1 theory Fv

     2 imports "../Nominal-General/Nominal2_Atoms" 

     3         "Abs" "Perm" "Rsp" "Nominal2_FSet"

     4 begin

     5 

     6 (* The bindings data structure:

     7 

     8   Bindings are a list of lists of lists of triples.

     9 

    10    The first list represents the datatypes defined.

    11    The second list represents the constructors.

    12    The internal list is a list of all the bndings that

    13    concern the constructor.

    14 

    15    Every triple consists of a function, the binding and

    16    the body.

    17 

    18   Eg:

    19 nominal_datatype

    20 

    21    C1

    22  | C2 x y z bind x in z

    23  | C3 x y z bind f x in z bind g y in z

    24 

    25 yields:

    26 [

    27  [],

    28  [(NONE, 0, 2)],

    29  [(SOME (Const f), 0, 2), (Some (Const g), 1, 2)]]

    30 

    31 A SOME binding has to have a function which takes an appropriate

    32 argument and returns an atom set. A NONE binding has to be on an

    33 argument that is an atom or an atom set.

    34 *)

    35 

    36 (*

    37 An overview of the generation of free variables:

    38 

    39 1) fv_bn functions are generated only for the non-recursive binds.

    40 

    41    An fv_bn for a constructor is a union of values for the arguments:

    42 

    43    For an argument x that is in the bn function

    44    - if it is a recursive argument bn' we return: fv_bn' x

    45    - otherwise empty

    46 

    47    For an argument x that is not in the bn function

    48    - for atom we return: {atom x}

    49    - for atom set we return: atom ` x

    50    - for a recursive call to type ty' we return: fv_ty' x

    51      with fv of the appropriate type

    52    - otherwise empty

    53 

    54 2) fv_ty functions generated for all types being defined:

    55 

    56    fv_ty for a constructor is a union of values for the arguments.

    57 

    58    For an argument that is bound in a shallow binding we return empty.

    59 

    60    For an argument x that bound in a non-recursive deep binding

    61    we return: fv_bn x.

    62 

    63    Otherwise we return the free variables of the argument minus the

    64    bound variables of the argument.

    65 

    66    The free variables for an argument x are:

    67    - for an atom: {atom x}

    68    - for atom set: atom ` x

    69    - for recursive call to type ty' return: fv_ty' x

    70    - for nominal datatype ty' return: fv_ty' x

    71 

    72    The bound variables are a union of results of all bindings that

    73    involve the given argument. For a paricular binding:

    74 

    75    - for a binding function bn: bn x

    76    - for a recursive argument of type ty': fv_fy' x

    77    - for nominal datatype ty' return: fv_ty' x

    78 *)

    79 

    80 (*

    81 An overview of the generation of alpha-equivalence:

    82 

    83 1) alpha_bn relations are generated for binding functions.

    84 

    85    An alpha_bn for a constructor is true if a conjunction of

    86    propositions for each argument holds.

    87 

    88    For an argument a proposition is build as follows from

    89    th:

    90 

    91    - for a recursive argument in the bn function, we return: alpha_bn argl argr

    92    - for a recursive argument for type ty not in bn, we return: alpha_ty argl argr

    93    - for other arguments in the bn function we return: True

    94    - for other arguments not in the bn function we return: argl = argr

    95 

    96 2) alpha_ty relations are generated for all the types being defined:

    97 

    98    For each constructor we gather all the arguments that are bound,

    99    and for each of those we add a permutation. We associate those

   100    permutations with the bindings. Note that two bindings can have

   101    the same permutation if the arguments being bound are the same.

   102 

   103    An alpha_ty for a constructor is true if there exist permutations

   104    as above such that a conjunction of propositions for all arguments holds.

   105 

   106    For an argument we allow bindings where only one of the following

   107    holds:

   108 

   109    - Argument is bound in some shallow bindings: We return true

   110    - Argument of type ty is bound recursively in some other

   111      arguments [i1, .. in] with one binding function bn.

   112      We return:

   113 

   114      (bn argl, (argl, argl_i1, ..., argl_in)) \<approx>gen

   115      \<lambda>(argl,argl1,..,argln) (argr,argr1,..,argrn). 

