(*  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_equiv : 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_spaces skip_comments is_evil =

  let

    fun strip_c_style_comment [] accum = accum

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

        strip_spaces_in_list true cs accum

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

    and strip_spaces_in_list _ [] accum = rev accum

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

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

                             accum

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

        strip_c_style_comment cs accum

      | strip_spaces_in_list _ [c1] accum =

        accum |> not (Char.isSpace c1) ? cons c1

      | strip_spaces_in_list skip_comments (cs as [_, _]) accum =

        accum |> fold (strip_spaces_in_list skip_comments o single) cs

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

        if Char.isSpace c1 then

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

        else if Char.isSpace c2 then

          if Char.isSpace c3 then

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

          else

            strip_spaces_in_list skip_comments (c3 :: cs)

                (c1 :: accum |> forall is_evil [c1, c3] ? cons #" ")

        else

          strip_spaces_in_list skip_comments (c2 :: c3 :: cs) (cons c1 accum)

  in

    String.explode

    #> rpair [] #-> strip_spaces_in_list skip_comments

    #> rev #> String.implode

  end

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' =

  can (Sign.typ_unify (Proof_Context.theory_of ctxt)

                      (T, Logic.incr_tvar (maxidx_of_typ T + 1) T'))

      (Vartab.empty, 0)

val type_equiv = Sign.typ_equiv o Proof_Context.theory_of

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_equiv 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, ...}, _) :: _ =>

               if not sound then

                 (* 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. *)

                 (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)

               else

                 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 ((s, i), T) => fn t' =>

           HOLogic.all_const T

           $ Abs (s, T, abstract_over (Var ((s, 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;