(*  Title:      HOL/Tools/ATP/atp_util.ML

     Author:     Jasmin Blanchette, TU Muenchen

 General-purpose functions used by the ATP module.

 *)

 signature ATP_UTIL =

 sig

   val timestamp : unit -> string

   val hash_string : string -> int

   val hash_term : term -> int

   val strip_spaces : bool -> (char -> bool) -> string -> string

   val strip_spaces_except_between_idents : string -> string

   val nat_subscript : int -> string

   val unyxml : string -> string

   val maybe_quote : string -> string

   val string_from_ext_time : bool * Time.time -> string

   val string_from_time : Time.time -> string

   val type_instance : Proof.context -> typ -> typ -> bool

   val type_generalization : Proof.context -> typ -> typ -> bool

   val type_intersect : Proof.context -> typ -> typ -> bool

   val type_aconv : Proof.context -> typ * typ -> bool

   val varify_type : Proof.context -> typ -> typ

   val instantiate_type : theory -> typ -> typ -> typ -> typ

   val varify_and_instantiate_type : Proof.context -> typ -> typ -> typ -> typ

   val typ_of_dtyp :

     Datatype_Aux.descr -> (Datatype_Aux.dtyp * typ) list -> Datatype_Aux.dtyp

     -> typ

   val is_type_surely_finite : Proof.context -> typ -> bool

   val is_type_surely_infinite : Proof.context -> bool -> typ list -> typ -> bool

   val s_not : term -> term

   val s_conj : term * term -> term

   val s_disj : term * term -> term

   val s_imp : term * term -> term

   val s_iff : term * term -> term

   val close_form : term -> term

   val monomorphic_term : Type.tyenv -> term -> term

   val eta_expand : typ list -> term -> int -> term

   val transform_elim_prop : term -> term

   val specialize_type : theory -> (string * typ) -> term -> term

   val strip_subgoal :

     Proof.context -> thm -> int -> (string * typ) list * term list * term

 end;

 structure ATP_Util : ATP_UTIL =

 struct

 val timestamp = Date.fmt "%Y-%m-%d %H:%M:%S" o Date.fromTimeLocal o Time.now

 (* This hash function is recommended in "Compilers: Principles, Techniques, and

    Tools" by Aho, Sethi, and Ullman. The "hashpjw" function, which they

    particularly recommend, triggers a bug in versions of Poly/ML up to 4.2.0. *)

 fun hashw (u, w) = Word.+ (u, Word.* (0w65599, w))

 fun hashw_char (c, w) = hashw (Word.fromInt (Char.ord c), w)

 fun hashw_string (s : string, w) = CharVector.foldl hashw_char w s

 fun hashw_term (t1 $ t2) = hashw (hashw_term t1, hashw_term t2)

   | hashw_term (Const (s, _)) = hashw_string (s, 0w0)

   | hashw_term (Free (s, _)) = hashw_string (s, 0w0)

   | hashw_term _ = 0w0

 fun hash_string s = Word.toInt (hashw_string (s, 0w0))

 val hash_term = Word.toInt o hashw_term

 fun strip_c_style_comment _ [] = []

   | strip_c_style_comment is_evil (#"*" :: #"/" :: cs) =

     strip_spaces_in_list true is_evil cs

   | strip_c_style_comment is_evil (_ :: cs) = strip_c_style_comment is_evil cs

 and strip_spaces_in_list _ _ [] = []

   | strip_spaces_in_list true is_evil (#"%" :: cs) =

     strip_spaces_in_list true is_evil

                          (cs |> chop_while (not_equal #"\n") |> snd)

   | strip_spaces_in_list true is_evil (#"/" :: #"*" :: cs) =

     strip_c_style_comment is_evil cs

   | strip_spaces_in_list _ _ [c1] = if Char.isSpace c1 then [] else [str c1]

