wneuper/isa: src/HOL/Tools/Nitpick/nitpick

     1 (*  Title:      HOL/Tools/Nitpick/nitpick_preproc.ML

     2     Author:     Jasmin Blanchette, TU Muenchen

     3     Copyright   2008, 2009, 2010

     5 Nitpick's HOL preprocessor.

     6 *)

     8 signature NITPICK_PREPROC =

     9 sig

    10   type hol_context = Nitpick_HOL.hol_context

    11   val preprocess_term :

    12     hol_context -> term

    13     -> ((term list * term list) * (bool * bool)) * term * bool

    14 end

    16 structure Nitpick_Preproc : NITPICK_PREPROC =

    17 struct

    19 open Nitpick_Util

    20 open Nitpick_HOL

    22 (* polarity -> string -> bool *)

    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})

    27 (** Binary coding of integers **)

    29 (* If a formula contains a numeral whose absolute value is more than this

    30    threshold, the unary coding is likely not to work well and we prefer the

    31    binary coding. *)

    32 val binary_int_threshold = 3

    34 (* term -> bool *)

    35 fun may_use_binary_ints (t1 $ t2) =

    36     may_use_binary_ints t1 andalso may_use_binary_ints t2

    37   | may_use_binary_ints (t as Const (s, _)) =

    38     t <> @{const Suc} andalso

    39     not (member (op =) [@{const_name Abs_Frac}, @{const_name Rep_Frac},

    40                         @{const_name nat_gcd}, @{const_name nat_lcm},

    41                         @{const_name Frac}, @{const_name norm_frac}] s)

    42   | may_use_binary_ints (Abs (_, _, t')) = may_use_binary_ints t'

    43   | may_use_binary_ints _ = true

    44 fun should_use_binary_ints (t1 $ t2) =

    45     should_use_binary_ints t1 orelse should_use_binary_ints t2

    46   | should_use_binary_ints (Const (s, T)) =

    47     ((s = @{const_name times} orelse s = @{const_name div}) andalso

    48      is_integer_type (body_type T)) orelse

    49     (String.isPrefix numeral_prefix s andalso

    50      let val n = the (Int.fromString (unprefix numeral_prefix s)) in

    51        n < ~ binary_int_threshold orelse n > binary_int_threshold

    52      end)

    53   | should_use_binary_ints (Abs (_, _, t')) = should_use_binary_ints t'

    54   | should_use_binary_ints _ = false

    56 (** Uncurrying **)

    58 (* theory -> term -> int Termtab.tab -> int Termtab.tab *)

    59 fun add_to_uncurry_table thy t =

    60   let

    61     (* term -> term list -> int Termtab.tab -> int Termtab.tab *)

    62     fun aux (t1 $ t2) args table =

    63         let val table = aux t2 [] table in aux t1 (t2 :: args) table end

    64       | aux (Abs (_, _, t')) _ table = aux t' [] table

    65       | aux (t as Const (x as (s, _))) args table =

    66         if is_built_in_const thy [(NONE, true)] true x orelse

    67            is_constr_like thy x orelse

    68            is_sel s orelse s = @{const_name Sigma} then

    69           table

    70         else

    71           Termtab.map_default (t, 65536) (curry Int.min (length args)) table

    72       | aux _ _ table = table

    73   in aux t [] end

    75 (* int -> int -> string *)

    76 fun uncurry_prefix_for k j =

    77   uncurry_prefix ^ string_of_int k ^ "@" ^ string_of_int j ^ name_sep

    79 (* int Termtab.tab term -> term *)

    80 fun uncurry_term table t =

    81   let

    82     (* term -> term list -> term *)

    83     fun aux (t1 $ t2) args = aux t1 (aux t2 [] :: args)

    84       | aux (Abs (s, T, t')) args = betapplys (Abs (s, T, aux t' []), args)

    85       | aux (t as Const (s, T)) args =

    86         (case Termtab.lookup table t of

    87            SOME n =>

    88            if n >= 2 then

    89              let

    90                val arg_Ts = strip_n_binders n T |> fst

    91                val j =

    92                  if is_iterator_type (hd arg_Ts) then

    93                    1

    94                  else case find_index (not_equal bool_T) arg_Ts of

    95                    ~1 => n

    96                  | j => j

    97                val ((before_args, tuple_args), after_args) =

    98                  args |> chop n |>> chop j

    99                val ((before_arg_Ts, tuple_arg_Ts), rest_T) =

   100                  T |> strip_n_binders n |>> chop j

   101                val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts

   102              in

   103                if n - j < 2 then

   104                  betapplys (t, args)

   105                else

   106                  betapplys (Const (uncurry_prefix_for (n - j) j ^ s,

   107                                    before_arg_Ts ---> tuple_T --> rest_T),

   108                             before_args @ [mk_flat_tuple tuple_T tuple_args] @

   109                             after_args)

   110              end

   111            else

   112              betapplys (t, args)

   113          | NONE => betapplys (t, args))

   114       | aux t args = betapplys (t, args)

   115   in aux t [] end

   117 (** Boxing **)

   119 (* hol_context -> typ -> term -> term *)

   120 fun constr_expand (hol_ctxt as {thy, stds, ...}) T t =

   121   (case head_of t of

   122      Const x => if is_constr_like thy x then t else raise SAME ()

   123    | _ => raise SAME ())

   124   handle SAME () =>

   125          let

   126            val x' as (_, T') =

   127              if is_pair_type T then

   128                let val (T1, T2) = HOLogic.dest_prodT T in

   129                  (@{const_name Pair}, T1 --> T2 --> T)

   130                end

   131              else

   132                datatype_constrs hol_ctxt T |> hd

   133            val arg_Ts = binder_types T'

   134          in

   135            list_comb (Const x', map2 (select_nth_constr_arg thy stds x' t)

   136                                      (index_seq 0 (length arg_Ts)) arg_Ts)

   137          end

   139 (* hol_context -> bool -> term -> term *)

   140 fun box_fun_and_pair_in_term (hol_ctxt as {thy, stds, fast_descrs, ...}) def

   141                              orig_t =

   142   let

   143     (* typ -> typ *)

   144     fun box_relational_operator_type (Type ("fun", Ts)) =

   145         Type ("fun", map box_relational_operator_type Ts)

   146       | box_relational_operator_type (Type ("*", Ts)) =

   147         Type ("*", map (box_type hol_ctxt InPair) Ts)

   148       | box_relational_operator_type T = T

   149     (* (term -> term) -> int -> term -> term *)

   150     fun coerce_bound_no f j t =

   151       case t of

   152         t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2

   153       | Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')

   154       | Bound j' => if j' = j then f t else t

   155       | _ => t

   156     (* typ -> typ -> term -> term *)

   157     fun coerce_bound_0_in_term new_T old_T =

   158       old_T <> new_T ? coerce_bound_no (coerce_term [new_T] old_T new_T) 0

   159     (* typ list -> typ -> term -> term *)

   160     and coerce_term Ts new_T old_T t =

   161       if old_T = new_T then

   162         t

   163       else

   164         case (new_T, old_T) of

   165           (Type (new_s, new_Ts as [new_T1, new_T2]),

   166            Type ("fun", [old_T1, old_T2])) =>

   167           (case eta_expand Ts t 1 of

   168              Abs (s, _, t') =>

   169              Abs (s, new_T1,

   170                   t' |> coerce_bound_0_in_term new_T1 old_T1

   171                      |> coerce_term (new_T1 :: Ts) new_T2 old_T2)

   172              |> Envir.eta_contract

   173              |> new_s <> "fun"

   174                 ? construct_value thy stds

   175                       (@{const_name FunBox}, Type ("fun", new_Ts) --> new_T)

   176                       o single

   177            | t' => raise TERM ("Nitpick_Preproc.box_fun_and_pair_in_term.\

   178                                \coerce_term", [t']))

   179         | (Type (new_s, new_Ts as [new_T1, new_T2]),

   180            Type (old_s, old_Ts as [old_T1, old_T2])) =>

   181           if old_s = @{type_name fun_box} orelse

   182              old_s = @{type_name pair_box} orelse old_s = "*" then

   183             case constr_expand hol_ctxt old_T t of

   184               Const (@{const_name FunBox}, _) $ t1 =>

   185               if new_s = "fun" then

   186                 coerce_term Ts new_T (Type ("fun", old_Ts)) t1

   187               else

   188                 construct_value thy stds

   189                     (@{const_name FunBox}, Type ("fun", new_Ts) --> new_T)

   190                     [coerce_term Ts (Type ("fun", new_Ts))

   191                                  (Type ("fun", old_Ts)) t1]

   192             | Const _ $ t1 $ t2 =>

   193               construct_value thy stds

   194                   (if new_s = "*" then @{const_name Pair}

   195                    else @{const_name PairBox}, new_Ts ---> new_T)

   196                   [coerce_term Ts new_T1 old_T1 t1,

   197                    coerce_term Ts new_T2 old_T2 t2]

   198             | t' => raise TERM ("Nitpick_Preproc.box_fun_and_pair_in_term.\

   199                                 \coerce_term", [t'])

