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 ML {*

   140 datatype alpha_mode = AlphaGen | AlphaRes | AlphaLst;

   141 *}

   142 

   143 ML {*

   144 fun atyp_const AlphaGen = @{const_name alpha_gen}

   145   | atyp_const AlphaRes = @{const_name alpha_res}

   146   | atyp_const AlphaLst = @{const_name alpha_lst}

   147 *}

   148 

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

   150 ML {*

   151 fun alpha_const_for_binds [] = atyp_const AlphaGen

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

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

   154 *}

   155 

   156 ML {*

   157 fun is_atom thy typ =

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

   159 

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

   161   | is_atom_set _ _ = false;

   162 

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

   164   | is_atom_fset _ _ = false;

   165 *}

   166 

   167 

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

   169 ML {*

   170 fun map2i _ [] [] = []

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

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

   173   | map2i _ _ _ = raise UnequalLengths;

   174 *}

   175 

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

   177    bodys for such pairs

   178  *)

   179 ML {*

   180 fun gather_binds binds =

   181 let

   182   fun gather_binds_cons binds =

   183     let

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

   185       val nodups = distinct (op =) common

   186       fun find_bodys (sf, sbi, sty) =

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

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

   189     in

   190       nodups ~~ bodys

   191     end

   192 in

   193   map (map gather_binds_cons) binds

   194 end

   195 *}

   196 

   197 ML {*

   198 fun un_gather_binds_cons binds =

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

   200 *}

   201 

   202 ML {*

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

   204 *}

   205 ML {*

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

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

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

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

   210   fun mk_union sets =

   211     fold (fn a => fn b =>

   212       if a = noatoms then b else

   213       if b = noatoms then a else

   214       if a = b then a else

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

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

   217   fun mk_diff a b =

   218     if b = noatoms then a else

   219     if b = a then noatoms else

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

   221   fun mk_atom_set t =

   222     let

   223       val ty = fastype_of t;

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

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

   226     in

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

   228     end;

   229   fun mk_atom_fset t =

   230     let

   231       val ty = fastype_of t;

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

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

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

   235     in

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

   237     end;

   238   (* Similar to one in USyntax *)

   239   fun mk_pair (fst, snd) =

   240     let val ty1 = fastype_of fst

   241       val ty2 = fastype_of snd

   242       val c = HOLogic.pair_const ty1 ty2

   243     in c $ fst $ snd

   244     end;

   245 *}

   246 

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

   248 ML {*

   249 fun mk_compound_fv fvs =

   250 let

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

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

   253   val fvs_union = mk_union fvs_applied;

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

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

   256 in

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

   258 end;

   259 *}

   260 

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

   262 ML {*

   263 fun mk_compound_alpha Rs =

   264 let

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

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

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

   268   val Rs_conj = mk_conjl Rs_applied;

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

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

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

   272 in

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

   274 end;

   275 *}

   276 

   277 

   278 ML {*

   279 fun non_rec_binds l =

   280 let

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

   282     | is_non_rec _ = NONE

   283 in

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

   285 end

   286 *}

   287 

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

   289 ML {*

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

   291 let

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

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

   294   fun fv_bn_constr (cname, dts) args_in_bn =

   295   let

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

   297     val names = Datatype_Prop.make_tnames Ts;

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

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

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

   301       let

   302         val ty = fastype_of x

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

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

   305       in

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

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

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

   309         | NONE =>

   310             if is_atom thy ty then mk_single_atom x else

   311             if is_atom_set thy ty then mk_atom_set x else

   312             if is_atom_fset thy ty then mk_atom_fset x else

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

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

   315       end;

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

   317   in

   318     HOLogic.mk_Trueprop (HOLogic.mk_eq

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

   320   end;

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

   322   val eqs = map2i fv_bn_constr constrs args_in_bns

   323 in

   324   ((bn, fvbn), eqs)

   325 end

   326 *}

   327 

   328 ML {* print_depth 100 *}

   329 ML {*

   330 fun fv_bns thy dt_info fv_frees rel_bns =

   331 let

   332   fun mk_fvbn_free (bn, ith, _) =

   333     let

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

   335     in

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

   337     end;

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

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

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

   341 in

   342   (l1, (fvbn_names ~~ l2))

   343 end

   344 *}

   345 

   346 

   347 ML {*

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

   349 let

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

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

   352   fun alpha_bn_constr (cname, dts) args_in_bn =

   353   let

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

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

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

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

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

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

   360     val rhs = HOLogic.mk_Trueprop

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

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

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

   364         SOME NONE => @{term True}

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

   366       | NONE =>

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

   368           else HOLogic.mk_eq (arg, arg2)

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

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

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

   372   in

   373     eq

   374   end

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

   376   val eqs = map2i alpha_bn_constr constrs args_in_bns

   377 in

   378   ((bn, alpha_bn_free), eqs)

   379 end

   380 *}

   381 

   382 ML {*

   383 fun alpha_bns dt_info alpha_frees rel_bns bns_rec =

   384 let

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

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

   387   fun mk_alphabn_free (bn, ith, _) =

   388     let

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

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

   391       val alphabn_free = Free(alphabn_name, alphabn_type);

   392     in

   393       (alphabn_name, alphabn_free)

   394     end;

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

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

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

   398     (rel_bns ~~ (alphabn_frees ~~ bns_rec)))

   399 in

   400   (alphabn_names, pair)

   401 end

   402 *}

   403 

   404 

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

   406 ML {*

   407 fun bns_same l =

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

   409 *}

   410 

   411 ML {*

   412 fun setify x =

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

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

   415 *}

   416 

   417 ML {*

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

   419 let

   420   val thy = ProofContext.theory_of lthy;

