src/HOL/Tools/Nitpick/nitpick_preproc.ML
author blanchet
Tue Dec 07 11:56:53 2010 +0100 (2010-12-07)
changeset 41052 3db267a01c1d
parent 41001 11715564e2ad
child 41791 01d722707a36
permissions -rw-r--r--
remove the "fin_fun" optimization in Nitpick -- it was always a hack and didn't help much
     1 (*  Title:      HOL/Tools/Nitpick/nitpick_preproc.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2008, 2009, 2010
     4 
     5 Nitpick's HOL preprocessor.
     6 *)
     7 
     8 signature NITPICK_PREPROC =
     9 sig
    10   type hol_context = Nitpick_HOL.hol_context
    11   val preprocess_formulas :
    12     hol_context -> term list -> term
    13     -> term list * term list * bool * bool * bool
    14 end;
    15 
    16 structure Nitpick_Preproc : NITPICK_PREPROC =
    17 struct
    18 
    19 open Nitpick_Util
    20 open Nitpick_HOL
    21 open Nitpick_Mono
    22 
    23 fun is_positive_existential polar quant_s =
    24   (polar = Pos andalso quant_s = @{const_name Ex}) orelse
    25   (polar = Neg andalso quant_s <> @{const_name Ex})
    26 
    27 val is_descr =
    28   member (op =) [@{const_name The}, @{const_name Eps}, @{const_name safe_The}]
    29 
    30 (** Binary coding of integers **)
    31 
    32 (* If a formula contains a numeral whose absolute value is more than this
    33    threshold, the unary coding is likely not to work well and we prefer the
    34    binary coding. *)
    35 val binary_int_threshold = 3
    36 
    37 val may_use_binary_ints =
    38   let
    39     fun aux def (Const (@{const_name "=="}, _) $ t1 $ t2) =
    40         aux def t1 andalso aux false t2
    41       | aux def (@{const "==>"} $ t1 $ t2) = aux false t1 andalso aux def t2
    42       | aux def (Const (@{const_name HOL.eq}, _) $ t1 $ t2) =
    43         aux def t1 andalso aux false t2
    44       | aux def (@{const HOL.implies} $ t1 $ t2) = aux false t1 andalso aux def t2
    45       | aux def (t1 $ t2) = aux def t1 andalso aux def t2
    46       | aux def (t as Const (s, _)) =
    47         (not def orelse t <> @{const Suc}) andalso
    48         not (member (op =) [@{const_name Abs_Frac}, @{const_name Rep_Frac},
    49                             @{const_name nat_gcd}, @{const_name nat_lcm},
    50                             @{const_name Frac}, @{const_name norm_frac}] s)
    51       | aux def (Abs (_, _, t')) = aux def t'
    52       | aux _ _ = true
    53   in aux end
    54 val should_use_binary_ints =
    55   let
    56     fun aux (t1 $ t2) = aux t1 orelse aux t2
    57       | aux (Const (s, T)) =
    58         ((s = @{const_name times} orelse s = @{const_name div}) andalso
    59          is_integer_type (body_type T)) orelse
    60         (String.isPrefix numeral_prefix s andalso
    61          let val n = the (Int.fromString (unprefix numeral_prefix s)) in
    62            n < ~ binary_int_threshold orelse n > binary_int_threshold
    63          end)
    64       | aux (Abs (_, _, t')) = aux t'
    65       | aux _ = false
    66   in aux end
    67 
    68 (** Uncurrying **)
    69 
    70 fun add_to_uncurry_table ctxt t =
    71   let
    72     val thy = ProofContext.theory_of ctxt
    73     fun aux (t1 $ t2) args table =
    74         let val table = aux t2 [] table in aux t1 (t2 :: args) table end
    75       | aux (Abs (_, _, t')) _ table = aux t' [] table
    76       | aux (t as Const (x as (s, _))) args table =
    77         if is_built_in_const thy [(NONE, true)] x orelse
    78            is_constr_like ctxt x orelse
    79            is_sel s orelse s = @{const_name Sigma} then
    80           table
    81         else
    82           Termtab.map_default (t, 65536) (Integer.min (length args)) table
    83       | aux _ _ table = table
    84   in aux t [] end
    85 
    86 fun uncurry_prefix_for k j =
    87   uncurry_prefix ^ string_of_int k ^ "@" ^ string_of_int j ^ name_sep
    88 
    89 fun uncurry_term table t =
    90   let
    91     fun aux (t1 $ t2) args = aux t1 (aux t2 [] :: args)
    92       | aux (Abs (s, T, t')) args = s_betapplys [] (Abs (s, T, aux t' []), args)
    93       | aux (t as Const (s, T)) args =
    94         (case Termtab.lookup table t of
    95            SOME n =>
    96            if n >= 2 then
    97              let
    98                val arg_Ts = strip_n_binders n T |> fst
    99                val j =
   100                  if is_iterator_type (hd arg_Ts) then
   101                    1
   102                  else case find_index (not_equal bool_T) arg_Ts of
   103                    ~1 => n
   104                  | j => j
   105                val ((before_args, tuple_args), after_args) =
   106                  args |> chop n |>> chop j
   107                val ((before_arg_Ts, tuple_arg_Ts), rest_T) =
   108                  T |> strip_n_binders n |>> chop j
   109                val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
   110              in
   111                if n - j < 2 then
   112                  s_betapplys [] (t, args)
   113                else
   114                  s_betapplys []
   115                      (Const (uncurry_prefix_for (n - j) j ^ s,
   116                              before_arg_Ts ---> tuple_T --> rest_T),
   117                       before_args @ [mk_flat_tuple tuple_T tuple_args] @
   118                       after_args)
   119              end
   120            else
   121              s_betapplys [] (t, args)
   122          | NONE => s_betapplys [] (t, args))
   123       | aux t args = s_betapplys [] (t, args)
   124   in aux t [] end
   125 
   126 (** Boxing **)
   127 
   128 fun box_fun_and_pair_in_term (hol_ctxt as {ctxt, thy, stds, ...}) def orig_t =
   129   let
   130     fun box_relational_operator_type (Type (@{type_name fun}, Ts)) =
   131         Type (@{type_name fun}, map box_relational_operator_type Ts)
   132       | box_relational_operator_type (Type (@{type_name prod}, Ts)) =
   133         Type (@{type_name prod}, map (box_type hol_ctxt InPair) Ts)
   134       | box_relational_operator_type T = T
   135     fun add_boxed_types_for_var (z as (_, T)) (T', t') =
   136       case t' of
   137         Var z' => z' = z ? insert (op =) T'
   138       | Const (@{const_name Pair}, _) $ t1 $ t2 =>
   139         (case T' of
   140            Type (_, [T1, T2]) =>
   141            fold (add_boxed_types_for_var z) [(T1, t1), (T2, t2)]
   142          | _ => raise TYPE ("Nitpick_Preproc.box_fun_and_pair_in_term.\
   143                             \add_boxed_types_for_var", [T'], []))
   144       | _ => exists_subterm (curry (op =) (Var z)) t' ? insert (op =) T
   145     fun box_var_in_def new_Ts old_Ts t (z as (_, T)) =
   146       case t of
   147         @{const Trueprop} $ t1 => box_var_in_def new_Ts old_Ts t1 z
   148       | Const (s0, _) $ t1 $ _ =>
   149         if s0 = @{const_name "=="} orelse s0 = @{const_name HOL.eq} then
   150           let
   151             val (t', args) = strip_comb t1
   152             val T' = fastype_of1 (new_Ts, do_term new_Ts old_Ts Neut t')
   153           in
   154             case fold (add_boxed_types_for_var z)
   155                       (fst (strip_n_binders (length args) T') ~~ args) [] of
   156               [T''] => T''
   157             | _ => T
   158           end
   159         else
   160           T
   161       | _ => T
   162     and do_quantifier new_Ts old_Ts polar quant_s quant_T abs_s abs_T t =
   163       let
   164         val abs_T' =
   165           if polar = Neut orelse is_positive_existential polar quant_s then
   166             box_type hol_ctxt InFunLHS abs_T
   167           else
   168             abs_T
   169         val body_T = body_type quant_T
   170       in
   171         Const (quant_s, (abs_T' --> body_T) --> body_T)
   172         $ Abs (abs_s, abs_T',
   173                t |> do_term (abs_T' :: new_Ts) (abs_T :: old_Ts) polar)
   174       end
   175     and do_equals new_Ts old_Ts s0 T0 t1 t2 =
   176       let
   177         val (t1, t2) = pairself (do_term new_Ts old_Ts Neut) (t1, t2)
   178         val (T1, T2) = pairself (curry fastype_of1 new_Ts) (t1, t2)
