src/HOL/Tools/Nitpick/nitpick_preproc.ML
author blanchet
Tue Apr 19 14:04:58 2011 +0200 (2011-04-19)
changeset 42415 10accf397ab6
parent 42361 23f352990944
child 44241 7943b69f0188
permissions -rw-r--r--
use "Spec_Rules" for finding axioms -- more reliable and cleaner
     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 * 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 = Proof_Context.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 strong 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 Ts careful ((t0 as Const (@{const_name "=="}, _)) $ t1 $ t2) =
   396         aux_eq Ts careful true t0 t1 t2
   397       | aux Ts careful ((t0 as @{const "==>"}) $ t1 $ t2) =
   398         t0 $ aux Ts false t1 $ aux Ts careful t2
   399       | aux Ts careful ((t0 as Const (@{const_name HOL.eq}, _)) $ t1 $ t2) =
   400         aux_eq Ts careful true t0 t1 t2
   401       | aux Ts careful ((t0 as @{const HOL.implies}) $ t1 $ t2) =
   402         t0 $ aux Ts false t1 $ aux Ts careful t2
   403       | aux Ts careful (Abs (s, T, t')) = Abs (s, T, aux (T :: Ts) careful t')
   404       | aux Ts careful (t1 $ t2) = aux Ts careful t1 $ aux Ts careful t2
   405       | aux _ _ t = t
   406     and aux_eq Ts careful pass1 t0 t1 t2 =
   407       ((if careful orelse
   408            not (strong orelse forall (is_constr_pattern ctxt) [t1, t2]) then
   409           raise SAME ()
   410         else if axiom andalso is_Var t2 andalso
   411                 num_occs_of_var (dest_Var t2) = 1 then
   412           @{const True}
   413         else case strip_comb t2 of
   414           (* The first case is not as general as it could be. *)
   415           (Const (@{const_name PairBox}, _),
   416                   [Const (@{const_name fst}, _) $ Var z1,
   417                    Const (@{const_name snd}, _) $ Var z2]) =>
   418           if z1 = z2 andalso num_occs_of_var z1 = 2 then @{const True}
   419           else raise SAME ()
   420         | (Const (x as (s, T)), args) =>
   421           let
   422             val (arg_Ts, dataT) = strip_type T
   423             val n = length arg_Ts
   424           in
   425             if length args = n andalso
   426                (is_constr ctxt stds x orelse s = @{const_name Pair} orelse
   427                 x = (@{const_name Suc}, nat_T --> nat_T)) andalso
   428                (not careful orelse not (is_Var t1) orelse
   429                 String.isPrefix val_var_prefix (fst (fst (dest_Var t1)))) then
   430                 s_let Ts "l" (n + 1) dataT bool_T
   431                       (fn t1 =>
   432                           discriminate_value hol_ctxt x t1 ::
   433                           map3 (sel_eq Ts x t1) (index_seq 0 n) arg_Ts args
   434                           |> foldr1 s_conj) t1
   435             else
   436               raise SAME ()
   437           end
   438         | _ => raise SAME ())
   439        |> body_type (type_of t0) = prop_T ? HOLogic.mk_Trueprop)
   440       handle SAME () => if pass1 then aux_eq Ts careful false t0 t2 t1
   441                         else t0 $ aux Ts false t2 $ aux Ts false t1
   442     and sel_eq Ts x t n nth_T nth_t =
   443       HOLogic.eq_const nth_T $ nth_t
   444                              $ select_nth_constr_arg ctxt stds x t n nth_T
   445       |> aux Ts false
   446   in aux [] axiom t end
   447 
   448 (** Destruction of universal and existential equalities **)
   449 
   450 fun curry_assms (@{const "==>"} $ (@{const Trueprop}
   451                                    $ (@{const HOL.conj} $ t1 $ t2)) $ t3) =
   452     curry_assms (Logic.list_implies ([t1, t2] |> map HOLogic.mk_Trueprop, t3))
   453   | curry_assms (@{const "==>"} $ t1 $ t2) =
   454     @{const "==>"} $ curry_assms t1 $ curry_assms t2
   455   | curry_assms t = t
   456 
   457 val destroy_universal_equalities =
   458   let
   459     fun aux prems zs t =
   460       case t of
   461         @{const "==>"} $ t1 $ t2 => aux_implies prems zs t1 t2
   462       | _ => Logic.list_implies (rev prems, t)
   463     and aux_implies prems zs t1 t2 =
   464       case t1 of
   465         Const (@{const_name "=="}, _) $ Var z $ t' => aux_eq prems zs z t' t1 t2
   466       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ Var z $ t') =>
   467         aux_eq prems zs z t' t1 t2
   468       | @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ t' $ Var z) =>
   469         aux_eq prems zs z t' t1 t2
   470       | _ => aux (t1 :: prems) (Term.add_vars t1 zs) t2
   471     and aux_eq prems zs z t' t1 t2 =
   472       if not (member (op =) zs z) andalso
   473          not (exists_subterm (curry (op =) (Var z)) t') then
   474         aux prems zs (subst_free [(Var z, t')] t2)
   475       else
   476         aux (t1 :: prems) (Term.add_vars t1 zs) t2
   477   in aux [] [] end
   478 
   479 fun find_bound_assign ctxt stds j =
   480   let
   481     fun do_term _ [] = NONE
   482       | do_term seen (t :: ts) =
   483         let
   484           fun do_eq pass1 t1 t2 =
   485             (if loose_bvar1 (t2, j) then
   486                if pass1 then do_eq false t2 t1 else raise SAME ()
   487              else case t1 of
   488                Bound j' => if j' = j then SOME (t2, ts @ seen) else raise SAME ()
   489              | Const (s, Type (@{type_name fun}, [T1, T2])) $ Bound j' =>
   490                if j' = j andalso
   491                   s = nth_sel_name_for_constr_name @{const_name FunBox} 0 then
   492                  SOME (construct_value ctxt stds
   493                                        (@{const_name FunBox}, T2 --> T1) [t2],
   494                        ts @ seen)
   495                else
   496                  raise SAME ()
   497              | _ => raise SAME ())
   498             handle SAME () => do_term (t :: seen) ts
   499         in
   500           case t of
