_some pos) T t'
      | Const (@{const_name Ball}, _) $ t1 $ Abs (_, T, t') =>
        do_quantifier (pos = SOME false) T
                      (HOLogic.mk_imp (incr_boundvars 1 t1 $ Bound 0, t'))
      | Const (@{const_name Bex}, _) $ t1 $ Abs (_, T, t') =>
        do_quantifier (pos = SOME true) T
                      (HOLogic.mk_conj (incr_boundvars 1 t1 $ Bound 0, t'))
      | (t0 as Const (_, @{typ bool})) $ t1 =>
        do_term t0 #> do_formula pos t1  (* theory constant *)
      | _ => do_term t
  in Symtab.empty |> fold (do_formula pos) ts end

(*Inserts a dummy "constant" referring to the theory name, so that relevance
  takes the given theory into account.*)
fun theory_const_prop_of theory_relevant th =
  if theory_relevant then
    let
      val name = Context.theory_name (theory_of_thm th)
      val t = Const (name ^ theory_const_suffix, @{typ bool})
    in t $ prop_of th end
  else
    prop_of th

(**** Constant / Type Frequencies ****)

(* A two-dimensional symbol table counts frequencies of constants. It's keyed
   first by constant name and second by its list of type instantiations. For the
   latter, we need a linear ordering on "pattern list". *)

fun pattern_ord p =
  case p of
    (PVar, PVar) => EQUAL
  | (PVar, PApp _) => LESS
  | (PApp _, PVar) => GREATER
  | (PApp q1, PApp q2) =>
    prod_ord fast_string_ord (dict_ord pattern_ord) (q1, q2)
fun ptype_ord (PType p, PType q) =
  prod_ord (dict_ord pattern_ord) int_ord (swap p, swap q)

structure PType_Tab = Table(type key = ptype val ord = ptype_ord)

fun count_axiom_consts theory_relevant thy =
  let
    fun do_const const (s, T) ts =
      (* Two-dimensional table update. Constant maps to types maps to count. *)
      PType_Tab.map_default (rich_ptype thy const (s, T) ts, 0) (Integer.add 1)
      |> Symtab.map_default (s, PType_Tab.empty)
      #> fold do_term ts
    and do_term t =
      case strip_comb t of
        (Const x, ts) => do_const true x ts
      | (Free x, ts) => do_const false x ts
      | (Abs (_, _, t'), ts) => fold do_term (t' :: ts)
      | (_, ts) => fold do_term ts
  in do_term o theory_const_prop_of theory_relevant o snd end


(**** Actual Filtering Code ****)

fun pow_int x 0 = 1.0
  | pow_int x 1 = x
  | pow_int x n = if n > 0 then x * pow_int x (n - 1) else pow_int x (n + 1) / x

(*The frequency of a constant is the sum of those of all instances of its type.*)
fun pconst_freq match const_tab (c, ps) =
  PType_Tab.fold (fn (qs, m) => match (ps, qs) ? Integer.add m)
                 (the (Symtab.lookup const_tab c)) 0


(* A surprising number of theorems contain only a few significant constants.
   These include all induction rules, and other general theorems. *)

(* "log" seems best in practice. A constant function of one ignores the constant
   frequencies. Rare constants give more points if they are relevant than less
   rare ones. *)
fun rel_weight_for order freq = 1.0 + 2.0 / Math.ln (Real.fromInt freq + 1.0)

(* FUDGE *)
val worse_irrel_freq = Unsynchronized.ref 100.0
val higher_order_irrel_weight = Unsynchronized.ref 1.05

(* Irrelevant constants are treated differently. We associate lower penalties to
   very rare constants and very common ones -- the former because they can't
   lead to the inclusion of too many new facts, and the latter because they are
   so common as to be of little interest. *)
fun irrel_weight_for order freq =
  let val (k, x) = !worse_irrel_freq |> `Real.ceil in
    (if freq < k then Math.ln (Real.fromInt (freq + 1)) / Math.ln x
     else rel_weight_for order freq / rel_weight_for order k)
    * pow_int (!higher_order_irrel_weight) (order - 1)
  end

(* FUDGE *)
val abs_rel_weight = Unsynchronized.ref 0.5
val abs_irrel_weight = Unsynchronized.ref 2.0
val skolem_irrel_weight = Unsynchronized.ref 0.75
val theory_const_rel_weight = Unsynchronized.ref 0.5
val theory_const_irrel_weight = Unsynchronized.ref 0.25