   179         val T = if def then T1
   180                 else [T1, T2] |> sort (int_ord o pairself size_of_typ) |> hd
   181       in
   182         list_comb (Const (s0, T --> T --> body_type T0),
   183                    map2 (coerce_term hol_ctxt new_Ts T) [T1, T2] [t1, t2])
   184       end
   185     and do_descr s T =
   186       let val T1 = box_type hol_ctxt InFunLHS (range_type T) in
   187         Const (s, (T1 --> bool_T) --> T1)
   188       end
   189     and do_term new_Ts old_Ts polar t =
   190       case t of
   191         Const (s0 as @{const_name all}, T0) $ Abs (s1, T1, t1) =>
   192         do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1
   193       | Const (s0 as @{const_name "=="}, T0) $ t1 $ t2 =>
   194         do_equals new_Ts old_Ts s0 T0 t1 t2
   195       | @{const "==>"} $ t1 $ t2 =>
   196         @{const "==>"} $ do_term new_Ts old_Ts (flip_polarity polar) t1
   197         $ do_term new_Ts old_Ts polar t2
   198       | @{const Pure.conjunction} $ t1 $ t2 =>
   199         @{const Pure.conjunction} $ do_term new_Ts old_Ts polar t1
   200         $ do_term new_Ts old_Ts polar t2
   201       | @{const Trueprop} $ t1 =>
   202         @{const Trueprop} $ do_term new_Ts old_Ts polar t1
   203       | @{const Not} $ t1 =>
   204         @{const Not} $ do_term new_Ts old_Ts (flip_polarity polar) t1
   205       | Const (s0 as @{const_name All}, T0) $ Abs (s1, T1, t1) =>
   206         do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1
   207       | Const (s0 as @{const_name Ex}, T0) $ Abs (s1, T1, t1) =>
   208         do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1
   209       | Const (s0 as @{const_name HOL.eq}, T0) $ t1 $ t2 =>
   210         do_equals new_Ts old_Ts s0 T0 t1 t2
   211       | @{const HOL.conj} $ t1 $ t2 =>
   212         @{const HOL.conj} $ do_term new_Ts old_Ts polar t1
   213         $ do_term new_Ts old_Ts polar t2
   214       | @{const HOL.disj} $ t1 $ t2 =>
   215         @{const HOL.disj} $ do_term new_Ts old_Ts polar t1
   216         $ do_term new_Ts old_Ts polar t2
   217       | @{const HOL.implies} $ t1 $ t2 =>
   218         @{const HOL.implies} $ do_term new_Ts old_Ts (flip_polarity polar) t1
   219         $ do_term new_Ts old_Ts polar t2
   220       | Const (x as (s, T)) =>
   221         if is_descr s then
   222           do_descr s T
   223         else
   224           Const (s, if s = @{const_name converse} orelse
   225                        s = @{const_name trancl} then
   226                       box_relational_operator_type T
   227                     else if String.isPrefix quot_normal_prefix s then
   228                       let val T' = box_type hol_ctxt InFunLHS (domain_type T) in
   229                         T' --> T'
   230                       end
   231                     else if is_built_in_const thy stds x orelse
   232                             s = @{const_name Sigma} then
   233                       T
   234                     else if is_constr_like ctxt x then
   235                       box_type hol_ctxt InConstr T
   236                     else if is_sel s orelse is_rep_fun ctxt x then
   237                       box_type hol_ctxt InSel T
   238                     else
   239                       box_type hol_ctxt InExpr T)
   240       | t1 $ Abs (s, T, t2') =>
   241         let
   242           val t1 = do_term new_Ts old_Ts Neut t1
   243           val T1 = fastype_of1 (new_Ts, t1)
   244           val (s1, Ts1) = dest_Type T1
   245           val T' = hd (snd (dest_Type (hd Ts1)))
   246           val t2 = Abs (s, T', do_term (T' :: new_Ts) (T :: old_Ts) Neut t2')
   247           val T2 = fastype_of1 (new_Ts, t2)
   248           val t2 = coerce_term hol_ctxt new_Ts (hd Ts1) T2 t2
   249         in
   250           s_betapply new_Ts (if s1 = @{type_name fun} then
   251                                t1
   252                              else
   253                                select_nth_constr_arg ctxt stds
   254                                    (@{const_name FunBox},
   255                                     Type (@{type_name fun}, Ts1) --> T1) t1 0
   256                                    (Type (@{type_name fun}, Ts1)), t2)
   257         end
   258       | t1 $ t2 =>
   259         let
   260           val t1 = do_term new_Ts old_Ts Neut t1
   261           val T1 = fastype_of1 (new_Ts, t1)
   262           val (s1, Ts1) = dest_Type T1
   263           val t2 = do_term new_Ts old_Ts Neut t2
   264           val T2 = fastype_of1 (new_Ts, t2)
   265           val t2 = coerce_term hol_ctxt new_Ts (hd Ts1) T2 t2
   266         in
   267           s_betapply new_Ts (if s1 = @{type_name fun} then
   268                                t1
   269                              else
   270                                select_nth_constr_arg ctxt stds
   271                                    (@{const_name FunBox},
   272                                     Type (@{type_name fun}, Ts1) --> T1) t1 0
   273                                    (Type (@{type_name fun}, Ts1)), t2)
   274         end
   275       | Free (s, T) => Free (s, box_type hol_ctxt InExpr T)
   276       | Var (z as (x, T)) =>
   277         Var (x, if def then box_var_in_def new_Ts old_Ts orig_t z
   278                 else box_type hol_ctxt InExpr T)
   279       | Bound _ => t
   280       | Abs (s, T, t') =>
   281         Abs (s, T, do_term (T :: new_Ts) (T :: old_Ts) Neut t')
   282   in do_term [] [] Pos orig_t end
   283 
   284 (** Destruction of constructors **)
   285 
   286 val val_var_prefix = nitpick_prefix ^ "v"
   287 
   288 fun fresh_value_var Ts k n j t =
   289   Var ((val_var_prefix ^ nat_subscript (n - j), k), fastype_of1 (Ts, t))
   290 
   291 fun has_heavy_bounds_or_vars Ts t =
   292   let
   293     fun aux [] = false
   294       | aux [T] = is_fun_type T orelse is_pair_type T
   295       | aux _ = true
   296   in aux (map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t)) end
   297 
   298 fun pull_out_constr_comb ({ctxt, stds, ...} : hol_context) Ts relax k level t
   299                          args seen =
   300   let val t_comb = list_comb (t, args) in
   301     case t of
   302       Const x =>
   303       if not relax andalso is_constr ctxt stds x andalso
   304          not (is_fun_type (fastype_of1 (Ts, t_comb))) andalso
   305          has_heavy_bounds_or_vars Ts t_comb andalso
   306          not (loose_bvar (t_comb, level)) then
   307         let
   308           val (j, seen) = case find_index (curry (op =) t_comb) seen of
   309                             ~1 => (0, t_comb :: seen)
   310                           | j => (j, seen)
   311         in (fresh_value_var Ts k (length seen) j t_comb, seen) end
   312       else
   313         (t_comb, seen)
   314     | _ => (t_comb, seen)
   315   end
   316 
   317 fun equations_for_pulled_out_constrs mk_eq Ts k seen =
   318   let val n = length seen in
   319     map2 (fn j => fn t => mk_eq (fresh_value_var Ts k n j t, t))
   320          (index_seq 0 n) seen
   321   end
   322 
   323 fun pull_out_universal_constrs hol_ctxt def t =
   324   let
   325     val k = maxidx_of_term t + 1
   326     fun do_term Ts def t args seen =
   327       case t of
   328         (t0 as Const (@{const_name "=="}, _)) $ t1 $ t2 =>
   329         do_eq_or_imp Ts true def t0 t1 t2 seen
   330       | (t0 as @{const "==>"}) $ t1 $ t2 =>
   331         if def then (t, []) else do_eq_or_imp Ts false def t0 t1 t2 seen
   332       | (t0 as Const (@{const_name HOL.eq}, _)) $ t1 $ t2 =>
   333         do_eq_or_imp Ts true def t0 t1 t2 seen
   334       | (t0 as @{const HOL.implies}) $ t1 $ t2 =>
   335         do_eq_or_imp Ts false def t0 t1 t2 seen
   336       | Abs (s, T, t') =>
   337         let val (t', seen) = do_term (T :: Ts) def t' [] seen in
   338           (list_comb (Abs (s, T, t'), args), seen)
   339         end
   340       | t1 $ t2 =>
   341         let val (t2, seen) = do_term Ts def t2 [] seen in
   342           do_term Ts def t1 (t2 :: args) seen
   343         end