   501             Const (@{const_name HOL.eq}, _) $ t1 $ t2 => do_eq true t1 t2
   502           | _ => do_term (t :: seen) ts
   503         end
   504   in do_term end
   505 
   506 fun subst_one_bound j arg t =
   507   let
   508     fun aux (Bound i, lev) =
   509         if i < lev then raise SAME ()
   510         else if i = lev then incr_boundvars (lev - j) arg
   511         else Bound (i - 1)
   512       | aux (Abs (a, T, body), lev) = Abs (a, T, aux (body, lev + 1))
   513       | aux (f $ t, lev) =
   514         (aux (f, lev) $ (aux (t, lev) handle SAME () => t)
   515          handle SAME () => f $ aux (t, lev))
   516       | aux _ = raise SAME ()
   517   in aux (t, j) handle SAME () => t end
   518 
   519 fun destroy_existential_equalities ({ctxt, stds, ...} : hol_context) =
   520   let
   521     fun kill [] [] ts = foldr1 s_conj ts
   522       | kill (s :: ss) (T :: Ts) ts =
   523         (case find_bound_assign ctxt stds (length ss) [] ts of
   524            SOME (_, []) => @{const True}
   525          | SOME (arg_t, ts) =>
   526            kill ss Ts (map (subst_one_bound (length ss)
   527                                 (incr_bv (~1, length ss + 1, arg_t))) ts)
   528          | NONE =>
   529            Const (@{const_name Ex}, (T --> bool_T) --> bool_T)
   530            $ Abs (s, T, kill ss Ts ts))
   531       | kill _ _ _ = raise ListPair.UnequalLengths
   532     fun gather ss Ts (Const (@{const_name Ex}, _) $ Abs (s1, T1, t1)) =
   533         gather (ss @ [s1]) (Ts @ [T1]) t1
   534       | gather [] [] (Abs (s, T, t1)) = Abs (s, T, gather [] [] t1)
   535       | gather [] [] (t1 $ t2) = gather [] [] t1 $ gather [] [] t2
   536       | gather [] [] t = t
   537       | gather ss Ts t = kill ss Ts (conjuncts_of (gather [] [] t))
   538   in gather [] [] end
   539 
   540 (** Skolemization **)
   541 
   542 fun skolem_prefix_for k j =
   543   skolem_prefix ^ string_of_int k ^ "@" ^ string_of_int j ^ name_sep
   544 
   545 fun skolemize_term_and_more (hol_ctxt as {thy, def_tables, skolems, ...})
   546                             skolem_depth =
   547   let
   548     val incrs = map (Integer.add 1)
   549     fun aux ss Ts js skolemizable polar t =
   550       let
   551         fun do_quantifier quant_s quant_T abs_s abs_T t =
   552           (if not (loose_bvar1 (t, 0)) then
   553              aux ss Ts js skolemizable polar (incr_boundvars ~1 t)
   554            else if is_positive_existential polar quant_s then
   555              let
   556                val j = length (!skolems) + 1
   557                val (js', (ss', Ts')) =
   558                  js ~~ (ss ~~ Ts)
   559                  |> filter (fn (j, _) => loose_bvar1 (t, j + 1))
   560                  |> ListPair.unzip ||> ListPair.unzip
   561              in
   562                if skolemizable andalso length js' <= skolem_depth then
   563                  let
   564                    val sko_s = skolem_prefix_for (length js') j ^ abs_s
   565                    val _ = Unsynchronized.change skolems (cons (sko_s, ss'))
   566                    val sko_t = list_comb (Const (sko_s, rev Ts' ---> abs_T),
   567                                           map Bound (rev js'))
   568                    val abs_t = Abs (abs_s, abs_T,
   569                                     aux ss Ts (incrs js) skolemizable polar t)
   570                  in
   571                    if null js' then
   572                      s_betapply Ts (abs_t, sko_t)
   573                    else
   574                      Const (@{const_name Let}, abs_T --> quant_T) $ sko_t
   575                      $ abs_t
   576                  end
   577                else
   578                  raise SAME ()
   579              end
   580            else
   581              raise SAME ())
   582           handle SAME () =>
   583                  Const (quant_s, quant_T)
   584                  $ Abs (abs_s, abs_T,
   585                         aux (abs_s :: ss) (abs_T :: Ts) (0 :: incrs js)
   586                             (skolemizable andalso
   587                              not (is_higher_order_type abs_T)) polar t)
   588       in
   589         case t of
   590           Const (s0 as @{const_name all}, T0) $ Abs (s1, T1, t1) =>
   591           do_quantifier s0 T0 s1 T1 t1
   592         | @{const "==>"} $ t1 $ t2 =>
   593           @{const "==>"} $ aux ss Ts js skolemizable (flip_polarity polar) t1
   594           $ aux ss Ts js skolemizable polar t2
   595         | @{const Pure.conjunction} $ t1 $ t2 =>
   596           @{const Pure.conjunction} $ aux ss Ts js skolemizable polar t1
   597           $ aux ss Ts js skolemizable polar t2
   598         | @{const Trueprop} $ t1 =>
   599           @{const Trueprop} $ aux ss Ts js skolemizable polar t1
   600         | @{const Not} $ t1 =>
   601           @{const Not} $ aux ss Ts js skolemizable (flip_polarity polar) t1
   602         | Const (s0 as @{const_name All}, T0) $ Abs (s1, T1, t1) =>
   603           do_quantifier s0 T0 s1 T1 t1
   604         | Const (s0 as @{const_name Ex}, T0) $ Abs (s1, T1, t1) =>
   605           do_quantifier s0 T0 s1 T1 t1
   606         | @{const HOL.conj} $ t1 $ t2 =>
   607           s_conj (pairself (aux ss Ts js skolemizable polar) (t1, t2))
   608         | @{const HOL.disj} $ t1 $ t2 =>
   609           s_disj (pairself (aux ss Ts js skolemizable polar) (t1, t2))
   610         | @{const HOL.implies} $ t1 $ t2 =>
   611           @{const HOL.implies} $ aux ss Ts js skolemizable (flip_polarity polar) t1
   612           $ aux ss Ts js skolemizable polar t2
   613         | (t0 as Const (@{const_name Let}, _)) $ t1 $ t2 =>
   614           t0 $ t1 $ aux ss Ts js skolemizable polar t2
   615         | Const (x as (s, T)) =>
   616           if is_real_inductive_pred hol_ctxt x andalso
