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_type = 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 add_perm (p1, p2) = Const(@{const_name plus}, @{typ "perm \<Rightarrow> perm \<Rightarrow> perm"}) $ p1 $ p2 *}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 *)(* TODO: currently works only with current fv_bn function *)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*}(* TODO: Notice datatypes without bindings and replace alpha with equality *)ML {*fun define_fv_alpha (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; 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 fv_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 bindcs = map fst bind_pis; 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 fv_c = nth fv_frees 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 we do not know what to do with non-atomizable things *) @{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) 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 we do not know what to do with non-atomizable things *) @{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)))) 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 add_perm (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 add_perm (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 (fv_eq, alpha_eq) end; fun fv_alpha_eq (i, (_, _, constrs)) binds = map2i (fv_alpha_constr i) constrs binds; val fveqs_alphaeqs = map2i fv_alpha_eq descr (gather_binds bindsall) val (fv_eqs_perfv, alpha_eqs) = apsnd flat (split_list (map split_list fveqs_alphaeqs)) 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 (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'') val ordered_fvs = fv_frees @ fvbns; val all_fvs = (fst fvs @ fst fvs2, snd fvs @ snd fvs2)in (((all_fvs, ordered_fvs), alphas), lthy''')end*}ML {*fun build_alpha_sym_trans_gl alphas (x, y, z) =let fun build_alpha alpha = let val ty = domain_type (fastype_of alpha); val var = Free(x, ty); val var2 = Free(y, ty); val var3 = Free(z, ty); val symp = HOLogic.mk_imp (alpha $ var $ var2, alpha $ var2 $ var); val transp = HOLogic.mk_imp (alpha $ var $ var2, HOLogic.mk_all (z, ty, HOLogic.mk_imp (alpha $ var2 $ var3, alpha $ var $ var3))) in (symp, transp) end; val eqs = map build_alpha alphas val (sym_eqs, trans_eqs) = split_list eqs fun conj l = @{term Trueprop} $ foldr1 HOLogic.mk_conj lin (conj sym_eqs, conj trans_eqs)end*}ML {*fun build_alpha_refl_gl fv_alphas_lst alphas =let val (fvs_alphas, _) = split_list fv_alphas_lst; val (_, alpha_ts) = split_list fvs_alphas; val tys = map (domain_type o fastype_of) alpha_ts; val names = Datatype_Prop.make_tnames tys; val args = map Free (names ~~ tys); fun find_alphas ty x = domain_type (fastype_of x) = ty; fun refl_eq_arg (ty, arg) = let val rel_alphas = filter (find_alphas ty) alphas; in map (fn x => x $ arg $ arg) rel_alphas end; (* Flattening loses the induction structure *) val eqs = map (foldr1 HOLogic.mk_conj) (map refl_eq_arg (tys ~~ args))in (names, HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj eqs))end*}ML {*fun reflp_tac induct eq_iff = rtac induct THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps eq_iff) THEN_ALL_NEW split_conj_tac THEN_ALL_NEW REPEAT o rtac @{thm exI[of _ "0 :: perm"]} THEN_ALL_NEW split_conj_tac THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps @{thms alphas fresh_star_def fresh_zero_perm permute_zero ball_triv add_0_left supp_zero_perm Int_empty_left split_conv})*}ML {*fun build_alpha_refl fv_alphas_lst alphas induct eq_iff ctxt =let val (names, gl) = build_alpha_refl_gl fv_alphas_lst