   344       | _ => pull_out_constr_comb hol_ctxt Ts def k 0 t args seen
   345     and do_eq_or_imp Ts eq def t0 t1 t2 seen =
   346       let
   347         val (t2, seen) = if eq andalso def then (t2, seen)
   348                          else do_term Ts false t2 [] seen
   349         val (t1, seen) = do_term Ts false t1 [] seen
   350       in (t0 $ t1 $ t2, seen) end
   351     val (concl, seen) = do_term [] def t [] []
   352   in
   353     Logic.list_implies (equations_for_pulled_out_constrs Logic.mk_equals [] k
   354                                                          seen, concl)
   355   end
   356 
   357 fun mk_exists v t =
   358   HOLogic.exists_const (fastype_of v) $ lambda v (incr_boundvars 1 t)
   359 
   360 fun pull_out_existential_constrs hol_ctxt t =
   361   let
   362     val k = maxidx_of_term t + 1
   363     fun aux Ts num_exists t args seen =
   364       case t of
   365         (t0 as Const (@{const_name Ex}, _)) $ Abs (s1, T1, t1) =>
   366         let
   367           val (t1, seen') = aux (T1 :: Ts) (num_exists + 1) t1 [] []
   368           val n = length seen'
   369           fun vars () = map2 (fresh_value_var Ts k n) (index_seq 0 n) seen'
   370         in
   371           (equations_for_pulled_out_constrs HOLogic.mk_eq Ts k seen'
   372            |> List.foldl s_conj t1 |> fold mk_exists (vars ())
   373            |> curry3 Abs s1 T1 |> curry (op $) t0, seen)
   374         end
   375       | t1 $ t2 =>
   376         let val (t2, seen) = aux Ts num_exists t2 [] seen in
   377           aux Ts num_exists t1 (t2 :: args) seen
   378         end
   379       | Abs (s, T, t') =>
   380         let
   381           val (t', seen) = aux (T :: Ts) 0 t' [] (map (incr_boundvars 1) seen)
   382         in (list_comb (Abs (s, T, t'), args), map (incr_boundvars ~1) seen) end
   383       | _ =>
   384         if num_exists > 0 then
   385           pull_out_constr_comb hol_ctxt Ts false k num_exists t args seen
   386         else
   387           (list_comb (t, args), seen)
   388   in aux [] 0 t [] [] |> fst end
   389 
   390 fun destroy_pulled_out_constrs (hol_ctxt as {ctxt, stds, ...}) axiom t =
   391   let
   392     val num_occs_of_var =
   393       fold_aterms (fn Var z => (fn f => fn z' => f z' |> z = z' ? Integer.add 1)
   394                     | _ => I) t (K 0)
   395     fun aux careful ((t0 as Const (@{const_name "=="}, _)) $ t1 $ t2) =
   396         aux_eq careful true t0 t1 t2
   397       | aux careful ((t0 as @{const "==>"}) $ t1 $ t2) =
   398         t0 $ aux false t1 $ aux careful t2
   399       | aux careful ((t0 as Const (@{const_name HOL.eq}, _)) $ t1 $ t2) =
   400         aux_eq careful true t0 t1 t2
   401       | aux careful ((t0 as @{const HOL.implies}) $ t1 $ t2) =
   402         t0 $ aux false t1 $ aux careful t2
   403       | aux careful (Abs (s, T, t')) = Abs (s, T, aux careful t')
   404       | aux careful (t1 $ t2) = aux careful t1 $ aux careful t2
   405       | aux _ t = t
   406     and aux_eq careful pass1 t0 t1 t2 =
   407       ((if careful then
   408           raise SAME ()
   409         else if axiom andalso is_Var t2 andalso
   410                 num_occs_of_var (dest_Var t2) = 1 then
   411           @{const True}
   412         else case strip_comb t2 of
   413           (* The first case is not as general as it could be. *)
   414           (Const (@{const_name PairBox}, _),
   415                   [Const (@{const_name fst}, _) $ Var z1,
   416                    Const (@{const_name snd}, _) $ Var z2]) =>
   417           if z1 = z2 andalso num_occs_of_var z1 = 2 then @{const True}
   418           else raise SAME ()
   419         | (Const (x as (s, T)), args) =>
   420           let
   421             val (arg_Ts, dataT) = strip_type T
   422             val n = length arg_Ts
   423           in
   424             if length args = n andalso
   425                (is_constr ctxt stds x orelse s = @{const_name Pair} orelse
   426                 x = (@{const_name Suc}, nat_T --> nat_T)) andalso
   427                (not careful orelse not (is_Var t1) orelse
   428                 String.isPrefix val_var_prefix (fst (fst (dest_Var t1)))) then
   429                 s_let "l" (n + 1) dataT bool_T
   430                       (fn t1 => discriminate_value hol_ctxt x t1 ::
   431                                 map3 (sel_eq x t1) (index_seq 0 n) arg_Ts args
   432                                 |> foldr1 s_conj) t1
   433             else
   434               raise SAME ()
   435           end
   436         | _ => raise SAME ())
   437        |> body_type (type_of t0) = prop_T ? HOLogic.mk_Trueprop)
   438       handle SAME () => if pass1 then aux_eq careful false t0 t2 t1
   439                         else t0 $ aux false t2 $ aux false t1
   440     and sel_eq x t n nth_T nth_t =
   441       HOLogic.eq_const nth_T $ nth_t
   442                              $ select_nth_constr_arg ctxt stds x t n nth_T
   443       |> aux false
   444   in aux axiom t end
   445 
   446 (** Destruction of universal and existential equalities **)
   447 
   448 fun curry_assms (@{const "==>"} $ (@{const Trueprop}
   449                                    $ (@{const HOL.conj} $ t1 $ t2)) $ t3) =
   450     curry_assms (Logic.list_implies ([t1, t2] |> map HOLogic.mk_Trueprop, t3))
   451   | curry_assms (@{const "==>"} $ t1 $ t2) =
   452     @{const "==>"} $ curry_assms t1 $ curry_assms t2
   453   | curry_assms t = t
   454 
   455 val destroy_universal_equalities =
   456   let
   457     fun aux prems zs t =
   458       case t of
   459         @{const "==>"} $ t1 $ t2 => aux_implies prems zs t1 t2
   460       | _ => Logic.list_implies (rev prems, t)
   461     and aux_implies prems zs t1 t2 =
   462       case t1 of
   463         Const (@{const_name "=="}, _) $ Var z $ t' => aux_eq prems zs z t' t1 t2
   464       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ Var z $ t') =>
   465         aux_eq prems zs z t' t1 t2
   466       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t' $ Var z) =>
   467         aux_eq prems zs z t' t1 t2
   468       | _ => aux (t1 :: prems) (Term.add_vars t1 zs) t2
   469     and aux_eq prems zs z t' t1 t2 =
   470       if not (member (op =) zs z) andalso
   471          not (exists_subterm (curry (op =) (Var z)) t') then
   472         aux prems zs (subst_free [(Var z, t')] t2)
   473       else
   474         aux (t1 :: prems) (Term.add_vars t1 zs) t2
   475   in aux [] [] end
   476 
   477 fun find_bound_assign ctxt stds j =
   478   let
   479     fun do_term _ [] = NONE
   480       | do_term seen (t :: ts) =
   481         let
   482           fun do_eq pass1 t1 t2 =
   483             (if loose_bvar1 (t2, j) then
   484                if pass1 then do_eq false t2 t1 else raise SAME ()
   485              else case t1 of
   486                Bound j' => if j' = j then SOME (t2, ts @ seen) else raise SAME ()
   487              | Const (s, Type (@{type_name fun}, [T1, T2])) $ Bound j' =>
   488                if j' = j andalso
   489                   s = nth_sel_name_for_constr_name @{const_name FunBox} 0 then
   490                  SOME (construct_value ctxt stds
   491                                        (@{const_name FunBox}, T2 --> T1) [t2],
   492                        ts @ seen)
   493                else
   494                  raise SAME ()
   495              | _ => raise SAME ())
   496             handle SAME () => do_term (t :: seen) ts
   497         in
   498           case t of
   499             Const (@{const_name HOL.eq}, _) $ t1 $ t2 => do_eq true t1 t2
   500           | _ => do_term (t :: seen) ts
   501         end
   502   in do_term end
   503 
   504 fun subst_one_bound j arg t =
   505   let
   506     fun aux (Bound i, lev) =
   507         if i < lev then raise SAME ()
   508         else if i = lev then incr_boundvars (lev - j) arg
   509         else Bound (i - 1)
   510       | aux (Abs (a, T, body), lev) = Abs (a, T, aux (body, lev + 1))
   511       | aux (f $ t, lev) =
   512         (aux (f, lev) $ (aux (t, lev) handle SAME () => t)