   116          (alpha_ty argl argr) \<and> (alpha_i1 argl1 argr1) \<and> .. \<and> (alpha_in argln argrn)

   117      \<lambda>(arg,arg1,..,argn). (fv_ty arg) \<union> (fv_i1 arg1) \<union> .. \<union> (fv_in argn)

   118      pi

   119      (bn argr, (argr, argr_i1, ..., argr_in))

   120 

   121    - Argument is bound in some deep non-recursive bindings.

   122      We return: alpha_bn argl argr

   123    - Argument of type ty has some shallow bindings [b1..bn] and/or

   124      non-recursive bindings [f1 a1, .., fm am], where the bindings

   125      have the permutations p1..pl. We return:

   126 

   127      (b1l \<union>..\<union> bnl \<union> f1 a1l \<union>..\<union> fn anl, argl) \<approx>gen

   128      alpha_ty fv_ty (p1 +..+ pl)

   129      (b1r \<union>..\<union> bnr \<union> f1 a1r \<union>..\<union> fn anr, argr)

   130 

   131    - Argument has some recursive bindings. The bindings were

   132      already treated in 2nd case so we return: True

   133    - Argument has no bindings and is not bound.

   134      If it is recursive for type ty, we return: alpha_ty argl argr

   135      Otherwise we return: argl = argr

   136 

   137 *)

   138 

   139 

   140 ML {*

   141 datatype alpha_mode = AlphaGen | AlphaRes | AlphaLst;

   142 *}

   143 

   144 ML {*

   145 fun atyp_const AlphaGen = @{const_name alpha_gen}

   146   | atyp_const AlphaRes = @{const_name alpha_res}

   147   | atyp_const AlphaLst = @{const_name alpha_lst}

   148 *}

   149 

   150 (* TODO: make sure that parser checks that bindings are compatible *)

   151 ML {*

   152 fun alpha_const_for_binds [] = atyp_const AlphaGen

   153   | alpha_const_for_binds ((NONE, _, _, at) :: t) = atyp_const at

   154   | alpha_const_for_binds ((SOME (_, _), _, _, at) :: _) = atyp_const at

   155 *}

   156 

   157 ML {*

   158 fun is_atom thy typ =

   159   Sign.of_sort thy (typ, @{sort at})

   160 

   161 fun is_atom_set thy (Type ("fun", [t, @{typ bool}])) = is_atom thy t

   162   | is_atom_set _ _ = false;

   163 

   164 fun is_atom_fset thy (Type ("FSet.fset", [t])) = is_atom thy t

   165   | is_atom_fset _ _ = false;

   166 *}

   167 

   168 

   169 (* Like map2, only if the second list is empty passes empty lists insted of error *)

   170 ML {*

   171 fun map2i _ [] [] = []

   172   | map2i f (x :: xs) (y :: ys) = f x y :: map2i f xs ys

   173   | map2i f (x :: xs) [] = f x [] :: map2i f xs []

   174   | map2i _ _ _ = raise UnequalLengths;

   175 *}

   176 

   177 (* Finds bindings with the same function and binding, and gathers all

   178    bodys for such pairs

   179  *)

   180 ML {*

   181 fun gather_binds binds =

   182 let

   183   fun gather_binds_cons binds =

   184     let

   185       val common = map (fn (f, bi, _, aty) => (f, bi, aty)) binds

   186       val nodups = distinct (op =) common

   187       fun find_bodys (sf, sbi, sty) =

   188         filter (fn (f, bi, _, aty) => f = sf andalso bi = sbi andalso aty = sty) binds

   189       val bodys = map ((map (fn (_, _, bo, _) => bo)) o find_bodys) nodups

   190     in

   191       nodups ~~ bodys

   192     end

   193 in

   194   map (map gather_binds_cons) binds

   195 end

   196 *}

   197 

   198 ML {*

   199 fun un_gather_binds_cons binds =

   200   flat (map (fn (((f, bi, aty), bos), pi) => map (fn bo => ((f, bi, bo, aty), pi)) bos) binds)

   201 *}

   202 

   203 ML {*

   204   open Datatype_Aux; (* typ_of_dtyp, DtRec, ... *);

   205 *}

   206 ML {*

   207   (* TODO: It is the same as one in 'nominal_atoms' *)

   208   fun mk_atom ty = Const (@{const_name atom}, ty --> @{typ atom});