   617              not (is_real_equational_fun hol_ctxt x) andalso
   618              not (is_well_founded_inductive_pred hol_ctxt x) then
   619             let
   620               val gfp = (fixpoint_kind_of_const thy def_tables x = Gfp)
   621               val (pref, connective) =
   622                 if gfp then (lbfp_prefix, @{const HOL.disj})
   623                 else (ubfp_prefix, @{const HOL.conj})
   624               fun pos () = unrolled_inductive_pred_const hol_ctxt gfp x
   625                            |> aux ss Ts js skolemizable polar
   626               fun neg () = Const (pref ^ s, T)
   627             in
   628               case polar |> gfp ? flip_polarity of
   629                 Pos => pos ()
   630               | Neg => neg ()
   631               | Neut =>
   632                 let
   633                   val arg_Ts = binder_types T
   634                   fun app f =
   635                     list_comb (f (), map Bound (length arg_Ts - 1 downto 0))
   636                 in
   637                   List.foldr absdummy (connective $ app pos $ app neg) arg_Ts
   638                 end
   639             end
   640           else
   641             Const x
   642         | t1 $ t2 =>
   643           s_betapply Ts (aux ss Ts js false polar t1,
   644                          aux ss Ts js false Neut t2)
   645         | Abs (s, T, t1) =>
   646           Abs (s, T, aux ss Ts (incrs js) skolemizable polar t1)
   647         | _ => t
   648       end
   649   in aux [] [] [] true Pos end
   650 
   651 (** Function specialization **)
   652 
   653 fun params_in_equation (@{const "==>"} $ _ $ t2) = params_in_equation t2
   654   | params_in_equation (@{const Trueprop} $ t1) = params_in_equation t1
   655   | params_in_equation (Const (@{const_name HOL.eq}, _) $ t1 $ _) =
   656     snd (strip_comb t1)
   657   | params_in_equation _ = []
   658 
   659 fun specialize_fun_axiom x x' fixed_js fixed_args extra_args t =
   660   let
   661     val k = fold Integer.max (map maxidx_of_term (fixed_args @ extra_args)) 0
   662             + 1
   663     val t = map_aterms (fn Var ((s, i), T) => Var ((s, k + i), T) | t' => t') t
   664     val fixed_params = filter_indices fixed_js (params_in_equation t)
   665     fun aux args (Abs (s, T, t)) = list_comb (Abs (s, T, aux [] t), args)
   666       | aux args (t1 $ t2) = aux (aux [] t2 :: args) t1
   667       | aux args t =
   668         if t = Const x then
   669           list_comb (Const x', extra_args @ filter_out_indices fixed_js args)
   670         else
   671           let val j = find_index (curry (op =) t) fixed_params in
   672             list_comb (if j >= 0 then nth fixed_args j else t, args)
   673           end
   674   in aux [] t end
   675 
   676 fun static_args_in_term ({ersatz_table, ...} : hol_context) x t =
   677   let
   678     fun fun_calls (Abs (_, _, t)) _ = fun_calls t []
   679       | fun_calls (t1 $ t2) args = fun_calls t2 [] #> fun_calls t1 (t2 :: args)
   680       | fun_calls t args =
   681         (case t of
   682            Const (x' as (s', T')) =>
   683            x = x' orelse (case AList.lookup (op =) ersatz_table s' of
   684                             SOME s'' => x = (s'', T')
   685                           | NONE => false)
   686          | _ => false) ? cons args
   687     fun call_sets [] [] vs = [vs]
   688       | call_sets [] uss vs = vs :: call_sets uss [] []
   689       | call_sets ([] :: _) _ _ = []
   690       | call_sets ((t :: ts) :: tss) uss vs =
   691         Ord_List.insert Term_Ord.term_ord t vs |> call_sets tss (ts :: uss)
   692     val sets = call_sets (fun_calls t [] []) [] []
   693     val indexed_sets = sets ~~ (index_seq 0 (length sets))
   694   in
   695     fold_rev (fn (set, j) =>
   696                  case set of
   697                    [Var _] => AList.lookup (op =) indexed_sets set = SOME j
   698                               ? cons (j, NONE)
   699                  | [t as Const _] => cons (j, SOME t)
   700                  | [t as Free _] => cons (j, SOME t)
   701                  | _ => I) indexed_sets []
   702   end
   703 fun static_args_in_terms hol_ctxt x =
   704   map (static_args_in_term hol_ctxt x)
   705   #> fold1 (Ord_List.inter (prod_ord int_ord (option_ord Term_Ord.term_ord)))
   706 
   707 fun overlapping_indices [] _ = []
   708   | overlapping_indices _ [] = []
   709   | overlapping_indices (ps1 as (j1, t1) :: ps1') (ps2 as (j2, t2) :: ps2') =
   710     if j1 < j2 then overlapping_indices ps1' ps2
   711     else if j1 > j2 then overlapping_indices ps1 ps2'
   712     else overlapping_indices ps1' ps2' |> the_default t2 t1 = t2 ? cons j1
   713 
   714 fun special_prefix_for j = special_prefix ^ string_of_int j ^ name_sep
   715 
   716 (* If a constant's definition is picked up deeper than this threshold, we
   717    prevent excessive specialization by not specializing it. *)
   718 val special_max_depth = 20
   719 
   720 val bound_var_prefix = "b"
   721 
   722 fun special_fun_aconv ((x1, js1, ts1), (x2, js2, ts2)) =
   723   x1 = x2 andalso js1 = js2 andalso length ts1 = length ts2 andalso
   724   forall (op aconv) (ts1 ~~ ts2)
   725 
   726 fun specialize_consts_in_term
   727         (hol_ctxt as {ctxt, thy, stds, specialize, def_tables, simp_table,
   728                       special_funs, ...}) def depth t =
   729   if not specialize orelse depth > special_max_depth then
   730     t
   731   else
   732     let
   733       val blacklist =
   734         if def then case term_under_def t of Const x => [x] | _ => [] else []
   735       fun aux args Ts (Const (x as (s, T))) =
   736           ((if not (member (op =) blacklist x) andalso not (null args) andalso
   737                not (String.isPrefix special_prefix s) andalso
   738                not (is_built_in_const thy stds x) andalso