   513          handle SAME () => f $ aux (t, lev))
   514       | aux _ = raise SAME ()
   515   in aux (t, j) handle SAME () => t end
   516 
   517 fun destroy_existential_equalities ({ctxt, stds, ...} : hol_context) =
   518   let
   519     fun kill [] [] ts = foldr1 s_conj ts
   520       | kill (s :: ss) (T :: Ts) ts =
   521         (case find_bound_assign ctxt stds (length ss) [] ts of
   522            SOME (_, []) => @{const True}
   523          | SOME (arg_t, ts) =>
   524            kill ss Ts (map (subst_one_bound (length ss)
   525                                 (incr_bv (~1, length ss + 1, arg_t))) ts)
   526          | NONE =>
   527            Const (@{const_name Ex}, (T --> bool_T) --> bool_T)
   528            $ Abs (s, T, kill ss Ts ts))
   529       | kill _ _ _ = raise ListPair.UnequalLengths
   530     fun gather ss Ts (Const (@{const_name Ex}, _) $ Abs (s1, T1, t1)) =
   531         gather (ss @ [s1]) (Ts @ [T1]) t1
   532       | gather [] [] (Abs (s, T, t1)) = Abs (s, T, gather [] [] t1)
   533       | gather [] [] (t1 $ t2) = gather [] [] t1 $ gather [] [] t2
   534       | gather [] [] t = t
   535       | gather ss Ts t = kill ss Ts (conjuncts_of (gather [] [] t))
   536   in gather [] [] end
   537 
   538 (** Skolemization **)
   539 
   540 fun skolem_prefix_for k j =
   541   skolem_prefix ^ string_of_int k ^ "@" ^ string_of_int j ^ name_sep
   542 
   543 fun skolemize_term_and_more (hol_ctxt as {thy, def_table, skolems, ...})
   544                             skolem_depth =
   545   let
   546     val incrs = map (Integer.add 1)
   547     fun aux ss Ts js skolemizable polar t =
   548       let
   549         fun do_quantifier quant_s quant_T abs_s abs_T t =
   550           (if not (loose_bvar1 (t, 0)) then
   551              aux ss Ts js skolemizable polar (incr_boundvars ~1 t)
   552            else if is_positive_existential polar quant_s then
   553              let
   554                val j = length (!skolems) + 1
   555                val (js', (ss', Ts')) =
   556                  js ~~ (ss ~~ Ts)
   557                  |> filter (fn (j, _) => loose_bvar1 (t, j + 1))
   558                  |> ListPair.unzip ||> ListPair.unzip
   559              in
   560                if skolemizable andalso length js' <= skolem_depth then
   561                  let
   562                    val sko_s = skolem_prefix_for (length js') j ^ abs_s
   563                    val _ = Unsynchronized.change skolems (cons (sko_s, ss'))
   564                    val sko_t = list_comb (Const (sko_s, rev Ts' ---> abs_T),
   565                                           map Bound (rev js'))
   566                    val abs_t = Abs (abs_s, abs_T,
   567                                     aux ss Ts (incrs js) skolemizable polar t)
   568                  in
   569                    if null js' then
   570                      s_betapply Ts (abs_t, sko_t)
   571                    else
   572                      Const (@{const_name Let}, abs_T --> quant_T) $ sko_t
   573                      $ abs_t
   574                  end
   575                else
   576                  raise SAME ()
   577              end
   578            else
   579              raise SAME ())
   580           handle SAME () =>
   581                  Const (quant_s, quant_T)
   582                  $ Abs (abs_s, abs_T,
   583                         aux (abs_s :: ss) (abs_T :: Ts) (0 :: incrs js)
   584                             (skolemizable andalso
   585                              not (is_higher_order_type abs_T)) polar t)
   586       in
   587         case t of
   588           Const (s0 as @{const_name all}, T0) $ Abs (s1, T1, t1) =>
   589           do_quantifier s0 T0 s1 T1 t1
   590         | @{const "==>"} $ t1 $ t2 =>
   591           @{const "==>"} $ aux ss Ts js skolemizable (flip_polarity polar) t1
   592           $ aux ss Ts js skolemizable polar t2
   593         | @{const Pure.conjunction} $ t1 $ t2 =>
   594           @{const Pure.conjunction} $ aux ss Ts js skolemizable polar t1
   595           $ aux ss Ts js skolemizable polar t2
   596         | @{const Trueprop} $ t1 =>
   597           @{const Trueprop} $ aux ss Ts js skolemizable polar t1
   598         | @{const Not} $ t1 =>
   599           @{const Not} $ aux ss Ts js skolemizable (flip_polarity polar) t1
   600         | Const (s0 as @{const_name All}, T0) $ Abs (s1, T1, t1) =>
   601           do_quantifier s0 T0 s1 T1 t1
   602         | Const (s0 as @{const_name Ex}, T0) $ Abs (s1, T1, t1) =>
   603           do_quantifier s0 T0 s1 T1 t1
   604         | @{const HOL.conj} $ t1 $ t2 =>
   605           s_conj (pairself (aux ss Ts js skolemizable polar) (t1, t2))
   606         | @{const HOL.disj} $ t1 $ t2 =>
   607           s_disj (pairself (aux ss Ts js skolemizable polar) (t1, t2))
   608         | @{const HOL.implies} $ t1 $ t2 =>
   609           @{const HOL.implies} $ aux ss Ts js skolemizable (flip_polarity polar) t1
   610           $ aux ss Ts js skolemizable polar t2
   611         | (t0 as Const (@{const_name Let}, _)) $ t1 $ t2 =>
   612           t0 $ t1 $ aux ss Ts js skolemizable polar t2
   613         | Const (x as (s, T)) =>
   614           if is_real_inductive_pred hol_ctxt x andalso
   615              not (is_real_equational_fun hol_ctxt x) andalso
   616              not (is_well_founded_inductive_pred hol_ctxt x) then
   617             let
   618               val gfp = (fixpoint_kind_of_const thy def_table x = Gfp)
   619               val (pref, connective) =
   620                 if gfp then (lbfp_prefix, @{const HOL.disj})
   621                 else (ubfp_prefix, @{const HOL.conj})
   622               fun pos () = unrolled_inductive_pred_const hol_ctxt gfp x
   623                            |> aux ss Ts js skolemizable polar
   624               fun neg () = Const (pref ^ s, T)
   625             in
   626               case polar |> gfp ? flip_polarity of
   627                 Pos => pos ()
   628               | Neg => neg ()
   629               | Neut =>
   630                 let
   631                   val arg_Ts = binder_types T
   632                   fun app f =
   633                     list_comb (f (), map Bound (length arg_Ts - 1 downto 0))
   634                 in
   635                   List.foldr absdummy (connective $ app pos $ app neg) arg_Ts
   636                 end
   637             end
   638           else
   639             Const x
   640         | t1 $ t2 =>
   641           s_betapply Ts (aux ss Ts js false polar t1,
   642                          aux ss Ts js false Neut t2)
   643         | Abs (s, T, t1) =>
   644           Abs (s, T, aux ss Ts (incrs js) skolemizable polar t1)
   645         | _ => t
   646       end
   647   in aux [] [] [] true Pos end
   648 
   649 (** Function specialization **)
   650 
   651 fun params_in_equation (@{const "==>"} $ _ $ t2) = params_in_equation t2
   652   | params_in_equation (@{const Trueprop} $ t1) = params_in_equation t1
   653   | params_in_equation (Const (@{const_name HOL.eq}, _) $ t1 $ _) =
   654     snd (strip_comb t1)
   655   | params_in_equation _ = []
   656 
   657 fun specialize_fun_axiom x x' fixed_js fixed_args extra_args t =
   658   let
   659     val k = fold Integer.max (map maxidx_of_term (fixed_args @ extra_args)) 0
   660             + 1
   661     val t = map_aterms (fn Var ((s, i), T) => Var ((s, k + i), T) | t' => t') t
   662     val fixed_params = filter_indices fixed_js (params_in_equation t)
   663     fun aux args (Abs (s, T, t)) = list_comb (Abs (s, T, aux [] t), args)
   664       | aux args (t1 $ t2) = aux (aux [] t2 :: args) t1
   665       | aux args t =
   666         if t = Const x then
   667           list_comb (Const x', extra_args @ filter_out_indices fixed_js args)
   668         else
   669           let val j = find_index (curry (op =) t) fixed_params in
   670             list_comb (if j >= 0 then nth fixed_args j else t, args)
   671           end
   672   in aux [] t end
   673 
   674 fun static_args_in_term ({ersatz_table, ...} : hol_context) x t =
   675   let
