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 -> (typ option * bool option) list

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

    14     -> term list * term list * bool * bool * bool

    15 end;

    17 structure Nitpick_Preproc : NITPICK_PREPROC =

    18 struct

    20 open Nitpick_Util

    21 open Nitpick_HOL

    22 open Nitpick_Mono

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

    25 fun is_positive_existential polar quant_s =

    26   (polar = Pos andalso quant_s = @{const_name Ex}) orelse

    27   (polar = Neg andalso quant_s <> @{const_name Ex})

    29 (** Binary coding of integers **)

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

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

    33    binary coding. *)

    34 val binary_int_threshold = 3

    36 (* bool -> term -> bool *)

    37 val may_use_binary_ints =

    38   let

    39     (* bool -> term -> bool *)

    40     fun aux def (Const (@{const_name "=="}, _) $ t1 $ t2) =

    41         aux def t1 andalso aux false t2

    42       | aux def (@{const "==>"} $ t1 $ t2) = aux false t1 andalso aux def t2

    43       | aux def (Const (@{const_name "op ="}, _) $ t1 $ t2) =

    44         aux def t1 andalso aux false t2

    45       | aux def (@{const "op -->"} $ t1 $ t2) = aux false t1 andalso aux def t2

    46       | aux def (t1 $ t2) = aux def t1 andalso aux def t2

    47       | aux def (t as Const (s, _)) =

    48         (not def orelse t <> @{const Suc}) andalso

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

    50                             @{const_name nat_gcd}, @{const_name nat_lcm},

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

    52       | aux def (Abs (_, _, t')) = aux def t'

    53       | aux _ _ = true

    54   in aux end

    55 (* term -> bool *)

    56 val should_use_binary_ints =

    57   let

    58     (* term -> bool *)

    59     fun aux (t1 $ t2) = aux t1 orelse aux t2

    60       | aux (Const (s, T)) =

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

    62          is_integer_type (body_type T)) orelse

    63         (String.isPrefix numeral_prefix s andalso

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

    65            n < ~ binary_int_threshold orelse n > binary_int_threshold

    66          end)

    67       | aux (Abs (_, _, t')) = aux t'

    68       | aux _ = false

    69   in aux end

    71 (** Uncurrying **)

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

    74 fun add_to_uncurry_table thy t =

    75   let

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

    77     fun aux (t1 $ t2) args table =

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

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

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

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

    82            is_constr_like thy x orelse

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

    84           table

    85         else

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

    87       | aux _ _ table = table

    88   in aux t [] end

    90 (* int -> int -> string *)

    91 fun uncurry_prefix_for k j =

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

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

    95 fun uncurry_term table t =

    96   let

    97     (* term -> term list -> term *)

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

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

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

   101         (case Termtab.lookup table t of

   102            SOME n =>

   103            if n >= 2 then

   104              let

   105                val arg_Ts = strip_n_binders n T |> fst

   106                val j =

   107                  if is_iterator_type (hd arg_Ts) then

   108                    1

   109                  else case find_index (not_equal bool_T) arg_Ts of

   110                    ~1 => n

   111                  | j => j

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

   113                  args |> chop n |>> chop j

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

   115                  T |> strip_n_binders n |>> chop j

   116                val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts

   117              in

   118                if n - j < 2 then

   119                  betapplys (t, args)

   120                else

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

   122                                    before_arg_Ts ---> tuple_T --> rest_T),

   123                             before_args @ [mk_flat_tuple tuple_T tuple_args] @

   124                             after_args)

   125              end

   126            else

   127              betapplys (t, args)

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

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

   130   in aux t [] end

   132 (** Boxing **)

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

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

   136                              orig_t =

   137   let

   138     (* typ -> typ *)

   139     fun box_relational_operator_type (Type (@{type_name fun}, Ts)) =

   140         Type (@{type_name fun}, map box_relational_operator_type Ts)

   141       | box_relational_operator_type (Type (@{type_name "*"}, Ts)) =

   142         Type (@{type_name "*"}, map (box_type hol_ctxt InPair) Ts)

   143       | box_relational_operator_type T = T

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

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

   146       case t' of

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

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

   149         (case T' of

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

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

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

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

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

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

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

   157       case t of

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

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

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

   161           let

   162             val (t', args) = strip_comb t1

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

   164           in

   165             case fold (add_boxed_types_for_var z)

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

   167               [T''] => T''

   168             | _ => T

   169           end

   170         else

   171           T

   172       | _ => T

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

   174        -> term -> term *)

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

   176       let

   177         val abs_T' =

   178           if polar = Neut orelse is_positive_existential polar quant_s then

   179             box_type hol_ctxt InFunLHS abs_T

   180           else

   181             abs_T

   182         val body_T = body_type quant_T

   183       in

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

   185         $ Abs (abs_s, abs_T',

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

   187       end

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

   189     and do_equals new_Ts old_Ts s0 T0 t1 t2 =

   190       let

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

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

   193         val T = [T1, T2] |> sort Term_Ord.typ_ord |> List.last

   194       in

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

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

   197       end

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

   199     and do_description_operator s T =

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

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

   202       end

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

   204     and do_term new_Ts old_Ts polar t =

   205       case t of

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

   207         do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1

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

   209         do_equals new_Ts old_Ts s0 T0 t1 t2

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

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

   212         $ do_term new_Ts old_Ts polar t2

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

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

   215         $ do_term new_Ts old_Ts polar t2

   216       | @{const Trueprop} $ t1 =>

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

   218       | @{const Not} $ t1 =>

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

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

   221         do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1

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

   223         do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1

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

   225         do_equals new_Ts old_Ts s0 T0 t1 t2

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

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

   228         $ do_term new_Ts old_Ts polar t2

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

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

   231         $ do_term new_Ts old_Ts polar t2

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

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

   234         $ do_term new_Ts old_Ts polar t2

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

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

   237       | Const (s as @{const_name safe_The}, T) => do_description_operator s T