(* Computes a constant's weight, as determined by its frequency. *)
fun generic_pconst_weight abs_weight skolem_weight theory_const_weight
                          weight_for f const_tab (c as (s, PType (m, _))) =
  if s = abs_name then abs_weight
  else if String.isPrefix skolem_prefix s then skolem_weight
  else if String.isSuffix theory_const_suffix s then theory_const_weight
  else weight_for m (pconst_freq (match_ptype o f) const_tab c)

fun rel_pconst_weight const_tab =
  generic_pconst_weight (!abs_rel_weight) 0.0 (!theory_const_rel_weight)
                        rel_weight_for I const_tab
fun irrel_pconst_weight const_tab =
  generic_pconst_weight (!abs_irrel_weight) (!skolem_irrel_weight)
                    (!theory_const_irrel_weight) irrel_weight_for swap const_tab

(* FUDGE *)
val intro_bonus = Unsynchronized.ref 0.15
val elim_bonus = Unsynchronized.ref 0.15
val simp_bonus = Unsynchronized.ref 0.15
val local_bonus = Unsynchronized.ref 0.55
val assum_bonus = Unsynchronized.ref 1.05
val chained_bonus = Unsynchronized.ref 1.5

fun locality_bonus General = 0.0
  | locality_bonus Intro = !intro_bonus
  | locality_bonus Elim = !elim_bonus
  | locality_bonus Simp = !simp_bonus
  | locality_bonus Local = !local_bonus
  | locality_bonus Assum = !assum_bonus
  | locality_bonus Chained = !chained_bonus

fun axiom_weight loc const_tab relevant_consts axiom_consts =
  case axiom_consts |> List.partition (pconst_hyper_mem I relevant_consts)
                    ||> filter_out (pconst_hyper_mem swap relevant_consts) of
    ([], _) => 0.0
  | (rel, irrel) =>
    let
      val irrel = irrel |> filter_out (pconst_mem swap rel)
      val rel_weight =
        0.0 |> fold (curry (op +) o rel_pconst_weight const_tab) rel
      val irrel_weight =
        ~ (locality_bonus loc)
        |> fold (curry (op +) o irrel_pconst_weight const_tab) irrel
      val res = rel_weight / (rel_weight + irrel_weight)
    in if Real.isFinite res then res else 0.0 end

(* FIXME: experiment
fun debug_axiom_weight loc const_tab relevant_consts axiom_consts =
  case axiom_consts |> List.partition (pconst_hyper_mem I relevant_consts)
                    ||> filter_out (pconst_hyper_mem swap relevant_consts) of
    ([], _) => 0.0
  | (rel, irrel) =>
    let
      val irrel = irrel |> filter_out (pconst_mem swap rel)
      val rels_weight =
        0.0 |> fold (curry (op +) o rel_pconst_weight const_tab) rel
      val irrels_weight =
        ~ (locality_bonus loc)
        |> fold (curry (op +) o irrel_pconst_weight const_tab) irrel
val _ = tracing (PolyML.makestring ("REL: ", map (`(rel_pconst_weight const_tab)) rel))
val _ = tracing (PolyML.makestring ("IRREL: ", map (`(irrel_pconst_weight const_tab)) irrel))
      val res = rels_weight / (rels_weight + irrels_weight)
    in if Real.isFinite res then res else 0.0 end
*)

fun pconsts_in_axiom thy t =
  Symtab.fold (fn (s, pss) => fold (cons o pair s) pss)
              (pconsts_in_terms thy true (SOME true) [t]) []
fun pair_consts_axiom theory_relevant thy axiom =
  case axiom |> snd |> theory_const_prop_of theory_relevant
             |> pconsts_in_axiom thy of
    [] => NONE
  | consts => SOME ((axiom, consts), NONE)

type annotated_thm =
  (((unit -> string) * locality) * thm) * (string * ptype) list