   676     fun fun_calls (Abs (_, _, t)) _ = fun_calls t []
   677       | fun_calls (t1 $ t2) args = fun_calls t2 [] #> fun_calls t1 (t2 :: args)
   678       | fun_calls t args =
   679         (case t of
   680            Const (x' as (s', T')) =>
   681            x = x' orelse (case AList.lookup (op =) ersatz_table s' of
   682                             SOME s'' => x = (s'', T')
   683                           | NONE => false)
   684          | _ => false) ? cons args
   685     fun call_sets [] [] vs = [vs]
   686       | call_sets [] uss vs = vs :: call_sets uss [] []
   687       | call_sets ([] :: _) _ _ = []
   688       | call_sets ((t :: ts) :: tss) uss vs =
   689         Ord_List.insert Term_Ord.term_ord t vs |> call_sets tss (ts :: uss)
   690     val sets = call_sets (fun_calls t [] []) [] []
   691     val indexed_sets = sets ~~ (index_seq 0 (length sets))
   692   in
   693     fold_rev (fn (set, j) =>
   694                  case set of
   695                    [Var _] => AList.lookup (op =) indexed_sets set = SOME j
   696                               ? cons (j, NONE)
   697                  | [t as Const _] => cons (j, SOME t)
   698                  | [t as Free _] => cons (j, SOME t)
   699                  | _ => I) indexed_sets []
   700   end
   701 fun static_args_in_terms hol_ctxt x =
   702   map (static_args_in_term hol_ctxt x)
   703   #> fold1 (Ord_List.inter (prod_ord int_ord (option_ord Term_Ord.term_ord)))
   704 
   705 fun overlapping_indices [] _ = []
   706   | overlapping_indices _ [] = []
   707   | overlapping_indices (ps1 as (j1, t1) :: ps1') (ps2 as (j2, t2) :: ps2') =
   708     if j1 < j2 then overlapping_indices ps1' ps2
   709     else if j1 > j2 then overlapping_indices ps1 ps2'
   710     else overlapping_indices ps1' ps2' |> the_default t2 t1 = t2 ? cons j1
   711 
   712 fun is_eligible_arg Ts t =
   713   let val bad_Ts = map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t) in
   714     null bad_Ts orelse
   715     (is_higher_order_type (fastype_of1 (Ts, t)) andalso
   716      forall (not o is_higher_order_type) bad_Ts)
   717   end
   718 
   719 fun special_prefix_for j = special_prefix ^ string_of_int j ^ name_sep
   720 
   721 (* If a constant's definition is picked up deeper than this threshold, we
   722    prevent excessive specialization by not specializing it. *)
   723 val special_max_depth = 20
   724 
   725 val bound_var_prefix = "b"
   726 
   727 fun special_fun_aconv ((x1, js1, ts1), (x2, js2, ts2)) =
   728   x1 = x2 andalso js1 = js2 andalso length ts1 = length ts2 andalso
   729   forall (op aconv) (ts1 ~~ ts2)
   730 
   731 fun specialize_consts_in_term
   732         (hol_ctxt as {ctxt, thy, stds, specialize, def_table, simp_table,
   733                       special_funs, ...}) def depth t =
   734   if not specialize orelse depth > special_max_depth then
   735     t
   736   else
   737     let
   738       val blacklist =
   739         if def then case term_under_def t of Const x => [x] | _ => [] else []
   740       fun aux args Ts (Const (x as (s, T))) =
   741           ((if not (member (op =) blacklist x) andalso not (null args) andalso
   742                not (String.isPrefix special_prefix s) andalso
   743                not (is_built_in_const thy stds x) andalso
   744                (is_equational_fun_but_no_plain_def hol_ctxt x orelse
   745                 (is_some (def_of_const thy def_table x) andalso
   746                  not (is_of_class_const thy x) andalso
   747                  not (is_constr ctxt stds x) andalso
   748                  not (is_choice_spec_fun hol_ctxt x))) then
   749               let
   750                 val eligible_args = filter (is_eligible_arg Ts o snd)
   751                                            (index_seq 0 (length args) ~~ args)
   752                 val _ = not (null eligible_args) orelse raise SAME ()
   753                 val old_axs = equational_fun_axioms hol_ctxt x
   754                               |> map (destroy_existential_equalities hol_ctxt)
   755                 val static_params = static_args_in_terms hol_ctxt x old_axs
   756                 val fixed_js = overlapping_indices static_params eligible_args
   757                 val _ = not (null fixed_js) orelse raise SAME ()
   758                 val fixed_args = filter_indices fixed_js args
   759                 val vars = fold Term.add_vars fixed_args []
   760                            |> sort (Term_Ord.fast_indexname_ord o pairself fst)
   761                 val bound_js = fold (fn t => fn js => add_loose_bnos (t, 0, js))
   762                                     fixed_args []
   763                                |> sort int_ord
   764                 val live_args = filter_out_indices fixed_js args
   765                 val extra_args = map Var vars @ map Bound bound_js @ live_args
   766                 val extra_Ts = map snd vars @ filter_indices bound_js Ts
   767                 val k = maxidx_of_term t + 1
   768                 fun var_for_bound_no j =
   769                   Var ((bound_var_prefix ^
   770                         nat_subscript (find_index (curry (op =) j) bound_js
   771                                        + 1), k),
   772                        nth Ts j)
   773                 val fixed_args_in_axiom =
   774                   map (curry subst_bounds
   775                              (map var_for_bound_no (index_seq 0 (length Ts))))
   776                       fixed_args
   777               in
   778                 case AList.lookup special_fun_aconv (!special_funs)
   779                                   (x, fixed_js, fixed_args_in_axiom) of
   780                   SOME x' => list_comb (Const x', extra_args)
   781                 | NONE =>
   782                   let
   783                     val extra_args_in_axiom =
   784                       map Var vars @ map var_for_bound_no bound_js
   785                     val x' as (s', _) =
   786                       (special_prefix_for (length (!special_funs) + 1) ^ s,
   787                        extra_Ts @ filter_out_indices fixed_js (binder_types T)
   788                        ---> body_type T)
   789                     val new_axs =
   790                       map (specialize_fun_axiom x x' fixed_js
   791                                fixed_args_in_axiom extra_args_in_axiom) old_axs
   792                     val _ =
   793                       Unsynchronized.change special_funs
   794                           (cons ((x, fixed_js, fixed_args_in_axiom), x'))
   795                     val _ = add_simps simp_table s' new_axs
   796                   in list_comb (Const x', extra_args) end
   797               end
   798             else
   799               raise SAME ())
   800            handle SAME () => list_comb (Const x, args))
   801         | aux args Ts (Abs (s, T, t)) =
   802           list_comb (Abs (s, T, aux [] (T :: Ts) t), args)
   803         | aux args Ts (t1 $ t2) = aux (aux [] Ts t2 :: args) Ts t1
   804         | aux args _ t = list_comb (t, args)
   805     in aux [] [] t end
   806 
   807 type special_triple = int list * term list * styp
   808 
   809 val cong_var_prefix = "c"
   810 
   811 fun special_congruence_axiom T (js1, ts1, x1) (js2, ts2, x2) =
   812   let
   813     val (bounds1, bounds2) = pairself (map Var o special_bounds) (ts1, ts2)
   814     val Ts = binder_types T
   815     val max_j = fold (fold Integer.max) [js1, js2] ~1
   816     val (eqs, (args1, args2)) =
   817       fold (fn j => case pairself (fn ps => AList.lookup (op =) ps j)
   818                                   (js1 ~~ ts1, js2 ~~ ts2) of
   819                       (SOME t1, SOME t2) => apfst (cons (t1, t2))
   820                     | (SOME t1, NONE) => apsnd (apsnd (cons t1))
   821                     | (NONE, SOME t2) => apsnd (apfst (cons t2))
   822                     | (NONE, NONE) =>
   823                       let val v = Var ((cong_var_prefix ^ nat_subscript j, 0),
   824                                        nth Ts j) in
   825                         apsnd (pairself (cons v))
