theory Fv+ −
imports "../Nominal-General/Nominal2_Atoms" + −
"Abs" "Perm" "Rsp" "Nominal2_FSet"+ −
begin+ −
+ −
(* The bindings data structure:+ −
+ −
Bindings are a list of lists of lists of triples.+ −
+ −
The first list represents the datatypes defined.+ −
The second list represents the constructors.+ −
The internal list is a list of all the bndings that+ −
concern the constructor.+ −
+ −
Every triple consists of a function, the binding and+ −
the body.+ −
+ −
Eg:+ −
nominal_datatype+ −
+ −
C1+ −
| C2 x y z bind x in z+ −
| C3 x y z bind f x in z bind g y in z+ −
+ −
yields:+ −
[+ −
[],+ −
[(NONE, 0, 2)],+ −
[(SOME (Const f), 0, 2), (Some (Const g), 1, 2)]]+ −
+ −
A SOME binding has to have a function which takes an appropriate+ −
argument and returns an atom set. A NONE binding has to be on an+ −
argument that is an atom or an atom set.+ −
*)+ −
+ −
(*+ −
An overview of the generation of free variables:+ −
+ −
1) fv_bn functions are generated only for the non-recursive binds.+ −
+ −
An fv_bn for a constructor is a union of values for the arguments:+ −
+ −
For an argument x that is in the bn function+ −
- if it is a recursive argument bn' we return: fv_bn' x+ −
- otherwise empty+ −
+ −
For an argument x that is not in the bn function+ −
- for atom we return: {atom x}+ −
- for atom set we return: atom ` x+ −
- for a recursive call to type ty' we return: fv_ty' x+ −
with fv of the appropriate type+ −
- otherwise empty+ −
+ −
2) fv_ty functions generated for all types being defined:+ −
+ −
fv_ty for a constructor is a union of values for the arguments.+ −
+ −
For an argument that is bound in a shallow binding we return empty.+ −
+ −
For an argument x that bound in a non-recursive deep binding+ −
we return: fv_bn x.+ −
+ −
Otherwise we return the free variables of the argument minus the+ −
bound variables of the argument.+ −
+ −
The free variables for an argument x are:+ −
- for an atom: {atom x}+ −
- for atom set: atom ` x+ −
- for recursive call to type ty' return: fv_ty' x+ −
- for nominal datatype ty' return: fv_ty' x+ −
+ −
The bound variables are a union of results of all bindings that+ −
involve the given argument. For a paricular binding:+ −
+ −
- for a binding function bn: bn x+ −
- for a recursive argument of type ty': fv_fy' x+ −
- for nominal datatype ty' return: fv_ty' x+ −
*)+ −
+ −
(*+ −
An overview of the generation of alpha-equivalence:+ −
+ −
1) alpha_bn relations are generated for binding functions.+ −
+ −
An alpha_bn for a constructor is true if a conjunction of+ −
propositions for each argument holds.+ −
+ −
For an argument a proposition is build as follows from+ −
th:+ −
+ −
- for a recursive argument in the bn function, we return: alpha_bn argl argr+ −
- for a recursive argument for type ty not in bn, we return: alpha_ty argl argr+ −
- for other arguments in the bn function we return: True+ −
- for other arguments not in the bn function we return: argl = argr+ −
+ −
2) alpha_ty relations are generated for all the types being defined:+ −
+ −
For each constructor we gather all the arguments that are bound,+ −
and for each of those we add a permutation. We associate those+ −
permutations with the bindings. Note that two bindings can have+ −