alphas; val refl_conj = Goal.prove ctxt names [] gl (fn _ => reflp_tac induct eq_iff 1);in HOLogic.conj_elims refl_conjend*}lemma exi_neg: "\<exists>(pi :: perm). P pi \<Longrightarrow> (\<And>(p :: perm). P p \<Longrightarrow> Q (- p)) \<Longrightarrow> \<exists>pi. Q pi"apply (erule exE)apply (rule_tac x="-pi" in exI)by autoML {*fun symp_tac induct inj eqvt ctxt = rel_indtac induct THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps inj) THEN_ALL_NEW split_conj_tac THEN_ALL_NEW REPEAT o etac @{thm exi_neg} THEN_ALL_NEW split_conj_tac THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps @{thms supp_minus_perm minus_add[symmetric]}) THEN_ALL_NEW TRY o (resolve_tac @{thms alphas_compose_sym2} ORELSE' resolve_tac @{thms alphas_compose_sym}) THEN_ALL_NEW (asm_full_simp_tac (HOL_ss addsimps (eqvt @ all_eqvts ctxt)))*}lemma exi_sum: "\<exists>(pi :: perm). P pi \<Longrightarrow> \<exists>(pi :: perm). Q pi \<Longrightarrow> (\<And>(p :: perm) (pi :: perm). P p \<Longrightarrow> Q pi \<Longrightarrow> R (pi + p)) \<Longrightarrow> \<exists>pi. R pi"apply (erule exE)+apply (rule_tac x="pia + pi" in exI)by autoML {*fun eetac rule = Subgoal.FOCUS_PARAMS (fn focus => let val concl = #concl focus val prems = Logic.strip_imp_prems (term_of concl) val exs = filter (fn x => is_ex (HOLogic.dest_Trueprop x)) prems val cexs = map (SOME o (cterm_of (ProofContext.theory_of (#context focus)))) exs val thins = map (fn cex => Drule.instantiate' [] [cex] Drule.thin_rl) cexs in (etac rule THEN' RANGE[atac, eresolve_tac thins]) 1 end )*}ML {*fun transp_tac ctxt induct alpha_inj term_inj distinct cases eqvt = rel_indtac induct THEN_ALL_NEW (TRY o rtac allI THEN' imp_elim_tac cases ctxt) THEN_ALL_NEW asm_full_simp_tac (HOL_basic_ss addsimps alpha_inj) THEN_ALL_NEW split_conj_tac THEN_ALL_NEW REPEAT o (eetac @{thm exi_sum} ctxt) THEN_ALL_NEW split_conj_tac THEN_ALL_NEW (asm_full_simp_tac (HOL_ss addsimps (term_inj @ distinct))) THEN_ALL_NEW split_conj_tac THEN_ALL_NEW TRY o (eresolve_tac @{thms alphas_compose_trans2} ORELSE' eresolve_tac @{thms alphas_compose_trans}) THEN_ALL_NEW (asm_full_simp_tac (HOL_ss addsimps (all_eqvts ctxt @ eqvt @ term_inj @ distinct)))*}lemma transpI: "(\<And>xa ya. R xa ya \<longrightarrow> (\<forall>z. R ya z \<longrightarrow> R xa z)) \<Longrightarrow> transp R" unfolding transp_def by blastML {*fun equivp_tac reflps symps transps = (*let val _ = tracing (PolyML.makestring (reflps, symps, transps)) in *) simp_tac (HOL_ss addsimps @{thms equivp_reflp_symp_transp reflp_def symp_def}) THEN' rtac conjI THEN' rtac allI THEN' resolve_tac reflps THEN' rtac conjI THEN' rtac allI THEN' rtac allI THEN' resolve_tac symps THEN' rtac @{thm transpI} THEN' resolve_tac transps*}ML {*fun build_equivps alphas reflps alpha_induct term_inj alpha_inj distinct cases eqvt ctxt =let val ([x, y, z], ctxt') = Variable.variant_fixes ["x","y","z"] ctxt; val (symg, transg) = build_alpha_sym_trans_gl alphas (x, y, z) fun symp_tac' _ = symp_tac alpha_induct alpha_inj eqvt ctxt 1; fun transp_tac' _ = transp_tac ctxt alpha_induct alpha_inj term_inj distinct cases eqvt 1; val symp_loc = Goal.prove ctxt' [] [] symg symp_tac'; val transp_loc = Goal.prove ctxt' [] [] transg transp_tac'; val [symp, transp] = Variable.export ctxt' ctxt [symp_loc, transp_loc] val symps = HOLogic.conj_elims symp val transps = HOLogic.conj_elims transp fun equivp alpha = let val equivp = Const (@{const_name equivp}, fastype_of alpha --> @{typ bool}) val goal = @{term Trueprop} $ (equivp $ alpha) fun tac _ = equivp_tac reflps symps transps 1 in Goal.prove ctxt [] [] goal tac endin map equivp alphasend*}lemma not_in_union: "c \<notin> a \<union> b \<equiv> (c \<notin> a \<and> c \<notin> b)"by autoML {*fun supports_tac perm = simp_tac (HOL_ss addsimps @{thms supports_def not_in_union} @ perm) THEN_ALL_NEW ( REPEAT o rtac allI THEN' REPEAT o rtac impI THEN' split_conj_tac THEN' asm_full_simp_tac (HOL_ss addsimps @{thms fresh_def[symmetric] swap_fresh_fresh fresh_atom swap_at_base_simps(3) swap_atom_image_fresh supp_fset_to_set supp_fmap_atom}))*}ML {*fun mk_supp ty x = Const (@{const_name supp}, ty --> @{typ "atom set"}) $ x*}ML {*fun mk_supports_eq thy cnstr =let val (tys, ty) = (strip_type o fastype_of) cnstr val names = Datatype_Prop.make_tnames tys val frees = map Free (names ~~ tys) val rhs = list_comb (cnstr, frees) fun mk_supp_arg (x, ty) = if is_atom thy ty then mk_supp @{typ atom} (mk_atom ty $ x) else if is_atom_set thy ty then mk_supp @{typ "atom set"} (mk_atom_set x) else if is_atom_fset thy ty then mk_supp @{typ "atom set"} (mk_atom_fset x) else mk_supp ty x val lhss = map mk_supp_arg (frees ~~ tys) val supports = Const(@{const_name "supports"}, @{typ "atom set"} --> ty --> @{typ bool}) val eq = HOLogic.mk_Trueprop (supports $ mk_union lhss $ rhs)in (names, eq)end*}ML {*fun prove_supports ctxt perms cnst =let val (names, eq) = mk_supports_eq (ProofContext.theory_of ctxt) cnstin Goal.prove ctxt names [] eq (fn _ => supports_tac perms 1)end*}ML {*fun mk_fs tys =let val names = Datatype_Prop.make_tnames tys val frees = map Free (names ~~ tys) val supps = map2 mk_supp tys frees val fin_supps = map (fn x => @{term "finite :: atom set \<Rightarrow> bool"} $ x) suppsin (names, HOLogic.mk_Trueprop (mk_conjl fin_supps))end*}ML {*fun fs_tac induct supports = rel_indtac induct THEN_ALL_NEW ( rtac @{thm supports_finite} THEN' resolve_tac supports) THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps @{thms supp_atom supp_atom_image supp_fset_to_set supp_fmap_atom finite_insert finite.emptyI finite_Un finite_supp})*}ML {*fun prove_fs ctxt induct supports tys =let val (names, eq) = mk_fs tysin Goal.prove ctxt names [] eq (fn _ => fs_tac induct supports 1)end*}ML {*fun mk_supp x = Const (@{const_name supp}, fastype_of x --> @{typ "atom set"}) $ x;fun mk_supp_neq arg (fv, alpha) =let val collect = Const ("Collect", @{typ "(atom \<Rightarrow> bool) \<Rightarrow> atom \<Rightarrow> bool"}); val ty = fastype_of arg; val perm = Const ("Nominal2_Base.pt_class.permute", @{typ perm} --> ty --> ty); val finite = @{term "finite :: atom set \<Rightarrow> bool"} val rhs = collect $ Abs ("a", @{typ atom}, HOLogic.mk_not (finite $ (collect $ Abs ("b", @{typ atom}, HOLogic.mk_not (alpha $ (perm $ (@{term swap} $ Bound 1 $ Bound 0) $ arg) $ arg)))))in HOLogic.mk_eq (fv $ arg, rhs)end;fun supp_eq fv_alphas_lst =let val (fvs_alphas, ls) = split_list fv_alphas_lst; val (fv_ts, _) = split_list fvs_alphas; val tys = map (domain_type o fastype_of) fv_ts; val names = Datatype_Prop.make_tnames tys; val args = map Free (names ~~ tys); fun supp_eq_arg ((fv, arg), l) = mk_conjl ((HOLogic.mk_eq (fv $ arg, mk_supp arg)) :: (map (mk_supp_neq arg) l)) val eqs = mk_conjl (map supp_eq_arg ((fv_ts ~~ args) ~~ ls))in (names, HOLogic.mk_Trueprop eqs)end*}ML {*fun combine_fv_alpha_bns (fv_ts_nobn, fv_ts_bn) (alpha_ts_nobn, alpha_ts_bn) bn_nos =if length fv_ts_bn < length alpha_ts_bn then (fv_ts_nobn ~~ alpha_ts_nobn) ~~ (replicate (length fv_ts_nobn) [])else let val fv_alpha_nos = 0 upto (length fv_ts_nobn - 1); fun filter_fn i (x, j) = if j = i then SOME x else NONE; val fv_alpha_bn_nos = (fv_ts_bn ~~ alpha_ts_bn) ~~ bn_nos; val fv_alpha_bn_all = map (fn i => map_filter (filter_fn i) fv_alpha_bn_nos) fv_alpha_nos;in (fv_ts_nobn ~~ alpha_ts_nobn) ~~ fv_alpha_bn_allend*}(* TODO: this is a hack, it assumes that only one type of Abs's is present in the type and chooses this supp_abs. Additionally single atoms are treated properly. *)ML {*fun choose_alpha_abs eqiff =let fun exists_subterms f ts = true mem (map (exists_subterm f) ts); val terms = map prop_of eqiff; fun check cname = exists_subterms (fn x => fst(dest_Const x) = cname handle _ => false) terms val no = if check @{const_name alpha_lst} then 2 else if check @{const_name alpha_res} then 1 else if check @{const_name alpha_gen} then 0 else error "Failure choosing supp_abs"in nth @{thms supp_abs[symmetric]} noend*}lemma supp_abs_atom: "supp (Abs {atom a} (x :: 'a :: fs)) = supp x - {atom a}"by (rule supp_abs(1))lemma supp_abs_sum: "supp (Abs x (a :: 'a :: fs)) \<union> supp (Abs x (b :: 'b :: fs)) = supp (Abs x (a, b))" "supp (Abs_res x (a :: 'a :: fs)) \<union> supp (Abs_res x (b :: 'b :: fs)) = supp (Abs_res x (a, b))" "supp (Abs_lst y (a :: 'a :: fs)) \<union> supp (Abs_lst y (b :: 'b :: fs)) = supp (Abs_lst y (a, b))" apply (simp_all add: supp_abs supp_Pair) apply blast+ doneML {*fun supp_eq_tac ind fv perm eqiff ctxt = rel_indtac ind THEN_ALL_NEW asm_full_simp_tac (HOL_basic_ss addsimps fv) THEN_ALL_NEW asm_full_simp_tac (HOL_basic_ss addsimps @{thms supp_abs_atom[symmetric]}) THEN_ALL_NEW asm_full_simp_tac (HOL_basic_ss addsimps [choose_alpha_abs eqiff]) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps @{thms supp_abs_sum}) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps @{thms supp_def}) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps (@{thms permute_abs} @ perm)) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps (@{thms Abs_eq_iff} @ eqiff)) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps @{thms alphas3 alphas2}) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps @{thms alphas}) THEN_ALL_NEW asm_full_simp_tac (HOL_basic_ss addsimps (@{thm supp_Pair} :: sym_eqvts ctxt)) THEN_ALL_NEW asm_full_simp_tac (HOL_basic_ss addsimps (@{thm Pair_eq} :: all_eqvts ctxt)) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps @{thms supp_at_base[symmetric,simplified supp_def]}) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps @{thms Collect_disj_eq[symmetric]}) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps @{thms infinite_Un[symmetric]}) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps @{thms Collect_disj_eq[symmetric]}) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps @{thms de_Morgan_conj[symmetric]}) THEN_ALL_NEW simp_tac (HOL_basic_ss addsimps @{thms ex_simps(1,2)[symmetric]}) THEN_ALL_NEW simp_tac (HOL_ss addsimps @{thms Collect_const finite.emptyI})*}ML {*fun build_eqvt_gl pi frees fnctn ctxt =let val typ = domain_type (fastype_of fnctn); val arg = the (AList.lookup (op=) frees typ);in ([HOLogic.mk_eq ((perm_arg (fnctn $ arg) $ pi $ (fnctn $ arg)), (fnctn $ (perm_arg arg $ pi $ arg)))], ctxt)end*}ML {*fun prove_eqvt tys ind simps funs ctxt =let val ([pi], ctxt') = Variable.variant_fixes ["p"] ctxt; val pi = Free (pi, @{typ perm}); val tac = asm_full_simp_tac (HOL_ss addsimps (@{thms atom_eqvt permute_list.simps} @ simps @ all_eqvts ctxt')) val ths_loc = prove_by_induct tys (build_eqvt_gl pi) ind tac funs ctxt' val ths = Variable.export ctxt' ctxt ths_loc val add_eqvt = Attrib.internal (fn _ => Nominal_ThmDecls.eqvt_add)in (ths, snd (Local_Theory.note ((Binding.empty, [add_eqvt]), ths) ctxt))end*}end