   826                       end) (max_j downto 0) ([], ([], []))
   827   in
   828     Logic.list_implies (eqs |> filter_out (op aconv) |> distinct (op =)
   829                             |> map Logic.mk_equals,
   830                         Logic.mk_equals (list_comb (Const x1, bounds1 @ args1),
   831                                          list_comb (Const x2, bounds2 @ args2)))
   832   end
   833 
   834 fun special_congruence_axioms (hol_ctxt as {special_funs, ...}) ts =
   835   let
   836     val groups =
   837       !special_funs
   838       |> map (fn ((x, js, ts), x') => (x, (js, ts, x')))
   839       |> AList.group (op =)
   840       |> filter_out (is_equational_fun_surely_complete hol_ctxt o fst)
   841       |> map (fn (x, zs) =>
   842                  (x, zs |> member (op =) ts (Const x) ? cons ([], [], x)))
   843     fun generality (js, _, _) = ~(length js)
   844     fun is_more_specific (j1, t1, x1) (j2, t2, x2) =
   845       x1 <> x2 andalso length j2 < length j1 andalso
   846       Ord_List.subset (prod_ord int_ord Term_Ord.term_ord) (j2 ~~ t2, j1 ~~ t1)
   847     fun do_pass_1 _ [] [_] [_] = I
   848       | do_pass_1 T skipped _ [] = do_pass_2 T skipped
   849       | do_pass_1 T skipped all (z :: zs) =
   850         case filter (is_more_specific z) all
   851              |> sort (int_ord o pairself generality) of
   852           [] => do_pass_1 T (z :: skipped) all zs
   853         | (z' :: _) => cons (special_congruence_axiom T z z')
   854                        #> do_pass_1 T skipped all zs
   855     and do_pass_2 _ [] = I
   856       | do_pass_2 T (z :: zs) =
   857         fold (cons o special_congruence_axiom T z) zs #> do_pass_2 T zs
   858   in fold (fn ((_, T), zs) => do_pass_1 T [] zs zs) groups [] end
   859 
   860 (** Axiom selection **)
   861 
   862 fun defined_free_by_assumption t =
   863   let
   864     fun do_equals x def =
   865       if exists_subterm (curry (op aconv) (Free x)) def then NONE else SOME x
   866   in
   867     case t of
   868       Const (@{const_name "=="}, _) $ Free x $ def => do_equals x def
   869     | @{const Trueprop} $ (Const (@{const_name "=="}, _) $ Free x $ def) =>
   870       do_equals x def
   871     | _ => NONE
   872   end
   873 
   874 fun assumption_exclusively_defines_free assm_ts t =
   875   case defined_free_by_assumption t of
   876     SOME x =>
   877     length (filter ((fn SOME x' => x = x' | NONE => false)
   878                      o defined_free_by_assumption) assm_ts) = 1
   879   | NONE => false
   880 
   881 fun all_table_entries table = Symtab.fold (append o snd) table []
   882 fun extra_table table s = Symtab.make [(s, all_table_entries table)]
   883 
   884 fun eval_axiom_for_term j t =
   885   Logic.mk_equals (Const (eval_prefix ^ string_of_int j, fastype_of t), t)
   886 
   887 val is_trivial_equation = the_default false o try (op aconv o Logic.dest_equals)
   888 
   889 (* Prevents divergence in case of cyclic or infinite axiom dependencies. *)
   890 val axioms_max_depth = 255
   891 
   892 fun axioms_for_term
   893         (hol_ctxt as {thy, ctxt, max_bisim_depth, stds, user_axioms,
   894                       evals, def_table, nondef_table, choice_spec_table,
   895                       user_nondefs, ...}) assm_ts neg_t =
   896   let
   897     val (def_assm_ts, nondef_assm_ts) =
   898       List.partition (assumption_exclusively_defines_free assm_ts) assm_ts
   899     val def_assm_table = map (`(the o defined_free_by_assumption)) def_assm_ts
   900     type accumulator = styp list * (term list * term list)
   901     fun add_axiom get app def depth t (accum as (seen, axs)) =
   902       let
   903         val t = t |> unfold_defs_in_term hol_ctxt
   904                   |> skolemize_term_and_more hol_ctxt ~1 (* FIXME: why ~1? *)
   905       in
   906         if is_trivial_equation t then
   907           accum
   908         else
   909           let val t' = t |> specialize_consts_in_term hol_ctxt def depth in
   910             if exists (member (op aconv) (get axs)) [t, t'] then accum
   911             else add_axioms_for_term (depth + 1) t' (seen, app (cons t') axs)
   912           end
   913       end
   914     and add_def_axiom depth = add_axiom fst apfst true depth
   915     and add_nondef_axiom depth = add_axiom snd apsnd false depth
   916     and add_maybe_def_axiom depth t =
   917       (if head_of t <> @{const "==>"} then add_def_axiom
   918        else add_nondef_axiom) depth t
   919     and add_eq_axiom depth t =
   920       (if is_constr_pattern_formula ctxt t then add_def_axiom
   921        else add_nondef_axiom) depth t
   922     and add_axioms_for_term depth t (accum as (seen, axs)) =
   923       case t of
   924         t1 $ t2 => accum |> fold (add_axioms_for_term depth) [t1, t2]
   925       | Const (x as (s, T)) =>
   926         (if member (op aconv) seen t orelse is_built_in_const thy stds x then
   927            accum
   928          else
   929            let val accum = (t :: seen, axs) in
   930              if depth > axioms_max_depth then
   931                raise TOO_LARGE ("Nitpick_Preproc.axioms_for_term.\
   932                                 \add_axioms_for_term",
   933                                 "too many nested axioms (" ^
   934                                 string_of_int depth ^ ")")
   935              else if is_of_class_const thy x then
   936                let
   937                  val class = Logic.class_of_const s
   938                  val of_class = Logic.mk_of_class (TVar (("'a", 0), [class]),
   939                                                    class)
   940                  val ax1 = try (specialize_type thy x) of_class
   941                  val ax2 = Option.map (specialize_type thy x o snd)
   942                                       (get_class_def thy class)
   943                in
   944                  fold (add_maybe_def_axiom depth) (map_filter I [ax1, ax2])
   945                       accum
   946                end
   947              else if is_constr ctxt stds x then
   948                accum
   949              else if is_descr (original_name s) then
   950                fold (add_nondef_axiom depth) (equational_fun_axioms hol_ctxt x)
   951                     accum
   952              else if is_equational_fun_but_no_plain_def hol_ctxt x then
   953                fold (add_eq_axiom depth) (equational_fun_axioms hol_ctxt x)
   954                     accum
   955              else if is_choice_spec_fun hol_ctxt x then
   956                fold (add_nondef_axiom depth)
   957                     (nondef_props_for_const thy true choice_spec_table x) accum
   958              else if is_abs_fun ctxt x then
   959                accum |> fold (add_nondef_axiom depth)
   960                              (nondef_props_for_const thy false nondef_table x)
   961                      |> (is_funky_typedef ctxt (range_type T) orelse
   962                          range_type T = nat_T)
   963                         ? fold (add_maybe_def_axiom depth)
   964                                (nondef_props_for_const thy true
   965                                                     (extra_table def_table s) x)
   966              else if is_rep_fun ctxt x then
   967                accum |> fold (add_nondef_axiom depth)
   968                              (nondef_props_for_const thy false nondef_table x)
   969                      |> (is_funky_typedef ctxt (range_type T) orelse
   970                          range_type T = nat_T)
   971                         ? fold (add_maybe_def_axiom depth)
   972                                (nondef_props_for_const thy true
   973                                                     (extra_table def_table s) x)
   974                      |> add_axioms_for_term depth
   975                                             (Const (mate_of_rep_fun ctxt x))
   976                      |> fold (add_def_axiom depth)
   977                              (inverse_axioms_for_rep_fun ctxt x)
   978              else if s = @{const_name TYPE} then