   209   val noatoms = @{term "{} :: atom set"};

   210   fun mk_single_atom x = HOLogic.mk_set @{typ atom} [mk_atom (type_of x) $ x];

   211   fun mk_union sets =

   212     fold (fn a => fn b =>

   213       if a = noatoms then b else

   214       if b = noatoms then a else

   215       if a = b then a else

   216       HOLogic.mk_binop @{const_name sup} (a, b)) (rev sets) noatoms;

   217   val mk_inter = foldr1 (HOLogic.mk_binop @{const_name inf})

   218   fun mk_diff a b =

   219     if b = noatoms then a else

   220     if b = a then noatoms else

   221     HOLogic.mk_binop @{const_name minus} (a, b);

   222   fun mk_atom_set t =

   223     let

   224       val ty = fastype_of t;

   225       val atom_ty = HOLogic.dest_setT ty --> @{typ atom};

   226       val img_ty = atom_ty --> ty --> @{typ "atom set"};

   227     in

   228       (Const (@{const_name image}, img_ty) $ Const (@{const_name atom}, atom_ty) $ t)

   229     end;

   230   fun mk_atom_fset t =

   231     let

   232       val ty = fastype_of t;

   233       val atom_ty = dest_fsetT ty --> @{typ atom};

   234       val fmap_ty = atom_ty --> ty --> @{typ "atom fset"};

   235       val fset_to_set = @{term "fset_to_set :: atom fset \<Rightarrow> atom set"}

   236     in

   237       fset_to_set $ ((Const (@{const_name fmap}, fmap_ty) $ Const (@{const_name atom}, atom_ty) $ t))

   238     end;

   239   (* Similar to one in USyntax *)

   240   fun mk_pair (fst, snd) =

   241     let val ty1 = fastype_of fst

   242       val ty2 = fastype_of snd

   243       val c = HOLogic.pair_const ty1 ty2

   244     in c $ fst $ snd

   245     end;

   246 *}

   247 

   248 (* Given [fv1, fv2, fv3] creates %(x, y, z). fv1 x u fv2 y u fv3 z *)

   249 ML {*

   250 fun mk_compound_fv fvs =

   251 let

   252   val nos = (length fvs - 1) downto 0;

   253   val fvs_applied = map (fn (fv, no) => fv $ Bound no) (fvs ~~ nos);

   254   val fvs_union = mk_union fvs_applied;

   255   val (tyh :: tys) = rev (map (domain_type o fastype_of) fvs);

   256   fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))

   257 in

   258   fold fold_fun tys (Abs ("", tyh, fvs_union))

   259 end;

   260 *}

   261 

   262 (* Given [R1, R2, R3] creates %(x,x'). %(y,y'). %(z,z'). R x x' \<and> R y y' \<and> R z z' *)

   263 ML {*

   264 fun mk_compound_alpha Rs =

   265 let

   266   val nos = (length Rs - 1) downto 0;

   267   val nos2 = (2 * length Rs - 1) downto length Rs;

   268   val Rs_applied = map (fn (R, (no2, no)) => R $ Bound no2 $ Bound no) (Rs ~~ (nos2 ~~ nos));

   269   val Rs_conj = mk_conjl Rs_applied;

   270   val (tyh :: tys) = rev (map (domain_type o fastype_of) Rs);

   271   fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))

   272   val abs_rhs = fold fold_fun tys (Abs ("", tyh, Rs_conj))

   273 in

   274   fold fold_fun tys (Abs ("", tyh, abs_rhs))

   275 end;

   276 *}

   277 

   278 

   279 ML {*

   280 fun non_rec_binds l =

   281 let

   282   fun is_non_rec (SOME (f, false), _, _, _) = SOME f

   283     | is_non_rec _ = NONE

   284 in

   285   distinct (op =) (map_filter is_non_rec (flat (flat l)))

   286 end

   287 *}

   288 

   289 (* We assume no bindings in the type on which bn is defined *)

   290 ML {*

   291 fun fv_bn thy (dt_info : Datatype_Aux.info) fv_frees bn_fvbn (fvbn, (bn, ith_dtyp, args_in_bns)) =

   292 let

   293   val {descr, sorts, ...} = dt_info;

   294   fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);