   | strip_spaces_in_list skip_comments is_evil [c1, c2] =

     strip_spaces_in_list skip_comments is_evil [c1] @

     strip_spaces_in_list skip_comments is_evil [c2]

   | strip_spaces_in_list skip_comments is_evil (c1 :: c2 :: c3 :: cs) =

     if Char.isSpace c1 then

       strip_spaces_in_list skip_comments is_evil (c2 :: c3 :: cs)

     else if Char.isSpace c2 then

       if Char.isSpace c3 then

         strip_spaces_in_list skip_comments is_evil (c1 :: c3 :: cs)

       else

         str c1 :: (if forall is_evil [c1, c3] then [" "] else []) @

         strip_spaces_in_list skip_comments is_evil (c3 :: cs)

     else

       str c1 :: strip_spaces_in_list skip_comments is_evil (c2 :: c3 :: cs)

 fun strip_spaces skip_comments is_evil =

   implode o strip_spaces_in_list skip_comments is_evil o String.explode

 fun is_ident_char c = Char.isAlphaNum c orelse c = #"_"

 val strip_spaces_except_between_idents = strip_spaces true is_ident_char

 val subscript = implode o map (prefix "\<^isub>") o raw_explode  (* FIXME Symbol.explode (?) *)

 fun nat_subscript n =

   n |> string_of_int |> print_mode_active Symbol.xsymbolsN ? subscript

 val unyxml = XML.content_of o YXML.parse_body

 val is_long_identifier = forall Lexicon.is_identifier o space_explode "."

 fun maybe_quote y =

   let val s = unyxml y in

     y |> ((not (is_long_identifier (perhaps (try (unprefix "'")) s)) andalso

            not (is_long_identifier (perhaps (try (unprefix "?")) s))) orelse

            Keyword.is_keyword s) ? quote

   end

 fun string_from_ext_time (plus, time) =

   let val ms = Time.toMilliseconds time in

     (if plus then "> " else "") ^

     (if plus andalso ms mod 1000 = 0 then

        signed_string_of_int (ms div 1000) ^ " s"

      else if ms < 1000 then

        signed_string_of_int ms ^ " ms"

      else

        string_of_real (0.01 * Real.fromInt (ms div 10)) ^ " s")

   end

 val string_from_time = string_from_ext_time o pair false

 fun type_instance ctxt T T' =

   Sign.typ_instance (Proof_Context.theory_of ctxt) (T, T')

 fun type_generalization ctxt T T' = type_instance ctxt T' T

 fun type_intersect ctxt T T' =

   type_instance ctxt T T' orelse type_generalization ctxt T T'

 fun type_aconv ctxt (T, T') =

   type_instance ctxt T T' andalso type_instance ctxt T' T

 fun varify_type ctxt T =

   Variable.polymorphic_types ctxt [Const (@{const_name undefined}, T)]

   |> snd |> the_single |> dest_Const |> snd

 (* TODO: use "Term_Subst.instantiateT" instead? *)

 fun instantiate_type thy T1 T1' T2 =

   Same.commit (Envir.subst_type_same

                    (Sign.typ_match thy (T1, T1') Vartab.empty)) T2

   handle Type.TYPE_MATCH => raise TYPE ("instantiate_type", [T1, T1'], [])

 fun varify_and_instantiate_type ctxt T1 T1' T2 =

   let val thy = Proof_Context.theory_of ctxt in

     instantiate_type thy (varify_type ctxt T1) T1' (varify_type ctxt T2)

   end

 fun typ_of_dtyp _ typ_assoc (Datatype_Aux.DtTFree a) =

     the (AList.lookup (op =) typ_assoc (Datatype_Aux.DtTFree a))

   | typ_of_dtyp descr typ_assoc (Datatype_Aux.DtType (s, Us)) =

     Type (s, map (typ_of_dtyp descr typ_assoc) Us)