   200           else

   201             raise TYPE ("coerce_term", [new_T, old_T], [t])

   202         | _ => raise TYPE ("coerce_term", [new_T, old_T], [t])

   203     (* indexname * typ -> typ * term -> typ option list -> typ option list *)

   204     fun add_boxed_types_for_var (z as (_, T)) (T', t') =

   205       case t' of

   206         Var z' => z' = z ? insert (op =) T'

   207       | Const (@{const_name Pair}, _) $ t1 $ t2 =>

   208         (case T' of

   209            Type (_, [T1, T2]) =>

   210            fold (add_boxed_types_for_var z) [(T1, t1), (T2, t2)]

   211          | _ => raise TYPE ("Nitpick_Preproc.box_fun_and_pair_in_term.\

   212                             \add_boxed_types_for_var", [T'], []))

   213       | _ => exists_subterm (curry (op =) (Var z)) t' ? insert (op =) T

   214     (* typ list -> typ list -> term -> indexname * typ -> typ *)

   215     fun box_var_in_def new_Ts old_Ts t (z as (_, T)) =

   216       case t of

   217         @{const Trueprop} $ t1 => box_var_in_def new_Ts old_Ts t1 z

   218       | Const (s0, _) $ t1 $ _ =>

   219         if s0 = @{const_name "=="} orelse s0 = @{const_name "op ="} then

   220           let

   221             val (t', args) = strip_comb t1

   222             val T' = fastype_of1 (new_Ts, do_term new_Ts old_Ts Neut t')

   223           in

   224             case fold (add_boxed_types_for_var z)

   225                       (fst (strip_n_binders (length args) T') ~~ args) [] of

   226               [T''] => T''

   227             | _ => T

   228           end

   229         else

   230           T

   231       | _ => T

   232     (* typ list -> typ list -> polarity -> string -> typ -> string -> typ

   233        -> term -> term *)

   234     and do_quantifier new_Ts old_Ts polar quant_s quant_T abs_s abs_T t =

   235       let

   236         val abs_T' =

   237           if polar = Neut orelse is_positive_existential polar quant_s then

   238             box_type hol_ctxt InFunLHS abs_T

   239           else

   240             abs_T

   241         val body_T = body_type quant_T

   242       in

   243         Const (quant_s, (abs_T' --> body_T) --> body_T)

   244         $ Abs (abs_s, abs_T',

   245                t |> do_term (abs_T' :: new_Ts) (abs_T :: old_Ts) polar)

   246       end

   247     (* typ list -> typ list -> string -> typ -> term -> term -> term *)

   248     and do_equals new_Ts old_Ts s0 T0 t1 t2 =

   249       let

   250         val (t1, t2) = pairself (do_term new_Ts old_Ts Neut) (t1, t2)

   251         val (T1, T2) = pairself (curry fastype_of1 new_Ts) (t1, t2)

   252         val T = [T1, T2] |> sort TermOrd.typ_ord |> List.last

   253       in

   254         list_comb (Const (s0, T --> T --> body_type T0),

   255                    map2 (coerce_term new_Ts T) [T1, T2] [t1, t2])

   256       end

   257     (* string -> typ -> term *)

   258     and do_description_operator s T =

   259       let val T1 = box_type hol_ctxt InFunLHS (range_type T) in

   260         Const (s, (T1 --> bool_T) --> T1)

   261       end

   262     (* typ list -> typ list -> polarity -> term -> term *)

   263     and do_term new_Ts old_Ts polar t =

   264       case t of

   265         Const (s0 as @{const_name all}, T0) $ Abs (s1, T1, t1) =>

   266         do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1

   267       | Const (s0 as @{const_name "=="}, T0) $ t1 $ t2 =>

   268         do_equals new_Ts old_Ts s0 T0 t1 t2

   269       | @{const "==>"} $ t1 $ t2 =>

   270         @{const "==>"} $ do_term new_Ts old_Ts (flip_polarity polar) t1

   271         $ do_term new_Ts old_Ts polar t2

   272       | @{const Pure.conjunction} $ t1 $ t2 =>

   273         @{const Pure.conjunction} $ do_term new_Ts old_Ts polar t1

   274         $ do_term new_Ts old_Ts polar t2

   275       | @{const Trueprop} $ t1 =>

   276         @{const Trueprop} $ do_term new_Ts old_Ts polar t1

   277       | @{const Not} $ t1 =>

   278         @{const Not} $ do_term new_Ts old_Ts (flip_polarity polar) t1

   279       | Const (s0 as @{const_name All}, T0) $ Abs (s1, T1, t1) =>

   280         do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1

   281       | Const (s0 as @{const_name Ex}, T0) $ Abs (s1, T1, t1) =>

   282         do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1

   283       | Const (s0 as @{const_name "op ="}, T0) $ t1 $ t2 =>

   284         do_equals new_Ts old_Ts s0 T0 t1 t2

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

   286         @{const "op &"} $ do_term new_Ts old_Ts polar t1

   287         $ do_term new_Ts old_Ts polar t2

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

   289         @{const "op |"} $ do_term new_Ts old_Ts polar t1

   290         $ do_term new_Ts old_Ts polar t2

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

   292         @{const "op -->"} $ do_term new_Ts old_Ts (flip_polarity polar) t1

   293         $ do_term new_Ts old_Ts polar t2

   294       | Const (s as @{const_name The}, T) => do_description_operator s T

   295       | Const (s as @{const_name Eps}, T) => do_description_operator s T

   296       | Const (s as @{const_name Tha}, T) => do_description_operator s T

   297       | Const (x as (s, T)) =>

   298         Const (s, if s = @{const_name converse} orelse

   299                      s = @{const_name trancl} then

   300                     box_relational_operator_type T

   301                   else if String.isPrefix quot_normal_prefix s then

   302                     let val T' = box_type hol_ctxt InFunLHS (domain_type T) in

   303                       T' --> T'

   304                     end

   305                   else if is_built_in_const thy stds fast_descrs x orelse

   306                           s = @{const_name Sigma} then

   307                     T

   308                   else if is_constr_like thy x then

   309                     box_type hol_ctxt InConstr T

   310                   else if is_sel s

   311                        orelse is_rep_fun thy x then

   312                     box_type hol_ctxt InSel T

   313                   else

   314                     box_type hol_ctxt InExpr T)

   315       | t1 $ Abs (s, T, t2') =>

   316         let

   317           val t1 = do_term new_Ts old_Ts Neut t1

   318           val T1 = fastype_of1 (new_Ts, t1)

   319           val (s1, Ts1) = dest_Type T1

   320           val T' = hd (snd (dest_Type (hd Ts1)))

   321           val t2 = Abs (s, T', do_term (T' :: new_Ts) (T :: old_Ts) Neut t2')

   322           val T2 = fastype_of1 (new_Ts, t2)

   323           val t2 = coerce_term new_Ts (hd Ts1) T2 t2

   324         in

   325           betapply (if s1 = "fun" then

   326                       t1

   327                     else

   328                       select_nth_constr_arg thy stds

   329                           (@{const_name FunBox}, Type ("fun", Ts1) --> T1) t1 0

   330                           (Type ("fun", Ts1)), t2)

   331         end

   332       | t1 $ t2 =>

   333         let

   334           val t1 = do_term new_Ts old_Ts Neut t1

   335           val T1 = fastype_of1 (new_Ts, t1)

   336           val (s1, Ts1) = dest_Type T1

   337           val t2 = do_term new_Ts old_Ts Neut t2

   338           val T2 = fastype_of1 (new_Ts, t2)

   339           val t2 = coerce_term new_Ts (hd Ts1) T2 t2

   340         in

   341           betapply (if s1 = "fun" then

   342                       t1

   343                     else

   344                       select_nth_constr_arg thy stds

   345                           (@{const_name FunBox}, Type ("fun", Ts1) --> T1) t1 0

   346                           (Type ("fun", Ts1)), t2)

   347         end

   348       | Free (s, T) => Free (s, box_type hol_ctxt InExpr T)

   349       | Var (z as (x, T)) =>

   350         Var (x, if def then box_var_in_def new_Ts old_Ts orig_t z

   351                 else box_type hol_ctxt InExpr T)

   352       | Bound _ => t

   353       | Abs (s, T, t') =>

   354         Abs (s, T, do_term (T :: new_Ts) (T :: old_Ts) Neut t')

   355   in do_term [] [] Pos orig_t end

   357 (** Destruction of constructors **)

   359 val val_var_prefix = nitpick_prefix ^ "v"

   361 (* typ list -> int -> int -> int -> term -> term *)

   362 fun fresh_value_var Ts k n j t =

   363   Var ((val_var_prefix ^ nat_subscript (n - j), k), fastype_of1 (Ts, t))

   365 (* typ list -> term -> bool *)

   366 fun has_heavy_bounds_or_vars Ts t =

   367   let

   368     (* typ list -> bool *)

   369     fun aux [] = false

   370       | aux [T] = is_fun_type T orelse is_pair_type T

   371       | aux _ = true

   372   in aux (map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t)) end

   374 (* hol_context -> typ list -> bool -> int -> int -> term -> term list

   375    -> term list -> term * term list *)