   295   fun fv_bn_constr (cname, dts) args_in_bn =

   296   let

   297     val Ts = map (typ_of_dtyp descr sorts) dts;

   298     val names = Datatype_Prop.make_tnames Ts;

   299     val args = map Free (names ~~ Ts);

   300     val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));

   301     fun fv_arg ((dt, x), arg_no) =

   302       let

   303         val ty = fastype_of x

   304 (*        val _ = tracing ("B 1" ^ PolyML.makestring args_in_bn);*)

   305 (*        val _ = tracing ("B 2" ^ PolyML.makestring bn_fvbn);*)

   306       in

   307         case AList.lookup (op=) args_in_bn arg_no of

   308           SOME NONE => @{term "{} :: atom set"}

   309         | SOME (SOME (f : term)) => (the (AList.lookup (op=) bn_fvbn f)) $ x

   310         | NONE =>

   311             if is_atom thy ty then mk_single_atom x else

   312             if is_atom_set thy ty then mk_atom_set x else

   313             if is_atom_fset thy ty then mk_atom_fset x else

   314             if is_rec_type dt then nth fv_frees (body_index dt) $ x else

   315             @{term "{} :: atom set"}

   316       end;

   317     val arg_nos = 0 upto (length dts - 1)

   318   in

   319     HOLogic.mk_Trueprop (HOLogic.mk_eq

   320       (fvbn $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ arg_nos))))

   321   end;

   322   val (_, (_, _, constrs)) = nth descr ith_dtyp;

   323   val eqs = map2i fv_bn_constr constrs args_in_bns

   324 in

   325   ((bn, fvbn), eqs)

   326 end

   327 *}

   328 

   329 ML {* print_depth 100 *}

   330 ML {*

   331 fun fv_bns thy dt_info fv_frees rel_bns =

   332 let

   333   fun mk_fvbn_free (bn, ith, _) =

   334     let

   335       val fvbn_name = "fv_" ^ (Long_Name.base_name (fst (dest_Const bn)));

   336     in

   337       (fvbn_name, Free (fvbn_name, fastype_of (nth fv_frees ith)))

   338     end;

   339   val (fvbn_names, fvbn_frees) = split_list (map mk_fvbn_free rel_bns);

   340   val bn_fvbn = (map (fn (bn, _, _) => bn) rel_bns) ~~ fvbn_frees

   341   val (l1, l2) = split_list (map (fv_bn thy dt_info fv_frees bn_fvbn) (fvbn_frees ~~ rel_bns));

   342 in

   343   (l1, (fvbn_names ~~ l2))

   344 end

   345 *}

   346 

   347 

   348 ML {*

   349 fun alpha_bn (dt_info : Datatype_Aux.info) alpha_frees bn_alphabn ((bn, ith_dtyp, args_in_bns), (alpha_bn_free, _ (*is_rec*) )) =

   350 let

   351   val {descr, sorts, ...} = dt_info;

   352   fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);

   353   fun alpha_bn_constr (cname, dts) args_in_bn =

   354   let

   355     val Ts = map (typ_of_dtyp descr sorts) dts;

   356     val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);

   357     val names2 = Name.variant_list ("pi" :: names) (Datatype_Prop.make_tnames Ts);

   358     val args = map Free (names ~~ Ts);

   359     val args2 = map Free (names2 ~~ Ts);

   360     val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));

   361     val rhs = HOLogic.mk_Trueprop

   362       (alpha_bn_free $ (list_comb (c, args)) $ (list_comb (c, args2)));

   363     fun lhs_arg ((dt, arg_no), (arg, arg2)) =

   364       case AList.lookup (op=) args_in_bn arg_no of

   365         SOME NONE => @{term True}

   366       | SOME (SOME f) => (the (AList.lookup (op=) bn_alphabn f)) $ arg $ arg2

   367       | NONE =>

   368           if is_rec_type dt then (nth alpha_frees (body_index dt)) $ arg $ arg2

   369           else HOLogic.mk_eq (arg, arg2)

   370     val arg_nos = 0 upto (length dts - 1)

   371     val lhss = mk_conjl (map lhs_arg (dts ~~ arg_nos ~~ (args ~~ args2)))

   372     val eq = Logic.mk_implies (HOLogic.mk_Trueprop lhss, rhs)