   | typ_of_dtyp descr typ_assoc (Datatype_Aux.DtRec i) =

     let val (s, ds, _) = the (AList.lookup (op =) descr i) in

       Type (s, map (typ_of_dtyp descr typ_assoc) ds)

     end

 fun datatype_constrs thy (T as Type (s, Ts)) =

     (case Datatype.get_info thy s of

        SOME {index, descr, ...} =>

        let val (_, dtyps, constrs) = AList.lookup (op =) descr index |> the in

          map (apsnd (fn Us => map (typ_of_dtyp descr (dtyps ~~ Ts)) Us ---> T))

              constrs

        end

      | NONE => [])

   | datatype_constrs _ _ = []

 (* Similar to "Nitpick_HOL.bounded_exact_card_of_type".

    0 means infinite type, 1 means singleton type (e.g., "unit"), and 2 means

    cardinality 2 or more. The specified default cardinality is returned if the

    cardinality of the type can't be determined. *)

 fun tiny_card_of_type ctxt sound assigns default_card T =

   let

     val thy = Proof_Context.theory_of ctxt

     val max = 2 (* 1 would be too small for the "fun" case *)

     fun aux slack avoid T =

       if member (op =) avoid T then

         0

       else case AList.lookup (type_aconv ctxt) assigns T of

         SOME k => k

       | NONE =>

         case T of

           Type (@{type_name fun}, [T1, T2]) =>

           (case (aux slack avoid T1, aux slack avoid T2) of

              (k, 1) => if slack andalso k = 0 then 0 else 1

            | (0, _) => 0

            | (_, 0) => 0

            | (k1, k2) =>

              if k1 >= max orelse k2 >= max then max

              else Int.min (max, Integer.pow k2 k1))

         | @{typ prop} => 2

         | @{typ bool} => 2 (* optimization *)

         | @{typ nat} => 0 (* optimization *)

         | Type ("Int.int", []) => 0 (* optimization *)

         | Type (s, _) =>

           (case datatype_constrs thy T of

              constrs as _ :: _ =>

              let

                val constr_cards =

                  map (Integer.prod o map (aux slack (T :: avoid)) o binder_types

                       o snd) constrs

              in

                if exists (curry (op =) 0) constr_cards then 0

                else Int.min (max, Integer.sum constr_cards)

              end

            | [] =>

              case Typedef.get_info ctxt s of

                ({abs_type, rep_type, ...}, _) :: _ =>

                (* We cheat here by assuming that typedef types are infinite if

                   their underlying type is infinite. This is unsound in general

                   but it's hard to think of a realistic example where this would

                   not be the case. We are also slack with representation types:

                   If a representation type has the form "sigma => tau", we

                   consider it enough to check "sigma" for infiniteness. (Look

                   for "slack" in this function.) *)

                (case varify_and_instantiate_type ctxt

                          (Logic.varifyT_global abs_type) T

                          (Logic.varifyT_global rep_type)

                      |> aux true avoid of

                   0 => 0

                 | 1 => 1

                 | _ => default_card)

              | [] => default_card)

           (* Very slightly unsound: Type variables are assumed not to be

              constrained to cardinality 1. (In practice, the user would most

              likely have used "unit" directly anyway.) *)

         | TFree _ =>

           if not sound andalso default_card = 1 then 2 else default_card

         | TVar _ => default_card

   in Int.min (max, aux false [] T) end

 fun is_type_surely_finite ctxt T = tiny_card_of_type ctxt true [] 0 T <> 0

 fun is_type_surely_infinite ctxt sound infinite_Ts T =

   tiny_card_of_type ctxt sound (map (rpair 0) infinite_Ts) 1 T = 0

 (* Simple simplifications to ensure that sort annotations don't leave a trail of

    spurious "True"s. *)

 fun s_not (Const (@{const_name All}, T) $ Abs (s, T', t')) =

     Const (@{const_name Ex}, T) $ Abs (s, T', s_not t')