   739                (is_equational_fun_but_no_plain_def hol_ctxt x orelse
   740                 (is_some (def_of_const thy def_tables x) andalso
   741                  not (is_of_class_const thy x) andalso
   742                  not (is_constr ctxt stds x) andalso
   743                  not (is_choice_spec_fun hol_ctxt x))) then
   744               let
   745                 val eligible_args =
   746                   filter (is_special_eligible_arg true Ts o snd)
   747                          (index_seq 0 (length args) ~~ args)
   748                 val _ = not (null eligible_args) orelse raise SAME ()
   749                 val old_axs = equational_fun_axioms hol_ctxt x
   750                               |> map (destroy_existential_equalities hol_ctxt)
   751                 val static_params = static_args_in_terms hol_ctxt x old_axs
   752                 val fixed_js = overlapping_indices static_params eligible_args
   753                 val _ = not (null fixed_js) orelse raise SAME ()
   754                 val fixed_args = filter_indices fixed_js args
   755                 val vars = fold Term.add_vars fixed_args []
   756                            |> sort (Term_Ord.fast_indexname_ord o pairself fst)
   757                 val bound_js = fold (fn t => fn js => add_loose_bnos (t, 0, js))
   758                                     fixed_args []
   759                                |> sort int_ord
   760                 val live_args = filter_out_indices fixed_js args
   761                 val extra_args = map Var vars @ map Bound bound_js @ live_args
   762                 val extra_Ts = map snd vars @ filter_indices bound_js Ts
   763                 val k = maxidx_of_term t + 1
   764                 fun var_for_bound_no j =
   765                   Var ((bound_var_prefix ^
   766                         nat_subscript (find_index (curry (op =) j) bound_js
   767                                        + 1), k),
   768                        nth Ts j)
   769                 val fixed_args_in_axiom =
   770                   map (curry subst_bounds
   771                              (map var_for_bound_no (index_seq 0 (length Ts))))
   772                       fixed_args
   773               in
   774                 case AList.lookup special_fun_aconv (!special_funs)
   775                                   (x, fixed_js, fixed_args_in_axiom) of
   776                   SOME x' => list_comb (Const x', extra_args)
   777                 | NONE =>
   778                   let
   779                     val extra_args_in_axiom =
   780                       map Var vars @ map var_for_bound_no bound_js
   781                     val x' as (s', _) =
   782                       (special_prefix_for (length (!special_funs) + 1) ^ s,
   783                        extra_Ts @ filter_out_indices fixed_js (binder_types T)
   784                        ---> body_type T)
   785                     val new_axs =
   786                       map (specialize_fun_axiom x x' fixed_js
   787                                fixed_args_in_axiom extra_args_in_axiom) old_axs
   788                     val _ =
   789                       Unsynchronized.change special_funs
   790                           (cons ((x, fixed_js, fixed_args_in_axiom), x'))
   791                     val _ = add_simps simp_table s' new_axs
   792                   in list_comb (Const x', extra_args) end
   793               end
   794             else
   795               raise SAME ())
   796            handle SAME () => list_comb (Const x, args))
   797         | aux args Ts (Abs (s, T, t)) =
   798           list_comb (Abs (s, T, aux [] (T :: Ts) t), args)
   799         | aux args Ts (t1 $ t2) = aux (aux [] Ts t2 :: args) Ts t1
   800         | aux args _ t = list_comb (t, args)
   801     in aux [] [] t end
   802 
   803 type special_triple = int list * term list * styp
   804 
   805 val cong_var_prefix = "c"
   806 
   807 fun special_congruence_axiom T (js1, ts1, x1) (js2, ts2, x2) =
   808   let
   809     val (bounds1, bounds2) = pairself (map Var o special_bounds) (ts1, ts2)
   810     val Ts = binder_types T
   811     val max_j = fold (fold Integer.max) [js1, js2] ~1
   812     val (eqs, (args1, args2)) =
   813       fold (fn j => case pairself (fn ps => AList.lookup (op =) ps j)
   814                                   (js1 ~~ ts1, js2 ~~ ts2) of
   815                       (SOME t1, SOME t2) => apfst (cons (t1, t2))
   816                     | (SOME t1, NONE) => apsnd (apsnd (cons t1))
   817                     | (NONE, SOME t2) => apsnd (apfst (cons t2))
   818                     | (NONE, NONE) =>
   819                       let val v = Var ((cong_var_prefix ^ nat_subscript j, 0),
   820                                        nth Ts j) in
   821                         apsnd (pairself (cons v))
   822                       end) (max_j downto 0) ([], ([], []))
   823   in
   824     Logic.list_implies (eqs |> filter_out (op aconv) |> distinct (op =)
   825                             |> map Logic.mk_equals,
   826                         Logic.mk_equals (list_comb (Const x1, bounds1 @ args1),
   827                                          list_comb (Const x2, bounds2 @ args2)))
   828   end
   829 
   830 fun special_congruence_axioms (hol_ctxt as {special_funs, ...}) ts =
   831   let
   832     val groups =
   833       !special_funs
   834       |> map (fn ((x, js, ts), x') => (x, (js, ts, x')))
   835       |> AList.group (op =)
   836       |> filter_out (is_equational_fun_surely_complete hol_ctxt o fst)
   837       |> map (fn (x, zs) =>
   838                  (x, zs |> member (op =) ts (Const x) ? cons ([], [], x)))
   839     fun generality (js, _, _) = ~(length js)
   840     fun is_more_specific (j1, t1, x1) (j2, t2, x2) =
   841       x1 <> x2 andalso length j2 < length j1 andalso
