theory Fvimports "../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 zyields:[ [], [(NONE, 0, 2)], [(SOME (Const f), 0, 2), (Some (Const g), 1, 2)]]A SOME binding has to have a function which takes an appropriateargument and returns an atom set. A NONE binding has to be on anargument 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 empty2) 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 = argr2) 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 endin map (map gather_binds_cons) bindsend*}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 _ = NONEin 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_bnsin ((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_bnsin ((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_locin ((((fv_ts, ord_fv_ts), fv_def), ((alpha_ts, alpha_intros), (alpha_cases, alpha_induct))), ctxt'')end;*}end