   373   in

   374     eq

   375   end

   376   val (_, (_, _, constrs)) = nth descr ith_dtyp;

   377   val eqs = map2i alpha_bn_constr constrs args_in_bns

   378 in

   379   ((bn, alpha_bn_free), eqs)

   380 end

   381 *}

   382 

   383 ML {*

   384 fun alpha_bns dt_info alpha_frees rel_bns bns_rec =

   385 let

   386   val {descr, sorts, ...} = dt_info;

   387   fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);

   388   fun mk_alphabn_free (bn, ith, _) =

   389     let

   390       val alphabn_name = "alpha_" ^ (Long_Name.base_name (fst (dest_Const bn)));

   391       val alphabn_type = nth_dtyp ith --> nth_dtyp ith --> @{typ bool};

   392       val alphabn_free = Free(alphabn_name, alphabn_type);

   393     in

   394       (alphabn_name, alphabn_free)

   395     end;

   396   val (alphabn_names, alphabn_frees) = split_list (map mk_alphabn_free rel_bns);

   397   val bn_alphabn = (map (fn (bn, _, _) => bn) rel_bns) ~~ alphabn_frees;

   398   val pair = split_list (map (alpha_bn dt_info alpha_frees bn_alphabn)

   399     (rel_bns ~~ (alphabn_frees ~~ bns_rec)))

   400 in

   401   (alphabn_names, pair)

   402 end

   403 *}

   404 

   405 

   406 (* Checks that a list of bindings contains only compatible ones *)

   407 ML {*

   408 fun bns_same l =

   409   length (distinct (op =) (map (fn ((b, _, _, atyp), _) => (b, atyp)) l)) = 1

   410 *}

   411 

   412 ML {*

   413 fun setify x =

   414   if fastype_of x = @{typ "atom list"} then

   415   Const (@{const_name set}, @{typ "atom list \<Rightarrow> atom set"}) $ x else x

   416 *}

   417 

   418 ML {*

   419 fun define_fv (dt_info : Datatype_Aux.info) bindsall bns lthy =

   420 let

   421   val thy = ProofContext.theory_of lthy;

   422   val {descr, sorts, ...} = dt_info;

   423   fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);

   424   val fv_names = Datatype_Prop.indexify_names (map (fn (i, _) =>

   425     "fv_" ^ name_of_typ (nth_dtyp i)) descr);

   426   val fv_types = map (fn (i, _) => nth_dtyp i --> @{typ "atom set"}) descr;

   427   val fv_frees = map Free (fv_names ~~ fv_types);

   428 (* TODO: We need a transitive closure, but instead we do this hack considering

   429    all binding functions as recursive or not *)

   430   val nr_bns =

   431     if (non_rec_binds bindsall) = [] then []

   432     else map (fn (bn, _, _) => bn) bns;

   433   val rel_bns = filter (fn (bn, _, _) => bn mem nr_bns) bns;

   434   val (bn_fv_bns, fv_bn_names_eqs) = fv_bns thy dt_info fv_frees rel_bns;

   435   val fvbns = map snd bn_fv_bns;

   436   val (fv_bn_names, fv_bn_eqs) = split_list fv_bn_names_eqs;

   437 

   438   fun fv_constr ith_dtyp (cname, dts) bindcs =

   439     let

   440       val Ts = map (typ_of_dtyp descr sorts) dts;

   441       val bindslen = length bindcs

   442       val pi_strs_same = replicate bindslen "pi"

   443       val pi_strs = Name.variant_list [] pi_strs_same;

   444       val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs;

   445       val bind_pis_gath = bindcs ~~ pis;

   446       val bind_pis = un_gather_binds_cons bind_pis_gath;

   447       val bindcs = map fst bind_pis;

   448       val names = Name.variant_list pi_strs (Datatype_Prop.make_tnames Ts);

   449       val args = map Free (names ~~ Ts);

   450       val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));

   451       val fv_c = nth fv_frees ith_dtyp;

   452       val arg_nos = 0 upto (length dts - 1)

   453       fun fv_bind args (NONE, i, _, _) =

   454             if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else

   455             if ((is_atom thy) o fastype_of) (nth args i) then mk_single_atom (nth args i) else

   456             if ((is_atom_set thy) o fastype_of) (nth args i) then mk_atom_set (nth args i) else