   238       | Const (s as @{const_name safe_Eps}, T) => do_description_operator s T

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

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

   241                      s = @{const_name trancl} then

   242                     box_relational_operator_type T

   243                   else if String.isPrefix quot_normal_prefix s then

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

   245                       T' --> T'

   246                     end

   247                   else if is_built_in_const thy stds fast_descrs x orelse

   248                           s = @{const_name Sigma} then

   249                     T

   250                   else if is_constr_like thy x then

   251                     box_type hol_ctxt InConstr T

   252                   else if is_sel s

   253                        orelse is_rep_fun thy x then

   254                     box_type hol_ctxt InSel T

   255                   else

   256                     box_type hol_ctxt InExpr T)

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

   258         let

   259           val t1 = do_term new_Ts old_Ts Neut t1

   260           val T1 = fastype_of1 (new_Ts, t1)

   261           val (s1, Ts1) = dest_Type T1

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

   263           val t2 = Abs (s, T', do_term (T' :: new_Ts) (T :: 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           betapply (if s1 = @{type_name fun} then

   268                       t1

   269                     else

   270                       select_nth_constr_arg thy stds

   271                           (@{const_name FunBox},

   272                            Type (@{type_name fun}, Ts1) --> T1) t1 0

   273                           (Type (@{type_name fun}, Ts1)), t2)

   274         end

   275       | t1 $ t2 =>

   276         let

   277           val t1 = do_term new_Ts old_Ts Neut t1

   278           val T1 = fastype_of1 (new_Ts, t1)

   279           val (s1, Ts1) = dest_Type T1

   280           val t2 = do_term new_Ts old_Ts Neut t2

   281           val T2 = fastype_of1 (new_Ts, t2)

   282           val t2 = coerce_term hol_ctxt new_Ts (hd Ts1) T2 t2

   283         in

   284           betapply (if s1 = @{type_name fun} then

   285                       t1

   286                     else

   287                       select_nth_constr_arg thy stds

   288                           (@{const_name FunBox},

   289                            Type (@{type_name fun}, Ts1) --> T1) t1 0

   290                           (Type (@{type_name fun}, Ts1)), t2)

   291         end

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

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

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

   295                 else box_type hol_ctxt InExpr T)

   296       | Bound _ => t

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

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

   299   in do_term [] [] Pos orig_t end

   301 (** Destruction of constructors **)

   303 val val_var_prefix = nitpick_prefix ^ "v"

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

   306 fun fresh_value_var Ts k n j t =

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

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

   310 fun has_heavy_bounds_or_vars Ts t =

   311   let

   312     (* typ list -> bool *)

   313     fun aux [] = false

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

   315       | aux _ = true

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

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

   319    -> term list -> term * term list *)

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

   321                          args seen =

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

   323     case t of

   324       Const x =>

   325       if not relax andalso is_constr thy stds x andalso

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

   327          has_heavy_bounds_or_vars Ts t_comb andalso

   328          not (loose_bvar (t_comb, level)) then

   329         let

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

   331                             ~1 => (0, t_comb :: seen)

   332                           | j => (j, seen)

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

   334       else

   335         (t_comb, seen)

   336     | _ => (t_comb, seen)

   337   end

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

   340 fun equations_for_pulled_out_constrs mk_eq Ts k seen =

   341   let val n = length seen in

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

   343          (index_seq 0 n) seen

   344   end

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

   347 fun pull_out_universal_constrs hol_ctxt def t =

   348   let

   349     val k = maxidx_of_term t + 1

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

   351     fun do_term Ts def t args seen =

   352       case t of

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

   354         do_eq_or_imp Ts true def t0 t1 t2 seen

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

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

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

   358         do_eq_or_imp Ts true def t0 t1 t2 seen

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

   360         do_eq_or_imp Ts false def t0 t1 t2 seen

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

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

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

   364         end

   365       | t1 $ t2 =>

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

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

   368         end

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

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

   371        -> term * term list *)

   372     and do_eq_or_imp Ts eq def t0 t1 t2 seen =

   373       let

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

   375                          else do_term Ts false t2 [] seen

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

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

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

   379   in

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

   381                                                          seen, concl)

   382   end

   384 (* term -> term -> term *)

   385 fun mk_exists v t =

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

   388 (* hol_context -> term -> term *)

   389 fun pull_out_existential_constrs hol_ctxt t =

   390   let

   391     val k = maxidx_of_term t + 1

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

   393     fun aux Ts num_exists t args seen =

   394       case t of

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

   396         let

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

   398           val n = length seen'

   399           (* unit -> term list *)

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

   401         in

   402           (equations_for_pulled_out_constrs HOLogic.mk_eq Ts k seen'

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

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

   405         end

   406       | t1 $ t2 =>

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

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

   409         end

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

   411         let

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

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

   414       | _ =>

   415         if num_exists > 0 then

   416           pull_out_constr_comb hol_ctxt Ts false k num_exists t args seen

   417         else

   418           (list_comb (t, args), seen)

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

   421 val let_var_prefix = nitpick_prefix ^ "l"

   422 val let_inline_threshold = 32

   424 (* int -> typ -> term -> (term -> term) -> term *)

   425 fun hol_let n abs_T body_T f t =

   426   if n * size_of_term t <= let_inline_threshold then

   427     f t

   428   else

   429     let val z = ((let_var_prefix, 0), abs_T) in

   430       Const (@{const_name Let}, abs_T --> (abs_T --> body_T) --> body_T)

   431       $ t $ abs_var z (incr_boundvars 1 (f (Var z)))

   432     end

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

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

   436   let

   437     (* styp -> int *)

   438     val num_occs_of_var =

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

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

   441     (* bool -> term -> term *)