   842       Ord_List.subset (prod_ord int_ord Term_Ord.term_ord) (j2 ~~ t2, j1 ~~ t1)
   843     fun do_pass_1 _ [] [_] [_] = I
   844       | do_pass_1 T skipped _ [] = do_pass_2 T skipped
   845       | do_pass_1 T skipped all (z :: zs) =
   846         case filter (is_more_specific z) all
   847              |> sort (int_ord o pairself generality) of
   848           [] => do_pass_1 T (z :: skipped) all zs
   849         | (z' :: _) => cons (special_congruence_axiom T z z')
   850                        #> do_pass_1 T skipped all zs
   851     and do_pass_2 _ [] = I
   852       | do_pass_2 T (z :: zs) =
   853         fold (cons o special_congruence_axiom T z) zs #> do_pass_2 T zs
   854   in fold (fn ((_, T), zs) => do_pass_1 T [] zs zs) groups [] end
   855 
   856 (** Axiom selection **)
   857 
   858 fun defined_free_by_assumption t =
   859   let
   860     fun do_equals x def =
   861       if exists_subterm (curry (op aconv) (Free x)) def then NONE else SOME x
   862   in
   863     case t of
   864       Const (@{const_name "=="}, _) $ Free x $ def => do_equals x def
   865     | @{const Trueprop} $ (Const (@{const_name "=="}, _) $ Free x $ def) =>
   866       do_equals x def
   867     | _ => NONE
   868   end
   869 
   870 fun assumption_exclusively_defines_free assm_ts t =
   871   case defined_free_by_assumption t of
   872     SOME x =>
   873     length (filter ((fn SOME x' => x = x' | NONE => false)
   874                      o defined_free_by_assumption) assm_ts) = 1
   875   | NONE => false
   876 
   877 fun all_table_entries table = Symtab.fold (append o snd) table []
   878 fun extra_table tables s =
   879   Symtab.make [(s, pairself all_table_entries tables |> op @)]
   880 
   881 fun eval_axiom_for_term j t =
   882   Logic.mk_equals (Const (eval_prefix ^ string_of_int j, fastype_of t), t)
   883 
   884 val is_trivial_equation = the_default false o try (op aconv o Logic.dest_equals)
   885 
   886 (* Prevents divergence in case of cyclic or infinite axiom dependencies. *)
   887 val axioms_max_depth = 255
   888 
   889 fun axioms_for_term
   890         (hol_ctxt as {thy, ctxt, max_bisim_depth, stds, user_axioms,
   891                       evals, def_tables, nondef_table, choice_spec_table,
   892                       nondefs, ...}) assm_ts neg_t =
   893   let
   894     val (def_assm_ts, nondef_assm_ts) =
   895       List.partition (assumption_exclusively_defines_free assm_ts) assm_ts
   896     val def_assm_table = map (`(the o defined_free_by_assumption)) def_assm_ts
   897     type accumulator = styp list * (term list * term list)
   898     fun add_axiom get app def depth t (accum as (seen, axs)) =
   899       let
   900         val t = t |> unfold_defs_in_term hol_ctxt
   901                   |> skolemize_term_and_more hol_ctxt ~1 (* FIXME: why ~1? *)
   902       in
   903         if is_trivial_equation t then
   904           accum
   905         else
   906           let val t' = t |> specialize_consts_in_term hol_ctxt def depth in
   907             if exists (member (op aconv) (get axs)) [t, t'] then accum
   908             else add_axioms_for_term (depth + 1) t' (seen, app (cons t') axs)
   909           end
   910       end
   911     and add_def_axiom depth = add_axiom fst apfst true depth
   912     and add_nondef_axiom depth = add_axiom snd apsnd false depth
   913     and add_maybe_def_axiom depth t =
   914       (if head_of t <> @{const "==>"} then add_def_axiom
   915        else add_nondef_axiom) depth t
   916     and add_eq_axiom depth t =
   917       (if is_constr_pattern_formula ctxt t then add_def_axiom
   918        else add_nondef_axiom) depth t
   919     and add_axioms_for_term depth t (accum as (seen, axs)) =
   920       case t of
   921         t1 $ t2 => accum |> fold (add_axioms_for_term depth) [t1, t2]
   922       | Const (x as (s, T)) =>
   923         (if member (op aconv) seen t orelse is_built_in_const thy stds x then
   924            accum
   925          else
   926            let val accum = (t :: seen, axs) in
   927              if depth > axioms_max_depth then
   928                raise TOO_LARGE ("Nitpick_Preproc.axioms_for_term.\
   929                                 \add_axioms_for_term",
   930                                 "too many nested axioms (" ^
   931                                 string_of_int depth ^ ")")
   932              else if is_of_class_const thy x then
   933                let
   934                  val class = Logic.class_of_const s
   935                  val of_class = Logic.mk_of_class (TVar (("'a", 0), [class]),
   936                                                    class)
   937                  val ax1 = try (specialize_type thy x) of_class
   938                  val ax2 = Option.map (specialize_type thy x o snd)
   939                                       (get_class_def thy class)
   940                in
   941                  fold (add_maybe_def_axiom depth) (map_filter I [ax1, ax2])
   942                       accum
   943                end
   944              else if is_constr ctxt stds x then
   945                accum
   946              else if is_descr (original_name s) then
   947                fold (add_nondef_axiom depth) (equational_fun_axioms hol_ctxt x)
   948                     accum
   949              else if is_equational_fun_but_no_plain_def hol_ctxt x then
   950                fold (add_eq_axiom depth) (equational_fun_axioms hol_ctxt x)
   951                     accum
   952              else if is_choice_spec_fun hol_ctxt x then
   953                fold (add_nondef_axiom depth)
   954                     (nondef_props_for_const thy true choice_spec_table x) accum
   955              else if is_abs_fun ctxt x then
   956                accum |> fold (add_nondef_axiom depth)
   957                              (nondef_props_for_const thy false nondef_table x)