the same permutation if the arguments being bound are the same.+ −
+ −
An alpha_ty for a constructor is true if there exist permutations+ −
as above such that a conjunction of propositions for all arguments holds.+ −
+ −
For an argument we allow bindings where only one of the following+ −
holds:+ −
+ −
- Argument is bound in some shallow bindings: We return true+ −
- Argument of type ty is bound recursively in some other+ −
arguments [i1, .. in] with one binding function bn.+ −
We return:+ −
+ −
(bn argl, (argl, argl_i1, ..., argl_in)) \<approx>gen+ −
\<lambda>(argl,argl1,..,argln) (argr,argr1,..,argrn). + −
(alpha_ty argl argr) \<and> (alpha_i1 argl1 argr1) \<and> .. \<and> (alpha_in argln argrn)+ −
\<lambda>(arg,arg1,..,argn). (fv_ty arg) \<union> (fv_i1 arg1) \<union> .. \<union> (fv_in argn)+ −
pi+ −
(bn argr, (argr, argr_i1, ..., argr_in))+ −
+ −
- Argument is bound in some deep non-recursive bindings.+ −
We return: alpha_bn argl argr+ −
- Argument of type ty has some shallow bindings [b1..bn] and/or+ −
non-recursive bindings [f1 a1, .., fm am], where the bindings+ −
have the permutations p1..pl. We return:+ −
+ −
(b1l \<union>..\<union> bnl \<union> f1 a1l \<union>..\<union> fn anl, argl) \<approx>gen+ −
alpha_ty fv_ty (p1 +..+ pl)+ −
(b1r \<union>..\<union> bnr \<union> f1 a1r \<union>..\<union> fn anr, argr)+ −
+ −
- Argument has some recursive bindings. The bindings were+ −
already treated in 2nd case so we return: True+ −
- Argument has no bindings and is not bound.+ −
If it is recursive for type ty, we return: alpha_ty argl argr+ −
Otherwise we return: argl = argr+ −
+ −
*)+ −
+ −
+ −
ML {*+ −
datatype alpha_mode = AlphaGen | AlphaRes | AlphaLst;+ −
*}+ −
+ −
ML {*+ −
fun atyp_const AlphaGen = @{const_name alpha_gen}+ −
| atyp_const AlphaRes = @{const_name alpha_res}+ −
| atyp_const AlphaLst = @{const_name alpha_lst}+ −
*}+ −
+ −
(* TODO: make sure that parser checks that bindings are compatible *)+ −
ML {*+ −
fun alpha_const_for_binds [] = atyp_const AlphaGen+ −
| alpha_const_for_binds ((NONE, _, _, at) :: t) = atyp_const at+ −
| alpha_const_for_binds ((SOME (_, _), _, _, at) :: _) = atyp_const at+ −
*}+ −
+ −
ML {*+ −
fun is_atom thy typ =+ −
Sign.of_sort thy (typ, @{sort at})+ −
+ −
fun is_atom_set thy (Type ("fun", [t, @{typ bool}])) = is_atom thy t+ −
| is_atom_set _ _ = false;+ −
+ −
fun is_atom_fset thy (Type ("FSet.fset", [t])) = is_atom thy t+ −
| is_atom_fset _ _ = false;+ −
*}+ −
+ −
+ −
(* Like map2, only if the second list is empty passes empty lists insted of error *)+ −
ML {*+ −
fun map2i _ [] [] = []+ −
| map2i f (x :: xs) (y :: ys) = f x y :: map2i f xs ys+ −
| map2i f (x :: xs) [] = f x [] :: map2i f xs []+ −
| map2i _ _ _ = raise UnequalLengths;+ −
*}+ −
+ −
(* Finds bindings with the same function and binding, and gathers all+ −
bodys for such pairs+ −
*)+ −
ML {*+ −
fun gather_binds binds =+ −
let+ −
fun gather_binds_cons binds =+ −
let+ −
val common = map (fn (f, bi, _, aty) => (f, bi, aty)) binds+ −
val nodups = distinct (op =) common+ −
fun find_bodys (sf, sbi, sty) =+ −
filter (fn (f, bi, _, aty) => f = sf andalso bi = sbi andalso aty = sty) binds+ −
val bodys = map ((map (fn (_, _, bo, _) => bo)) o find_bodys) nodups+ −
in+ −
nodups ~~ bodys+ −
end+ −
in+ −
map (map gather_binds_cons) binds+ −