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

   443         aux_eq careful true t0 t1 t2

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

   445         t0 $ aux false t1 $ aux careful t2

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

   447         aux_eq careful true t0 t1 t2

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

   449         t0 $ aux false t1 $ aux careful t2

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

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

   452       | aux _ t = t

   453     (* bool -> bool -> term -> term -> term -> term *)

   454     and aux_eq careful pass1 t0 t1 t2 =

   455       ((if careful then

   456           raise SAME ()

   457         else if axiom andalso is_Var t2 andalso

   458                 num_occs_of_var (dest_Var t2) = 1 then

   459           @{const True}

   460         else case strip_comb t2 of

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

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

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

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

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

   466           else raise SAME ()

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

   468           let

   469             val (arg_Ts, dataT) = strip_type T

   470             val n = length arg_Ts

   471           in

   472             if length args = n andalso

   473                (is_constr thy stds x orelse s = @{const_name Pair} orelse

   474                 x = (@{const_name Suc}, nat_T --> nat_T)) andalso

   475                (not careful orelse not (is_Var t1) orelse

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

   477                 hol_let (n + 1) dataT bool_T

   478                     (fn t1 => discriminate_value hol_ctxt x t1 ::

   479                               map3 (sel_eq x t1) (index_seq 0 n) arg_Ts args

   480                               |> foldr1 s_conj) t1

   481             else

   482               raise SAME ()

   483           end

   484         | _ => raise SAME ())

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

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

   487                         else t0 $ aux false t2 $ aux false t1

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

   489     and sel_eq x t n nth_T nth_t =

   490       HOLogic.eq_const nth_T $ nth_t

   491                              $ select_nth_constr_arg thy stds x t n nth_T

   492       |> aux false

   493   in aux axiom t end

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

   497 (* term -> term *)

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

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

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

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

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

   503   | curry_assms t = t

   505 (* term -> term *)

   506 val destroy_universal_equalities =

   507   let

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

   509     fun aux prems zs t =

   510       case t of

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

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

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

   514     and aux_implies prems zs t1 t2 =

   515       case t1 of

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

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

   518         aux_eq prems zs z t' t1 t2

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

   520         aux_eq prems zs z t' t1 t2

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

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

   523        -> term -> term *)

   524     and aux_eq prems zs z t' t1 t2 =

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

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

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

   528       else

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

   530   in aux [] [] end

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

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

   534 fun find_bound_assign thy stds j =

   535   let

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

   537     fun do_term _ [] = NONE

   538       | do_term seen (t :: ts) =

   539         let

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

   541           fun do_eq pass1 t1 t2 =

   542             (if loose_bvar1 (t2, j) then

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

   544              else case t1 of

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

   546              | Const (s, Type (@{type_name fun}, [T1, T2])) $ Bound j' =>

   547                if j' = j andalso

   548                   s = nth_sel_name_for_constr_name @{const_name FunBox} 0 then

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

   550                                        [t2], ts @ seen)

   551                else

   552                  raise SAME ()

   553              | _ => raise SAME ())

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

   555         in

   556           case t of

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

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

   559         end

   560   in do_term end

   562 (* int -> term -> term -> term *)

   563 fun subst_one_bound j arg t =

   564   let

   565     (* term * int -> term *)

   566     fun aux (Bound i, lev) =

   567         if i < lev then raise SAME ()

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

   569         else Bound (i - 1)

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

   571       | aux (f $ t, lev) =

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

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

   574       | aux _ = raise SAME ()

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

   577 (* hol_context -> term -> term *)

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

   579   let

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

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

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

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

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

   585          | SOME (arg_t, ts) =>

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

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

   588          | NONE =>

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

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

   591       | kill _ _ _ = raise UnequalLengths

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

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

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

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

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

   597       | gather [] [] t = t

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

   599   in gather [] [] end

   601 (** Skolemization **)

   603 (* int -> int -> string *)

   604 fun skolem_prefix_for k j =

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

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

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

   609                             skolem_depth =

   610   let

   611     (* int list -> int list *)

   612     val incrs = map (Integer.add 1)

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

   614     fun aux ss Ts js depth polar t =

   615       let

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

   617         fun do_quantifier quant_s quant_T abs_s abs_T t =

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

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

   620           else if depth <= skolem_depth andalso

   621                   is_positive_existential polar quant_s then

   622             let

   623               val j = length (!skolems) + 1

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

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

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

   627                                      map Bound (rev js))

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

   629             in

   630               if null js then betapply (abs_t, sko_t)

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

   632             end

   633           else

   634             Const (quant_s, quant_T)

   635             $ Abs (abs_s, abs_T,

   636                    if is_higher_order_type abs_T then

   637                      t

   638                    else

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

   640                          (depth + 1) polar t)

   641       in

   642         case t of

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

   644           do_quantifier s0 T0 s1 T1 t1

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

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

   647           $ aux ss Ts js depth polar t2

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

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

   650           $ aux ss Ts js depth polar t2

   651         | @{const Trueprop} $ t1 =>

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

   653         | @{const Not} $ t1 =>

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

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

   656           do_quantifier s0 T0 s1 T1 t1

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

   658           do_quantifier s0 T0 s1 T1 t1

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

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

   661           $ aux ss Ts js depth polar t2

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

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

   664           $ aux ss Ts js depth polar t2

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

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

   667           $ aux ss Ts js depth polar t2

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

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

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

   671           if is_inductive_pred hol_ctxt x andalso

   672              not (is_well_founded_inductive_pred hol_ctxt x) then

   673             let

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

   675               val (pref, connective, set_oper) =

   676                 if gfp then

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

   678                    @{const_name semilattice_sup_class.sup})

   679                 else

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

   681                    @{const_name semilattice_inf_class.inf})

   682               (* unit -> term *)