   376 fun pull_out_constr_comb ({thy, stds, ...} : hol_context) Ts relax k level t

   377                          args seen =

   378   let val t_comb = list_comb (t, args) in

   379     case t of

   380       Const x =>

   381       if not relax andalso is_constr thy stds x andalso

   382          not (is_fun_type (fastype_of1 (Ts, t_comb))) andalso

   383          has_heavy_bounds_or_vars Ts t_comb andalso

   384          not (loose_bvar (t_comb, level)) then

   385         let

   386           val (j, seen) = case find_index (curry (op =) t_comb) seen of

   387                             ~1 => (0, t_comb :: seen)

   388                           | j => (j, seen)

   389         in (fresh_value_var Ts k (length seen) j t_comb, seen) end

   390       else

   391         (t_comb, seen)

   392     | _ => (t_comb, seen)

   393   end

   395 (* (term -> term) -> typ list -> int -> term list -> term list *)

   396 fun equations_for_pulled_out_constrs mk_eq Ts k seen =

   397   let val n = length seen in

   398     map2 (fn j => fn t => mk_eq (fresh_value_var Ts k n j t, t))

   399          (index_seq 0 n) seen

   400   end

   402 (* hol_context -> bool -> term -> term *)

   403 fun pull_out_universal_constrs hol_ctxt def t =

   404   let

   405     val k = maxidx_of_term t + 1

   406     (* typ list -> bool -> term -> term list -> term list -> term * term list *)

   407     fun do_term Ts def t args seen =

   408       case t of

   409         (t0 as Const (@{const_name "=="}, _)) $ t1 $ t2 =>

   410         do_eq_or_imp Ts true def t0 t1 t2 seen

   411       | (t0 as @{const "==>"}) $ t1 $ t2 =>

   412         if def then (t, []) else do_eq_or_imp Ts false def t0 t1 t2 seen

   413       | (t0 as Const (@{const_name "op ="}, _)) $ t1 $ t2 =>

   414         do_eq_or_imp Ts true def t0 t1 t2 seen

   415       | (t0 as @{const "op -->"}) $ t1 $ t2 =>

   416         do_eq_or_imp Ts false def t0 t1 t2 seen

   417       | Abs (s, T, t') =>

   418         let val (t', seen) = do_term (T :: Ts) def t' [] seen in

   419           (list_comb (Abs (s, T, t'), args), seen)

   420         end

   421       | t1 $ t2 =>

   422         let val (t2, seen) = do_term Ts def t2 [] seen in

   423           do_term Ts def t1 (t2 :: args) seen

   424         end

   425       | _ => pull_out_constr_comb hol_ctxt Ts def k 0 t args seen

   426     (* typ list -> bool -> bool -> term -> term -> term -> term list

   427        -> term * term list *)

   428     and do_eq_or_imp Ts eq def t0 t1 t2 seen =

   429       let

   430         val (t2, seen) = if eq andalso def then (t2, seen)

   431                          else do_term Ts false t2 [] seen

   432         val (t1, seen) = do_term Ts false t1 [] seen

   433       in (t0 $ t1 $ t2, seen) end

   434     val (concl, seen) = do_term [] def t [] []

   435   in

   436     Logic.list_implies (equations_for_pulled_out_constrs Logic.mk_equals [] k

   437                                                          seen, concl)

   438   end

   440 (* term -> term -> term *)

   441 fun mk_exists v t =

   442   HOLogic.exists_const (fastype_of v) $ lambda v (incr_boundvars 1 t)

   444 (* hol_context -> term -> term *)

   445 fun pull_out_existential_constrs hol_ctxt t =

   446   let

   447     val k = maxidx_of_term t + 1

   448     (* typ list -> int -> term -> term list -> term list -> term * term list *)

   449     fun aux Ts num_exists t args seen =

   450       case t of

   451         (t0 as Const (@{const_name Ex}, _)) $ Abs (s1, T1, t1) =>

   452         let

   453           val (t1, seen') = aux (T1 :: Ts) (num_exists + 1) t1 [] []

   454           val n = length seen'

   455           (* unit -> term list *)

   456           fun vars () = map2 (fresh_value_var Ts k n) (index_seq 0 n) seen'

   457         in

   458           (equations_for_pulled_out_constrs HOLogic.mk_eq Ts k seen'

   459            |> List.foldl s_conj t1 |> fold mk_exists (vars ())

   460            |> curry3 Abs s1 T1 |> curry (op $) t0, seen)

   461         end

   462       | t1 $ t2 =>

   463         let val (t2, seen) = aux Ts num_exists t2 [] seen in

   464           aux Ts num_exists t1 (t2 :: args) seen

   465         end

   466       | Abs (s, T, t') =>

   467         let

   468           val (t', seen) = aux (T :: Ts) 0 t' [] (map (incr_boundvars 1) seen)

   469         in (list_comb (Abs (s, T, t'), args), map (incr_boundvars ~1) seen) end

   470       | _ =>

   471         if num_exists > 0 then

   472           pull_out_constr_comb hol_ctxt Ts false k num_exists t args seen

   473         else

   474           (list_comb (t, args), seen)

   475   in aux [] 0 t [] [] |> fst end

   477 (* hol_context -> bool -> term -> term *)

   478 fun destroy_pulled_out_constrs (hol_ctxt as {thy, stds, ...}) axiom t =

   479   let

   480     (* styp -> int *)

   481     val num_occs_of_var =

   482       fold_aterms (fn Var z => (fn f => fn z' => f z' |> z = z' ? Integer.add 1)

   483                     | _ => I) t (K 0)

   484     (* bool -> term -> term *)

   485     fun aux careful ((t0 as Const (@{const_name "=="}, _)) $ t1 $ t2) =

   486         aux_eq careful true t0 t1 t2

   487       | aux careful ((t0 as @{const "==>"}) $ t1 $ t2) =

   488         t0 $ aux false t1 $ aux careful t2

   489       | aux careful ((t0 as Const (@{const_name "op ="}, _)) $ t1 $ t2) =

   490         aux_eq careful true t0 t1 t2

   491       | aux careful ((t0 as @{const "op -->"}) $ t1 $ t2) =

   492         t0 $ aux false t1 $ aux careful t2

   493       | aux careful (Abs (s, T, t')) = Abs (s, T, aux careful t')

   494       | aux careful (t1 $ t2) = aux careful t1 $ aux careful t2

   495       | aux _ t = t

   496     (* bool -> bool -> term -> term -> term -> term *)

   497     and aux_eq careful pass1 t0 t1 t2 =

   498       ((if careful then

   499           raise SAME ()

   500         else if axiom andalso is_Var t2 andalso

   501                 num_occs_of_var (dest_Var t2) = 1 then

   502           @{const True}

   503         else case strip_comb t2 of

   504           (* The first case is not as general as it could be. *)

   505           (Const (@{const_name PairBox}, _),

   506                   [Const (@{const_name fst}, _) $ Var z1,

   507                    Const (@{const_name snd}, _) $ Var z2]) =>

   508           if z1 = z2 andalso num_occs_of_var z1 = 2 then @{const True}

   509           else raise SAME ()

   510         | (Const (x as (s, T)), args) =>

   511           let val arg_Ts = binder_types T in

   512             if length arg_Ts = length args andalso

   513                (is_constr thy stds x orelse s = @{const_name Pair}) andalso

   514                (not careful orelse not (is_Var t1) orelse

   515                 String.isPrefix val_var_prefix (fst (fst (dest_Var t1)))) then

   516               discriminate_value hol_ctxt x t1 ::

   517               map3 (sel_eq x t1) (index_seq 0 (length args)) arg_Ts args

   518               |> foldr1 s_conj

   519             else

   520               raise SAME ()

   521           end

   522         | _ => raise SAME ())

   523        |> body_type (type_of t0) = prop_T ? HOLogic.mk_Trueprop)

   524       handle SAME () => if pass1 then aux_eq careful false t0 t2 t1

   525                         else t0 $ aux false t2 $ aux false t1

   526     (* styp -> term -> int -> typ -> term -> term *)

   527     and sel_eq x t n nth_T nth_t =

   528       HOLogic.eq_const nth_T $ nth_t

   529                              $ select_nth_constr_arg thy stds x t n nth_T

   530       |> aux false

   531   in aux axiom t end

   533 (** Destruction of universal and existential equalities **)

   535 (* term -> term *)

   536 fun curry_assms (@{const "==>"} $ (@{const Trueprop}

   537                                    $ (@{const "op &"} $ t1 $ t2)) $ t3) =

   538     curry_assms (Logic.list_implies ([t1, t2] |> map HOLogic.mk_Trueprop, t3))

   539   | curry_assms (@{const "==>"} $ t1 $ t2) =

   540     @{const "==>"} $ curry_assms t1 $ curry_assms t2

   541   | curry_assms t = t

   543 (* term -> term *)

   544 val destroy_universal_equalities =

   545   let

   546     (* term list -> (indexname * typ) list -> term -> term *)