end+ −
*}+ −
+ −
ML {*+ −
fun un_gather_binds_cons binds =+ −
flat (map (fn (((f, bi, aty), bos), pi) => map (fn bo => ((f, bi, bo, aty), pi)) bos) binds)+ −
*}+ −
+ −
ML {*+ −
open Datatype_Aux; (* typ_of_dtyp, DtRec, ... *);+ −
*}+ −
ML {*+ −
(* TODO: It is the same as one in 'nominal_atoms' *)+ −
fun mk_atom ty = Const (@{const_name atom}, ty --> @{typ atom});+ −
val noatoms = @{term "{} :: atom set"};+ −
fun mk_single_atom x = HOLogic.mk_set @{typ atom} [mk_atom (type_of x) $ x];+ −
fun mk_union sets =+ −
fold (fn a => fn b =>+ −
if a = noatoms then b else+ −
if b = noatoms then a else+ −
if a = b then a else+ −
HOLogic.mk_binop @{const_name sup} (a, b)) (rev sets) noatoms;+ −
val mk_inter = foldr1 (HOLogic.mk_binop @{const_name inf})+ −
fun mk_diff a b =+ −
if b = noatoms then a else+ −
if b = a then noatoms else+ −
HOLogic.mk_binop @{const_name minus} (a, b);+ −
fun mk_atom_set t =+ −
let+ −
val ty = fastype_of t;+ −
val atom_ty = HOLogic.dest_setT ty --> @{typ atom};+ −
val img_ty = atom_ty --> ty --> @{typ "atom set"};+ −
in+ −
(Const (@{const_name image}, img_ty) $ Const (@{const_name atom}, atom_ty) $ t)+ −
end;+ −
fun mk_atom_fset t =+ −
let+ −
val ty = fastype_of t;+ −
val atom_ty = dest_fsetT ty --> @{typ atom};+ −
val fmap_ty = atom_ty --> ty --> @{typ "atom fset"};+ −
val fset_to_set = @{term "fset_to_set :: atom fset \<Rightarrow> atom set"}+ −
in+ −
fset_to_set $ ((Const (@{const_name fmap}, fmap_ty) $ Const (@{const_name atom}, atom_ty) $ t))+ −
end;+ −
(* Similar to one in USyntax *)+ −
fun mk_pair (fst, snd) =+ −
let val ty1 = fastype_of fst+ −
val ty2 = fastype_of snd+ −
val c = HOLogic.pair_const ty1 ty2+ −
in c $ fst $ snd+ −
end;+ −
*}+ −
+ −
(* Given [fv1, fv2, fv3] creates %(x, y, z). fv1 x u fv2 y u fv3 z *)+ −
ML {*+ −
fun mk_compound_fv fvs =+ −
let+ −
val nos = (length fvs - 1) downto 0;+ −
val fvs_applied = map (fn (fv, no) => fv $ Bound no) (fvs ~~ nos);+ −
val fvs_union = mk_union fvs_applied;+ −
val (tyh :: tys) = rev (map (domain_type o fastype_of) fvs);+ −
fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))+ −
in+ −
fold fold_fun tys (Abs ("", tyh, fvs_union))+ −
end;+ −
*}+ −
+ −
(* Given [R1, R2, R3] creates %(x,x'). %(y,y'). %(z,z'). R x x' \<and> R y y' \<and> R z z' *)+ −
ML {*+ −
fun mk_compound_alpha Rs =+ −
let+ −
val nos = (length Rs - 1) downto 0;+ −
val nos2 = (2 * length Rs - 1) downto length Rs;+ −
val Rs_applied = map (fn (R, (no2, no)) => R $ Bound no2 $ Bound no) (Rs ~~ (nos2 ~~ nos));+ −
val Rs_conj = mk_conjl Rs_applied;+ −
val (tyh :: tys) = rev (map (domain_type o fastype_of) Rs);+ −
fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))+ −
val abs_rhs = fold fold_fun tys (Abs ("", tyh, Rs_conj))+ −
in+ −
fold fold_fun tys (Abs ("", tyh, abs_rhs))+ −
end;+ −
*}+ −
+ −
+ −
ML {*+ −
fun non_rec_binds l =+ −
let+ −
fun is_non_rec (SOME (f, false), _, _, _) = SOME f+ −
| is_non_rec _ = NONE+ −
in+ −
distinct (op =) (map_filter is_non_rec (flat (flat l)))+ −
end+ −
*}+ −
+ −
(* We assume no bindings in the type on which bn is defined *)+ −
ML {*+ −
fun fv_bn thy (dt_info : Datatype_Aux.info) fv_frees bn_fvbn (fvbn, (bn, ith_dtyp, args_in_bns)) =+ −
let+ −
val {descr, sorts, ...} = dt_info;+ −
fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+ −