   683               fun pos () = unrolled_inductive_pred_const hol_ctxt gfp x

   684                            |> aux ss Ts js depth polar

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

   686             in

   687               (case polar |> gfp ? flip_polarity of

   688                  Pos => pos ()

   689                | Neg => neg ()

   690                | Neut =>

   691                  if is_fun_type T then

   692                    let

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

   694                        T |> strip_type |>> split_last

   695                      val set_T = rump_arg_T --> body_T

   696                      (* (unit -> term) -> term *)

   697                      fun app f =

   698                        list_comb (f (),

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

   700                    in

   701                      List.foldr absdummy

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

   703                                         $ app pos $ app neg) trunk_arg_Ts

   704                    end

   705                  else

   706                    connective $ pos () $ neg ())

   707             end

   708           else

   709             Const x

   710         | t1 $ t2 =>

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

   712                     aux ss Ts [] depth Neut t2)

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

   714         | _ => t

   715       end

   716   in aux [] [] [] 0 Pos end

   718 (** Function specialization **)

   720 (* term -> term list *)

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

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

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

   724     snd (strip_comb t1)

   725   | params_in_equation _ = []

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

   728 fun specialize_fun_axiom x x' fixed_js fixed_args extra_args t =

   729   let

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

   731             + 1

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

   733     val fixed_params = filter_indices fixed_js (params_in_equation t)

   734     (* term list -> term -> term *)

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

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

   737       | aux args t =

   738         if t = Const x then

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

   740         else

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

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

   743           end

   744   in aux [] t end

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

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

   748   let

   749     (* term -> term list -> term list -> term list list *)

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

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

   752       | fun_calls t args =

   753         (case t of

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

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

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

   757                           | NONE => false)

   758          | _ => false) ? cons args

   759     (* term list list -> term list list -> term list -> term list list *)

   760     fun call_sets [] [] vs = [vs]

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

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

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

   764         OrdList.insert Term_Ord.term_ord t vs |> call_sets tss (ts :: uss)

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

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

   767   in

   768     fold_rev (fn (set, j) =>

   769                  case set of

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

   771                               ? cons (j, NONE)

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

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

   774                  | _ => I) indexed_sets []

   775   end

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

   777 fun static_args_in_terms hol_ctxt x =

   778   map (static_args_in_term hol_ctxt x)

   779   #> fold1 (OrdList.inter (prod_ord int_ord (option_ord Term_Ord.term_ord)))

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

   782 fun overlapping_indices [] _ = []

   783   | overlapping_indices _ [] = []

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

   785     if j1 < j2 then overlapping_indices ps1' ps2

   786     else if j1 > j2 then overlapping_indices ps1 ps2'

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

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

   790 fun is_eligible_arg Ts t =

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

   792     null bad_Ts orelse

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

   794      forall (not o is_higher_order_type) bad_Ts)

   795   end

   797 (* int -> string *)

   798 fun special_prefix_for j = special_prefix ^ string_of_int j ^ name_sep

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

   801    prevent excessive specialization by not specializing it. *)

   802 val special_max_depth = 20

   804 val bound_var_prefix = "b"

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

   807 fun specialize_consts_in_term (hol_ctxt as {specialize, simp_table,

   808                                             special_funs, ...}) depth t =

   809   if not specialize orelse depth > special_max_depth then

   810     t

   811   else

   812     let

   813       val blacklist = if depth = 0 then []

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

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

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

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

   818                not (String.isPrefix special_prefix s) andalso

   819                is_equational_fun hol_ctxt x then

   820               let

   821                 val eligible_args = filter (is_eligible_arg Ts o snd)

   822                                            (index_seq 0 (length args) ~~ args)

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

   824                 val old_axs = equational_fun_axioms hol_ctxt x

   825                               |> map (destroy_existential_equalities hol_ctxt)

   826                 val static_params = static_args_in_terms hol_ctxt x old_axs

   827                 val fixed_js = overlapping_indices static_params eligible_args

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

   829                 val fixed_args = filter_indices fixed_js args

   830                 val vars = fold Term.add_vars fixed_args []

   831                            |> sort (Term_Ord.fast_indexname_ord o pairself fst)

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

   833                                     fixed_args []

   834                                |> sort int_ord

   835                 val live_args = filter_out_indices fixed_js args

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

   837                 val extra_Ts = map snd vars @ filter_indices bound_js Ts

   838                 val k = maxidx_of_term t + 1

   839                 (* int -> term *)

   840                 fun var_for_bound_no j =

   841                   Var ((bound_var_prefix ^

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

   843                                        + 1), k),

   844                        nth Ts j)

   845                 val fixed_args_in_axiom =

   846                   map (curry subst_bounds

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

   848                       fixed_args

   849               in

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

   851                                   (x, fixed_js, fixed_args_in_axiom) of

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

   853                 | NONE =>

   854                   let

   855                     val extra_args_in_axiom =

   856                       map Var vars @ map var_for_bound_no bound_js

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

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

   859                        extra_Ts @ filter_out_indices fixed_js (binder_types T)

   860                        ---> body_type T)

   861                     val new_axs =

   862                       map (specialize_fun_axiom x x' fixed_js

   863                                fixed_args_in_axiom extra_args_in_axiom) old_axs

   864                     val _ =

   865                       Unsynchronized.change special_funs

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

   867                     val _ = add_simps simp_table s' new_axs

   868                   in list_comb (Const x', extra_args) end

   869               end

   870             else

   871               raise SAME ())

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

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

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

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

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

   877     in aux [] [] t end

   879 type special_triple = int list * term list * styp

   881 val cong_var_prefix = "c"

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

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

   885   let

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

   887     val Ts = binder_types T

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

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

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

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

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

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

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

   895                     | (NONE, NONE) =>

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

   897                                        nth Ts j) in

   898                         apsnd (pairself (cons v))

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

   900   in

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

   902                             |> map Logic.mk_equals,

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

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

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

   906   end

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

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

   910   let

   911     val groups =

   912       !special_funs

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

   914       |> AList.group (op =)