(* FUDGE *)
val max_imperfect = Unsynchronized.ref 11.5
val max_imperfect_exp = Unsynchronized.ref 1.0

fun take_most_relevant max_relevant remaining_max
                       (candidates : (annotated_thm * real) list) =
  let
    val max_imperfect =
      Real.ceil (Math.pow (!max_imperfect,
                    Math.pow (Real.fromInt remaining_max
                              / Real.fromInt max_relevant, !max_imperfect_exp)))
    val (perfect, imperfect) =
      candidates |> sort (Real.compare o swap o pairself snd)
                 |> take_prefix (fn (_, w) => w > 0.99999)
    val ((accepts, more_rejects), rejects) =
      chop max_imperfect imperfect |>> append perfect |>> chop remaining_max
  in
    trace_msg (fn () =>
        "Actually passed (" ^ Int.toString (length accepts) ^ " of " ^
        Int.toString (length candidates) ^ "): " ^
        (accepts |> map (fn ((((name, _), _), _), weight) =>
                            name () ^ " [" ^ Real.toString weight ^ "]")
                 |> commas));
    (accepts, more_rejects @ rejects)
  end

fun if_empty_replace_with_locality thy axioms loc tab =
  if Symtab.is_empty tab then
    pconsts_in_terms thy false (SOME false)
        (map_filter (fn ((_, loc'), th) =>
                        if loc' = loc then SOME (prop_of th) else NONE) axioms)
  else
    tab

(* FUDGE *)
val threshold_divisor = Unsynchronized.ref 2.0
val ridiculous_threshold = Unsynchronized.ref 0.1

fun relevance_filter ctxt threshold0 decay max_relevant theory_relevant
                     ({add, del, ...} : relevance_override) axioms goal_ts =
  let
    val thy = ProofContext.theory_of ctxt
    val const_tab =
      fold (count_axiom_consts theory_relevant thy) axioms Symtab.empty
    val goal_const_tab =
      pconsts_in_terms thy false (SOME false) goal_ts
      |> fold (if_empty_replace_with_locality thy axioms)
              [Chained, Assum, Local]
    val add_thms = maps (ProofContext.get_fact ctxt) add
    val del_thms = maps (ProofContext.get_fact ctxt) del
    fun iter j remaining_max threshold rel_const_tab hopeless hopeful =
      let
        fun game_over rejects =
          (* Add "add:" facts. *)
          if null add_thms then
            []
          else
            map_filter (fn ((p as (_, th), _), _) =>
                           if member Thm.eq_thm add_thms th then SOME p
                           else NONE) rejects
        fun relevant [] rejects [] =
            (* Nothing has been added this iteration. *)
            if j = 0 andalso threshold >= !ridiculous_threshold then
              (* First iteration? Try again. *)
              iter 0 max_relevant (threshold / !threshold_divisor) rel_const_tab
                   hopeless hopeful
            else
              game_over (rejects @ hopeless)
          | relevant candidates rejects [] =
            let
              val (accepts, more_rejects) =
                take_most_relevant max_relevant remaining_max candidates
              val rel_const_tab' =
                rel_const_tab
                |> fold (add_pconst_to_table false) (maps (snd o fst) accepts)
              fun is_dirty (c, _) =
                Symtab.lookup rel_const_tab' c <> Symtab.lookup rel_const_tab c
              val (hopeful_rejects, hopeless_rejects) =
                 (rejects @ hopeless, ([], []))
                 |-> fold (fn (ax as (_, consts), old_weight) =>
                              if exists is_dirty consts then
                                apfst (cons (ax, NONE))
                              else
                                apsnd (cons (ax, old_weight)))
                 |>> append (more_rejects
                             |> map (fn (ax as (_, consts), old_weight) =>
                                        (ax, if exists is_dirty consts then NONE
                                             else SOME old_weight)))
              val threshold =
                1.0 - (1.0 - threshold)
                      * Math.pow (decay, Real.fromInt (length accepts))
              val remaining_max = remaining_max - length accepts
            in
              trace_msg (fn () => "New or updated constants: " ^
                  commas (rel_const_tab' |> Symtab.dest
                          |> subtract (op =) (rel_const_tab |> Symtab.dest)
                          |> map string_for_hyper_pconst));
              map (fst o fst) accepts @
              (if remaining_max = 0 then
                 game_over (hopeful_rejects @ map (apsnd SOME) hopeless_rejects)
               else
                 iter (j + 1) remaining_max threshold rel_const_tab'
                      hopeless_rejects hopeful_rejects)
            end
          | relevant candidates rejects
                     (((ax as (((_, loc), th), axiom_consts)), cached_weight)
                      :: hopeful) =
            let
              val weight =
                case cached_weight of
                  SOME w => w
                | NONE => axiom_weight loc const_tab rel_const_tab axiom_consts
(* FIXME: experiment
val name = fst (fst (fst ax)) ()
val _ = if String.isSubstring "positive_minus" name orelse String.isSubstring "not_exp_le_zero" name then
tracing ("*** " ^ name ^ PolyML.makestring (debug_axiom_weight loc const_tab rel_const_tab axiom_consts))
else
()
*)
            in
              if weight >= threshold then
                relevant ((ax, weight) :: candidates) rejects hopeful
              else
                relevant candidates ((ax, weight) :: rejects) hopeful
            end
        in
          trace_msg (fn () =>
              "ITERATION " ^ string_of_int j ^ ": current threshold: " ^
              Real.toString threshold ^ ", constants: " ^
              commas (rel_const_tab |> Symtab.dest
                      |> filter (curry (op <>) [] o snd)
                      |> map string_for_hyper_pconst));
          relevant [] [] hopeful
        end
  in
    axioms |> filter_out (member Thm.eq_thm del_thms o snd)
           |> map_filter (pair_consts_axiom theory_relevant thy)
           |> iter 0 max_relevant threshold0 goal_const_tab []
           |> tap (fn res => trace_msg (fn () =>
                                "Total relevant: " ^ Int.toString (length res)))
  end


