(*  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;