   979                accum
   980              else case def_of_const thy def_table x of
   981                SOME _ =>
   982                fold (add_eq_axiom depth) (equational_fun_axioms hol_ctxt x)
   983                     accum
   984              | NONE =>
   985                accum |> user_axioms <> SOME false
   986                         ? fold (add_nondef_axiom depth)
   987                                (nondef_props_for_const thy false nondef_table x)
   988            end)
   989         |> add_axioms_for_type depth T
   990       | Free (x as (_, T)) =>
   991         (if member (op aconv) seen t then
   992            accum
   993          else case AList.lookup (op =) def_assm_table x of
   994            SOME t => add_def_axiom depth t accum
   995          | NONE => accum)
   996         |> add_axioms_for_type depth T
   997       | Var (_, T) => add_axioms_for_type depth T accum
   998       | Bound _ => accum
   999       | Abs (_, T, t) => accum |> add_axioms_for_term depth t
  1000                                |> add_axioms_for_type depth T
  1001     and add_axioms_for_type depth T =
  1002       case T of
  1003         Type (@{type_name fun}, Ts) => fold (add_axioms_for_type depth) Ts
  1004       | Type (@{type_name prod}, Ts) => fold (add_axioms_for_type depth) Ts
  1005       | @{typ prop} => I
  1006       | @{typ bool} => I
  1007       | TFree (_, S) => add_axioms_for_sort depth T S
  1008       | TVar (_, S) => add_axioms_for_sort depth T S
  1009       | Type (z as (_, Ts)) =>
  1010         fold (add_axioms_for_type depth) Ts
  1011         #> (if is_pure_typedef ctxt T then
  1012               fold (add_maybe_def_axiom depth) (optimized_typedef_axioms ctxt z)
  1013             else if is_quot_type ctxt T then
  1014               fold (add_def_axiom depth)
  1015                    (optimized_quot_type_axioms ctxt stds z)
  1016             else if max_bisim_depth >= 0 andalso is_codatatype ctxt T then
  1017               fold (add_maybe_def_axiom depth)
  1018                    (codatatype_bisim_axioms hol_ctxt T)
  1019             else
  1020               I)
  1021     and add_axioms_for_sort depth T S =
  1022       let
  1023         val supers = Sign.complete_sort thy S
  1024         val class_axioms =
  1025           maps (fn class => map prop_of (AxClass.get_info thy class |> #axioms
  1026                                          handle ERROR _ => [])) supers
  1027         val monomorphic_class_axioms =
  1028           map (fn t => case Term.add_tvars t [] of
  1029                          [] => t
  1030                        | [(x, S)] =>
  1031                          monomorphic_term (Vartab.make [(x, (S, T))]) t
  1032                        | _ => raise TERM ("Nitpick_Preproc.axioms_for_term.\
  1033                                           \add_axioms_for_sort", [t]))
  1034               class_axioms
  1035       in fold (add_nondef_axiom depth) monomorphic_class_axioms end
  1036     val (mono_user_nondefs, poly_user_nondefs) =
  1037       List.partition (null o Term.hidden_polymorphism) user_nondefs
  1038     val eval_axioms = map2 eval_axiom_for_term (index_seq 0 (length evals))
  1039                            evals
  1040     val (seen, (defs, nondefs)) =
  1041       ([], ([], []))
  1042       |> add_axioms_for_term 1 neg_t
  1043       |> fold_rev (add_nondef_axiom 1) nondef_assm_ts
  1044       |> fold_rev (add_def_axiom 1) eval_axioms
  1045       |> user_axioms = SOME true ? fold (add_nondef_axiom 1) mono_user_nondefs
  1046     val defs = defs @ special_congruence_axioms hol_ctxt seen
  1047     val got_all_mono_user_axioms =
  1048       (user_axioms = SOME true orelse null mono_user_nondefs)
  1049   in
  1050     (neg_t :: nondefs, defs, got_all_mono_user_axioms, null poly_user_nondefs)
  1051   end
  1052 
  1053 (** Simplification of constructor/selector terms **)
  1054 
  1055 fun simplify_constrs_and_sels ctxt t =
  1056   let
  1057     fun is_nth_sel_on t' n (Const (s, _) $ t) =
  1058         (t = t' andalso is_sel_like_and_no_discr s andalso
  1059          sel_no_from_name s = n)
  1060       | is_nth_sel_on _ _ _ = false
  1061     fun do_term (Const (@{const_name Rep_Frac}, _)
  1062                  $ (Const (@{const_name Abs_Frac}, _) $ t1)) [] = do_term t1 []
  1063       | do_term (Const (@{const_name Abs_Frac}, _)
  1064                  $ (Const (@{const_name Rep_Frac}, _) $ t1)) [] = do_term t1 []
  1065       | do_term (t1 $ t2) args = do_term t1 (do_term t2 [] :: args)
  1066       | do_term (t as Const (x as (s, T))) (args as _ :: _) =
  1067         ((if is_constr_like ctxt x then
  1068             if length args = num_binder_types T then
  1069               case hd args of
  1070                 Const (_, T') $ t' =>
  1071                 if domain_type T' = body_type T andalso
  1072                    forall (uncurry (is_nth_sel_on t'))
  1073                           (index_seq 0 (length args) ~~ args) then
  1074                   t'
  1075                 else
  1076                   raise SAME ()
  1077               | _ => raise SAME ()
  1078             else
  1079               raise SAME ()
  1080           else if is_sel_like_and_no_discr s then
  1081             case strip_comb (hd args) of
  1082               (Const (x' as (s', T')), ts') =>
  1083               if is_constr_like ctxt x' andalso
  1084                  constr_name_for_sel_like s = s' andalso
  1085                  not (exists is_pair_type (binder_types T')) then
  1086                 list_comb (nth ts' (sel_no_from_name s), tl args)
  1087               else
  1088                 raise SAME ()
  1089             | _ => raise SAME ()
  1090           else
  1091             raise SAME ())
  1092          handle SAME () => s_betapplys [] (t, args))
  1093       | do_term (Abs (s, T, t')) args =
  1094         s_betapplys [] (Abs (s, T, do_term t' []), args)
  1095       | do_term t args = s_betapplys [] (t, args)
  1096   in do_term t [] end
  1097 
  1098 (** Quantifier massaging: Distributing quantifiers **)
  1099 
  1100 fun distribute_quantifiers t =
  1101   case t of
  1102     (t0 as Const (@{const_name All}, T0)) $ Abs (s, T1, t1) =>
  1103     (case t1 of
  1104        (t10 as @{const HOL.conj}) $ t11 $ t12 =>
  1105        t10 $ distribute_quantifiers (t0 $ Abs (s, T1, t11))
  1106            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))
  1107      | (t10 as @{const Not}) $ t11 =>
  1108        t10 $ distribute_quantifiers (Const (@{const_name Ex}, T0)
  1109                                      $ Abs (s, T1, t11))
  1110      | t1 =>
  1111        if not (loose_bvar1 (t1, 0)) then
  1112          distribute_quantifiers (incr_boundvars ~1 t1)
  1113        else
  1114          t0 $ Abs (s, T1, distribute_quantifiers t1))
  1115   | (t0 as Const (@{const_name Ex}, T0)) $ Abs (s, T1, t1) =>
  1116     (case distribute_quantifiers t1 of
  1117        (t10 as @{const HOL.disj}) $ t11 $ t12 =>
  1118        t10 $ distribute_quantifiers (t0 $ Abs (s, T1, t11))
  1119            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))
  1120      | (t10 as @{const HOL.implies}) $ t11 $ t12 =>
  1121        t10 $ distribute_quantifiers (Const (@{const_name All}, T0)
  1122                                      $ Abs (s, T1, t11))
  1123            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))
  1124      | (t10 as @{const Not}) $ t11 =>
  1125        t10 $ distribute_quantifiers (Const (@{const_name All}, T0)
  1126                                      $ Abs (s, T1, t11))
  1127      | t1 =>
  1128        if not (loose_bvar1 (t1, 0)) then
  1129          distribute_quantifiers (incr_boundvars ~1 t1)
  1130        else
  1131          t0 $ Abs (s, T1, distribute_quantifiers t1))
  1132   | t1 $ t2 => distribute_quantifiers t1 $ distribute_quantifiers t2
  1133   | Abs (s, T, t') => Abs (s, T, distribute_quantifiers t')
  1134   | _ => t
  1135 
  1136 (** Quantifier massaging: Pushing quantifiers inward **)
  1137 
  1138 fun renumber_bounds j n f t =
  1139   case t of