   457             if ((is_atom_fset thy) o fastype_of) (nth args i) then mk_atom_fset (nth args i) else

   458             (* TODO goes the code for preiously defined nominal datatypes *)

   459             @{term "{} :: atom set"}

   460         | fv_bind args (SOME (f, _), i, _, _) = f $ (nth args i)

   461       fun fv_binds_as_set args relevant = mk_union (map (setify o fv_bind args) relevant)

   462       fun find_nonrec_binder j (SOME (f, false), i, _, _) = if i = j then SOME f else NONE

   463         | find_nonrec_binder _ _ = NONE

   464       fun fv_arg ((dt, x), arg_no) =

   465         case get_first (find_nonrec_binder arg_no) bindcs of

   466           SOME f =>

   467             (case get_first (fn (x, y) => if x = f then SOME y else NONE) bn_fv_bns of

   468                 SOME fv_bn => fv_bn $ x

   469               | NONE => error "bn specified in a non-rec binding but not in bn list")

   470         | NONE =>

   471             let

   472               val arg =

   473                 if is_rec_type dt then nth fv_frees (body_index dt) $ x else

   474                 if ((is_atom thy) o fastype_of) x then mk_single_atom x else

   475                 if ((is_atom_set thy) o fastype_of) x then mk_atom_set x else

   476                 if ((is_atom_fset thy) o fastype_of) x then mk_atom_fset x else

   477                 (* TODO goes the code for preiously defined nominal datatypes *)

   478                 @{term "{} :: atom set"};

   479               (* If i = j then we generate it only once *)

   480               val relevant = filter (fn (_, i, j, _) => ((i = arg_no) orelse (j = arg_no))) bindcs;

   481               val sub = fv_binds_as_set args relevant

   482             in

   483               mk_diff arg sub

   484             end;

   485       val fv_eq = HOLogic.mk_Trueprop (HOLogic.mk_eq

   486         (fv_c $ list_comb (c, args), mk_union (map fv_arg  (dts ~~ args ~~ arg_nos))))

   487     in

   488       fv_eq

   489     end;

   490   fun fv_eq (i, (_, _, constrs)) binds = map2i (fv_constr i) constrs binds;

   491   val fveqs = map2i fv_eq descr (gather_binds bindsall)

   492   val fv_eqs_perfv = fveqs

   493   val rel_bns_nos = map (fn (_, i, _) => i) rel_bns;

   494   fun filter_fun (_, b) = b mem rel_bns_nos;

   495   val all_fvs = (fv_names ~~ fv_eqs_perfv) ~~ (0 upto (length fv_names - 1))

   496   val (fv_names_fst, fv_eqs_fst) = apsnd flat (split_list (map fst (filter_out filter_fun all_fvs)))

   497   val (fv_names_snd, fv_eqs_snd) = apsnd flat (split_list (map fst (filter filter_fun all_fvs)))

   498   val fv_eqs_all = fv_eqs_fst @ (flat fv_bn_eqs);

   499   val fv_names_all = fv_names_fst @ fv_bn_names;

   500   val add_binds = map (fn x => (Attrib.empty_binding, x))

   501 (* Function_Fun.add_fun Function_Common.default_config ... true *)

   502   val (fvs, lthy') = (Primrec.add_primrec

   503     (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_all) (add_binds fv_eqs_all) lthy)

   504   val (fvs2, lthy'') =

   505     if fv_eqs_snd = [] then (([], []), lthy') else

   506    (Primrec.add_primrec

   507     (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_snd) (add_binds fv_eqs_snd) lthy')

   508   val ordered_fvs = fv_frees @ fvbns;

   509   val all_fvs = (fst fvs @ fst fvs2, snd fvs @ snd fvs2)

   510 in

   511   ((all_fvs, ordered_fvs), lthy'')

   512 end

   513 *}

   514 

   515 ML {*

   516 fun define_alpha (dt_info : Datatype_Aux.info) bindsall bns fv_frees lthy =

   517 let

   518   val thy = ProofContext.theory_of lthy;

   519   val {descr, sorts, ...} = dt_info;

   520   fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);

   521 (* TODO: We need a transitive closure, but instead we do this hack considering

   522    all binding functions as recursive or not *)