   958                      |> (is_funky_typedef ctxt (range_type T) orelse
   959                          range_type T = nat_T)
   960                         ? fold (add_maybe_def_axiom depth)
   961                                (nondef_props_for_const thy true
   962                                     (extra_table def_tables s) x)
   963              else if is_rep_fun ctxt x then
   964                accum |> fold (add_nondef_axiom depth)
   965                              (nondef_props_for_const thy false nondef_table x)
   966                      |> (is_funky_typedef ctxt (range_type T) orelse
   967                          range_type T = nat_T)
   968                         ? fold (add_maybe_def_axiom depth)
   969                                (nondef_props_for_const thy true
   970                                     (extra_table def_tables s) x)
   971                      |> add_axioms_for_term depth
   972                                             (Const (mate_of_rep_fun ctxt x))
   973                      |> fold (add_def_axiom depth)
   974                              (inverse_axioms_for_rep_fun ctxt x)
   975              else if s = @{const_name TYPE} then
   976                accum
   977              else case def_of_const thy def_tables x of
   978                SOME _ =>
   979                fold (add_eq_axiom depth) (equational_fun_axioms hol_ctxt x)
   980                     accum
   981              | NONE =>
   982                accum |> user_axioms <> SOME false
   983                         ? fold (add_nondef_axiom depth)
   984                                (nondef_props_for_const thy false nondef_table x)
   985            end)
   986         |> add_axioms_for_type depth T
   987       | Free (x as (_, T)) =>
   988         (if member (op aconv) seen t then
   989            accum
   990          else case AList.lookup (op =) def_assm_table x of
   991            SOME t => add_def_axiom depth t accum
   992          | NONE => accum)
   993         |> add_axioms_for_type depth T
   994       | Var (_, T) => add_axioms_for_type depth T accum
   995       | Bound _ => accum
   996       | Abs (_, T, t) => accum |> add_axioms_for_term depth t
   997                                |> add_axioms_for_type depth T
   998     and add_axioms_for_type depth T =
   999       case T of
  1000         Type (@{type_name fun}, Ts) => fold (add_axioms_for_type depth) Ts
  1001       | Type (@{type_name prod}, Ts) => fold (add_axioms_for_type depth) Ts
  1002       | @{typ prop} => I
  1003       | @{typ bool} => I
  1004       | TFree (_, S) => add_axioms_for_sort depth T S
  1005       | TVar (_, S) => add_axioms_for_sort depth T S
  1006       | Type (z as (_, Ts)) =>
  1007         fold (add_axioms_for_type depth) Ts
  1008         #> (if is_pure_typedef ctxt T then
  1009               fold (add_maybe_def_axiom depth) (optimized_typedef_axioms ctxt z)
  1010             else if is_quot_type ctxt T then
  1011               fold (add_def_axiom depth)
  1012                    (optimized_quot_type_axioms ctxt stds z)
  1013             else if max_bisim_depth >= 0 andalso is_codatatype ctxt T then
  1014               fold (add_maybe_def_axiom depth)
  1015                    (codatatype_bisim_axioms hol_ctxt T)
  1016             else
  1017               I)
  1018     and add_axioms_for_sort depth T S =
  1019       let
  1020         val supers = Sign.complete_sort thy S
  1021         val class_axioms =
  1022           maps (fn class => map prop_of (AxClass.get_info thy class |> #axioms
  1023                                          handle ERROR _ => [])) supers
  1024         val monomorphic_class_axioms =
  1025           map (fn t => case Term.add_tvars t [] of
  1026                          [] => t
  1027                        | [(x, S)] =>
  1028                          monomorphic_term (Vartab.make [(x, (S, T))]) t
  1029                        | _ => raise TERM ("Nitpick_Preproc.axioms_for_term.\
  1030                                           \add_axioms_for_sort", [t]))
  1031               class_axioms
  1032       in fold (add_nondef_axiom depth) monomorphic_class_axioms end
  1033     val (mono_nondefs, poly_nondefs) =
  1034       List.partition (null o Term.hidden_polymorphism) nondefs
  1035     val eval_axioms = map2 eval_axiom_for_term (index_seq 0 (length evals))
  1036                            evals
  1037     val (seen, (defs, nondefs)) =
  1038       ([], ([], []))
  1039       |> add_axioms_for_term 1 neg_t
  1040       |> fold_rev (add_nondef_axiom 1) nondef_assm_ts
  1041       |> fold_rev (add_def_axiom 1) eval_axioms
  1042       |> user_axioms = SOME true ? fold (add_nondef_axiom 1) mono_nondefs
  1043     val defs = defs @ special_congruence_axioms hol_ctxt seen
  1044     val got_all_mono_user_axioms =
  1045       (user_axioms = SOME true orelse null mono_nondefs)
  1046   in (neg_t :: nondefs, defs, got_all_mono_user_axioms, null poly_nondefs) end
  1047 
  1048 (** Simplification of constructor/selector terms **)
  1049 
  1050 fun simplify_constrs_and_sels ctxt t =
  1051   let
  1052     fun is_nth_sel_on t' n (Const (s, _) $ t) =
  1053         (t = t' andalso is_sel_like_and_no_discr s andalso
  1054          sel_no_from_name s = n)
  1055       | is_nth_sel_on _ _ _ = false
  1056     fun do_term (Const (@{const_name Rep_Frac}, _)
  1057                  $ (Const (@{const_name Abs_Frac}, _) $ t1)) [] = do_term t1 []
  1058       | do_term (Const (@{const_name Abs_Frac}, _)
  1059                  $ (Const (@{const_name Rep_Frac}, _) $ t1)) [] = do_term t1 []
  1060       | do_term (t1 $ t2) args = do_term t1 (do_term t2 [] :: args)
  1061       | do_term (t as Const (x as (s, T))) (args as _ :: _) =
  1062         ((if is_constr_like ctxt x then
  1063             if length args = num_binder_types T then
  1064               case hd args of
