(*  Title:      HOL/Tools/Sledgehammer/sledgehammer_mepo.ML

    Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA

    Author:     Jasmin Blanchette, TU Muenchen

Sledgehammer's iterative relevance filter (MePo = Meng-Paulson).

*)

signature SLEDGEHAMMER_MEPO =

sig

  type stature = ATP_Problem_Generate.stature

  type fact = Sledgehammer_Fact.fact

  type params = Sledgehammer_Provers.params

  type relevance_fudge = Sledgehammer_Provers.relevance_fudge

  val trace : bool Config.T

  val pseudo_abs_name : string

  val pseudo_skolem_prefix : string

  val const_names_in_fact :

    theory -> (string * typ -> term list -> bool * term list) -> term

    -> string list

  val iterative_relevant_facts :

    Proof.context -> params -> string -> int -> relevance_fudge option

    -> term list -> term -> fact list -> fact list

end;

structure Sledgehammer_MePo : SLEDGEHAMMER_MEPO =

struct

open ATP_Problem_Generate

open Sledgehammer_Fact

open Sledgehammer_Provers

val trace =

  Attrib.setup_config_bool @{binding sledgehammer_filter_iter_trace} (K false)

fun trace_msg ctxt msg = if Config.get ctxt trace then tracing (msg ()) else ()

val sledgehammer_prefix = "Sledgehammer" ^ Long_Name.separator

val pseudo_abs_name = sledgehammer_prefix ^ "abs"

val pseudo_skolem_prefix = sledgehammer_prefix ^ "sko"

val theory_const_suffix = Long_Name.separator ^ " 1"

fun 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

(* Pairs a constant with the list of its type instantiations. *)

fun ptype thy const x =

  (if const then map pattern_for_type (these (try (Sign.const_typargs thy) x))

   else [])

fun rich_ptype thy const (s, T) =

  PType (order_of_type T, ptype thy const (s, T))

fun rich_pconst thy const (s, T) = (s, rich_ptype thy const (s, T))

fun string_for_hyper_pconst (s, ps) =

  s ^ "{" ^ commas (map string_for_ptype ps) ^ "}"

(* Add a pconstant to the table, but a [] entry means a standard

   connective, which we ignore.*)

fun add_pconst_to_table also_skolem (s, p) =

  if (not also_skolem andalso String.isPrefix pseudo_skolem_prefix s) then I

  else Symtab.map_default (s, [p]) (insert (op =) p)

(* Set constants tend to pull in too many irrelevant facts. We limit the damage

   by treating them more or less as if they were built-in but add their

   axiomatization at the end. *)

val set_consts = [@{const_name Collect}, @{const_name Set.member}]

val set_thms = @{thms Collect_mem_eq mem_Collect_eq Collect_cong}

fun add_pconsts_in_term thy is_built_in_const also_skolems pos =

  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 automatic

       prover, we introduce a fresh constant to simulate the effect of

       Skolemization. *)

    fun do_const const ext_arg (x as (s, _)) ts =

      let val (built_in, ts) = is_built_in_const x ts in

        if member (op =) set_consts s then

          fold (do_term ext_arg) ts

        else

          (not built_in

           ? add_pconst_to_table also_skolems (rich_pconst thy const x))

          #> fold (do_term false) ts

      end

    and do_term ext_arg t =

      case strip_comb t of

        (Const x, ts) => do_const true ext_arg x ts

      | (Free x, ts) => do_const false ext_arg x ts

      | (Abs (_, T, t'), ts) =>

        ((null ts andalso not ext_arg)

         (* Since lambdas on the right-hand side of equalities are usually

            extensionalized later by "abs_extensionalize_term", we don't

            penalize them here. *)

         ? add_pconst_to_table true (pseudo_abs_name,

                                     PType (order_of_type T + 1, [])))

        #> fold (do_term false) (t' :: ts)

      | (_, ts) => fold (do_term false) 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 (pseudo_skolem_prefix ^ serial_string (),

                                      PType (order_of_type abs_T, []))

          else

            I)

    and do_term_or_formula ext_arg T =

      if T = HOLogic.boolT then do_formula NONE else do_term ext_arg

    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 =>

        do_term_or_formula false T t1 #> do_term_or_formula true T t2

      | @{const Trueprop} $ t1 => do_formula pos t1

      | @{const False} => I

      | @{const True} => I

      | @{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 =>

        do_term_or_formula false T t1 #> do_term_or_formula true T t2

      | Const (@{const_name If}, Type (_, [_, Type (_, [T, _])]))