   523   val nr_bns =

   524     if (non_rec_binds bindsall) = [] then []

   525     else map (fn (bn, _, _) => bn) bns;

   526   val alpha_names = Datatype_Prop.indexify_names (map (fn (i, _) =>

   527     "alpha_" ^ name_of_typ (nth_dtyp i)) descr);

   528   val alpha_types = map (fn (i, _) => nth_dtyp i --> nth_dtyp i --> @{typ bool}) descr;

   529   val alpha_frees = map Free (alpha_names ~~ alpha_types);

   530   (* We assume that a bn is either recursive or not *)

   531   val bns_rec = map (fn (bn, _, _) => not (bn mem nr_bns)) bns;

   532   val (alpha_bn_names, (bn_alpha_bns, alpha_bn_eqs)) =

   533     alpha_bns dt_info alpha_frees bns bns_rec

   534   val alpha_bn_frees = map snd bn_alpha_bns;

   535   val alpha_bn_types = map fastype_of alpha_bn_frees;

   536 

   537   fun alpha_constr ith_dtyp (cname, dts) bindcs =

   538     let

   539       val Ts = map (typ_of_dtyp descr sorts) dts;

   540       val bindslen = length bindcs

   541       val pi_strs_same = replicate bindslen "pi"

   542       val pi_strs = Name.variant_list [] pi_strs_same;

   543       val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs;

   544       val bind_pis_gath = bindcs ~~ pis;

   545       val bind_pis = un_gather_binds_cons bind_pis_gath;

   546       val names = Name.variant_list pi_strs (Datatype_Prop.make_tnames Ts);

   547       val args = map Free (names ~~ Ts);

   548       val names2 = Name.variant_list (pi_strs @ names) (Datatype_Prop.make_tnames Ts);

   549       val args2 = map Free (names2 ~~ Ts);

   550       val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));

   551       val alpha = nth alpha_frees ith_dtyp;

   552       val arg_nos = 0 upto (length dts - 1)

   553       fun fv_bind args (NONE, i, _, _) =

   554             if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else

   555             if ((is_atom thy) o fastype_of) (nth args i) then mk_single_atom (nth args i) else

   556             if ((is_atom_set thy) o fastype_of) (nth args i) then mk_atom_set (nth args i) else

   557             if ((is_atom_fset thy) o fastype_of) (nth args i) then mk_atom_fset (nth args i) else

   558             (* TODO goes the code for preiously defined nominal datatypes *)

   559             @{term "{} :: atom set"}

   560         | fv_bind args (SOME (f, _), i, _, _) = f $ (nth args i)

   561       fun fv_binds args relevant = mk_union (map (fv_bind args) relevant)

   562       val alpha_rhs =

   563         HOLogic.mk_Trueprop (alpha $ (list_comb (c, args)) $ (list_comb (c, args2)));

   564       fun alpha_arg ((dt, arg_no), (arg, arg2)) =

   565         let

   566           val rel_in_simp_binds = filter (fn ((NONE, i, _, _), _) => i = arg_no | _ => false) bind_pis;

   567           val rel_in_comp_binds = filter (fn ((SOME _, i, _, _), _) => i = arg_no | _ => false) bind_pis;

   568           val rel_has_binds = filter (fn ((NONE, _, j, _), _) => j = arg_no

   569                                        | ((SOME (_, false), _, j, _), _) => j = arg_no

   570                                        | _ => false) bind_pis;

   571           val rel_has_rec_binds = filter

   572             (fn ((SOME (_, true), _, j, _), _) => j = arg_no | _ => false) bind_pis;

   573         in

   574           case (rel_in_simp_binds, rel_in_comp_binds, rel_has_binds, rel_has_rec_binds) of

   575             ([], [], [], []) =>

   576               if is_rec_type dt then (nth alpha_frees (body_index dt) $ arg $ arg2)

   577               else (HOLogic.mk_eq (arg, arg2))

   578           | (_, [], [], []) => @{term True}

   579           | ([], [], [], _) => @{term True}

   580           | ([], ((((SOME (bn, is_rec)), _, _, atyp), _) :: _), [], []) =>

   581             if not (bns_same rel_in_comp_binds) then error "incompatible bindings for an argument" else

