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

     Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA

     Author:     Jasmin Blanchette, TU Muenchen

 *)

 signature SLEDGEHAMMER_FACT_FILTER =

 sig

   type relevance_override =

     {add: Facts.ref list,

      del: Facts.ref list,

      only: bool}

   val trace : bool Unsynchronized.ref

   val chained_prefix : string

   val name_thms_pair_from_ref :

     Proof.context -> thm list -> Facts.ref -> string * thm list

   val relevant_facts :

     bool -> real -> real -> bool -> int -> bool -> relevance_override

     -> Proof.context * (thm list * 'a) -> term list -> term

     -> (string * thm) list

 end;

 structure Sledgehammer_Fact_Filter : SLEDGEHAMMER_FACT_FILTER =

 struct

 val trace = Unsynchronized.ref false

 fun trace_msg msg = if !trace then tracing (msg ()) else ()

 val respect_no_atp = true

 type relevance_override =

   {add: Facts.ref list,

    del: Facts.ref list,

    only: bool}

 val sledgehammer_prefix = "Sledgehammer" ^ Long_Name.separator

 (* Used to label theorems chained into the goal. *)

 val chained_prefix = sledgehammer_prefix ^ "chained_"

 fun name_thms_pair_from_ref ctxt chained_ths xref =

   let

     val ths = ProofContext.get_fact ctxt xref

     val name = Facts.string_of_ref xref

                |> forall (member Thm.eq_thm chained_ths) ths

                   ? prefix chained_prefix

   in (name, ths) end

 (***************************************************************)

 (* Relevance Filtering                                         *)

 (***************************************************************)

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

     strip_Trueprop_and_all t

   | strip_Trueprop_and_all (Const (@{const_name all}, _) $ Abs (_, _, t)) =

     strip_Trueprop_and_all t

   | strip_Trueprop_and_all (Const (@{const_name All}, _) $ Abs (_, _, t)) =

     strip_Trueprop_and_all t

   | strip_Trueprop_and_all t = t

 (*** constants with types ***)

 (*An abstraction of Isabelle types*)

 datatype const_typ =  CTVar | CType of string * const_typ list

 (*Is the second type an instance of the first one?*)

 fun match_type (CType(con1,args1)) (CType(con2,args2)) =

       con1=con2 andalso match_types args1 args2

   | match_type CTVar _ = true

   | match_type _ CTVar = false

 and match_types [] [] = true

   | match_types (a1::as1) (a2::as2) = match_type a1 a2 andalso match_types as1 as2;

 (*Is there a unifiable constant?*)

 fun uni_mem goal_const_tab (c, c_typ) =

   exists (match_types c_typ) (these (Symtab.lookup goal_const_tab c))

 (*Maps a "real" type to a const_typ*)

 fun const_typ_of (Type (c,typs)) = CType (c, map const_typ_of typs)

   | const_typ_of (TFree _) = CTVar

   | const_typ_of (TVar _) = CTVar

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

 fun const_with_typ thy (c,typ) =

     let val tvars = Sign.const_typargs thy (c,typ)

     in (c, map const_typ_of tvars) end

     handle TYPE _ => (c,[]);   (*Variable (locale constant): monomorphic*)

 (*Add a const/type pair to the table, but a [] entry means a standard connective,

   which we ignore.*)

 fun add_const_type_to_table (c, ctyps) =

   Symtab.map_default (c, [ctyps])

                      (fn [] => [] | ctypss => insert (op =) ctyps ctypss)

 val fresh_prefix = "Sledgehammer.Fresh."

 val flip = Option.map not

 (* These are typically simplified away by "Meson.presimplify". *)

 val boring_consts = [@{const_name If}, @{const_name Let}]

 fun get_consts_typs thy pos ts =

   let

     (* 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_term t =

       case t of

         Const (@{const_name Let}, _) => I

       | Const x => add_const_type_to_table (const_with_typ thy x)

       | Free x => add_const_type_to_table (const_with_typ thy x)

       | t1 $ t2 => do_term t1 #> do_term t2

       | Abs (_, _, t) =>

         (* Abstractions lead to combinators, so we add a penalty for them. *)

         add_const_type_to_table (gensym fresh_prefix, [])

