(* Title: HOL/Nominal/nominal_package.ML ID: $Id: nominal_package.ML,v 1.87 2007/10/06 14:50:04 wenzelm Exp $ Author: Stefan Berghofer and Christian Urban, TU Muenchen Nominal datatype package for Isabelle/HOL. *) signature NOMINAL_PACKAGE = sig val add_nominal_datatype : bool -> string list -> (string list * bstring * mixfix * (bstring * string list * mixfix) list) list -> theory -> theory type descr type nominal_datatype_info val get_nominal_datatypes : theory -> nominal_datatype_info Symtab.table val get_nominal_datatype : theory -> string -> nominal_datatype_info option val mk_perm: typ list -> term -> term -> term val perm_of_pair: term * term -> term val mk_not_sym: thm list -> thm list val perm_simproc: simproc end structure NominalPackage : NOMINAL_PACKAGE = struct val finite_emptyI = thm "finite.emptyI"; val finite_Diff = thm "finite_Diff"; val finite_Un = thm "finite_Un"; val Un_iff = thm "Un_iff"; val In0_eq = thm "In0_eq"; val In1_eq = thm "In1_eq"; val In0_not_In1 = thm "In0_not_In1"; val In1_not_In0 = thm "In1_not_In0"; val Un_assoc = thm "Un_assoc"; val Collect_disj_eq = thm "Collect_disj_eq"; val empty_def = thm "empty_def"; val empty_iff = thm "empty_iff"; open DatatypeAux; open NominalAtoms; (** FIXME: DatatypePackage should export this function **) local fun dt_recs (DtTFree _) = [] | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts) | dt_recs (DtRec i) = [i]; fun dt_cases (descr: descr) (_, args, constrs) = let fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i))); val bnames = map the_bname (distinct op = (List.concat (map dt_recs args))); in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end; fun induct_cases descr = DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr))); fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i)); in fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr); fun mk_case_names_exhausts descr new = map (RuleCases.case_names o exhaust_cases descr o #1) (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr); end; (* theory data *) type descr = (int * (string * dtyp list * (string * (dtyp list * dtyp) list) list)) list; type nominal_datatype_info = {index : int, descr : descr, sorts : (string * sort) list, rec_names : string list, rec_rewrites : thm list, induction : thm, distinct : thm list, inject : thm list}; structure NominalDatatypesData = TheoryDataFun ( type T = nominal_datatype_info Symtab.table; val empty = Symtab.empty; val copy = I; val extend = I; fun merge _ tabs : T = Symtab.merge (K true) tabs; ); val get_nominal_datatypes = NominalDatatypesData.get; val put_nominal_datatypes = NominalDatatypesData.put; val map_nominal_datatypes = NominalDatatypesData.map; val get_nominal_datatype = Symtab.lookup o get_nominal_datatypes; (**** make datatype info ****) fun make_dt_info descr sorts induct reccomb_names rec_thms (((i, (_, (tname, _, _))), distinct), inject) = (tname, {index = i, descr = descr, sorts = sorts, rec_names = reccomb_names, rec_rewrites = rec_thms, induction = induct, distinct = distinct, inject = inject}); (*******************************) val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma); fun read_typ sign ((Ts, sorts), str) = let val T = Type.no_tvars (Sign.read_def_typ (sign, (AList.lookup op =) (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg in (Ts @ [T], add_typ_tfrees (T, sorts)) end; (** taken from HOL/Tools/datatype_aux.ML **) fun indtac indrule indnames i st = let val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule)); val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop (Logic.strip_imp_concl (List.nth (prems_of st, i - 1)))); val getP = if can HOLogic.dest_imp (hd ts) then (apfst