   582             if is_rec then

   583               let

   584                 val (rbinds, rpis) = split_list rel_in_comp_binds

   585                 val bound_in_nos = map (fn (_, _, i, _) => i) rbinds

   586                 val bound_in_ty_nos = map (fn i => body_index (nth dts i)) bound_in_nos;

   587                 val bound_args = arg :: map (nth args) bound_in_nos;

   588                 val bound_args2 = arg2 :: map (nth args2) bound_in_nos;

   589                 val lhs_binds = fv_binds args rbinds

   590                 val lhs_arg = foldr1 HOLogic.mk_prod bound_args

   591                 val lhs = mk_pair (lhs_binds, lhs_arg);

   592                 val rhs_binds = fv_binds args2 rbinds;

   593                 val rhs_arg = foldr1 HOLogic.mk_prod bound_args2;

   594                 val rhs = mk_pair (rhs_binds, rhs_arg);

   595                 val fvs = map (nth fv_frees) ((body_index dt) :: bound_in_ty_nos);

   596                 val fv = mk_compound_fv fvs;

   597                 val alphas = map (nth alpha_frees) ((body_index dt) :: bound_in_ty_nos);

   598                 val alpha = mk_compound_alpha alphas;

   599                 val pi = foldr1 (uncurry mk_plus) (distinct (op =) rpis);

   600                 val alpha_gen_pre = Const (atyp_const atyp, dummyT) $ lhs $ alpha $ fv $ pi $ rhs;

   601                 val alpha_gen = Syntax.check_term lthy alpha_gen_pre

   602               in

   603                 alpha_gen

   604               end

   605             else

   606               let

   607                 val alpha_bn_const =

   608                   nth alpha_bn_frees (find_index (fn (b, _, _) => b = bn) bns)

   609               in

   610                 alpha_bn_const $ arg $ arg2

   611               end

   612           | ([], [], relevant, []) =>

   613             let

   614               val (rbinds, rpis) = split_list relevant

   615               val lhs_binds = fv_binds args rbinds

   616               val lhs = mk_pair (lhs_binds, arg);

   617               val rhs_binds = fv_binds args2 rbinds;

   618               val rhs = mk_pair (rhs_binds, arg2);

   619               val alpha = nth alpha_frees (body_index dt);

   620               val fv = nth fv_frees (body_index dt);

   621               val pi = foldr1 (uncurry mk_plus) (distinct (op =) rpis);

   622               val alpha_const = alpha_const_for_binds rbinds;

   623               val alpha_gen_pre = Const (alpha_const, dummyT) $ lhs $ alpha $ fv $ pi $ rhs;

   624               val alpha_gen = Syntax.check_term lthy alpha_gen_pre

   625             in

   626               alpha_gen

   627             end

   628           | _ => error "Fv.alpha: not supported binding structure"

   629         end

   630       val alphas = map alpha_arg (dts ~~ arg_nos ~~ (args ~~ args2))

   631       val alpha_lhss = mk_conjl alphas

   632       val alpha_lhss_ex =

   633         fold (fn pi_str => fn t => HOLogic.mk_exists (pi_str, @{typ perm}, t)) pi_strs alpha_lhss

   634       val alpha_eq = Logic.mk_implies (HOLogic.mk_Trueprop alpha_lhss_ex, alpha_rhs)

   635     in

   636       alpha_eq

   637     end;

   638   fun alpha_eq (i, (_, _, constrs)) binds = map2i (alpha_constr i) constrs binds;

   639   val alphaeqs = map2i alpha_eq descr (gather_binds bindsall)

   640   val alpha_eqs = flat alphaeqs

   641   val add_binds = map (fn x => (Attrib.empty_binding, x))

   642   val (alphas, lthy') = (Inductive.add_inductive_i

   643      {quiet_mode = true, verbose = false, alt_name = Binding.empty,

   644       coind = false, no_elim = false, no_ind = false, skip_mono = true, fork_mono = false}

   645      (map2 (fn x => fn y => ((Binding.name x, y), NoSyn)) (alpha_names @ alpha_bn_names)

   646      (alpha_types @ alpha_bn_types)) []

   647      (add_binds (alpha_eqs @ flat alpha_bn_eqs)) [] lthy)

   648 in

   649   (alphas, lthy')

   650 end

   651 *}

   652 

   653 end