         #> do_term t

       | _ => I

     fun do_quantifier will_surely_be_skolemized body_t =

       do_formula pos body_t

       #> (if will_surely_be_skolemized then

             add_const_type_to_table (gensym fresh_prefix, [])

           else

             I)

     and do_term_or_formula T =

       if T = @{typ bool} orelse T = @{typ prop} then do_formula NONE

       else do_term

     and do_formula pos t =

       case t of

         Const (@{const_name all}, _) $ Abs (_, _, body_t) =>

         do_quantifier (pos = SOME false) body_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 (_, _, body_t) =>

         do_quantifier (pos = SOME false) body_t

       | Const (@{const_name Ex}, _) $ Abs (_, _, body_t) =>

         do_quantifier (pos = SOME true) body_t

       | @{const "op &"} $ t1 $ t2 => fold (do_formula pos) [t1, t2]

       | @{const "op |"} $ t1 $ t2 => fold (do_formula pos) [t1, t2]

       | @{const "op -->"} $ t1 $ t2 =>

         do_formula (flip pos) t1 #> do_formula pos t2

       | Const (@{const_name "op ="}, 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 (_, _, body_t) =>

         do_quantifier (is_some pos) body_t

       | Const (@{const_name Ball}, _) $ t1 $ Abs (_, _, body_t) =>

         do_quantifier (pos = SOME false)

                       (HOLogic.mk_imp (incr_boundvars 1 t1 $ Bound 0, body_t))

       | Const (@{const_name Bex}, _) $ t1 $ Abs (_, _, body_t) =>

         do_quantifier (pos = SOME true)

                       (HOLogic.mk_conj (incr_boundvars 1 t1 $ Bound 0, body_t))

       | (t0 as Const (_, @{typ bool})) $ t1 =>

         do_term t0 #> do_formula pos t1  (* theory constant *)

       | _ => do_term t

   in

     Symtab.empty |> fold (Symtab.update o rpair []) boring_consts

                  |> 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 ^ ". 1", @{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 type const_typ list.*)

 local

 fun cons_nr CTVar = 0

   | cons_nr (CType _) = 1;

 in

 fun const_typ_ord TU =

   case TU of

     (CType (a, Ts), CType (b, Us)) =>

       (case fast_string_ord(a,b) of EQUAL => dict_ord const_typ_ord (Ts,Us) | ord => ord)

   | (T, U) => int_ord (cons_nr T, cons_nr U);

 end;

 structure CTtab = Table(type key = const_typ list val ord = dict_ord const_typ_ord);

 fun count_axiom_consts theory_relevant thy (_, th) =

   let

     fun do_const (a, T) =

       let val (c, cts) = const_with_typ thy (a,T) in

         (* Two-dimensional table update. Constant maps to types maps to

            count. *)

         CTtab.map_default (cts, 0) (Integer.add 1)

         |> Symtab.map_default (c, CTtab.empty)

       end

     fun do_term (Const x) = do_const x

       | do_term (Free x) = do_const x

       | do_term (t $ u) = do_term t #> do_term u

       | do_term (Abs (_, _, t)) = do_term t

       | do_term _ = I

   in th |> theory_const_prop_of theory_relevant |> do_term end

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

 (*The frequency of a constant is the sum of those of all instances of its type.*)

 fun const_frequency const_tab (c, cts) =

   CTtab.fold (fn (cts', m) => match_types cts cts' ? Integer.add m)

              (the (Symtab.lookup const_tab c)

               handle Option.Option => raise Fail ("Const: " ^ 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. *)

 fun log_weight2 (x:real) = 1.0 + 2.0 / Math.ln (x + 1.0)

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

 val ct_weight = log_weight2 o real oo const_frequency

 (*Relevant constants are weighted according to frequency,

   but irrelevant constants are simply counted. Otherwise, Skolem functions,

   which are rare, would harm a formula's chances of being picked.*)

 fun formula_weight const_tab gctyps consts_typs =

   let

     val rel = filter (uni_mem gctyps) consts_typs

     val rel_weight = fold (curry Real.+ o ct_weight const_tab) rel 0.0

     val res = rel_weight / (rel_weight + real (length consts_typs - length rel))

   in if Real.isFinite res then res else 0.0 end

 (*Multiplies out to a list of pairs: 'a * 'b list -> ('a * 'b) list -> ('a * 'b) list*)

 fun add_expand_pairs (x,ys) xys = List.foldl (fn (y,acc) => (x,y)::acc) xys ys;

 fun consts_typs_of_term thy t =

   Symtab.fold add_expand_pairs (get_consts_typs thy (SOME true) [t]) []

 fun pair_consts_typs_axiom theory_relevant thy axiom =

   (axiom, axiom |> snd |> theory_const_prop_of theory_relevant

                 |> consts_typs_of_term thy)

 exception CONST_OR_FREE of unit

 fun dest_Const_or_Free (Const x) = x

   | dest_Const_or_Free (Free x) = x

   | dest_Const_or_Free _ = raise CONST_OR_FREE ()