fun fv_bn_constr (cname, dts) args_in_bn =+ −
let+ −
val Ts = map (typ_of_dtyp descr sorts) dts;+ −
val names = Datatype_Prop.make_tnames Ts;+ −
val args = map Free (names ~~ Ts);+ −
val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+ −
fun fv_arg ((dt, x), arg_no) =+ −
let+ −
val ty = fastype_of x+ −
(* val _ = tracing ("B 1" ^ PolyML.makestring args_in_bn);*)+ −
(* val _ = tracing ("B 2" ^ PolyML.makestring bn_fvbn);*)+ −
in+ −
case AList.lookup (op=) args_in_bn arg_no of+ −
SOME NONE => @{term "{} :: atom set"}+ −
| SOME (SOME (f : term)) => (the (AList.lookup (op=) bn_fvbn f)) $ x+ −
| NONE =>+ −
if is_atom thy ty then mk_single_atom x else+ −
if is_atom_set thy ty then mk_atom_set x else+ −
if is_atom_fset thy ty then mk_atom_fset x else+ −
if is_rec_type dt then nth fv_frees (body_index dt) $ x else+ −
@{term "{} :: atom set"}+ −
end;+ −
val arg_nos = 0 upto (length dts - 1)+ −
in+ −
HOLogic.mk_Trueprop (HOLogic.mk_eq+ −
(fvbn $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ arg_nos))))+ −
end;+ −
val (_, (_, _, constrs)) = nth descr ith_dtyp;+ −
val eqs = map2i fv_bn_constr constrs args_in_bns+ −
in+ −
((bn, fvbn), eqs)+ −
end+ −
*}+ −
+ −
ML {* print_depth 100 *}+ −
ML {*+ −
fun fv_bns thy dt_info fv_frees rel_bns =+ −
let+ −
fun mk_fvbn_free (bn, ith, _) =+ −
let+ −
val fvbn_name = "fv_" ^ (Long_Name.base_name (fst (dest_Const bn)));+ −
in+ −
(fvbn_name, Free (fvbn_name, fastype_of (nth fv_frees ith)))+ −
end;+ −
val (fvbn_names, fvbn_frees) = split_list (map mk_fvbn_free rel_bns);+ −
val bn_fvbn = (map (fn (bn, _, _) => bn) rel_bns) ~~ fvbn_frees+ −
val (l1, l2) = split_list (map (fv_bn thy dt_info fv_frees bn_fvbn) (fvbn_frees ~~ rel_bns));+ −
in+ −
(l1, (fvbn_names ~~ l2))+ −
end+ −
*}+ −
+ −
+ −
ML {*+ −
fun alpha_bn (dt_info : Datatype_Aux.info) alpha_frees bn_alphabn ((bn, ith_dtyp, args_in_bns), (alpha_bn_free, _ (*is_rec*) )) =+ −
let+ −
val {descr, sorts, ...} = dt_info;+ −
fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+ −
fun alpha_bn_constr (cname, dts) args_in_bn =+ −
let+ −
val Ts = map (typ_of_dtyp descr sorts) dts;+ −
val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);+ −
val names2 = Name.variant_list ("pi" :: names) (Datatype_Prop.make_tnames Ts);+ −
val args = map Free (names ~~ Ts);+ −
val args2 = map Free (names2 ~~ Ts);+ −
val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+ −
val rhs = HOLogic.mk_Trueprop+ −
(alpha_bn_free $ (list_comb (c, args)) $ (list_comb (c, args2)));+ −
fun lhs_arg ((dt, arg_no), (arg, arg2)) =+ −
case AList.lookup (op=) args_in_bn arg_no of+ −
SOME NONE => @{term True}+ −
| SOME (SOME f) => (the (AList.lookup (op=) bn_alphabn f)) $ arg $ arg2+ −
| NONE =>+ −
if is_rec_type dt then (nth alpha_frees (body_index dt)) $ arg $ arg2+ −
else HOLogic.mk_eq (arg, arg2)+ −
val arg_nos = 0 upto (length dts - 1)+ −
val lhss = mk_conjl (map lhs_arg (dts ~~ arg_nos ~~ (args ~~ args2)))+ −
val eq = Logic.mk_implies (HOLogic.mk_Trueprop lhss, rhs)+ −
in+ −
eq+ −
end+ −
val (_, (_, _, constrs)) = nth descr ith_dtyp;+ −
val eqs = map2i alpha_bn_constr constrs args_in_bns+ −
in+ −
((bn, alpha_bn_free), eqs)+ −
end+ −
*}+ −
+ −
ML {*+ −
fun alpha_bns dt_info alpha_frees rel_bns bns_rec =+ −
let+ −