(***************************************************************)
(* Retrieving and filtering lemmas                             *)
(***************************************************************)

(*** retrieve lemmas and filter them ***)

(*Reject theorems with names like "List.filter.filter_list_def" or
  "Accessible_Part.acc.defs", as these are definitions arising from packages.*)
fun is_package_def a =
  let val names = Long_Name.explode a
  in
     length names > 2 andalso
     not (hd names = "local") andalso
     String.isSuffix "_def" a  orelse  String.isSuffix "_defs" a
  end;

fun mk_fact_table f xs =
  fold (Termtab.update o `(prop_of o f)) xs Termtab.empty
fun uniquify xs = Termtab.fold (cons o snd) (mk_fact_table snd xs) []

(* FIXME: put other record thms here, or declare as "no_atp" *)
val multi_base_blacklist =
  ["defs", "select_defs", "update_defs", "induct", "inducts", "split", "splits",
   "split_asm", "cases", "ext_cases", "eq.simps", "eq.refl", "nchotomy",
   "case_cong", "weak_case_cong"]
  |> map (prefix ".")

val max_lambda_nesting = 3

fun term_has_too_many_lambdas max (t1 $ t2) =
    exists (term_has_too_many_lambdas max) [t1, t2]
  | term_has_too_many_lambdas max (Abs (_, _, t)) =
    max = 0 orelse term_has_too_many_lambdas (max - 1) t
  | term_has_too_many_lambdas _ _ = false

(* Don't count nested lambdas at the level of formulas, since they are
   quantifiers. *)
fun formula_has_too_many_lambdas Ts (Abs (_, T, t)) =
    formula_has_too_many_lambdas (T :: Ts) t
  | formula_has_too_many_lambdas Ts t =
    if is_formula_type (fastype_of1 (Ts, t)) then
      exists (formula_has_too_many_lambdas Ts) (#2 (strip_comb t))
    else
      term_has_too_many_lambdas max_lambda_nesting t

(* The max apply depth of any "metis" call in "Metis_Examples" (on 2007-10-31)
   was 11. *)
val max_apply_depth = 15

fun apply_depth (f $ t) = Int.max (apply_depth f, apply_depth t + 1)
  | apply_depth (Abs (_, _, t)) = apply_depth t
  | apply_depth _ = 0

fun is_formula_too_complex t =
  apply_depth t > max_apply_depth orelse formula_has_too_many_lambdas [] t

val exists_sledgehammer_const =
  exists_Const (fn (s, _) => String.isPrefix sledgehammer_prefix s)

(* FIXME: make more reliable *)
val exists_low_level_class_const =
  exists_Const (fn (s, _) =>
     String.isSubstring (Long_Name.separator ^ "class" ^ Long_Name.separator) s)

fun is_metastrange_theorem th =
  case head_of (concl_of th) of
      Const (a, _) => (a <> @{const_name Trueprop} andalso
                       a <> @{const_name "=="})
    | _ => false

fun is_that_fact th =
  String.isSuffix (Long_Name.separator ^ Obtain.thatN) (Thm.get_name_hint th)
  andalso exists_subterm (fn Free (s, _) => s = Name.skolem Auto_Bind.thesisN
                           | _ => false) (prop_of th)

val type_has_top_sort =
  exists_subtype (fn TFree (_, []) => true | TVar (_, []) => true | _ => false)

(**** Predicates to detect unwanted facts (prolific or likely to cause
      unsoundness) ****)

(* Too general means, positive equality literal with a variable X as one
   operand, when X does not occur properly in the other operand. This rules out
   clearly inconsistent facts such as X = a | X = b, though it by no means
   guarantees soundness. *)