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

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

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

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

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

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

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

   428    all binding functions as recursive or not *)

   429   val nr_bns =

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

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

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

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

   434   val fvbns = map snd bn_fv_bns;

   435   val (fv_bn_names, fv_bn_eqs) = split_list fv_bn_names_eqs;

   436 

   437   fun fv_constr ith_dtyp (cname, dts) bindcs =

   438     let

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

   440       val bindslen = length bindcs

   441       val pi_strs_same = replicate bindslen "pi"

   442       val pi_strs = Name.variant_list [] pi_strs_same;

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

   444       val bind_pis_gath = bindcs ~~ pis;

   445       val bind_pis = un_gather_binds_cons bind_pis_gath;

   446       val bindcs = map fst bind_pis;

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

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

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

   450       val fv_c = nth fv_frees ith_dtyp;

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

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

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

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

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

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

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

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

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

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

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

   462         | find_nonrec_binder _ _ = NONE

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

   464         case get_first (find_nonrec_binder arg_no) bindcs of

   465           SOME f =>

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

   467                 SOME fv_bn => fv_bn $ x

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

   469         | NONE =>

   470             let

   471               val arg =

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

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

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

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

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

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

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

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

   480               val sub = fv_binds_as_set args relevant

   481             in

   482               mk_diff arg sub

   483             end;

   484       val fv_eq = HOLogic.mk_Trueprop (HOLogic.mk_eq

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

   486     in

   487       fv_eq

   488     end;

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

   490   val fveqs = map2i fv_eq descr (gather_binds bindsall)

   491   val fv_eqs_perfv = fveqs

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

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

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

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

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

   497   val fv_eqs_all = fv_eqs_fst @ (flat fv_bn_eqs);

   498   val fv_names_all = fv_names_fst @ fv_bn_names;

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

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

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

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

   503   val (fvs2, lthy'') =

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

   505    (Primrec.add_primrec

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

   507   val ordered_fvs = fv_frees @ fvbns;

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

   509 in

   510   ((all_fvs, ordered_fvs), lthy'')

   511 end

   512 *}

   513 

   514 ML {*

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

   516 let

   517   val thy = ProofContext.theory_of lthy;

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

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

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

   521    all binding functions as recursive or not *)

   522   val nr_bns =

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

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

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

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

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

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

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

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

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

   532     alpha_bns dt_info alpha_frees bns bns_rec

   533   val alpha_bn_frees = map snd bn_alpha_bns;

   534   val alpha_bn_types = map fastype_of alpha_bn_frees;

   535 

   536   fun alpha_constr ith_dtyp (cname, dts) bindcs =

   537     let

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

   539       val bindslen = length bindcs

   540       val pi_strs_same = replicate bindslen "pi"

   541       val pi_strs = Name.variant_list [] pi_strs_same;

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

   543       val bind_pis_gath = bindcs ~~ pis;

   544       val bind_pis = un_gather_binds_cons bind_pis_gath;

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

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

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

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

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

   550       val alpha = nth alpha_frees ith_dtyp;

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

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

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

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

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

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

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

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

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

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

   561       val alpha_rhs =

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

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

   564         let

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

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

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

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

   569                                        | _ => false) bind_pis;

   570           val rel_has_rec_binds = filter

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

   572         in

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

   574             ([], [], [], []) =>

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

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

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

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

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

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

   581             if is_rec then

   582               let

   583                 val (rbinds, rpis) = split_list rel_in_comp_binds

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

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

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

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

   588                 val lhs_binds = fv_binds args rbinds

   589                 val lhs_arg = foldr1 HOLogic.mk_prod bound_args

   590                 val lhs = mk_pair (lhs_binds, lhs_arg);

   591                 val rhs_binds = fv_binds args2 rbinds;

   592                 val rhs_arg = foldr1 HOLogic.mk_prod bound_args2;

   593                 val rhs = mk_pair (rhs_binds, rhs_arg);

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

   595                 val fv = mk_compound_fv fvs;

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

   597                 val alpha = mk_compound_alpha alphas;

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

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

   600                 val alpha_gen = Syntax.check_term lthy alpha_gen_pre

   601               in

   602                 alpha_gen

   603               end

   604             else

   605               let

   606                 val alpha_bn_const =

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

   608               in

   609                 alpha_bn_const $ arg $ arg2

   610               end

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

   612             let

   613               val (rbinds, rpis) = split_list relevant

   614               val lhs_binds = fv_binds args rbinds

   615               val lhs = mk_pair (lhs_binds, arg);

   616               val rhs_binds = fv_binds args2 rbinds;

   617               val rhs = mk_pair (rhs_binds, arg2);

   618               val alpha = nth alpha_frees (body_index dt);

   619               val fv = nth fv_frees (body_index dt);

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

   621               val alpha_const = alpha_const_for_binds rbinds;

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

   623               val alpha_gen = Syntax.check_term lthy alpha_gen_pre

   624             in

   625               alpha_gen

   626             end

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

   628         end

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

   630       val alpha_lhss = mk_conjl alphas

   631       val alpha_lhss_ex =

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

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

   634     in

   635       alpha_eq

   636     end;

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

   638   val alphaeqs = map2i alpha_eq descr (gather_binds bindsall)

   639   val alpha_eqs = flat alphaeqs

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

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

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

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

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

   645      (alpha_types @ alpha_bn_types)) []

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

   647 in

   648   (alphas, lthy')

   649 end

   650 *}

   651 

   652 end