val {descr, sorts, ...} = dt_info;+ −
fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+ −
fun mk_alphabn_free (bn, ith, _) =+ −
let+ −
val alphabn_name = "alpha_" ^ (Long_Name.base_name (fst (dest_Const bn)));+ −
val alphabn_type = nth_dtyp ith --> nth_dtyp ith --> @{typ bool};+ −
val alphabn_free = Free(alphabn_name, alphabn_type);+ −
in+ −
(alphabn_name, alphabn_free)+ −
end;+ −
val (alphabn_names, alphabn_frees) = split_list (map mk_alphabn_free rel_bns);+ −
val bn_alphabn = (map (fn (bn, _, _) => bn) rel_bns) ~~ alphabn_frees;+ −
val pair = split_list (map (alpha_bn dt_info alpha_frees bn_alphabn)+ −
(rel_bns ~~ (alphabn_frees ~~ bns_rec)))+ −
in+ −
(alphabn_names, pair)+ −
end+ −
*}+ −
+ −
+ −
(* Checks that a list of bindings contains only compatible ones *)+ −
ML {*+ −
fun bns_same l =+ −
length (distinct (op =) (map (fn ((b, _, _, atyp), _) => (b, atyp)) l)) = 1+ −
*}+ −
+ −
ML {*+ −
fun setify x =+ −
if fastype_of x = @{typ "atom list"} then+ −
Const (@{const_name set}, @{typ "atom list \<Rightarrow> atom set"}) $ x else x+ −
*}+ −
+ −
ML {*+ −
fun define_fv (dt_info : Datatype_Aux.info) bindsall bns lthy =+ −
let+ −
val thy = ProofContext.theory_of lthy;+ −
val {descr, sorts, ...} = dt_info;+ −
fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+ −
val fv_names = Datatype_Prop.indexify_names (map (fn (i, _) =>+ −
"fv_" ^ name_of_typ (nth_dtyp i)) descr);+ −
val fv_types = map (fn (i, _) => nth_dtyp i --> @{typ "atom set"}) descr;+ −
val fv_frees = map Free (fv_names ~~ fv_types);+ −
(* TODO: We need a transitive closure, but instead we do this hack considering+ −
all binding functions as recursive or not *)+ −
val nr_bns =+ −
if (non_rec_binds bindsall) = [] then []+ −
else map (fn (bn, _, _) => bn) bns;+ −
val rel_bns = filter (fn (bn, _, _) => bn mem nr_bns) bns;+ −
val (bn_fv_bns, fv_bn_names_eqs) = fv_bns thy dt_info fv_frees rel_bns;+ −
val fvbns = map snd bn_fv_bns;+ −
val (fv_bn_names, fv_bn_eqs) = split_list fv_bn_names_eqs;+ −
+ −
fun fv_constr ith_dtyp (cname, dts) bindcs =+ −
let+ −
val Ts = map (typ_of_dtyp descr sorts) dts;+ −
val bindslen = length bindcs+ −
val pi_strs_same = replicate bindslen "pi"+ −
val pi_strs = Name.variant_list [] pi_strs_same;+ −
val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs;+ −
val bind_pis_gath = bindcs ~~ pis;+ −
val bind_pis = un_gather_binds_cons bind_pis_gath;+ −
val bindcs = map fst bind_pis;+ −
val names = Name.variant_list pi_strs (Datatype_Prop.make_tnames Ts);+ −
val args = map Free (names ~~ Ts);+ −
val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+ −
val fv_c = nth fv_frees ith_dtyp;+ −
val arg_nos = 0 upto (length dts - 1)+ −
fun fv_bind args (NONE, i, _, _) =+ −
if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else+ −
if ((is_atom thy) o fastype_of) (nth args i) then mk_single_atom (nth args i) else+ −
if ((is_atom_set thy) o fastype_of) (nth args i) then mk_atom_set (nth args i) else+ −
if ((is_atom_fset thy) o fastype_of) (nth args i) then mk_atom_fset (nth args i) else+ −
(* TODO goes the code for preiously defined nominal datatypes *)+ −
@{term "{} :: atom set"}+ −
| fv_bind args (SOME (f, _), i, _, _) = f $ (nth args i)+ −
fun fv_binds_as_set args relevant = mk_union (map (setify o fv_bind args) relevant)+ −