  1065                 Const (_, T') $ t' =>
  1066                 if domain_type T' = body_type T andalso
  1067                    forall (uncurry (is_nth_sel_on t'))
  1068                           (index_seq 0 (length args) ~~ args) then
  1069                   t'
  1070                 else
  1071                   raise SAME ()
  1072               | _ => raise SAME ()
  1073             else
  1074               raise SAME ()
  1075           else if is_sel_like_and_no_discr s then
  1076             case strip_comb (hd args) of
  1077               (Const (x' as (s', T')), ts') =>
  1078               if is_constr_like ctxt x' andalso
  1079                  constr_name_for_sel_like s = s' andalso
  1080                  not (exists is_pair_type (binder_types T')) then
  1081                 list_comb (nth ts' (sel_no_from_name s), tl args)
  1082               else
  1083                 raise SAME ()
  1084             | _ => raise SAME ()
  1085           else
  1086             raise SAME ())
  1087          handle SAME () => s_betapplys [] (t, args))
  1088       | do_term (Abs (s, T, t')) args =
  1089         s_betapplys [] (Abs (s, T, do_term t' []), args)
  1090       | do_term t args = s_betapplys [] (t, args)
  1091   in do_term t [] end
  1092 
  1093 (** Quantifier massaging: Distributing quantifiers **)
  1094 
  1095 fun distribute_quantifiers t =
  1096   case t of
  1097     (t0 as Const (@{const_name All}, T0)) $ Abs (s, T1, t1) =>
  1098     (case t1 of
  1099        (t10 as @{const HOL.conj}) $ t11 $ t12 =>
  1100        t10 $ distribute_quantifiers (t0 $ Abs (s, T1, t11))
  1101            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))
  1102      | (t10 as @{const Not}) $ t11 =>
  1103        t10 $ distribute_quantifiers (Const (@{const_name Ex}, T0)
  1104                                      $ Abs (s, T1, t11))
  1105      | t1 =>
  1106        if not (loose_bvar1 (t1, 0)) then
  1107          distribute_quantifiers (incr_boundvars ~1 t1)
  1108        else
  1109          t0 $ Abs (s, T1, distribute_quantifiers t1))
  1110   | (t0 as Const (@{const_name Ex}, T0)) $ Abs (s, T1, t1) =>
  1111     (case distribute_quantifiers t1 of
  1112        (t10 as @{const HOL.disj}) $ t11 $ t12 =>
  1113        t10 $ distribute_quantifiers (t0 $ Abs (s, T1, t11))
  1114            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))
  1115      | (t10 as @{const HOL.implies}) $ t11 $ t12 =>
  1116        t10 $ distribute_quantifiers (Const (@{const_name All}, T0)
  1117                                      $ Abs (s, T1, t11))
  1118            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))
  1119      | (t10 as @{const Not}) $ t11 =>
  1120        t10 $ distribute_quantifiers (Const (@{const_name All}, T0)
  1121                                      $ Abs (s, T1, t11))
  1122      | t1 =>
  1123        if not (loose_bvar1 (t1, 0)) then
  1124          distribute_quantifiers (incr_boundvars ~1 t1)
  1125        else
  1126          t0 $ Abs (s, T1, distribute_quantifiers t1))
  1127   | t1 $ t2 => distribute_quantifiers t1 $ distribute_quantifiers t2
  1128   | Abs (s, T, t') => Abs (s, T, distribute_quantifiers t')
  1129   | _ => t
  1130 
  1131 (** Quantifier massaging: Pushing quantifiers inward **)
  1132 
  1133 fun renumber_bounds j n f t =
  1134   case t of
  1135     t1 $ t2 => renumber_bounds j n f t1 $ renumber_bounds j n f t2
  1136   | Abs (s, T, t') => Abs (s, T, renumber_bounds (j + 1) n f t')
  1137   | Bound j' =>
  1138     Bound (if j' >= j andalso j' < j + n then f (j' - j) + j else j')
  1139   | _ => t
  1140 
  1141 (* Maximum number of quantifiers in a cluster for which the exponential
  1142    algorithm is used. Larger clusters use a heuristic inspired by Claessen &
  1143    Soerensson's polynomial binary splitting procedure (p. 5 of their MODEL 2003
  1144    paper). *)
  1145 val quantifier_cluster_threshold = 7
  1146 
  1147 val push_quantifiers_inward =
  1148   let
  1149     fun aux quant_s ss Ts t =
  1150       (case t of
  1151          Const (s0, _) $ Abs (s1, T1, t1 as _ $ _) =>
  1152          if s0 = quant_s then
  1153            aux s0 (s1 :: ss) (T1 :: Ts) t1
  1154          else if quant_s = "" andalso
  1155                  (s0 = @{const_name All} orelse s0 = @{const_name Ex}) then
  1156            aux s0 [s1] [T1] t1
  1157          else
  1158            raise SAME ()
  1159        | _ => raise SAME ())
  1160       handle SAME () =>
  1161              case t of
  1162                t1 $ t2 =>
  1163                if quant_s = "" then
  1164                  aux "" [] [] t1 $ aux "" [] [] t2
  1165                else
  1166                  let
  1167                    fun big_union proj ps =
  1168                      fold (fold (insert (op =)) o proj) ps []
  1169                    val (ts, connective) = strip_any_connective t
  1170                    val T_costs = map typical_card_of_type Ts
  1171                    val t_costs = map size_of_term ts
  1172                    val num_Ts = length Ts
  1173                    val flip = curry (op -) (num_Ts - 1)
  1174                    val t_boundss = map (map flip o loose_bnos) ts
  1175                    fun merge costly_boundss [] = costly_boundss
  1176                      | merge costly_boundss (j :: js) =
  1177                        let
  1178                          val (yeas, nays) =
  1179                            List.partition (fn (bounds, _) =>
  1180                                               member (op =) bounds j)
  1181                                           costly_boundss
  1182                          val yeas_bounds = big_union fst yeas
  1183                          val yeas_cost = Integer.sum (map snd yeas)
  1184                                          * nth T_costs j