   547     fun aux prems zs t =

   548       case t of

   549         @{const "==>"} $ t1 $ t2 => aux_implies prems zs t1 t2

   550       | _ => Logic.list_implies (rev prems, t)

   551     (* term list -> (indexname * typ) list -> term -> term -> term *)

   552     and aux_implies prems zs t1 t2 =

   553       case t1 of

   554         Const (@{const_name "=="}, _) $ Var z $ t' => aux_eq prems zs z t' t1 t2

   555       | @{const Trueprop} $ (Const (@{const_name "op ="}, _) $ Var z $ t') =>

   556         aux_eq prems zs z t' t1 t2

   557       | @{const Trueprop} $ (Const (@{const_name "op ="}, _) $ t' $ Var z) =>

   558         aux_eq prems zs z t' t1 t2

   559       | _ => aux (t1 :: prems) (Term.add_vars t1 zs) t2

   560     (* term list -> (indexname * typ) list -> indexname * typ -> term -> term

   561        -> term -> term *)

   562     and aux_eq prems zs z t' t1 t2 =

   563       if not (member (op =) zs z) andalso

   564          not (exists_subterm (curry (op =) (Var z)) t') then

   565         aux prems zs (subst_free [(Var z, t')] t2)

   566       else

   567         aux (t1 :: prems) (Term.add_vars t1 zs) t2

   568   in aux [] [] end

   570 (* theory -> (typ option * bool) list -> int -> term list -> term list

   571    -> (term * term list) option *)

   572 fun find_bound_assign thy stds j =

   573   let

   574     (* term list -> term list -> (term * term list) option *)

   575     fun do_term _ [] = NONE

   576       | do_term seen (t :: ts) =

   577         let

   578           (* bool -> term -> term -> (term * term list) option *)

   579           fun do_eq pass1 t1 t2 =

   580             (if loose_bvar1 (t2, j) then

   581                if pass1 then do_eq false t2 t1 else raise SAME ()

   582              else case t1 of

   583                Bound j' => if j' = j then SOME (t2, ts @ seen) else raise SAME ()

   584              | Const (s, Type ("fun", [T1, T2])) $ Bound j' =>

   585                if j' = j andalso

   586                   s = nth_sel_name_for_constr_name @{const_name FunBox} 0 then

   587                  SOME (construct_value thy stds (@{const_name FunBox}, T2 --> T1)

   588                                        [t2], ts @ seen)

   589                else

   590                  raise SAME ()

   591              | _ => raise SAME ())

   592             handle SAME () => do_term (t :: seen) ts

   593         in

   594           case t of

   595             Const (@{const_name "op ="}, _) $ t1 $ t2 => do_eq true t1 t2

   596           | _ => do_term (t :: seen) ts

   597         end

   598   in do_term end

   600 (* int -> term -> term -> term *)

   601 fun subst_one_bound j arg t =

   602   let

   603     (* term * int -> term *)

   604     fun aux (Bound i, lev) =

   605         if i < lev then raise SAME ()

   606         else if i = lev then incr_boundvars (lev - j) arg

   607         else Bound (i - 1)

   608       | aux (Abs (a, T, body), lev) = Abs (a, T, aux (body, lev + 1))

   609       | aux (f $ t, lev) =

   610         (aux (f, lev) $ (aux (t, lev) handle SAME () => t)

   611          handle SAME () => f $ aux (t, lev))

   612       | aux _ = raise SAME ()

   613   in aux (t, j) handle SAME () => t end

   615 (* hol_context -> term -> term *)

   616 fun destroy_existential_equalities ({thy, stds, ...} : hol_context) =

   617   let

   618     (* string list -> typ list -> term list -> term *)

   619     fun kill [] [] ts = foldr1 s_conj ts

   620       | kill (s :: ss) (T :: Ts) ts =

   621         (case find_bound_assign thy stds (length ss) [] ts of

   622            SOME (_, []) => @{const True}

   623          | SOME (arg_t, ts) =>

   624            kill ss Ts (map (subst_one_bound (length ss)

   625                                 (incr_bv (~1, length ss + 1, arg_t))) ts)

   626          | NONE =>

   627            Const (@{const_name Ex}, (T --> bool_T) --> bool_T)

   628            $ Abs (s, T, kill ss Ts ts))

   629       | kill _ _ _ = raise UnequalLengths

   630     (* string list -> typ list -> term -> term *)

   631     fun gather ss Ts (Const (@{const_name Ex}, _) $ Abs (s1, T1, t1)) =

   632         gather (ss @ [s1]) (Ts @ [T1]) t1

   633       | gather [] [] (Abs (s, T, t1)) = Abs (s, T, gather [] [] t1)

   634       | gather [] [] (t1 $ t2) = gather [] [] t1 $ gather [] [] t2

   635       | gather [] [] t = t

   636       | gather ss Ts t = kill ss Ts (conjuncts_of (gather [] [] t))

   637   in gather [] [] end

   639 (** Skolemization **)

   641 (* int -> int -> string *)

   642 fun skolem_prefix_for k j =

   643   skolem_prefix ^ string_of_int k ^ "@" ^ string_of_int j ^ name_sep

   645 (* hol_context -> int -> term -> term *)

   646 fun skolemize_term_and_more (hol_ctxt as {thy, def_table, skolems, ...})

   647                             skolem_depth =

   648   let

   649     (* int list -> int list *)

   650     val incrs = map (Integer.add 1)

   651     (* string list -> typ list -> int list -> int -> polarity -> term -> term *)

   652     fun aux ss Ts js depth polar t =

   653       let

   654         (* string -> typ -> string -> typ -> term -> term *)

   655         fun do_quantifier quant_s quant_T abs_s abs_T t =

   656           if not (loose_bvar1 (t, 0)) then

   657             aux ss Ts js depth polar (incr_boundvars ~1 t)

   658           else if depth <= skolem_depth andalso

   659                   is_positive_existential polar quant_s then

   660             let

   661               val j = length (!skolems) + 1

   662               val sko_s = skolem_prefix_for (length js) j ^ abs_s

   663               val _ = Unsynchronized.change skolems (cons (sko_s, ss))

   664               val sko_t = list_comb (Const (sko_s, rev Ts ---> abs_T),

   665                                      map Bound (rev js))

   666               val abs_t = Abs (abs_s, abs_T, aux ss Ts (incrs js) depth polar t)

   667             in

   668               if null js then betapply (abs_t, sko_t)

   669               else Const (@{const_name Let}, abs_T --> quant_T) $ sko_t $ abs_t

   670             end

   671           else

   672             Const (quant_s, quant_T)

   673             $ Abs (abs_s, abs_T,

   674                    if is_higher_order_type abs_T then

   675                      t

   676                    else

   677                      aux (abs_s :: ss) (abs_T :: Ts) (0 :: incrs js)

   678                          (depth + 1) polar t)

   679       in

   680         case t of

   681           Const (s0 as @{const_name all}, T0) $ Abs (s1, T1, t1) =>

   682           do_quantifier s0 T0 s1 T1 t1

   683         | @{const "==>"} $ t1 $ t2 =>

   684           @{const "==>"} $ aux ss Ts js depth (flip_polarity polar) t1

   685           $ aux ss Ts js depth polar t2

   686         | @{const Pure.conjunction} $ t1 $ t2 =>

   687           @{const Pure.conjunction} $ aux ss Ts js depth polar t1

   688           $ aux ss Ts js depth polar t2

   689         | @{const Trueprop} $ t1 =>

   690           @{const Trueprop} $ aux ss Ts js depth polar t1

   691         | @{const Not} $ t1 =>

   692           @{const Not} $ aux ss Ts js depth (flip_polarity polar) t1

   693         | Const (s0 as @{const_name All}, T0) $ Abs (s1, T1, t1) =>

   694           do_quantifier s0 T0 s1 T1 t1

   695         | Const (s0 as @{const_name Ex}, T0) $ Abs (s1, T1, t1) =>

   696           do_quantifier s0 T0 s1 T1 t1

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

   698           @{const "op &"} $ aux ss Ts js depth polar t1

   699           $ aux ss Ts js depth polar t2

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

   701           @{const "op |"} $ aux ss Ts js depth polar t1

   702           $ aux ss Ts js depth polar t2

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

   704           @{const "op -->"} $ aux ss Ts js depth (flip_polarity polar) t1

   705           $ aux ss Ts js depth polar t2

   706         | (t0 as Const (@{const_name Let}, _)) $ t1 $ t2 =>

   707           t0 $ t1 $ aux ss Ts js depth polar t2

   708         | Const (x as (s, T)) =>

   709           if is_inductive_pred hol_ctxt x andalso

   710              not (is_well_founded_inductive_pred hol_ctxt x) then

   711             let

   712               val gfp = (fixpoint_kind_of_const thy def_table x = Gfp)

   713               val (pref, connective, set_oper) =

   714                 if gfp then

   715                   (lbfp_prefix, @{const "op |"},

   716                    @{const_name semilattice_sup_class.sup})

   717                 else

   718                   (ubfp_prefix, @{const "op &"},

   719                    @{const_name semilattice_inf_class.inf})

   720               (* unit -> term *)