(* Unwanted equalities are those between a (bound or schematic) variable that
   does not properly occur in the second operand. *)
val is_exhaustive_finite =
  let
    fun is_bad_equal (Var z) t =
        not (exists_subterm (fn Var z' => z = z' | _ => false) t)
      | is_bad_equal (Bound j) t = not (loose_bvar1 (t, j))
      | is_bad_equal _ _ = false
    fun do_equals t1 t2 = is_bad_equal t1 t2 orelse is_bad_equal t2 t1
    fun do_formula pos t =
      case (pos, t) of
        (_, @{const Trueprop} $ t1) => do_formula pos t1
      | (true, Const (@{const_name all}, _) $ Abs (_, _, t')) =>
        do_formula pos t'
      | (true, Const (@{const_name All}, _) $ Abs (_, _, t')) =>
        do_formula pos t'
      | (false, Const (@{const_name Ex}, _) $ Abs (_, _, t')) =>
        do_formula pos t'
      | (_, @{const "==>"} $ t1 $ t2) =>
        do_formula (not pos) t1 andalso
        (t2 = @{prop False} orelse do_formula pos t2)
      | (_, @{const HOL.implies} $ t1 $ t2) =>
        do_formula (not pos) t1 andalso
        (t2 = @{const False} orelse do_formula pos t2)
      | (_, @{const Not} $ t1) => do_formula (not pos) t1
      | (true, @{const HOL.disj} $ t1 $ t2) => forall (do_formula pos) [t1, t2]
      | (false, @{const HOL.conj} $ t1 $ t2) => forall (do_formula pos) [t1, t2]
      | (true, Const (@{const_name HOL.eq}, _) $ t1 $ t2) => do_equals t1 t2
      | (true, Const (@{const_name "=="}, _) $ t1 $ t2) => do_equals t1 t2
      | _ => false
  in do_formula true end

fun has_bound_or_var_of_type tycons =
  exists_subterm (fn Var (_, Type (s, _)) => member (op =) tycons s
                   | Abs (_, Type (s, _), _) => member (op =) tycons s
                   | _ => false)

(* Facts are forbidden to contain variables of these types. The typical reason
   is that they lead to unsoundness. Note that "unit" satisfies numerous
   equations like "?x = ()". The resulting clauses will have no type constraint,
   yielding false proofs. Even "bool" leads to many unsound proofs, though only
   for higher-order problems. *)
val dangerous_types = [@{type_name unit}, @{type_name bool}, @{type_name prop}];

(* Facts containing variables of type "unit" or "bool" or of the form
   "ALL x. x = A | x = B | x = C" are likely to lead to unsound proofs if types
   are omitted. *)
fun is_dangerous_term full_types t =
  not full_types andalso
  let val t = transform_elim_term t in
    has_bound_or_var_of_type dangerous_types t orelse
    is_exhaustive_finite t
  end

fun is_theorem_bad_for_atps full_types thm =
  let val t = prop_of thm in
    is_formula_too_complex t orelse exists_type type_has_top_sort t orelse
    is_dangerous_term full_types t orelse exists_sledgehammer_const t orelse
    exists_low_level_class_const t orelse is_metastrange_theorem thm orelse
    is_that_fact thm
  end

fun clasimpset_rules_of ctxt =
  let
    val {safeIs, safeEs, hazIs, hazEs, ...} = ctxt |> claset_of |> rep_cs
    val intros = safeIs @ hazIs
    val elims = map Classical.classical_rule (safeEs @ hazEs)
    val simps = ctxt |> simpset_of |> dest_ss |> #simps |> map snd
  in (mk_fact_table I intros, mk_fact_table I elims, mk_fact_table I simps) end