  1185                        in merge ((yeas_bounds, yeas_cost) :: nays) js end
  1186                    val cost = Integer.sum o map snd oo merge
  1187                    fun heuristically_best_permutation _ [] = []
  1188                      | heuristically_best_permutation costly_boundss js =
  1189                        let
  1190                          val (costly_boundss, (j, js)) =
  1191                            js |> map (`(merge costly_boundss o single))
  1192                               |> sort (int_ord
  1193                                        o pairself (Integer.sum o map snd o fst))
  1194                               |> split_list |>> hd ||> pairf hd tl
  1195                        in
  1196                          j :: heuristically_best_permutation costly_boundss js
  1197                        end
  1198                    val js =
  1199                      if length Ts <= quantifier_cluster_threshold then
  1200                        all_permutations (index_seq 0 num_Ts)
  1201                        |> map (`(cost (t_boundss ~~ t_costs)))
  1202                        |> sort (int_ord o pairself fst) |> hd |> snd
  1203                      else
  1204                        heuristically_best_permutation (t_boundss ~~ t_costs)
  1205                                                       (index_seq 0 num_Ts)
  1206                    val back_js = map (fn j => find_index (curry (op =) j) js)
  1207                                      (index_seq 0 num_Ts)
  1208                    val ts = map (renumber_bounds 0 num_Ts (nth back_js o flip))
  1209                                 ts
  1210                    fun mk_connection [] =
  1211                        raise ARG ("Nitpick_Preproc.push_quantifiers_inward.aux.\
  1212                                   \mk_connection", "")
  1213                      | mk_connection ts_cum_bounds =
  1214                        ts_cum_bounds |> map fst
  1215                        |> foldr1 (fn (t1, t2) => connective $ t1 $ t2)
  1216                    fun build ts_cum_bounds [] = ts_cum_bounds |> mk_connection
  1217                      | build ts_cum_bounds (j :: js) =
  1218                        let
  1219                          val (yeas, nays) =
  1220                            List.partition (fn (_, bounds) =>
  1221                                               member (op =) bounds j)
  1222                                           ts_cum_bounds
  1223                            ||> map (apfst (incr_boundvars ~1))
  1224                        in
  1225                          if null yeas then
  1226                            build nays js
  1227                          else
  1228                            let val T = nth Ts (flip j) in
  1229                              build ((Const (quant_s, (T --> bool_T) --> bool_T)
  1230                                      $ Abs (nth ss (flip j), T,
  1231                                             mk_connection yeas),
  1232                                       big_union snd yeas) :: nays) js
  1233                            end
  1234                        end
  1235                  in build (ts ~~ t_boundss) js end
  1236              | Abs (s, T, t') => Abs (s, T, aux "" [] [] t')
  1237              | _ => t
  1238   in aux "" [] [] end
  1239 
  1240 (** Preprocessor entry point **)
  1241 
  1242 val max_skolem_depth = 3
  1243 
  1244 fun preprocess_formulas
  1245         (hol_ctxt as {thy, ctxt, stds, binary_ints, destroy_constrs, boxes,
  1246                       needs, ...}) assm_ts neg_t =
  1247   let
  1248     val (nondef_ts, def_ts, got_all_mono_user_axioms, no_poly_user_axioms) =
  1249       neg_t |> unfold_defs_in_term hol_ctxt
  1250             |> close_form
  1251             |> skolemize_term_and_more hol_ctxt max_skolem_depth
  1252             |> specialize_consts_in_term hol_ctxt false 0
  1253             |> axioms_for_term hol_ctxt assm_ts
  1254     val binarize =
  1255       is_standard_datatype thy stds nat_T andalso
  1256       case binary_ints of
  1257         SOME false => false
  1258       | _ => forall (may_use_binary_ints false) nondef_ts andalso
  1259              forall (may_use_binary_ints true) def_ts andalso
  1260              (binary_ints = SOME true orelse
  1261               exists should_use_binary_ints (nondef_ts @ def_ts))
  1262     val box = exists (not_equal (SOME false) o snd) boxes
  1263     val table =
  1264       Termtab.empty
  1265       |> box ? fold (add_to_uncurry_table ctxt) (nondef_ts @ def_ts)
  1266     fun do_middle def =
  1267       binarize ? binarize_nat_and_int_in_term
  1268       #> box ? uncurry_term table
  1269       #> box ? box_fun_and_pair_in_term hol_ctxt def
  1270     fun do_tail def =
  1271       destroy_constrs ? (pull_out_universal_constrs hol_ctxt def
  1272                          #> pull_out_existential_constrs hol_ctxt)
  1273       #> destroy_pulled_out_constrs hol_ctxt def destroy_constrs
  1274       #> curry_assms
  1275       #> destroy_universal_equalities
  1276       #> destroy_existential_equalities hol_ctxt
  1277       #> simplify_constrs_and_sels ctxt
  1278       #> distribute_quantifiers
  1279       #> push_quantifiers_inward
  1280       #> close_form
  1281       #> Term.map_abs_vars shortest_name
  1282     val nondef_ts = nondef_ts |> map (do_middle false)
  1283     val need_ts =
  1284       case needs of
  1285         SOME needs =>
  1286         needs |> map (unfold_defs_in_term hol_ctxt #> do_middle false)
  1287       | NONE => [] (* FIXME: Implement inference. *)
  1288     val nondef_ts = nondef_ts |> map (do_tail false)
  1289     val def_ts = def_ts |> map (do_middle true #> do_tail true)
  1290   in
  1291     (nondef_ts, def_ts, need_ts, got_all_mono_user_axioms, no_poly_user_axioms,
  1292      binarize)
  1293   end
  1294 
  1295 end;