fun find_nonrec_binder j (SOME (f, false), i, _, _) = if i = j then SOME f else NONE+ −
| find_nonrec_binder _ _ = NONE+ −
fun fv_arg ((dt, x), arg_no) =+ −
case get_first (find_nonrec_binder arg_no) bindcs of+ −
SOME f =>+ −
(case get_first (fn (x, y) => if x = f then SOME y else NONE) bn_fv_bns of+ −
SOME fv_bn => fv_bn $ x+ −
| NONE => error "bn specified in a non-rec binding but not in bn list")+ −
| NONE =>+ −
let+ −
val arg =+ −
if is_rec_type dt then nth fv_frees (body_index dt) $ x else+ −
if ((is_atom thy) o fastype_of) x then mk_single_atom x else+ −
if ((is_atom_set thy) o fastype_of) x then mk_atom_set x else+ −
if ((is_atom_fset thy) o fastype_of) x then mk_atom_fset x else+ −
(* TODO goes the code for preiously defined nominal datatypes *)+ −
@{term "{} :: atom set"};+ −
(* If i = j then we generate it only once *)+ −
val relevant = filter (fn (_, i, j, _) => ((i = arg_no) orelse (j = arg_no))) bindcs;+ −
val sub = fv_binds_as_set args relevant+ −
in+ −
mk_diff arg sub+ −
end;+ −
val fv_eq = HOLogic.mk_Trueprop (HOLogic.mk_eq+ −
(fv_c $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ arg_nos))))+ −
in+ −
fv_eq+ −
end;+ −
fun fv_eq (i, (_, _, constrs)) binds = map2i (fv_constr i) constrs binds;+ −
val fveqs = map2i fv_eq descr (gather_binds bindsall)+ −
val fv_eqs_perfv = fveqs+ −
val rel_bns_nos = map (fn (_, i, _) => i) rel_bns;+ −
fun filter_fun (_, b) = b mem rel_bns_nos;+ −
val all_fvs = (fv_names ~~ fv_eqs_perfv) ~~ (0 upto (length fv_names - 1))+ −
val (fv_names_fst, fv_eqs_fst) = apsnd flat (split_list (map fst (filter_out filter_fun all_fvs)))+ −
val (fv_names_snd, fv_eqs_snd) = apsnd flat (split_list (map fst (filter filter_fun all_fvs)))+ −
val fv_eqs_all = fv_eqs_fst @ (flat fv_bn_eqs);+ −
val fv_names_all = fv_names_fst @ fv_bn_names;+ −
val add_binds = map (fn x => (Attrib.empty_binding, x))+ −
(* Function_Fun.add_fun Function_Common.default_config ... true *)+ −
val (fvs, lthy') = (Primrec.add_primrec+ −
(map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_all) (add_binds fv_eqs_all) lthy)+ −
val (fvs2, lthy'') =+ −
if fv_eqs_snd = [] then (([], []), lthy') else+ −
(Primrec.add_primrec+ −
(map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_snd) (add_binds fv_eqs_snd) lthy')+ −
val ordered_fvs = fv_frees @ fvbns;+ −
val all_fvs = (fst fvs @ fst fvs2, snd fvs @ snd fvs2)+ −
in+ −
((all_fvs, ordered_fvs), lthy'')+ −
end+ −
*}+ −
+ −
ML {*+ −
fun define_alpha (dt_info : Datatype_Aux.info) bindsall bns fv_frees lthy =+ −
let+ −
val thy = ProofContext.theory_of lthy;+ −
val {descr, sorts, ...} = dt_info;+ −
fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+ −
(* TODO: We need a transitive closure, but instead we do this hack considering+ −
all binding functions as recursive or not *)+ −
val nr_bns =+ −
if (non_rec_binds bindsall) = [] then []+ −
else map (fn (bn, _, _) => bn) bns;+ −
val alpha_names = Datatype_Prop.indexify_names (map (fn (i, _) =>+ −
"alpha_" ^ name_of_typ (nth_dtyp i)) descr);+ −
val alpha_types = map (fn (i, _) => nth_dtyp i --> nth_dtyp i --> @{typ bool}) descr;+ −
val alpha_frees = map Free (alpha_names ~~ alpha_types);+ −
(* We assume that a bn is either recursive or not *)+ −
val bns_rec = map (fn (bn, _, _) => not (bn mem nr_bns)) bns;+ −