        $ t1 $ t2 $ t3 =>

        do_formula NONE t1 #> fold (do_term_or_formula false 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 false t0 #> do_formula pos t1  (* theory constant *)

      | _ => do_term false t

  in do_formula pos end

fun pconsts_in_fact thy is_built_in_const t =

  Symtab.fold (fn (s, pss) => fold (cons o pair s) pss)

              (Symtab.empty |> add_pconsts_in_term thy is_built_in_const true

                                                   (SOME true) t) []

val const_names_in_fact = map fst ooo pconsts_in_fact

(* Inserts a dummy "constant" referring to the theory name, so that relevance

   takes the given theory into account. *)

fun theory_constify ({theory_const_rel_weight, theory_const_irrel_weight, ...}

                     : relevance_fudge) thy_name t =

  if exists (curry (op <) 0.0) [theory_const_rel_weight,

                                theory_const_irrel_weight] then

    Const (thy_name ^ theory_const_suffix, @{typ bool}) $ t

  else

    t

fun theory_const_prop_of fudge th =

  theory_constify fudge (Context.theory_name (theory_of_thm th)) (prop_of th)

fun pair_consts_fact thy is_built_in_const fudge fact =

  case fact |> snd |> theory_const_prop_of fudge

            |> pconsts_in_fact thy is_built_in_const of

    [] => NONE

  | consts => SOME ((fact, consts), NONE)

(* 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_fact_consts thy fudge =

  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), 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 fudge o snd end

fun pow_int _ 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 _ freq = 1.0 + 2.0 / Math.ln (Real.fromInt freq + 1.0)

(* 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 ({worse_irrel_freq, higher_order_irrel_weight, ...}

                      : relevance_fudge) 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

fun multiplier_for_const_name local_const_multiplier s =

  if String.isSubstring "." s then 1.0 else local_const_multiplier

(* Computes a constant's weight, as determined by its frequency. *)

fun generic_pconst_weight local_const_multiplier abs_weight skolem_weight

                          theory_const_weight chained_const_weight weight_for f

                          const_tab chained_const_tab (c as (s, PType (m, _))) =

  if s = pseudo_abs_name then

    abs_weight

  else if String.isPrefix pseudo_skolem_prefix s then

    skolem_weight

  else if String.isSuffix theory_const_suffix s then

    theory_const_weight

  else

    multiplier_for_const_name local_const_multiplier s

    * weight_for m (pconst_freq (match_ptype o f) const_tab c)

    |> (if chained_const_weight < 1.0 andalso

           pconst_hyper_mem I chained_const_tab c then

          curry (op *) chained_const_weight

        else

          I)

fun rel_pconst_weight ({local_const_multiplier, abs_rel_weight,

                        theory_const_rel_weight, ...} : relevance_fudge)

                      const_tab =

  generic_pconst_weight local_const_multiplier abs_rel_weight 0.0

                        theory_const_rel_weight 0.0 rel_weight_for I const_tab

                        Symtab.empty

fun irrel_pconst_weight (fudge as {local_const_multiplier, abs_irrel_weight,

                                   skolem_irrel_weight,

                                   theory_const_irrel_weight,

                                   chained_const_irrel_weight, ...})

                        const_tab chained_const_tab =

  generic_pconst_weight local_const_multiplier abs_irrel_weight

                        skolem_irrel_weight theory_const_irrel_weight

                        chained_const_irrel_weight (irrel_weight_for fudge) swap

                        const_tab chained_const_tab

fun stature_bonus ({intro_bonus, ...} : relevance_fudge) (_, Intro) =

    intro_bonus

  | stature_bonus {elim_bonus, ...} (_, Elim) = elim_bonus

  | stature_bonus {simp_bonus, ...} (_, Simp) = simp_bonus

  | stature_bonus {local_bonus, ...} (Local, _) = local_bonus

  | stature_bonus {assum_bonus, ...} (Assum, _) = assum_bonus

  | stature_bonus {chained_bonus, ...} (Chained, _) = chained_bonus

  | stature_bonus _ _ = 0.0

fun is_odd_const_name s =

  s = pseudo_abs_name orelse String.isPrefix pseudo_skolem_prefix s orelse

  String.isSuffix theory_const_suffix s

fun fact_weight fudge stature const_tab relevant_consts chained_consts

                fact_consts =

  case fact_consts |> List.partition (pconst_hyper_mem I relevant_consts)