   915       |> filter_out (is_equational_fun_surely_complete hol_ctxt o fst)

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

   917     (* special_triple -> int *)

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

   919     (* special_triple -> special_triple -> bool *)

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

   921       x1 <> x2 andalso OrdList.subset (prod_ord int_ord Term_Ord.term_ord)

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

   923     (* typ -> special_triple list -> special_triple list -> special_triple list

   924        -> term list -> term list *)

   925     fun do_pass_1 _ [] [_] [_] = I

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

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

   928         case filter (is_more_specific z) all

   929              |> sort (int_ord o pairself generality) of

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

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

   932                        #> do_pass_1 T skipped all zs

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

   934     and do_pass_2 _ [] = I

   935       | do_pass_2 T (z :: zs) =

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

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

   939 (** Axiom selection **)

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

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

   943 (* const_table -> string -> const_table *)

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

   946 (* int -> term -> term *)

   947 fun eval_axiom_for_term j t =

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

   950 (* term -> bool *)

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

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

   954 val axioms_max_depth = 255

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

   957 fun axioms_for_term

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

   959                       fast_descrs, evals, def_table, nondef_table,

   960                       choice_spec_table, user_nondefs, ...}) t =

   961   let

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

   963     (* (term list * term list -> term list)

   964        -> ((term list -> term list) -> term list * term list

   965            -> term list * term list)

   966        -> int -> term -> accumulator -> accumulator *)

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

   968       let

   969         val t = t |> unfold_defs_in_term hol_ctxt

   970                   |> skolemize_term_and_more hol_ctxt ~1

   971       in

   972         if is_trivial_equation t then

   973           accum

   974         else

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

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

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

   978           end

   979       end

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

   981     and add_def_axiom depth = add_axiom fst apfst depth

   982     and add_nondef_axiom depth = add_axiom snd apsnd depth

   983     and add_maybe_def_axiom depth t =

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

   985        else add_nondef_axiom) depth t

   986     and add_eq_axiom depth t =

   987       (if is_constr_pattern_formula thy t then add_def_axiom

   988        else add_nondef_axiom) depth t

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

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

   991       case t of

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

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

   994         (if member (op =) xs x orelse

   995             is_built_in_const thy stds fast_descrs x then

   996            accum

   997          else

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

   999              if depth > axioms_max_depth then

  1000                raise TOO_LARGE ("Nitpick_Preproc.axioms_for_term.\

  1001                                 \add_axioms_for_term",

  1002                                 "too many nested axioms (" ^

  1003                                 string_of_int depth ^ ")")

  1004              else if Refute.is_const_of_class thy x then

  1005                let

  1006                  val class = Logic.class_of_const s

  1007                  val of_class = Logic.mk_of_class (TVar (("'a", 0), [class]),

  1008                                                    class)

  1009                  val ax1 = try (Refute.specialize_type thy x) of_class

  1010                  val ax2 = Option.map (Refute.specialize_type thy x o snd)

  1011                                       (Refute.get_classdef thy class)

  1012                in

  1013                  fold (add_maybe_def_axiom depth) (map_filter I [ax1, ax2])

  1014                       accum

  1015                end

  1016              else if is_constr thy stds x then

  1017                accum

  1018              else if is_equational_fun hol_ctxt x then

  1019                fold (add_eq_axiom depth) (equational_fun_axioms hol_ctxt x)

  1020                     accum

  1021              else if is_choice_spec_fun hol_ctxt x then

  1022                fold (add_nondef_axiom depth)

  1023                     (nondef_props_for_const thy true choice_spec_table x) accum

  1024              else if is_abs_fun thy x then

  1025                if is_quot_type thy (range_type T) then

  1026                  raise NOT_SUPPORTED "\"Abs_\" function of quotient type"

  1027                else

  1028                  accum |> fold (add_nondef_axiom depth)

  1029                                (nondef_props_for_const thy false nondef_table x)

  1030                        |> (is_funky_typedef thy (range_type T) orelse

  1031                            range_type T = nat_T)

  1032                           ? fold (add_maybe_def_axiom depth)

  1033                                  (nondef_props_for_const thy true

  1034                                                     (extra_table def_table s) x)

  1035              else if is_rep_fun thy x then

  1036                if is_quot_type thy (domain_type T) then

  1037                  raise NOT_SUPPORTED "\"Rep_\" function of quotient type"

  1038                else

  1039                  accum |> fold (add_nondef_axiom depth)

  1040                                (nondef_props_for_const thy false nondef_table x)

  1041                        |> (is_funky_typedef thy (range_type T) orelse

  1042                            range_type T = nat_T)

  1043                           ? fold (add_maybe_def_axiom depth)

  1044                                  (nondef_props_for_const thy true

  1045                                                     (extra_table def_table s) x)

  1046                        |> add_axioms_for_term depth

  1047                                               (Const (mate_of_rep_fun thy x))

  1048                        |> fold (add_def_axiom depth)

  1049                                (inverse_axioms_for_rep_fun thy x)

  1050              else

  1051                accum |> user_axioms <> SOME false

  1052                         ? fold (add_nondef_axiom depth)

  1053                                (nondef_props_for_const thy false nondef_table x)

  1054            end)

  1055         |> add_axioms_for_type depth T

  1056       | Free (_, T) => add_axioms_for_type depth T accum

  1057       | Var (_, T) => add_axioms_for_type depth T accum

  1058       | Bound _ => accum

  1059       | Abs (_, T, t) => accum |> add_axioms_for_term depth t

  1060                                |> add_axioms_for_type depth T

  1061     (* int -> typ -> accumulator -> accumulator *)