   721               fun pos () = unrolled_inductive_pred_const hol_ctxt gfp x

   722                            |> aux ss Ts js depth polar

   723               fun neg () = Const (pref ^ s, T)

   724             in

   725               (case polar |> gfp ? flip_polarity of

   726                  Pos => pos ()

   727                | Neg => neg ()

   728                | Neut =>

   729                  if is_fun_type T then

   730                    let

   731                      val ((trunk_arg_Ts, rump_arg_T), body_T) =

   732                        T |> strip_type |>> split_last

   733                      val set_T = rump_arg_T --> body_T

   734                      (* (unit -> term) -> term *)

   735                      fun app f =

   736                        list_comb (f (),

   737                                   map Bound (length trunk_arg_Ts - 1 downto 0))

   738                    in

   739                      List.foldr absdummy

   740                                 (Const (set_oper, set_T --> set_T --> set_T)

   741                                         $ app pos $ app neg) trunk_arg_Ts

   742                    end

   743                  else

   744                    connective $ pos () $ neg ())

   745             end

   746           else

   747             Const x

   748         | t1 $ t2 =>

   749           betapply (aux ss Ts [] (skolem_depth + 1) polar t1,

   750                     aux ss Ts [] depth Neut t2)

   751         | Abs (s, T, t1) => Abs (s, T, aux ss Ts (incrs js) depth polar t1)

   752         | _ => t

   753       end

   754   in aux [] [] [] 0 Pos end

   756 (** Function specialization **)

   758 (* term -> term list *)

   759 fun params_in_equation (@{const "==>"} $ _ $ t2) = params_in_equation t2

   760   | params_in_equation (@{const Trueprop} $ t1) = params_in_equation t1

   761   | params_in_equation (Const (@{const_name "op ="}, _) $ t1 $ _) =

   762     snd (strip_comb t1)

   763   | params_in_equation _ = []

   765 (* styp -> styp -> int list -> term list -> term list -> term -> term *)

   766 fun specialize_fun_axiom x x' fixed_js fixed_args extra_args t =

   767   let

   768     val k = fold Integer.max (map maxidx_of_term (fixed_args @ extra_args)) 0

   769             + 1

   770     val t = map_aterms (fn Var ((s, i), T) => Var ((s, k + i), T) | t' => t') t

   771     val fixed_params = filter_indices fixed_js (params_in_equation t)

   772     (* term list -> term -> term *)

   773     fun aux args (Abs (s, T, t)) = list_comb (Abs (s, T, aux [] t), args)

   774       | aux args (t1 $ t2) = aux (aux [] t2 :: args) t1

   775       | aux args t =

   776         if t = Const x then

   777           list_comb (Const x', extra_args @ filter_out_indices fixed_js args)

   778         else

   779           let val j = find_index (curry (op =) t) fixed_params in

   780             list_comb (if j >= 0 then nth fixed_args j else t, args)

   781           end

   782   in aux [] t end

   784 (* hol_context -> styp -> (int * term option) list *)

   785 fun static_args_in_term ({ersatz_table, ...} : hol_context) x t =

   786   let

   787     (* term -> term list -> term list -> term list list *)

   788     fun fun_calls (Abs (_, _, t)) _ = fun_calls t []

   789       | fun_calls (t1 $ t2) args = fun_calls t2 [] #> fun_calls t1 (t2 :: args)

   790       | fun_calls t args =

   791         (case t of

   792            Const (x' as (s', T')) =>

   793            x = x' orelse (case AList.lookup (op =) ersatz_table s' of

   794                             SOME s'' => x = (s'', T')

   795                           | NONE => false)

   796          | _ => false) ? cons args

   797     (* term list list -> term list list -> term list -> term list list *)

   798     fun call_sets [] [] vs = [vs]

   799       | call_sets [] uss vs = vs :: call_sets uss [] []

   800       | call_sets ([] :: _) _ _ = []

   801       | call_sets ((t :: ts) :: tss) uss vs =

   802         OrdList.insert TermOrd.term_ord t vs |> call_sets tss (ts :: uss)

   803     val sets = call_sets (fun_calls t [] []) [] []

   804     val indexed_sets = sets ~~ (index_seq 0 (length sets))

   805   in

   806     fold_rev (fn (set, j) =>

   807                  case set of

   808                    [Var _] => AList.lookup (op =) indexed_sets set = SOME j

   809                               ? cons (j, NONE)

   810                  | [t as Const _] => cons (j, SOME t)

   811                  | [t as Free _] => cons (j, SOME t)

   812                  | _ => I) indexed_sets []

   813   end

   814 (* hol_context -> styp -> term list -> (int * term option) list *)

   815 fun static_args_in_terms hol_ctxt x =

   816   map (static_args_in_term hol_ctxt x)

   817   #> fold1 (OrdList.inter (prod_ord int_ord (option_ord TermOrd.term_ord)))

   819 (* (int * term option) list -> (int * term) list -> int list *)

   820 fun overlapping_indices [] _ = []

   821   | overlapping_indices _ [] = []

   822   | overlapping_indices (ps1 as (j1, t1) :: ps1') (ps2 as (j2, t2) :: ps2') =

   823     if j1 < j2 then overlapping_indices ps1' ps2

   824     else if j1 > j2 then overlapping_indices ps1 ps2'

   825     else overlapping_indices ps1' ps2' |> the_default t2 t1 = t2 ? cons j1

   827 (* typ list -> term -> bool *)

   828 fun is_eligible_arg Ts t =

   829   let val bad_Ts = map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t) in

   830     null bad_Ts orelse

   831     (is_higher_order_type (fastype_of1 (Ts, t)) andalso

   832      forall (not o is_higher_order_type) bad_Ts)

   833   end

   835 (* int -> string *)

   836 fun special_prefix_for j = special_prefix ^ string_of_int j ^ name_sep

   838 (* If a constant's definition is picked up deeper than this threshold, we

   839    prevent excessive specialization by not specializing it. *)

   840 val special_max_depth = 20

   842 val bound_var_prefix = "b"

   844 (* hol_context -> int -> term -> term *)

   845 fun specialize_consts_in_term (hol_ctxt as {specialize, simp_table,

   846                                             special_funs, ...}) depth t =

   847   if not specialize orelse depth > special_max_depth then

   848     t

   849   else

   850     let

   851       val blacklist = if depth = 0 then []

   852                       else case term_under_def t of Const x => [x] | _ => []

   853       (* term list -> typ list -> term -> term *)

   854       fun aux args Ts (Const (x as (s, T))) =

   855           ((if not (member (op =) blacklist x) andalso not (null args) andalso

   856                not (String.isPrefix special_prefix s) andalso

   857                is_equational_fun hol_ctxt x then

   858               let

   859                 val eligible_args = filter (is_eligible_arg Ts o snd)

   860                                            (index_seq 0 (length args) ~~ args)

   861                 val _ = not (null eligible_args) orelse raise SAME ()

   862                 val old_axs = equational_fun_axioms hol_ctxt x

   863                               |> map (destroy_existential_equalities hol_ctxt)

   864                 val static_params = static_args_in_terms hol_ctxt x old_axs

   865                 val fixed_js = overlapping_indices static_params eligible_args

   866                 val _ = not (null fixed_js) orelse raise SAME ()

   867                 val fixed_args = filter_indices fixed_js args

   868                 val vars = fold Term.add_vars fixed_args []

   869                            |> sort (TermOrd.fast_indexname_ord o pairself fst)

   870                 val bound_js = fold (fn t => fn js => add_loose_bnos (t, 0, js))

   871                                     fixed_args []

   872                                |> sort int_ord

   873                 val live_args = filter_out_indices fixed_js args

   874                 val extra_args = map Var vars @ map Bound bound_js @ live_args

   875                 val extra_Ts = map snd vars @ filter_indices bound_js Ts

   876                 val k = maxidx_of_term t + 1

   877                 (* int -> term *)

   878                 fun var_for_bound_no j =

   879                   Var ((bound_var_prefix ^

   880                         nat_subscript (find_index (curry (op =) j) bound_js

   881                                        + 1), k),

   882                        nth Ts j)

   883                 val fixed_args_in_axiom =

   884                   map (curry subst_bounds

   885                              (map var_for_bound_no (index_seq 0 (length Ts))))

   886                       fixed_args

   887               in

   888                 case AList.lookup (op =) (!special_funs)

   889                                   (x, fixed_js, fixed_args_in_axiom) of

   890                   SOME x' => list_comb (Const x', extra_args)

   891                 | NONE =>

   892                   let

   893                     val extra_args_in_axiom =

   894                       map Var vars @ map var_for_bound_no bound_js

   895                     val x' as (s', _) =

   896                       (special_prefix_for (length (!special_funs) + 1) ^ s,

   897                        extra_Ts @ filter_out_indices fixed_js (binder_types T)

   898                        ---> body_type T)

   899                     val new_axs =

   900                       map (specialize_fun_axiom x x' fixed_js

   901                                fixed_args_in_axiom extra_args_in_axiom) old_axs

   902                     val _ =

   903                       Unsynchronized.change special_funs

   904                           (cons ((x, fixed_js, fixed_args_in_axiom), x'))