                   ||> filter_out (pconst_hyper_mem swap relevant_consts) of

    ([], _) => 0.0

  | (rel, irrel) =>

    if forall (forall (is_odd_const_name o fst)) [rel, irrel] then

      0.0

    else

      let

        val irrel = irrel |> filter_out (pconst_mem swap rel)

        val rel_weight =

          0.0 |> fold (curry (op +) o rel_pconst_weight fudge const_tab) rel

        val irrel_weight =

          ~ (stature_bonus fudge stature)

          |> fold (curry (op +)

                   o irrel_pconst_weight fudge const_tab chained_consts) irrel

        val res = rel_weight / (rel_weight + irrel_weight)

      in if Real.isFinite res then res else 0.0 end

fun take_most_relevant ctxt max_facts remaining_max

        ({max_imperfect, max_imperfect_exp, ...} : relevance_fudge)

        (candidates : ((fact * (string * ptype) list) * real) list) =

  let

    val max_imperfect =

      Real.ceil (Math.pow (max_imperfect,

                    Math.pow (Real.fromInt remaining_max

                              / Real.fromInt max_facts, 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 ctxt (fn () =>

        "Actually passed (" ^ string_of_int (length accepts) ^ " of " ^

        string_of_int (length candidates) ^ "): " ^

        (accepts |> map (fn ((((name, _), _), _), weight) =>

                            name () ^ " [" ^ Real.toString weight ^ "]")

                 |> commas));

    (accepts, more_rejects @ rejects)

  end

fun if_empty_replace_with_scope thy is_built_in_const facts sc tab =

  if Symtab.is_empty tab then

    Symtab.empty

    |> fold (add_pconsts_in_term thy is_built_in_const false (SOME false))

            (map_filter (fn ((_, (sc', _)), th) =>

                            if sc' = sc then SOME (prop_of th) else NONE) facts)

  else

    tab

fun consider_arities is_built_in_const th =

  let

    fun aux _ _ NONE = NONE

      | aux t args (SOME tab) =

        case t of

          t1 $ t2 => SOME tab |> aux t1 (t2 :: args) |> aux t2 []

        | Const (x as (s, _)) =>

          (if is_built_in_const x args |> fst then

             SOME tab

           else case Symtab.lookup tab s of

             NONE => SOME (Symtab.update (s, length args) tab)

           | SOME n => if n = length args then SOME tab else NONE)

        | _ => SOME tab

  in aux (prop_of th) [] end

(* FIXME: This is currently only useful for polymorphic type encodings. *)

fun could_benefit_from_ext is_built_in_const facts =

  fold (consider_arities is_built_in_const o snd) facts (SOME Symtab.empty)

  |> is_none

(* High enough so that it isn't wrongly considered as very relevant (e.g., for E

   weights), but low enough so that it is unlikely to be truncated away if few

   facts are included. *)

val special_fact_index = 75

fun relevance_filter ctxt thres0 decay max_facts is_built_in_const

        (fudge as {threshold_divisor, ridiculous_threshold, ...}) facts hyp_ts

        concl_t =

  let

    val thy = Proof_Context.theory_of ctxt

    val const_tab = fold (count_fact_consts thy fudge) facts Symtab.empty

    val add_pconsts = add_pconsts_in_term thy is_built_in_const false o SOME

    val chained_ts =

      facts |> map_filter (fn ((_, (Chained, _)), th) => SOME (prop_of th)

                            | _ => NONE)

    val chained_const_tab = Symtab.empty |> fold (add_pconsts true) chained_ts

    val goal_const_tab =

      Symtab.empty |> fold (add_pconsts true) hyp_ts

                   |> add_pconsts false concl_t

      |> (fn tab => if Symtab.is_empty tab then chained_const_tab else tab)

      |> fold (if_empty_replace_with_scope thy is_built_in_const facts)