  1062     and add_axioms_for_type depth T =

  1063       case T of

  1064         Type (@{type_name fun}, Ts) => fold (add_axioms_for_type depth) Ts

  1065       | Type (@{type_name "*"}, Ts) => fold (add_axioms_for_type depth) Ts

  1066       | @{typ prop} => I

  1067       | @{typ bool} => I

  1068       | @{typ unit} => I

  1069       | TFree (_, S) => add_axioms_for_sort depth T S

  1070       | TVar (_, S) => add_axioms_for_sort depth T S

  1071       | Type (z as (_, Ts)) =>

  1072         fold (add_axioms_for_type depth) Ts

  1073         #> (if is_pure_typedef thy T then

  1074               fold (add_maybe_def_axiom depth) (optimized_typedef_axioms thy z)

  1075             else if is_quot_type thy T then

  1076               fold (add_def_axiom depth)

  1077                    (optimized_quot_type_axioms ctxt stds z)

  1078             else if max_bisim_depth >= 0 andalso is_codatatype thy T then

  1079               fold (add_maybe_def_axiom depth)

  1080                    (codatatype_bisim_axioms hol_ctxt T)

  1081             else

  1082               I)

  1083     (* int -> typ -> sort -> accumulator -> accumulator *)

  1084     and add_axioms_for_sort depth T S =

  1085       let

  1086         val supers = Sign.complete_sort thy S

  1087         val class_axioms =

  1088           maps (fn class => map prop_of (AxClass.get_info thy class |> #axioms

  1089                                          handle ERROR _ => [])) supers

  1090         val monomorphic_class_axioms =

  1091           map (fn t => case Term.add_tvars t [] of

  1092                          [] => t

  1093                        | [(x, S)] =>

  1094                          Refute.monomorphic_term (Vartab.make [(x, (S, T))]) t

  1095                        | _ => raise TERM ("Nitpick_Preproc.axioms_for_term.\

  1096                                           \add_axioms_for_sort", [t]))

  1097               class_axioms

  1098       in fold (add_nondef_axiom depth) monomorphic_class_axioms end

  1099     val (mono_user_nondefs, poly_user_nondefs) =

  1100       List.partition (null o Term.hidden_polymorphism) user_nondefs

  1101     val eval_axioms = map2 eval_axiom_for_term (index_seq 0 (length evals))

  1102                            evals

  1103     val (xs, (defs, nondefs)) =

  1104       ([], ([], [])) |> add_axioms_for_term 1 t

  1105                      |> fold_rev (add_def_axiom 1) eval_axioms

  1106                      |> user_axioms = SOME true

  1107                         ? fold (add_nondef_axiom 1) mono_user_nondefs

  1108     val defs = defs @ special_congruence_axioms hol_ctxt xs

  1109     val got_all_mono_user_axioms =

  1110       (user_axioms = SOME true orelse null mono_user_nondefs)

  1111   in (t :: nondefs, defs, got_all_mono_user_axioms, null poly_user_nondefs) end

  1113 (** Simplification of constructor/selector terms **)

  1115 (* theory -> term -> term *)

  1116 fun simplify_constrs_and_sels thy t =

  1117   let

  1118     (* term -> int -> term *)

  1119     fun is_nth_sel_on t' n (Const (s, _) $ t) =

  1120         (t = t' andalso is_sel_like_and_no_discr s andalso

  1121          sel_no_from_name s = n)

  1122       | is_nth_sel_on _ _ _ = false

  1123     (* term -> term list -> term *)

  1124     fun do_term (Const (@{const_name Rep_Frac}, _)

  1125                  $ (Const (@{const_name Abs_Frac}, _) $ t1)) [] = do_term t1 []

  1126       | do_term (Const (@{const_name Abs_Frac}, _)

  1127                  $ (Const (@{const_name Rep_Frac}, _) $ t1)) [] = do_term t1 []

  1128       | do_term (t1 $ t2) args = do_term t1 (do_term t2 [] :: args)

  1129       | do_term (t as Const (x as (s, T))) (args as _ :: _) =

  1130         ((if is_constr_like thy x then

  1131             if length args = num_binder_types T then

  1132               case hd args of

  1133                 Const (_, T') $ t' =>

  1134                 if domain_type T' = body_type T andalso

  1135                    forall (uncurry (is_nth_sel_on t'))

  1136                           (index_seq 0 (length args) ~~ args) then

  1137                   t'

  1138                 else

  1139                   raise SAME ()

  1140               | _ => raise SAME ()

  1141             else

  1142               raise SAME ()

  1143           else if is_sel_like_and_no_discr s then

  1144             case strip_comb (hd args) of

  1145               (Const (x' as (s', T')), ts') =>

  1146               if is_constr_like thy x' andalso

  1147                  constr_name_for_sel_like s = s' andalso

  1148                  not (exists is_pair_type (binder_types T')) then

  1149                 list_comb (nth ts' (sel_no_from_name s), tl args)

  1150               else

  1151                 raise SAME ()

  1152             | _ => raise SAME ()

  1153           else

  1154             raise SAME ())

  1155          handle SAME () => betapplys (t, args))

  1156       | do_term (Abs (s, T, t')) args =

  1157         betapplys (Abs (s, T, do_term t' []), args)

  1158       | do_term t args = betapplys (t, args)

  1159   in do_term t [] end

  1161 (** Quantifier massaging: Distributing quantifiers **)

  1163 (* term -> term *)

  1164 fun distribute_quantifiers t =

  1165   case t of

  1166     (t0 as Const (@{const_name All}, T0)) $ Abs (s, T1, t1) =>

  1167     (case t1 of

  1168        (t10 as @{const "op &"}) $ t11 $ t12 =>

  1169        t10 $ distribute_quantifiers (t0 $ Abs (s, T1, t11))

  1170            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))

  1171      | (t10 as @{const Not}) $ t11 =>

  1172        t10 $ distribute_quantifiers (Const (@{const_name Ex}, T0)

  1173                                      $ Abs (s, T1, t11))