   905                     val _ = add_simps simp_table s' new_axs

   906                   in list_comb (Const x', extra_args) end

   907               end

   908             else

   909               raise SAME ())

   910            handle SAME () => list_comb (Const x, args))

   911         | aux args Ts (Abs (s, T, t)) =

   912           list_comb (Abs (s, T, aux [] (T :: Ts) t), args)

   913         | aux args Ts (t1 $ t2) = aux (aux [] Ts t2 :: args) Ts t1

   914         | aux args _ t = list_comb (t, args)

   915     in aux [] [] t end

   917 type special_triple = int list * term list * styp

   919 val cong_var_prefix = "c"

   921 (* typ -> special_triple -> special_triple -> term *)

   922 fun special_congruence_axiom T (js1, ts1, x1) (js2, ts2, x2) =

   923   let

   924     val (bounds1, bounds2) = pairself (map Var o special_bounds) (ts1, ts2)

   925     val Ts = binder_types T

   926     val max_j = fold (fold Integer.max) [js1, js2] ~1

   927     val (eqs, (args1, args2)) =

   928       fold (fn j => case pairself (fn ps => AList.lookup (op =) ps j)

   929                                   (js1 ~~ ts1, js2 ~~ ts2) of

   930                       (SOME t1, SOME t2) => apfst (cons (t1, t2))

   931                     | (SOME t1, NONE) => apsnd (apsnd (cons t1))

   932                     | (NONE, SOME t2) => apsnd (apfst (cons t2))

   933                     | (NONE, NONE) =>

   934                       let val v = Var ((cong_var_prefix ^ nat_subscript j, 0),

   935                                        nth Ts j) in

   936                         apsnd (pairself (cons v))

   937                       end) (max_j downto 0) ([], ([], []))

   938   in

   939     Logic.list_implies (eqs |> filter_out (op =) |> distinct (op =)

   940                             |> map Logic.mk_equals,

   941                         Logic.mk_equals (list_comb (Const x1, bounds1 @ args1),

   942                                          list_comb (Const x2, bounds2 @ args2)))

   943     |> close_form (* TODO: needed? *)

   944   end

   946 (* hol_context -> styp list -> term list *)

   947 fun special_congruence_axioms (hol_ctxt as {special_funs, ...}) xs =

   948   let

   949     val groups =

   950       !special_funs

   951       |> map (fn ((x, js, ts), x') => (x, (js, ts, x')))

   952       |> AList.group (op =)

   953       |> filter_out (is_equational_fun_surely_complete hol_ctxt o fst)

   954       |> map (fn (x, zs) => (x, zs |> member (op =) xs x ? cons ([], [], x)))

   955     (* special_triple -> int *)

   956     fun generality (js, _, _) = ~(length js)

   957     (* special_triple -> special_triple -> bool *)

   958     fun is_more_specific (j1, t1, x1) (j2, t2, x2) =

   959       x1 <> x2 andalso OrdList.subset (prod_ord int_ord TermOrd.term_ord)

   960                                       (j2 ~~ t2, j1 ~~ t1)

   961     (* typ -> special_triple list -> special_triple list -> special_triple list

   962        -> term list -> term list *)

   963     fun do_pass_1 _ [] [_] [_] = I

   964       | do_pass_1 T skipped _ [] = do_pass_2 T skipped

   965       | do_pass_1 T skipped all (z :: zs) =

   966         case filter (is_more_specific z) all

   967              |> sort (int_ord o pairself generality) of

   968           [] => do_pass_1 T (z :: skipped) all zs

   969         | (z' :: _) => cons (special_congruence_axiom T z z')

   970                        #> do_pass_1 T skipped all zs

   971     (* typ -> special_triple list -> term list -> term list *)

   972     and do_pass_2 _ [] = I

   973       | do_pass_2 T (z :: zs) =

   974         fold (cons o special_congruence_axiom T z) zs #> do_pass_2 T zs

   975   in fold (fn ((_, T), zs) => do_pass_1 T [] zs zs) groups [] end

   977 (** Axiom selection **)

   979 (* Similar to "Refute.specialize_type" but returns all matches rather than only

   980    the first (preorder) match. *)

   981 (* theory -> styp -> term -> term list *)

   982 fun multi_specialize_type thy slack (s, T) t =

   983   let

   984     (* term -> (typ * term) list -> (typ * term) list *)

   985     fun aux (Const (s', T')) ys =

   986         if s = s' then

   987           ys |> (if AList.defined (op =) ys T' then

   988                    I

   989                  else

   990                   cons (T', Refute.monomorphic_term

   991                                 (Sign.typ_match thy (T', T) Vartab.empty) t)

   992                   handle Type.TYPE_MATCH => I

   993                        | Refute.REFUTE _ =>

   994                          if slack then

   995                            I

   996                          else

   997                            raise NOT_SUPPORTED ("too much polymorphism in \

   998                                                 \axiom involving " ^ quote s))

   999         else

  1000           ys

  1001       | aux _ ys = ys

  1002   in map snd (fold_aterms aux t []) end

  1004 (* theory -> bool -> const_table -> styp -> term list *)

  1005 fun nondef_props_for_const thy slack table (x as (s, _)) =

  1006   these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)

  1008 (* 'a Symtab.table -> 'a list *)

  1009 fun all_table_entries table = Symtab.fold (append o snd) table []

  1010 (* const_table -> string -> const_table *)

  1011 fun extra_table table s = Symtab.make [(s, all_table_entries table)]

  1013 (* int -> term -> term *)

  1014 fun eval_axiom_for_term j t =

  1015   Logic.mk_equals (Const (eval_prefix ^ string_of_int j, fastype_of t), t)

  1017 (* term -> bool *)

  1018 val is_trivial_equation = the_default false o try (op aconv o Logic.dest_equals)

  1020 (* Prevents divergence in case of cyclic or infinite axiom dependencies. *)

  1021 val axioms_max_depth = 255

  1023 (* hol_context -> term -> (term list * term list) * (bool * bool) *)

  1024 fun axioms_for_term

  1025         (hol_ctxt as {thy, ctxt, max_bisim_depth, stds, user_axioms,

  1026                       fast_descrs, evals, def_table, nondef_table, user_nondefs,

  1027                       ...}) t =

  1028   let

  1029     type accumulator = styp list * (term list * term list)

  1030     (* (term list * term list -> term list)

  1031        -> ((term list -> term list) -> term list * term list

  1032            -> term list * term list)

  1033        -> int -> term -> accumulator -> accumulator *)

  1034     fun add_axiom get app depth t (accum as (xs, axs)) =

  1035       let

  1036         val t = t |> unfold_defs_in_term hol_ctxt

  1037                   |> skolemize_term_and_more hol_ctxt ~1

  1038       in

  1039         if is_trivial_equation t then

  1040           accum

  1041         else

  1042           let val t' = t |> specialize_consts_in_term hol_ctxt depth in

  1043             if exists (member (op aconv) (get axs)) [t, t'] then accum

  1044             else add_axioms_for_term (depth + 1) t' (xs, app (cons t') axs)

  1045           end

  1046       end

  1047     (* int -> term -> accumulator -> accumulator *)

  1048     and add_def_axiom depth = add_axiom fst apfst depth

  1049     and add_nondef_axiom depth = add_axiom snd apsnd depth

  1050     and add_maybe_def_axiom depth t =

  1051       (if head_of t <> @{const "==>"} then add_def_axiom

  1052        else add_nondef_axiom) depth t

  1053     and add_eq_axiom depth t =

  1054       (if is_constr_pattern_formula thy t then add_def_axiom

  1055        else add_nondef_axiom) depth t

  1056     (* int -> term -> accumulator -> accumulator *)

  1057     and add_axioms_for_term depth t (accum as (xs, axs)) =

  1058       case t of

  1059         t1 $ t2 => accum |> fold (add_axioms_for_term depth) [t1, t2]

  1060       | Const (x as (s, T)) =>

  1061         (if member (op =) xs x orelse

  1062             is_built_in_const thy stds fast_descrs x then

  1063            accum

  1064          else

  1065            let val accum = (x :: xs, axs) in

  1066              if depth > axioms_max_depth then

  1067                raise TOO_LARGE ("Nitpick_Preproc.axioms_for_term.\

  1068                                 \add_axioms_for_term",

  1069                                 "too many nested axioms (" ^

  1070                                 string_of_int depth ^ ")")

  1071              else if Refute.is_const_of_class thy x then

  1072                let

  1073                  val class = Logic.class_of_const s

  1074                  val of_class = Logic.mk_of_class (TVar (("'a", 0), [class]),

  1075                                                    class)

  1076                  val ax1 = try (Refute.specialize_type thy x) of_class

  1077                  val ax2 = Option.map (Refute.specialize_type thy x o snd)

  1078                                       (Refute.get_classdef thy class)

  1079                in

  1080                  fold (add_maybe_def_axiom depth) (map_filter I [ax1, ax2])

  1081                       accum

  1082                end

  1083              else if is_constr thy stds x then

  1084                accum

  1085              else if is_equational_fun hol_ctxt x then

  1086                fold (add_eq_axiom depth) (equational_fun_axioms hol_ctxt x)