 (*Look for definitions of the form f ?x1 ... ?xn = t, but not reversed.*)

 fun defines thy thm gctypes =

     let val tm = prop_of thm

         fun defs lhs rhs =

             let val (rator,args) = strip_comb lhs

                 val ct = const_with_typ thy (dest_Const_or_Free rator)

             in

               forall is_Var args andalso uni_mem gctypes ct andalso

                 subset (op =) (Term.add_vars rhs [], Term.add_vars lhs [])

             end

             handle CONST_OR_FREE () => false

     in

         case tm of

           @{const Trueprop} $ (Const (@{const_name "op ="}, _) $ lhs $ rhs) =>

             defs lhs rhs

         | _ => false

     end;

 type annotated_thm = (string * thm) * (string * const_typ list) list

 (*For a reverse sort, putting the largest values first.*)

 fun compare_pairs ((_, w1), (_, w2)) = Real.compare (w2, w1)

 (* Limit the number of new facts, to prevent runaway acceptance. *)

 fun take_best max_new (newpairs : (annotated_thm * real) list) =

   let val nnew = length newpairs in

     if nnew <= max_new then

       (map #1 newpairs, [])

     else

       let

         val newpairs = sort compare_pairs newpairs

         val accepted = List.take (newpairs, max_new)

       in

         trace_msg (fn () => ("Number of candidates, " ^ Int.toString nnew ^

                        ", exceeds the limit of " ^ Int.toString max_new));

         trace_msg (fn () => ("Effective pass mark: " ^ Real.toString (#2 (List.last accepted))));

         trace_msg (fn () => "Actually passed: " ^

           space_implode ", " (map (fst o fst o fst) accepted));

         (map #1 accepted, map #1 (List.drop (newpairs, max_new)))

       end

   end;

 fun relevance_filter ctxt relevance_threshold relevance_convergence

                      defs_relevant max_new theory_relevant

                      ({add, del, ...} : relevance_override) axioms goal_ts =

   if relevance_threshold > 1.0 then

     []

   else if relevance_threshold < 0.0 then

     axioms

   else

     let

       val thy = ProofContext.theory_of ctxt

       val const_tab = fold (count_axiom_consts theory_relevant thy) axioms

                            Symtab.empty

       val goal_const_tab = get_consts_typs thy (SOME false) goal_ts

       val relevance_threshold = 0.8 * relevance_threshold (* FIXME *)

       val _ =

         trace_msg (fn () => "Initial constants: " ^

                             commas (goal_const_tab

                                     |> Symtab.dest

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

                                     |> map fst))

       val add_thms = maps (ProofContext.get_fact ctxt) add

       val del_thms = maps (ProofContext.get_fact ctxt) del

       fun iter threshold rel_const_tab =

         let

           fun relevant ([], rejects) [] =

               (* Nothing was added this iteration: Add "add:" facts. *)

               if null add_thms then

                 []

               else

                 map_filter (fn (p as (name, th), _) =>

                                if member Thm.eq_thm add_thms th then SOME p

                                else NONE) rejects

             | relevant (newpairs, rejects) [] =

               let

                 val (newrels, more_rejects) = take_best max_new newpairs

                 val new_consts = maps #2 newrels

                 val rel_const_tab =

                   rel_const_tab |> fold add_const_type_to_table new_consts

                 val threshold =

                   threshold + (1.0 - threshold) / relevance_convergence

               in

                 trace_msg (fn () => "relevant this iteration: " ^

                                     Int.toString (length newrels));

                 map #1 newrels @ iter threshold rel_const_tab

                     (more_rejects @ rejects)

               end

             | relevant (newrels, rejects)

                        ((ax as ((name, th), consts_typs)) :: axs) =

               let

                 val weight =

                   if member Thm.eq_thm del_thms th then 0.0

                   else formula_weight const_tab rel_const_tab consts_typs

               in

                 if weight >= threshold orelse

                    (defs_relevant andalso defines thy th rel_const_tab) then

                   (trace_msg (fn () =>

                        name ^ " passes: " ^ Real.toString weight

                        (* ^ " consts: " ^ commas (map fst consts_typs) *));

                    relevant ((ax, weight) :: newrels, rejects) axs)

                 else

                   relevant (newrels, ax :: rejects) axs

               end

           in

             trace_msg (fn () => "relevant_facts, current threshold: " ^

                                 Real.toString threshold);

             relevant ([], [])

           end

       val relevant = iter relevance_threshold goal_const_tab

                           (map (pair_consts_typs_axiom theory_relevant thy)

                                axioms)

     in

       trace_msg (fn () => "Total relevant: " ^ Int.toString (length relevant));

       relevant

     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 make_fact_table xs =

   fold (Termtab.update o `(prop_of o snd)) xs Termtab.empty

 fun make_unique xs = Termtab.fold (cons o snd) (make_fact_table 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"]