              [Chained, Assum, Local]

    fun iter j remaining_max thres rel_const_tab hopeless hopeful =

      let

        fun relevant [] _ [] =

            (* Nothing has been added this iteration. *)

            if j = 0 andalso thres >= ridiculous_threshold then

              (* First iteration? Try again. *)

              iter 0 max_facts (thres / threshold_divisor) rel_const_tab

                   hopeless hopeful

            else

              []

          | relevant candidates rejects [] =

            let

              val (accepts, more_rejects) =

                take_most_relevant ctxt max_facts remaining_max fudge 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 thres =

                1.0 - (1.0 - thres)

                      * Math.pow (decay, Real.fromInt (length accepts))

              val remaining_max = remaining_max - length accepts

            in

              trace_msg ctxt (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

                 []

               else

                 iter (j + 1) remaining_max thres rel_const_tab'

                      hopeless_rejects hopeful_rejects)

            end

          | relevant candidates rejects

                     (((ax as (((_, stature), _), fact_consts)), cached_weight)

                      :: hopeful) =

            let

              val weight =

                case cached_weight of

                  SOME w => w

                | NONE => fact_weight fudge stature const_tab rel_const_tab

                                      chained_const_tab fact_consts

            in

              if weight >= thres then

                relevant ((ax, weight) :: candidates) rejects hopeful

              else

                relevant candidates ((ax, weight) :: rejects) hopeful

            end

        in

          trace_msg ctxt (fn () =>

              "ITERATION " ^ string_of_int j ^ ": current threshold: " ^

              Real.toString thres ^ ", constants: " ^

              commas (rel_const_tab |> Symtab.dest

                      |> filter (curry (op <>) [] o snd)

                      |> map string_for_hyper_pconst));

          relevant [] [] hopeful

        end

    fun uses_const s t =

      fold_aterms (curry (fn (Const (s', _), false) => s' = s | (_, b) => b)) t

                  false

    fun uses_const_anywhere accepts s =

      exists (uses_const s o prop_of o snd) accepts orelse

      exists (uses_const s) (concl_t :: hyp_ts)

    fun add_set_const_thms accepts =

      exists (uses_const_anywhere accepts) set_consts ? append set_thms

    fun insert_into_facts accepts [] = accepts

      | insert_into_facts accepts ths =

        let

          val add = facts |> filter (member Thm.eq_thm_prop ths o snd)

          val (bef, after) =

            accepts |> filter_out (member Thm.eq_thm_prop ths o snd)

                    |> take (max_facts - length add)

                    |> chop special_fact_index

        in bef @ add @ after end

    fun insert_special_facts accepts =

       (* FIXME: get rid of "ext" here once it is treated as a helper *)

       [] |> could_benefit_from_ext is_built_in_const accepts ? cons @{thm ext}

          |> add_set_const_thms accepts

          |> insert_into_facts accepts

  in

    facts |> map_filter (pair_consts_fact thy is_built_in_const fudge)

          |> iter 0 max_facts thres0 goal_const_tab []

          |> insert_special_facts

          |> tap (fn accepts => trace_msg ctxt (fn () =>

                      "Total relevant: " ^ string_of_int (length accepts)))

  end

fun iterative_relevant_facts ctxt

        ({fact_thresholds = (thres0, thres1), ...} : params) prover

        max_facts fudge hyp_ts concl_t facts =

  let

    val thy = Proof_Context.theory_of ctxt

    val is_built_in_const =

      Sledgehammer_Provers.is_built_in_const_for_prover ctxt prover

    val fudge =

      case fudge of

        SOME fudge => fudge

      | NONE => Sledgehammer_Provers.relevance_fudge_for_prover ctxt prover

    val decay = Math.pow ((1.0 - thres1) / (1.0 - thres0),

                          1.0 / Real.fromInt (max_facts + 1))

  in

    trace_msg ctxt (fn () => "Considering " ^ string_of_int (length facts) ^

                             " facts");

    (if thres1 < 0.0 then

       facts

     else if thres0 > 1.0 orelse thres0 > thres1 then

       []

     else

       relevance_filter ctxt thres0 decay max_facts is_built_in_const fudge

           facts hyp_ts

           (concl_t |> theory_constify fudge (Context.theory_name thy)))

  end

end;