  1140     t1 $ t2 => renumber_bounds j n f t1 $ renumber_bounds j n f t2
  1141   | Abs (s, T, t') => Abs (s, T, renumber_bounds (j + 1) n f t')
  1142   | Bound j' =>
  1143     Bound (if j' >= j andalso j' < j + n then f (j' - j) + j else j')
  1144   | _ => t
  1145 
  1146 (* Maximum number of quantifiers in a cluster for which the exponential
  1147    algorithm is used. Larger clusters use a heuristic inspired by Claessen &
  1148    Soerensson's polynomial binary splitting procedure (p. 5 of their MODEL 2003
  1149    paper). *)
  1150 val quantifier_cluster_threshold = 7
  1151 
  1152 val push_quantifiers_inward =
  1153   let
  1154     fun aux quant_s ss Ts t =
  1155       (case t of
  1156          Const (s0, _) $ Abs (s1, T1, t1 as _ $ _) =>
  1157          if s0 = quant_s then
  1158            aux s0 (s1 :: ss) (T1 :: Ts) t1
  1159          else if quant_s = "" andalso
  1160                  (s0 = @{const_name All} orelse s0 = @{const_name Ex}) then
  1161            aux s0 [s1] [T1] t1
  1162          else
  1163            raise SAME ()
  1164        | _ => raise SAME ())
  1165       handle SAME () =>
  1166              case t of
  1167                t1 $ t2 =>
  1168                if quant_s = "" then
  1169                  aux "" [] [] t1 $ aux "" [] [] t2
  1170                else
  1171                  let
  1172                    val typical_card = 4
  1173                    fun big_union proj ps =
  1174                      fold (fold (insert (op =)) o proj) ps []
  1175                    val (ts, connective) = strip_any_connective t
  1176                    val T_costs =
  1177                      map (bounded_card_of_type 65536 typical_card []) Ts
  1178                    val t_costs = map size_of_term ts
  1179                    val num_Ts = length Ts
  1180                    val flip = curry (op -) (num_Ts - 1)
  1181                    val t_boundss = map (map flip o loose_bnos) ts
  1182                    fun merge costly_boundss [] = costly_boundss
  1183                      | merge costly_boundss (j :: js) =
  1184                        let
  1185                          val (yeas, nays) =
  1186                            List.partition (fn (bounds, _) =>
  1187                                               member (op =) bounds j)
  1188                                           costly_boundss
  1189                          val yeas_bounds = big_union fst yeas
  1190                          val yeas_cost = Integer.sum (map snd yeas)
  1191                                          * nth T_costs j
  1192                        in merge ((yeas_bounds, yeas_cost) :: nays) js end
  1193                    val cost = Integer.sum o map snd oo merge
  1194                    fun heuristically_best_permutation _ [] = []
  1195                      | heuristically_best_permutation costly_boundss js =
  1196                        let
  1197                          val (costly_boundss, (j, js)) =
  1198                            js |> map (`(merge costly_boundss o single))
  1199                               |> sort (int_ord
  1200                                        o pairself (Integer.sum o map snd o fst))
  1201                               |> split_list |>> hd ||> pairf hd tl
  1202                        in
  1203                          j :: heuristically_best_permutation costly_boundss js
  1204                        end
  1205                    val js =
  1206                      if length Ts <= quantifier_cluster_threshold then
  1207                        all_permutations (index_seq 0 num_Ts)
  1208                        |> map (`(cost (t_boundss ~~ t_costs)))
  1209                        |> sort (int_ord o pairself fst) |> hd |> snd
  1210                      else
  1211                        heuristically_best_permutation (t_boundss ~~ t_costs)
  1212                                                       (index_seq 0 num_Ts)
  1213                    val back_js = map (fn j => find_index (curry (op =) j) js)
  1214                                      (index_seq 0 num_Ts)
  1215                    val ts = map (renumber_bounds 0 num_Ts (nth back_js o flip))
  1216                                 ts
  1217                    fun mk_connection [] =
  1218                        raise ARG ("Nitpick_Preproc.push_quantifiers_inward.aux.\
  1219                                   \mk_connection", "")
  1220                      | mk_connection ts_cum_bounds =
  1221                        ts_cum_bounds |> map fst
  1222                        |> foldr1 (fn (t1, t2) => connective $ t1 $ t2)
  1223                    fun build ts_cum_bounds [] = ts_cum_bounds |> mk_connection
  1224                      | build ts_cum_bounds (j :: js) =
  1225                        let
  1226                          val (yeas, nays) =
  1227                            List.partition (fn (_, bounds) =>
  1228                                               member (op =) bounds j)
  1229                                           ts_cum_bounds
  1230                            ||> map (apfst (incr_boundvars ~1))
  1231                        in
  1232                          if null yeas then
  1233                            build nays js
  1234                          else
  1235                            let val T = nth Ts (flip j) in
  1236                              build ((Const (quant_s, (T --> bool_T) --> bool_T)
  1237                                      $ Abs (nth ss (flip j), T,
  1238                                             mk_connection yeas),
  1239                                       big_union snd yeas) :: nays) js
  1240                            end
  1241                        end
  1242                  in build (ts ~~ t_boundss) js end
  1243              | Abs (s, T, t') => Abs (s, T, aux "" [] [] t')
  1244              | _ => t
  1245   in aux "" [] [] end
  1246 
  1247 (** Preprocessor entry point **)
  1248 
  1249 val max_skolem_depth = 3
  1250 
  1251 fun preprocess_formulas
  1252         (hol_ctxt as {thy, ctxt, stds, binary_ints, destroy_constrs, boxes,
  1253                       ...}) assm_ts neg_t =
  1254   let
  1255     val (nondef_ts, def_ts, got_all_mono_user_axioms, no_poly_user_axioms) =
  1256       neg_t |> unfold_defs_in_term hol_ctxt
  1257             |> close_form
  1258             |> skolemize_term_and_more hol_ctxt max_skolem_depth
  1259             |> specialize_consts_in_term hol_ctxt false 0
  1260             |> axioms_for_term hol_ctxt assm_ts
  1261     val binarize =
  1262       is_standard_datatype thy stds nat_T andalso
  1263       case binary_ints of
  1264         SOME false => false
  1265       | _ => forall (may_use_binary_ints false) nondef_ts andalso
  1266              forall (may_use_binary_ints true) def_ts andalso
  1267              (binary_ints = SOME true orelse
  1268               exists should_use_binary_ints (nondef_ts @ def_ts))
  1269     val box = exists (not_equal (SOME false) o snd) boxes
  1270     val table =
  1271       Termtab.empty
  1272       |> box ? fold (add_to_uncurry_table ctxt) (nondef_ts @ def_ts)
  1273     fun do_rest def =
  1274       binarize ? binarize_nat_and_int_in_term
  1275       #> box ? uncurry_term table
  1276       #> box ? box_fun_and_pair_in_term hol_ctxt def
  1277       #> destroy_constrs ? (pull_out_universal_constrs hol_ctxt def
  1278                             #> pull_out_existential_constrs hol_ctxt
  1279                             #> destroy_pulled_out_constrs hol_ctxt def)
  1280       #> curry_assms
  1281       #> destroy_universal_equalities
  1282       #> destroy_existential_equalities hol_ctxt
  1283       #> simplify_constrs_and_sels ctxt
  1284       #> distribute_quantifiers
  1285       #> push_quantifiers_inward
  1286       #> close_form
  1287       #> Term.map_abs_vars shortest_name
  1288     val nondef_ts = map (do_rest false) nondef_ts
  1289     val def_ts = map (do_rest true) def_ts
  1290   in
  1291     (nondef_ts, def_ts, got_all_mono_user_axioms, no_poly_user_axioms, binarize)
  1292   end
  1293 
  1294 end;