val (alpha_bn_names, (bn_alpha_bns, alpha_bn_eqs)) =+ −
alpha_bns dt_info alpha_frees bns bns_rec+ −
val alpha_bn_frees = map snd bn_alpha_bns;+ −
val alpha_bn_types = map fastype_of alpha_bn_frees;+ −
+ −
fun alpha_constr ith_dtyp (cname, dts) bindcs =+ −
let+ −
val Ts = map (typ_of_dtyp descr sorts) dts;+ −
val bindslen = length bindcs+ −
val pi_strs_same = replicate bindslen "pi"+ −
val pi_strs = Name.variant_list [] pi_strs_same;+ −
val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs;+ −
val bind_pis_gath = bindcs ~~ pis;+ −
val bind_pis = un_gather_binds_cons bind_pis_gath;+ −
val names = Name.variant_list pi_strs (Datatype_Prop.make_tnames Ts);+ −
val args = map Free (names ~~ Ts);+ −
val names2 = Name.variant_list (pi_strs @ names) (Datatype_Prop.make_tnames Ts);+ −
val args2 = map Free (names2 ~~ Ts);+ −
val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+ −
val alpha = nth alpha_frees ith_dtyp;+ −
val arg_nos = 0 upto (length dts - 1)+ −
fun fv_bind args (NONE, i, _, _) =+ −
if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else+ −
if ((is_atom thy) o fastype_of) (nth args i) then mk_single_atom (nth args i) else+ −
if ((is_atom_set thy) o fastype_of) (nth args i) then mk_atom_set (nth args i) else+ −
if ((is_atom_fset thy) o fastype_of) (nth args i) then mk_atom_fset (nth args i) else+ −
(* TODO goes the code for preiously defined nominal datatypes *)+ −
@{term "{} :: atom set"}+ −
| fv_bind args (SOME (f, _), i, _, _) = f $ (nth args i)+ −
fun fv_binds args relevant = mk_union (map (fv_bind args) relevant)+ −
val alpha_rhs =+ −
HOLogic.mk_Trueprop (alpha $ (list_comb (c, args)) $ (list_comb (c, args2)));+ −
fun alpha_arg ((dt, arg_no), (arg, arg2)) =+ −
let+ −
val rel_in_simp_binds = filter (fn ((NONE, i, _, _), _) => i = arg_no | _ => false) bind_pis;+ −
val rel_in_comp_binds = filter (fn ((SOME _, i, _, _), _) => i = arg_no | _ => false) bind_pis;+ −
val rel_has_binds = filter (fn ((NONE, _, j, _), _) => j = arg_no+ −
| ((SOME (_, false), _, j, _), _) => j = arg_no+ −
| _ => false) bind_pis;+ −
val rel_has_rec_binds = filter+ −
(fn ((SOME (_, true), _, j, _), _) => j = arg_no | _ => false) bind_pis;+ −
in+ −
case (rel_in_simp_binds, rel_in_comp_binds, rel_has_binds, rel_has_rec_binds) of+ −
([], [], [], []) =>+ −
if is_rec_type dt then (nth alpha_frees (body_index dt) $ arg $ arg2)+ −
else (HOLogic.mk_eq (arg, arg2))+ −
| (_, [], [], []) => @{term True}+ −
| ([], [], [], _) => @{term True}+ −
| ([], ((((SOME (bn, is_rec)), _, _, atyp), _) :: _), [], []) =>+ −
if not (bns_same rel_in_comp_binds) then error "incompatible bindings for an argument" else+ −
if is_rec then+ −
let+ −
val (rbinds, rpis) = split_list rel_in_comp_binds+ −
val bound_in_nos = map (fn (_, _, i, _) => i) rbinds+ −
val bound_in_ty_nos = map (fn i => body_index (nth dts i)) bound_in_nos;+ −
val bound_args = arg :: map (nth args) bound_in_nos;+ −
val bound_args2 = arg2 :: map (nth args2) bound_in_nos;+ −
val lhs_binds = fv_binds args rbinds+ −
val lhs_arg = foldr1 HOLogic.mk_prod bound_args+ −
val lhs = mk_pair (lhs_binds, lhs_arg);+ −
val rhs_binds = fv_binds args2 rbinds;+ −
val rhs_arg = foldr1 HOLogic.mk_prod bound_args2;+ −
val rhs = mk_pair (rhs_binds, rhs_arg);+ −