   | s_not (Const (@{const_name Ex}, T) $ Abs (s, T', t')) =

     Const (@{const_name All}, T) $ Abs (s, T', s_not t')

   | s_not (@{const HOL.implies} $ t1 $ t2) = @{const HOL.conj} $ t1 $ s_not t2

   | s_not (@{const HOL.conj} $ t1 $ t2) =

     @{const HOL.disj} $ s_not t1 $ s_not t2

   | s_not (@{const HOL.disj} $ t1 $ t2) =

     @{const HOL.conj} $ s_not t1 $ s_not t2

   | s_not (@{const False}) = @{const True}

   | s_not (@{const True}) = @{const False}

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

   | s_not t = @{const Not} $ t

 fun s_conj (@{const True}, t2) = t2

   | s_conj (t1, @{const True}) = t1

   | s_conj p = HOLogic.mk_conj p

 fun s_disj (@{const False}, t2) = t2

   | s_disj (t1, @{const False}) = t1

   | s_disj p = HOLogic.mk_disj p

 fun s_imp (@{const True}, t2) = t2

   | s_imp (t1, @{const False}) = s_not t1

   | s_imp p = HOLogic.mk_imp p

 fun s_iff (@{const True}, t2) = t2

   | s_iff (t1, @{const True}) = t1

   | s_iff (t1, t2) = HOLogic.eq_const HOLogic.boolT $ t1 $ t2

 fun close_form t =

   fold (fn ((x, i), T) => fn t' =>

            HOLogic.all_const T $ Abs (x, T, abstract_over (Var ((x, i), T), t')))

        (Term.add_vars t []) t

 fun monomorphic_term subst =

   map_types (map_type_tvar (fn v =>

       case Type.lookup subst v of

         SOME typ => typ

       | NONE => TVar v))

 fun eta_expand _ t 0 = t

   | eta_expand Ts (Abs (s, T, t')) n =

     Abs (s, T, eta_expand (T :: Ts) t' (n - 1))

   | eta_expand Ts t n =

     fold_rev (fn T => fn t' => Abs ("x" ^ nat_subscript n, T, t'))

              (List.take (binder_types (fastype_of1 (Ts, t)), n))

              (list_comb (incr_boundvars n t, map Bound (n - 1 downto 0)))

 (* Converts an elim-rule into an equivalent theorem that does not have the

    predicate variable. Leaves other theorems unchanged. We simply instantiate

    the conclusion variable to "False". (Cf. "transform_elim_theorem" in

    "Meson_Clausify".) *)

 fun transform_elim_prop t =

   case Logic.strip_imp_concl t of

     @{const Trueprop} $ Var (z, @{typ bool}) =>

     subst_Vars [(z, @{const False})] t

   | Var (z, @{typ prop}) => subst_Vars [(z, @{prop False})] t

   | _ => t

 fun specialize_type thy (s, T) t =

   let

     fun subst_for (Const (s', T')) =

       if s = s' then

         SOME (Sign.typ_match thy (T', T) Vartab.empty)

         handle Type.TYPE_MATCH => NONE

       else

         NONE

     | subst_for (t1 $ t2) =

       (case subst_for t1 of SOME x => SOME x | NONE => subst_for t2)

     | subst_for (Abs (_, _, t')) = subst_for t'

     | subst_for _ = NONE

   in

     case subst_for t of

       SOME subst => monomorphic_term subst t

     | NONE => raise Type.TYPE_MATCH

   end

 fun strip_subgoal ctxt goal i =

   let

     val (t, (frees, params)) =

       Logic.goal_params (prop_of goal) i

       ||> (map dest_Free #> Variable.variant_frees ctxt [] #> `(map Free))

     val hyp_ts = t |> Logic.strip_assums_hyp |> map (curry subst_bounds frees)

     val concl_t = t |> Logic.strip_assums_concl |> curry subst_bounds frees

   in (rev params, hyp_ts, concl_t) end

 end;