  1087                     accum

  1088              else if is_abs_fun thy x then

  1089                if is_quot_type thy (range_type T) then

  1090                  raise NOT_SUPPORTED "\"Abs_\" function of quotient type"

  1091                else

  1092                  accum |> fold (add_nondef_axiom depth)

  1093                                (nondef_props_for_const thy false nondef_table x)

  1094                        |> is_funky_typedef thy (range_type T)

  1095                           ? fold (add_maybe_def_axiom depth)

  1096                                  (nondef_props_for_const thy true

  1097                                                     (extra_table def_table s) x)

  1098              else if is_rep_fun thy x then

  1099                if is_quot_type thy (domain_type T) then

  1100                  raise NOT_SUPPORTED "\"Rep_\" function of quotient type"

  1101                else

  1102                  accum |> fold (add_nondef_axiom depth)

  1103                                (nondef_props_for_const thy false nondef_table x)

  1104                        |> is_funky_typedef thy (range_type T)

  1105                           ? fold (add_maybe_def_axiom depth)

  1106                                  (nondef_props_for_const thy true

  1107                                                     (extra_table def_table s) x)

  1108                        |> add_axioms_for_term depth

  1109                                               (Const (mate_of_rep_fun thy x))

  1110                        |> fold (add_def_axiom depth)

  1111                                (inverse_axioms_for_rep_fun thy x)

  1112              else

  1113                accum |> user_axioms <> SOME false

  1114                         ? fold (add_nondef_axiom depth)

  1115                                (nondef_props_for_const thy false nondef_table x)

  1116            end)

  1117         |> add_axioms_for_type depth T

  1118       | Free (_, T) => add_axioms_for_type depth T accum

  1119       | Var (_, T) => add_axioms_for_type depth T accum

  1120       | Bound _ => accum

  1121       | Abs (_, T, t) => accum |> add_axioms_for_term depth t

  1122                                |> add_axioms_for_type depth T

  1123     (* int -> typ -> accumulator -> accumulator *)

  1124     and add_axioms_for_type depth T =

  1125       case T of

  1126         Type ("fun", Ts) => fold (add_axioms_for_type depth) Ts

  1127       | Type ("*", Ts) => fold (add_axioms_for_type depth) Ts

  1128       | @{typ prop} => I

  1129       | @{typ bool} => I

  1130       | @{typ unit} => I

  1131       | TFree (_, S) => add_axioms_for_sort depth T S

  1132       | TVar (_, S) => add_axioms_for_sort depth T S

  1133       | Type (z as (_, Ts)) =>

  1134         fold (add_axioms_for_type depth) Ts

  1135         #> (if is_pure_typedef thy T then

  1136               fold (add_maybe_def_axiom depth) (optimized_typedef_axioms thy z)

  1137             else if is_quot_type thy T then

  1138               fold (add_def_axiom depth)

  1139                    (optimized_quot_type_axioms ctxt stds z)

  1140             else if max_bisim_depth >= 0 andalso is_codatatype thy T then

  1141               fold (add_maybe_def_axiom depth)

  1142                    (codatatype_bisim_axioms hol_ctxt T)

  1143             else

  1144               I)

  1145     (* int -> typ -> sort -> accumulator -> accumulator *)

  1146     and add_axioms_for_sort depth T S =

  1147       let

  1148         val supers = Sign.complete_sort thy S

  1149         val class_axioms =

  1150           maps (fn class => map prop_of (AxClass.get_info thy class |> #axioms

  1151                                          handle ERROR _ => [])) supers

  1152         val monomorphic_class_axioms =

  1153           map (fn t => case Term.add_tvars t [] of

  1154                          [] => t

  1155                        | [(x, S)] =>

  1156                          Refute.monomorphic_term (Vartab.make [(x, (S, T))]) t

  1157                        | _ => raise TERM ("Nitpick_Preproc.axioms_for_term.\

  1158                                           \add_axioms_for_sort", [t]))

  1159               class_axioms

  1160       in fold (add_nondef_axiom depth) monomorphic_class_axioms end

  1161     val (mono_user_nondefs, poly_user_nondefs) =

  1162       List.partition (null o Term.hidden_polymorphism) user_nondefs

  1163     val eval_axioms = map2 eval_axiom_for_term (index_seq 0 (length evals))

  1164                            evals

  1165     val (xs, (defs, nondefs)) =

  1166       ([], ([], [])) |> add_axioms_for_term 1 t

  1167                      |> fold_rev (add_def_axiom 1) eval_axioms

  1168                      |> user_axioms = SOME true

  1169                         ? fold (add_nondef_axiom 1) mono_user_nondefs

  1170     val defs = defs @ special_congruence_axioms hol_ctxt xs

  1171   in

  1172     ((defs, nondefs), (user_axioms = SOME true orelse null mono_user_nondefs,

  1173                        null poly_user_nondefs))

  1174   end

  1176 (** Simplification of constructor/selector terms **)

  1178 (* theory -> term -> term *)

  1179 fun simplify_constrs_and_sels thy t =

  1180   let

  1181     (* term -> int -> term *)

  1182     fun is_nth_sel_on t' n (Const (s, _) $ t) =

  1183         (t = t' andalso is_sel_like_and_no_discr s andalso

  1184          sel_no_from_name s = n)

  1185       | is_nth_sel_on _ _ _ = false

  1186     (* term -> term list -> term *)

  1187     fun do_term (Const (@{const_name Rep_Frac}, _)

  1188                  $ (Const (@{const_name Abs_Frac}, _) $ t1)) [] = do_term t1 []

  1189       | do_term (Const (@{const_name Abs_Frac}, _)

  1190                  $ (Const (@{const_name Rep_Frac}, _) $ t1)) [] = do_term t1 []

  1191       | do_term (t1 $ t2) args = do_term t1 (do_term t2 [] :: args)

  1192       | do_term (t as Const (x as (s, T))) (args as _ :: _) =

  1193         ((if is_constr_like thy x then

  1194             if length args = num_binder_types T then

  1195               case hd args of

  1196                 Const (_, T') $ t' =>

  1197                 if domain_type T' = body_type T andalso

  1198                    forall (uncurry (is_nth_sel_on t'))

  1199                           (index_seq 0 (length args) ~~ args) then

  1200                   t'

  1201                 else

  1202                   raise SAME ()

  1203               | _ => raise SAME ()

  1204             else

  1205               raise SAME ()

  1206           else if is_sel_like_and_no_discr s then

  1207             case strip_comb (hd args) of

  1208               (Const (x' as (s', T')), ts') =>

  1209               if is_constr_like thy x' andalso

  1210                  constr_name_for_sel_like s = s' andalso

  1211                  not (exists is_pair_type (binder_types T')) then

  1212                 list_comb (nth ts' (sel_no_from_name s), tl args)

  1213               else

  1214                 raise SAME ()

  1215             | _ => raise SAME ()

  1216           else

  1217             raise SAME ())

  1218          handle SAME () => betapplys (t, args))

  1219       | do_term (Abs (s, T, t')) args =

  1220         betapplys (Abs (s, T, do_term t' []), args)

  1221       | do_term t args = betapplys (t, args)

  1222   in do_term t [] end

  1224 (** Quantifier massaging: Distributing quantifiers **)

  1226 (* term -> term *)

  1227 fun distribute_quantifiers t =

  1228   case t of

  1229     (t0 as Const (@{const_name All}, T0)) $ Abs (s, T1, t1) =>

  1230     (case t1 of

  1231        (t10 as @{const "op &"}) $ t11 $ t12 =>

  1232        t10 $ distribute_quantifiers (t0 $ Abs (s, T1, t11))

  1233            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))

  1234      | (t10 as @{const Not}) $ t11 =>

  1235        t10 $ distribute_quantifiers (Const (@{const_name Ex}, T0)

  1236                                      $ Abs (s, T1, t11))

  1237      | t1 =>

  1238        if not (loose_bvar1 (t1, 0)) then

  1239          distribute_quantifiers (incr_boundvars ~1 t1)

  1240        else

  1241          t0 $ Abs (s, T1, distribute_quantifiers t1))

  1242   | (t0 as Const (@{const_name Ex}, T0)) $ Abs (s, T1, t1) =>

  1243     (case distribute_quantifiers t1 of

  1244        (t10 as @{const "op |"}) $ t11 $ t12 =>

  1245        t10 $ distribute_quantifiers (t0 $ Abs (s, T1, t11))

  1246            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))

  1247      | (t10 as @{const "op -->"}) $ t11 $ t12 =>

  1248        t10 $ distribute_quantifiers (Const (@{const_name All}, T0)

  1249                                      $ Abs (s, T1, t11))

  1250            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))

  1251      | (t10 as @{const Not}) $ t11 =>

  1252        t10 $ distribute_quantifiers (Const (@{const_name All}, T0)

  1253                                      $ Abs (s, T1, t11))

  1254      | t1 =>

  1255        if not (loose_bvar1 (t1, 0)) then

  1256          distribute_quantifiers (incr_boundvars ~1 t1)