  1174      | t1 =>

  1175        if not (loose_bvar1 (t1, 0)) then

  1176          distribute_quantifiers (incr_boundvars ~1 t1)

  1177        else

  1178          t0 $ Abs (s, T1, distribute_quantifiers t1))

  1179   | (t0 as Const (@{const_name Ex}, T0)) $ Abs (s, T1, t1) =>

  1180     (case distribute_quantifiers t1 of

  1181        (t10 as @{const "op |"}) $ t11 $ t12 =>

  1182        t10 $ distribute_quantifiers (t0 $ Abs (s, T1, t11))

  1183            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))

  1184      | (t10 as @{const "op -->"}) $ t11 $ t12 =>

  1185        t10 $ distribute_quantifiers (Const (@{const_name All}, T0)

  1186                                      $ Abs (s, T1, t11))

  1187            $ distribute_quantifiers (t0 $ Abs (s, T1, t12))

  1188      | (t10 as @{const Not}) $ t11 =>

  1189        t10 $ distribute_quantifiers (Const (@{const_name All}, T0)

  1190                                      $ Abs (s, T1, t11))

  1191      | t1 =>

  1192        if not (loose_bvar1 (t1, 0)) then

  1193          distribute_quantifiers (incr_boundvars ~1 t1)

  1194        else

  1195          t0 $ Abs (s, T1, distribute_quantifiers t1))

  1196   | t1 $ t2 => distribute_quantifiers t1 $ distribute_quantifiers t2

  1197   | Abs (s, T, t') => Abs (s, T, distribute_quantifiers t')

  1198   | _ => t

  1200 (** Quantifier massaging: Pushing quantifiers inward **)

  1202 (* int -> int -> (int -> int) -> term -> term *)

  1203 fun renumber_bounds j n f t =

  1204   case t of

  1205     t1 $ t2 => renumber_bounds j n f t1 $ renumber_bounds j n f t2

  1206   | Abs (s, T, t') => Abs (s, T, renumber_bounds (j + 1) n f t')

  1207   | Bound j' =>

  1208     Bound (if j' >= j andalso j' < j + n then f (j' - j) + j else j')

  1209   | _ => t

  1211 (* Maximum number of quantifiers in a cluster for which the exponential

  1212    algorithm is used. Larger clusters use a heuristic inspired by Claessen &

  1213    Soerensson's polynomial binary splitting procedure (p. 5 of their MODEL 2003

  1214    paper). *)

  1215 val quantifier_cluster_threshold = 7

  1217 (* term -> term *)

  1218 val push_quantifiers_inward =

  1219   let

  1220     (* string -> string list -> typ list -> term -> term *)

  1221     fun aux quant_s ss Ts t =

  1222       (case t of

  1223          Const (s0, _) $ Abs (s1, T1, t1 as _ $ _) =>

  1224          if s0 = quant_s then

  1225            aux s0 (s1 :: ss) (T1 :: Ts) t1

  1226          else if quant_s = "" andalso

  1227                  (s0 = @{const_name All} orelse s0 = @{const_name Ex}) then

  1228            aux s0 [s1] [T1] t1

  1229          else

  1230            raise SAME ()

  1231        | _ => raise SAME ())

  1232       handle SAME () =>

  1233              case t of

  1234                t1 $ t2 =>

  1235                if quant_s = "" then

  1236                  aux "" [] [] t1 $ aux "" [] [] t2

  1237                else

  1238                  let

  1239                    val typical_card = 4

  1240                    (* ('a -> ''b list) -> 'a list -> ''b list *)

  1241                    fun big_union proj ps =

  1242                      fold (fold (insert (op =)) o proj) ps []

  1243                    val (ts, connective) = strip_any_connective t

  1244                    val T_costs =

  1245                      map (bounded_card_of_type 65536 typical_card []) Ts

  1246                    val t_costs = map size_of_term ts

  1247                    val num_Ts = length Ts

  1248                    (* int -> int *)

  1249                    val flip = curry (op -) (num_Ts - 1)

  1250                    val t_boundss = map (map flip o loose_bnos) ts

  1251                    (* (int list * int) list -> int list

  1252                       -> (int list * int) list *)

  1253                    fun merge costly_boundss [] = costly_boundss

  1254                      | merge costly_boundss (j :: js) =

  1255                        let

  1256                          val (yeas, nays) =

  1257                            List.partition (fn (bounds, _) =>

  1258                                               member (op =) bounds j)

  1259                                           costly_boundss

  1260                          val yeas_bounds = big_union fst yeas

  1261                          val yeas_cost = Integer.sum (map snd yeas)

  1262                                          * nth T_costs j

  1263                        in merge ((yeas_bounds, yeas_cost) :: nays) js end

  1264                    (* (int list * int) list -> int list -> int *)

  1265                    val cost = Integer.sum o map snd oo merge

  1266                    (* (int list * int) list -> int list -> int list *)

  1267                    fun heuristically_best_permutation _ [] = []

  1268                      | heuristically_best_permutation costly_boundss js =

  1269                        let

  1270                          val (costly_boundss, (j, js)) =

  1271                            js |> map (`(merge costly_boundss o single))

  1272                               |> sort (int_ord

  1273                                        o pairself (Integer.sum o map snd o fst))

  1274                               |> split_list |>> hd ||> pairf hd tl

  1275                        in

  1276                          j :: heuristically_best_permutation costly_boundss js

  1277                        end

  1278                    val js =

  1279                      if length Ts <= quantifier_cluster_threshold then

  1280                        all_permutations (index_seq 0 num_Ts)

  1281                        |> map (`(cost (t_boundss ~~ t_costs)))

  1282                        |> sort (int_ord o pairself fst) |> hd |> snd

  1283                      else

  1284                        heuristically_best_permutation (t_boundss ~~ t_costs)

  1285                                                       (index_seq 0 num_Ts)