 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

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

 (* 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 31-10-2007)

    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)

 fun is_strange_thm th =

   case head_of (concl_of th) of

       Const (a, _) => (a <> @{const_name Trueprop} andalso

                        a <> @{const_name "=="})

     | _ => false

 val type_has_top_sort =

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

 fun is_theorem_bad_for_atps thm =

   let val t = prop_of thm in

     is_formula_too_complex t orelse exists_type type_has_top_sort t orelse

     exists_sledgehammer_const t orelse is_strange_thm thm

   end

 fun all_name_thms_pairs ctxt chained_ths =

   let

     val global_facts = PureThy.facts_of (ProofContext.theory_of ctxt);

     val local_facts = ProofContext.facts_of ctxt;

     val full_space =

       Name_Space.merge (Facts.space_of global_facts, Facts.space_of local_facts);

     fun valid_facts facts =

       (facts, []) |-> Facts.fold_static (fn (name, ths0) =>

         if Facts.is_concealed facts name orelse

            (respect_no_atp andalso is_package_def name) orelse

            member (op =) multi_base_blacklist (Long_Name.base_name name) orelse

            String.isSuffix "_def_raw" (* FIXME: crude hack *) name then

           I

         else

           let

             fun check_thms a =

               (case try (ProofContext.get_thms ctxt) a of

                 NONE => false

               | SOME ths1 => Thm.eq_thms (ths0, ths1));

             val name1 = Facts.extern facts name;

             val name2 = Name_Space.extern full_space name;

             val ths = filter_out is_theorem_bad_for_atps ths0

           in

             case find_first check_thms [name1, name2, name] of

               NONE => I

             | SOME name' =>

               cons (name' |> forall (member Thm.eq_thm chained_ths) ths

                              ? prefix chained_prefix, ths)

           end)

   in valid_facts global_facts @ valid_facts local_facts end;

 fun multi_name a th (n, pairs) =

   (n + 1, (a ^ "(" ^ Int.toString n ^ ")", th) :: pairs);

 fun add_names (_, []) pairs = pairs

   | add_names (a, [th]) pairs = (a, th) :: pairs

   | add_names (a, ths) pairs = #2 (fold (multi_name a) ths (1, pairs))

 fun is_multi (a, ths) = length ths > 1 orelse String.isSuffix ".axioms" a;

 (* 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 name_thms_pairs =

   let

     val (mults, singles) = List.partition is_multi name_thms_pairs

     val ps = [] |> fold add_names singles |> fold add_names mults

   in ps |> respect_no_atp ? filter_out (No_ATPs.member ctxt o snd) end;

 fun is_named ("", th) =

     (warning ("No name for theorem " ^

               Display.string_of_thm_without_context th); false)

   | is_named _ = true

 fun checked_name_thm_pairs respect_no_atp ctxt =

   name_thm_pairs ctxt respect_no_atp

   #> tap (fn ps => trace_msg

                         (fn () => ("Considering " ^ Int.toString (length ps) ^

                                    " theorems")))

   #> filter is_named

 (***************************************************************)

 (* ATP invocation methods setup                                *)

 (***************************************************************)

 (**** 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. *)

 fun too_general_eqterms (Var z) t =

     not (exists_subterm (fn Var z' => z = z' | _ => false) t)

   | too_general_eqterms (Bound j) t = not (loose_bvar1 (t, j))

   | too_general_eqterms _ _ = false

 fun too_general_equality (Const (@{const_name "op ="}, _) $ t1 $ t2) =

     too_general_eqterms t1 t2 orelse too_general_eqterms t2 t1

   | too_general_equality _ = false

 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

     (has_bound_or_var_of_type dangerous_types t orelse

      forall too_general_equality (HOLogic.disjuncts (strip_Trueprop_and_all t)))

 fun relevant_facts full_types relevance_threshold relevance_convergence

                    defs_relevant max_new theory_relevant

                    (relevance_override as {add, del, only})

                    (ctxt, (chained_ths, _)) hyp_ts concl_t =

   let

     val add_thms = maps (ProofContext.get_fact ctxt) add

     val axioms =

       checked_name_thm_pairs (respect_no_atp andalso not only) ctxt

           (map (name_thms_pair_from_ref ctxt chained_ths) add @

            (if only then [] else all_name_thms_pairs ctxt chained_ths))

       |> filter (fn (_, th) => member Thm.eq_thm add_thms th orelse

                                not (is_dangerous_term full_types (prop_of th)))

   in

     relevance_filter ctxt relevance_threshold relevance_convergence

                      defs_relevant max_new theory_relevant relevance_override

                      axioms (concl_t :: hyp_ts)

     |> make_unique |> sort_wrt fst

   end

 end;