  1257        else

  1258          t0 $ Abs (s, T1, distribute_quantifiers t1))

  1259   | t1 $ t2 => distribute_quantifiers t1 $ distribute_quantifiers t2

  1260   | Abs (s, T, t') => Abs (s, T, distribute_quantifiers t')

  1261   | _ => t

  1263 (** Quantifier massaging: Pushing quantifiers inward **)

  1265 (* int -> int -> (int -> int) -> term -> term *)

  1266 fun renumber_bounds j n f t =

  1267   case t of

  1268     t1 $ t2 => renumber_bounds j n f t1 $ renumber_bounds j n f t2

  1269   | Abs (s, T, t') => Abs (s, T, renumber_bounds (j + 1) n f t')

  1270   | Bound j' =>

  1271     Bound (if j' >= j andalso j' < j + n then f (j' - j) + j else j')

  1272   | _ => t

  1274 (* Maximum number of quantifiers in a cluster for which the exponential

  1275    algorithm is used. Larger clusters use a heuristic inspired by Claessen &

  1276    Sörensson's polynomial binary splitting procedure (p. 5 of their MODEL 2003

  1277    paper). *)

  1278 val quantifier_cluster_threshold = 7

  1280 (* term -> term *)

  1281 val push_quantifiers_inward =

  1282   let

  1283     (* string -> string list -> typ list -> term -> term *)

  1284     fun aux quant_s ss Ts t =

  1285       (case t of

  1286          Const (s0, _) $ Abs (s1, T1, t1 as _ $ _) =>

  1287          if s0 = quant_s then

  1288            aux s0 (s1 :: ss) (T1 :: Ts) t1

  1289          else if quant_s = "" andalso

  1290                  (s0 = @{const_name All} orelse s0 = @{const_name Ex}) then

  1291            aux s0 [s1] [T1] t1

  1292          else

  1293            raise SAME ()

  1294        | _ => raise SAME ())

  1295       handle SAME () =>

  1296              case t of

  1297                t1 $ t2 =>

  1298                if quant_s = "" then

  1299                  aux "" [] [] t1 $ aux "" [] [] t2

  1300                else

  1301                  let

  1302                    val typical_card = 4

  1303                    (* ('a -> ''b list) -> 'a list -> ''b list *)

  1304                    fun big_union proj ps =

  1305                      fold (fold (insert (op =)) o proj) ps []

  1306                    val (ts, connective) = strip_any_connective t

  1307                    val T_costs =

  1308                      map (bounded_card_of_type 65536 typical_card []) Ts

  1309                    val t_costs = map size_of_term ts

  1310                    val num_Ts = length Ts

  1311                    (* int -> int *)

  1312                    val flip = curry (op -) (num_Ts - 1)

  1313                    val t_boundss = map (map flip o loose_bnos) ts

  1314                    (* (int list * int) list -> int list

  1315                       -> (int list * int) list *)

  1316                    fun merge costly_boundss [] = costly_boundss

  1317                      | merge costly_boundss (j :: js) =

  1318                        let

  1319                          val (yeas, nays) =

  1320                            List.partition (fn (bounds, _) =>

  1321                                               member (op =) bounds j)

  1322                                           costly_boundss

  1323                          val yeas_bounds = big_union fst yeas

  1324                          val yeas_cost = Integer.sum (map snd yeas)

  1325                                          * nth T_costs j

  1326                        in merge ((yeas_bounds, yeas_cost) :: nays) js end

  1327                    (* (int list * int) list -> int list -> int *)

  1328                    val cost = Integer.sum o map snd oo merge

  1329                    (* (int list * int) list -> int list -> int list *)

  1330                    fun heuristically_best_permutation _ [] = []

  1331                      | heuristically_best_permutation costly_boundss js =

  1332                        let

  1333                          val (costly_boundss, (j, js)) =

  1334                            js |> map (`(merge costly_boundss o single))

  1335                               |> sort (int_ord

  1336                                        o pairself (Integer.sum o map snd o fst))

  1337                               |> split_list |>> hd ||> pairf hd tl

  1338                        in

  1339                          j :: heuristically_best_permutation costly_boundss js

  1340                        end

  1341                    val js =

  1342                      if length Ts <= quantifier_cluster_threshold then

  1343                        all_permutations (index_seq 0 num_Ts)

  1344                        |> map (`(cost (t_boundss ~~ t_costs)))

  1345                        |> sort (int_ord o pairself fst) |> hd |> snd

  1346                      else

  1347                        heuristically_best_permutation (t_boundss ~~ t_costs)

  1348                                                       (index_seq 0 num_Ts)

  1349                    val back_js = map (fn j => find_index (curry (op =) j) js)

  1350                                      (index_seq 0 num_Ts)

  1351                    val ts = map (renumber_bounds 0 num_Ts (nth back_js o flip))

  1352                                 ts

  1353                    (* (term * int list) list -> term *)

  1354                    fun mk_connection [] =

  1355                        raise ARG ("Nitpick_Preproc.push_quantifiers_inward.aux.\

  1356                                   \mk_connection", "")

  1357                      | mk_connection ts_cum_bounds =

  1358                        ts_cum_bounds |> map fst

  1359                        |> foldr1 (fn (t1, t2) => connective $ t1 $ t2)

  1360                    (* (term * int list) list -> int list -> term *)

  1361                    fun build ts_cum_bounds [] = ts_cum_bounds |> mk_connection

  1362                      | build ts_cum_bounds (j :: js) =

  1363                        let

  1364                          val (yeas, nays) =

  1365                            List.partition (fn (_, bounds) =>

  1366                                               member (op =) bounds j)

  1367                                           ts_cum_bounds

  1368                            ||> map (apfst (incr_boundvars ~1))

  1369                        in

  1370                          if null yeas then

  1371                            build nays js

  1372                          else

  1373                            let val T = nth Ts (flip j) in

  1374                              build ((Const (quant_s, (T --> bool_T) --> bool_T)

  1375                                      $ Abs (nth ss (flip j), T,

  1376                                             mk_connection yeas),

  1377                                       big_union snd yeas) :: nays) js

  1378                            end

  1379                        end

  1380                  in build (ts ~~ t_boundss) js end

  1381              | Abs (s, T, t') => Abs (s, T, aux "" [] [] t')

  1382              | _ => t

  1383   in aux "" [] [] end

  1385 (** Preprocessor entry point **)

  1387 (* hol_context -> term

  1388    -> ((term list * term list) * (bool * bool)) * term * bool *)

  1389 fun preprocess_term (hol_ctxt as {thy, stds, binary_ints, destroy_constrs,

  1390                                   boxes, skolemize, uncurry, ...}) t =

  1391   let

  1392     val skolem_depth = if skolemize then 4 else ~1

  1393     val (((def_ts, nondef_ts), (got_all_mono_user_axioms, no_poly_user_axioms)),

  1394          core_t) = t |> unfold_defs_in_term hol_ctxt

  1395                      |> close_form

  1396                      |> skolemize_term_and_more hol_ctxt skolem_depth

  1397                      |> specialize_consts_in_term hol_ctxt 0

  1398                      |> `(axioms_for_term hol_ctxt)

  1399     val binarize =

  1400       is_standard_datatype thy stds nat_T andalso

  1401       case binary_ints of

  1402         SOME false => false

  1403       | _ => forall may_use_binary_ints (core_t :: def_ts @ nondef_ts) andalso

  1404              (binary_ints = SOME true orelse

  1405               exists should_use_binary_ints (core_t :: def_ts @ nondef_ts))

  1406     val box = exists (not_equal (SOME false) o snd) boxes

  1407     val table =

  1408       Termtab.empty |> uncurry

  1409         ? fold (add_to_uncurry_table thy) (core_t :: def_ts @ nondef_ts)

  1410     (* bool -> term -> term *)

  1411     fun do_rest def =

  1412       binarize ? binarize_nat_and_int_in_term

  1413       #> uncurry ? uncurry_term table

  1414       #> box ? box_fun_and_pair_in_term hol_ctxt def

  1415       #> destroy_constrs ? (pull_out_universal_constrs hol_ctxt def

  1416                             #> pull_out_existential_constrs hol_ctxt

  1417                             #> destroy_pulled_out_constrs hol_ctxt def)

  1418       #> curry_assms

  1419       #> destroy_universal_equalities

  1420       #> destroy_existential_equalities hol_ctxt

  1421       #> simplify_constrs_and_sels thy

  1422       #> distribute_quantifiers

  1423       #> push_quantifiers_inward

  1424       #> close_form

  1425       #> Term.map_abs_vars shortest_name

  1426   in

  1427     (((map (do_rest true) def_ts, map (do_rest false) nondef_ts),

  1428       (got_all_mono_user_axioms, no_poly_user_axioms)),

  1429      do_rest false core_t, binarize)

  1430   end

  1432 end;

author	blanchet
	Tue, 23 Feb 2010 11:05:32 +0100
changeset 35311	8f9a66fc9f80
parent 35284	9edc2bd6d2bd
child 35312	99cd1f96b400
permissions	-rw-r--r--