  1286                    val back_js = map (fn j => find_index (curry (op =) j) js)

  1287                                      (index_seq 0 num_Ts)

  1288                    val ts = map (renumber_bounds 0 num_Ts (nth back_js o flip))

  1289                                 ts

  1290                    (* (term * int list) list -> term *)

  1291                    fun mk_connection [] =

  1292                        raise ARG ("Nitpick_Preproc.push_quantifiers_inward.aux.\

  1293                                   \mk_connection", "")

  1294                      | mk_connection ts_cum_bounds =

  1295                        ts_cum_bounds |> map fst

  1296                        |> foldr1 (fn (t1, t2) => connective $ t1 $ t2)

  1297                    (* (term * int list) list -> int list -> term *)

  1298                    fun build ts_cum_bounds [] = ts_cum_bounds |> mk_connection

  1299                      | build ts_cum_bounds (j :: js) =

  1300                        let

  1301                          val (yeas, nays) =

  1302                            List.partition (fn (_, bounds) =>

  1303                                               member (op =) bounds j)

  1304                                           ts_cum_bounds

  1305                            ||> map (apfst (incr_boundvars ~1))

  1306                        in

  1307                          if null yeas then

  1308                            build nays js

  1309                          else

  1310                            let val T = nth Ts (flip j) in

  1311                              build ((Const (quant_s, (T --> bool_T) --> bool_T)

  1312                                      $ Abs (nth ss (flip j), T,

  1313                                             mk_connection yeas),

  1314                                       big_union snd yeas) :: nays) js

  1315                            end

  1316                        end

  1317                  in build (ts ~~ t_boundss) js end

  1318              | Abs (s, T, t') => Abs (s, T, aux "" [] [] t')

  1319              | _ => t

  1320   in aux "" [] [] end

  1322 (** Inference of finite functions **)

  1324 (* hol_context -> bool -> (typ option * bool option) list

  1325    -> (typ option * bool option) list -> term list * term list

  1326    -> term list * term list *)

  1327 fun finitize_all_types_of_funs (hol_ctxt as {thy, ...}) binarize finitizes monos

  1328                                (nondef_ts, def_ts) =

  1329   let

  1330     val Ts = ground_types_in_terms hol_ctxt binarize (nondef_ts @ def_ts)

  1331              |> filter_out (fn Type (@{type_name fun_box}, _) => true

  1332                              | @{typ signed_bit} => true

  1333                              | @{typ unsigned_bit} => true

  1334                              | T => is_small_finite_type hol_ctxt T orelse

  1335                                     triple_lookup (type_match thy) monos T

  1336                                     = SOME (SOME false))

  1337   in fold (finitize_funs hol_ctxt binarize finitizes) Ts (nondef_ts, def_ts) end

  1339 (** Preprocessor entry point **)

  1341 (* hol_context -> (typ option * bool option) list

  1342    -> (typ option * bool option) list -> term

  1343    -> term list * term list * bool * bool * bool *)

  1344 fun preprocess_term (hol_ctxt as {thy, stds, binary_ints, destroy_constrs,

  1345                                   boxes, skolemize, uncurry, ...})

  1346                     finitizes monos t =

  1347   let

  1348     val skolem_depth = if skolemize then 4 else ~1

  1349     val (nondef_ts, def_ts, got_all_mono_user_axioms, no_poly_user_axioms) =

  1350       t |> unfold_defs_in_term hol_ctxt

  1351         |> close_form

  1352         |> skolemize_term_and_more hol_ctxt skolem_depth

  1353         |> specialize_consts_in_term hol_ctxt 0

  1354         |> axioms_for_term hol_ctxt

  1355     val binarize =

  1356       is_standard_datatype thy stds nat_T andalso

  1357       case binary_ints of

  1358         SOME false => false

  1359       | _ => forall (may_use_binary_ints false) nondef_ts andalso

  1360              forall (may_use_binary_ints true) def_ts andalso

  1361              (binary_ints = SOME true orelse

  1362               exists should_use_binary_ints (nondef_ts @ def_ts))

  1363     val box = exists (not_equal (SOME false) o snd) boxes

  1364     val uncurry = uncurry andalso box

  1365     val table =

  1366       Termtab.empty

  1367       |> uncurry ? fold (add_to_uncurry_table thy) (nondef_ts @ def_ts)

  1368     (* bool -> term -> term *)

  1369     fun do_rest def =

  1370       binarize ? binarize_nat_and_int_in_term

  1371       #> uncurry ? uncurry_term table

  1372       #> box ? box_fun_and_pair_in_term hol_ctxt def

  1373       #> destroy_constrs ? (pull_out_universal_constrs hol_ctxt def

  1374                             #> pull_out_existential_constrs hol_ctxt

  1375                             #> destroy_pulled_out_constrs hol_ctxt def)

  1376       #> curry_assms

  1377       #> destroy_universal_equalities

  1378       #> destroy_existential_equalities hol_ctxt

  1379       #> simplify_constrs_and_sels thy

  1380       #> distribute_quantifiers

  1381       #> push_quantifiers_inward

  1382       #> close_form

  1383       #> Term.map_abs_vars shortest_name

  1384     val nondef_ts = map (do_rest false) nondef_ts

  1385     val def_ts = map (do_rest true) def_ts

  1386     val (nondef_ts, def_ts) =

  1387       finitize_all_types_of_funs hol_ctxt binarize finitizes monos

  1388                                  (nondef_ts, def_ts)

  1389   in

  1390     (nondef_ts, def_ts, got_all_mono_user_axioms, no_poly_user_axioms, binarize)

  1391   end

  1393 end;

author	blanchet
	Fri, 19 Mar 2010 15:07:44 +0100
changeset 35866	513074557e06
parent 35807	e4d1b5cbd429
child 36384	76d5fd5a45fb
permissions	-rw-r--r--