val fvs = map (nth fv_frees) ((body_index dt) :: bound_in_ty_nos);+ −
val fv = mk_compound_fv fvs;+ −
val alphas = map (nth alpha_frees) ((body_index dt) :: bound_in_ty_nos);+ −
val alpha = mk_compound_alpha alphas;+ −
val pi = foldr1 (uncurry mk_plus) (distinct (op =) rpis);+ −
val alpha_gen_pre = Const (atyp_const atyp, dummyT) $ lhs $ alpha $ fv $ pi $ rhs;+ −
val alpha_gen = Syntax.check_term lthy alpha_gen_pre+ −
in+ −
alpha_gen+ −
end+ −
else+ −
let+ −
val alpha_bn_const =+ −
nth alpha_bn_frees (find_index (fn (b, _, _) => b = bn) bns)+ −
in+ −
alpha_bn_const $ arg $ arg2+ −
end+ −
| ([], [], relevant, []) =>+ −
let+ −
val (rbinds, rpis) = split_list relevant+ −
val lhs_binds = fv_binds args rbinds+ −
val lhs = mk_pair (lhs_binds, arg);+ −
val rhs_binds = fv_binds args2 rbinds;+ −
val rhs = mk_pair (rhs_binds, arg2);+ −
val alpha = nth alpha_frees (body_index dt);+ −
val fv = nth fv_frees (body_index dt);+ −
val pi = foldr1 (uncurry mk_plus) (distinct (op =) rpis);+ −
val alpha_const = alpha_const_for_binds rbinds;+ −
val alpha_gen_pre = Const (alpha_const, dummyT) $ lhs $ alpha $ fv $ pi $ rhs;+ −
val alpha_gen = Syntax.check_term lthy alpha_gen_pre+ −
in+ −
alpha_gen+ −
end+ −
| _ => error "Fv.alpha: not supported binding structure"+ −
end+ −
val alphas = map alpha_arg (dts ~~ arg_nos ~~ (args ~~ args2))+ −
val alpha_lhss = mk_conjl alphas+ −
val alpha_lhss_ex =+ −
fold (fn pi_str => fn t => HOLogic.mk_exists (pi_str, @{typ perm}, t)) pi_strs alpha_lhss+ −
val alpha_eq = Logic.mk_implies (HOLogic.mk_Trueprop alpha_lhss_ex, alpha_rhs)+ −
in+ −
alpha_eq+ −
end;+ −
fun alpha_eq (i, (_, _, constrs)) binds = map2i (alpha_constr i) constrs binds;+ −
val alphaeqs = map2i alpha_eq descr (gather_binds bindsall)+ −
val alpha_eqs = flat alphaeqs+ −
val add_binds = map (fn x => (Attrib.empty_binding, x))+ −
val (alphas, lthy') = (Inductive.add_inductive_i+ −
{quiet_mode = true, verbose = false, alt_name = Binding.empty,+ −
coind = false, no_elim = false, no_ind = false, skip_mono = true, fork_mono = false}+ −
(map2 (fn x => fn y => ((Binding.name x, y), NoSyn)) (alpha_names @ alpha_bn_names)+ −
(alpha_types @ alpha_bn_types)) []+ −
(add_binds (alpha_eqs @ flat alpha_bn_eqs)) [] lthy)+ −
in+ −
(alphas, lthy')+ −
end+ −
*}+ −
+ −
+ −
ML {*+ −
fun define_fv_alpha_export dt binds bns ctxt =+ −
let+ −
val (((fv_ts_loc, fv_def_loc), ord_fv_ts_loc), ctxt') =+ −
define_fv dt binds bns ctxt;+ −
val (alpha, ctxt'') =+ −
define_alpha dt binds bns fv_ts_loc ctxt';+ −
val alpha_ts_loc = #preds alpha+ −
val alpha_induct_loc = #induct alpha+ −
val alpha_intros_loc = #intrs alpha;+ −
val alpha_cases_loc = #elims alpha+ −
val morphism = ProofContext.export_morphism ctxt'' ctxt;+ −
val fv_ts = map (Morphism.term morphism) fv_ts_loc;+ −
val ord_fv_ts = map (Morphism.term morphism) ord_fv_ts_loc;+ −
val fv_def = Morphism.fact morphism fv_def_loc;+ −
val alpha_ts = map (Morphism.term morphism) alpha_ts_loc;+ −
val alpha_induct = Morphism.thm morphism alpha_induct_loc;+ −
val alpha_intros = Morphism.fact morphism alpha_intros_loc+ −
val alpha_cases = Morphism.fact morphism alpha_cases_loc+ −
in+ −
((((fv_ts, ord_fv_ts), fv_def), ((alpha_ts, alpha_intros), (alpha_cases, alpha_induct))), ctxt'')+ −
end;+ −
*}+ −
+ −
end+ −