fun all_name_thms_pairs ctxt reserved full_types add_thms chained_ths =
  let
    val thy = ProofContext.theory_of ctxt
    val global_facts = PureThy.facts_of thy
    val local_facts = ProofContext.facts_of ctxt
    val named_locals = local_facts |> Facts.dest_static []
    val assms = Assumption.all_assms_of ctxt
    fun is_assum th = exists (fn ct => prop_of th aconv term_of ct) assms
    val is_chained = member Thm.eq_thm chained_ths
    val (intros, elims, simps) =
      if exists (curry (op <) 0.0) [!intro_bonus, !elim_bonus, !simp_bonus] then
        clasimpset_rules_of ctxt
      else
        (Termtab.empty, Termtab.empty, Termtab.empty)
    (* Unnamed nonchained formulas with schematic variables are omitted, because
       they are rejected by the backticks (`...`) parser for some reason. *)
    fun is_good_unnamed_local th =
      not (Thm.has_name_hint th) andalso
      (not (exists_subterm is_Var (prop_of th)) orelse (is_chained th)) andalso
      forall (fn (_, ths) => not (member Thm.eq_thm ths th)) named_locals
    val unnamed_locals =
      union Thm.eq_thm (Facts.props local_facts) chained_ths
      |> filter is_good_unnamed_local |> map (pair "" o single)
    val full_space =
      Name_Space.merge (Facts.space_of global_facts, Facts.space_of local_facts)
    fun add_facts global foldx facts =
      foldx (fn (name0, ths) =>
        if name0 <> "" andalso
           forall (not o member Thm.eq_thm add_thms) ths andalso
           (Facts.is_concealed facts name0 orelse
            (respect_no_atp andalso is_package_def name0) orelse
            exists (fn s => String.isSuffix s name0) multi_base_blacklist orelse
            String.isSuffix "_def_raw" (* FIXME: crude hack *) name0) then
          I
        else
          let
            val multi = length ths > 1
            fun backquotify th =
              "`" ^ Print_Mode.setmp [Print_Mode.input]
                                 (Syntax.string_of_term ctxt) (prop_of th) ^ "`"
              |> String.translate (fn c => if Char.isPrint c then str c else "")
              |> simplify_spaces
            fun check_thms a =
              case try (ProofContext.get_thms ctxt) a of
                NONE => false
              | SOME ths' => Thm.eq_thms (ths, ths')
          in
            pair 1
            #> fold (fn th => fn (j, rest) =>
                 (j + 1,
                  if is_theorem_bad_for_atps full_types th andalso
                     not (member Thm.eq_thm add_thms th) then
                    rest
                  else
                    (((fn () =>
                          if name0 = "" then
                            th |> backquotify
                          else
                            let
                              val name1 = Facts.extern facts name0
                              val name2 = Name_Space.extern full_space name0
                            in
                              case find_first check_thms [name1, name2, name0] of
                                SOME name => repair_name reserved multi j name
                              | NONE => ""
                            end),
                      let val t = prop_of th in
                        if is_chained th then Chained
                        else if global then
                          if Termtab.defined intros t then Intro
                          else if Termtab.defined elims t then Elim
                          else if Termtab.defined simps t then Simp
                          else General
                        else
                          if is_assum th then Assum else Local
                      end),
                      (multi, th)) :: rest)) ths
            #> snd
          end)
  in
    [] |> add_facts false fold local_facts (unnamed_locals @ named_locals)
       |> add_facts true Facts.fold_static global_facts global_facts
  end

(* The single-name theorems go after the multiple-name ones, so that single
   names are preferred when both are available. *)
fun name_thm_pairs ctxt respect_no_atp =
  List.partition (fst o snd) #> op @ #> map (apsnd snd)
  #> respect_no_atp ? filter_out (No_ATPs.member ctxt o snd)

(***************************************************************)
(* ATP invocation methods setup                                *)
(***************************************************************)

fun relevant_facts ctxt full_types (threshold0, threshold1) max_relevant
                   theory_relevant (relevance_override as {add, del, only})
                   chained_ths hyp_ts concl_t =
  let
    val decay = Math.pow ((1.0 - threshold1) / (1.0 - threshold0),
                          1.0 / Real.fromInt (max_relevant + 1))
    val add_thms = maps (ProofContext.get_fact ctxt) add
    val reserved = reserved_isar_keyword_table ()
    val axioms =
      (if only then
         maps (map (fn ((name, loc), th) => ((K name, loc), (true, th)))
               o name_thm_pairs_from_ref ctxt reserved chained_ths) add
       else
         all_name_thms_pairs ctxt reserved full_types add_thms chained_ths)
      |> name_thm_pairs ctxt (respect_no_atp andalso not only)
      |> uniquify
  in
    trace_msg (fn () => "Considering " ^ Int.toString (length axioms) ^
                        " theorems");
    (if threshold0 > 1.0 orelse threshold0 > threshold1 then
       []
     else if threshold0 < 0.0 then
       axioms
     else
       relevance_filter ctxt threshold0 decay max_relevant theory_relevant
                        relevance_override axioms (concl_t :: hyp_ts))
    |> map (apfst (apfst (fn f => f ())))
  end

end;