SOME) o HOLogic.dest_imp else pair NONE; fun abstr (t1, t2) = (case t1 of NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false) (term_frees t2) of [Free (s, T)] => absfree (s, T, t2) | _ => sys_error "indtac") | SOME (_ $ t') => Abs ("x", fastype_of t', abstract_over (t', t2))) val cert = cterm_of (Thm.theory_of_thm st); val Ps = map (cert o head_of o snd o getP) ts; val indrule' = cterm_instantiate (Ps ~~ (map (cert o abstr o getP) ts')) indrule in rtac indrule' i st end; fun mk_subgoal i f st = let val cg = List.nth (cprems_of st, i - 1); val g = term_of cg; val revcut_rl' = Thm.lift_rule cg revcut_rl; val v = head_of (Logic.strip_assums_concl (hd (prems_of revcut_rl'))); val ps = Logic.strip_params g; val cert = cterm_of (Thm.theory_of_thm st); in compose_tac (false, Thm.instantiate ([], [(cert v, cert (list_abs (ps, f (rev ps) (Logic.strip_assums_hyp g) (Logic.strip_assums_concl g))))]) revcut_rl', 2) i st end; (** simplification procedure for sorting permutations **) val dj_cp = thm "dj_cp"; fun dest_permT (Type ("fun", [Type ("List.list", [Type ("*", [T, _])]), Type ("fun", [_, U])])) = (T, U); fun permTs_of (Const ("Nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u | permTs_of _ = []; fun perm_simproc' thy ss (Const ("Nominal.perm", T) $ t $ (u as Const ("Nominal.perm", U) $ r $ s)) = let val (aT as Type (a, []), S) = dest_permT T; val (bT as Type (b, []), _) = dest_permT U in if aT mem permTs_of u andalso aT <> bT then let val a' = Sign.base_name a; val b' = Sign.base_name b; val cp = PureThy.get_thm thy (Name ("cp_" ^ a' ^ "_" ^ b' ^ "_inst")); val dj = PureThy.get_thm thy (Name ("dj_" ^ b' ^ "_" ^ a')); val dj_cp' = [cp, dj] MRS dj_cp; val cert = SOME o cterm_of thy in SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)] [cert t, cert r, cert s] dj_cp')) end else NONE end | perm_simproc' thy ss _ = NONE; val perm_simproc = Simplifier.simproc (the_context ()) "perm_simp" ["pi1 • (pi2 • x)"] perm_simproc'; val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE; val meta_spec = thm "meta_spec"; fun projections rule = ProjectRule.projections (ProofContext.init (Thm.theory_of_thm rule)) rule |> map (standard #> RuleCases.save rule); val supp_prod = thm "supp_prod"; val fresh_prod = thm "fresh_prod"; val supports_fresh = thm "supports_fresh"; val supports_def = thm "Nominal.supports_def"; val fresh_def = thm "fresh_def"; val supp_def = thm "supp_def"; val rev_simps = thms "rev.simps"; val app_simps = thms "append.simps"; val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq]; fun mk_perm Ts t u = let val T = fastype_of1 (Ts, t); val U = fastype_of1 (Ts, u) in Const ("Nominal.perm", T --> U --> U) $ t $ u end; fun perm_of_pair (x, y) = let val T = fastype_of x; val pT = mk_permT T in Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $ HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT) end; fun mk_not_sym ths = maps (fn th => case prop_of th of _ $ (Const ("Not", _) $ _) => [th, th RS not_sym] | _ => [th]) ths; fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy = let (* this theory is used just for parsing *) val tmp_thy = thy |> Theory.copy |> Sign.add_types (map (fn (tvs, tname, mx, _) => (tname, length tvs, mx)) dts); val atoms = atoms_of thy; val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms; val cp_classes = List.concat (map (fn atom1 => map (fn atom2 => Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2)) atoms) atoms); fun augment_sort S = S union classes; val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S)); fun prep_constr ((constrs, sorts), (cname, cargs, mx)) = let val (cargs', sorts') = Library.foldl (prep_typ tmp_thy) (([], sorts), cargs) in (constrs @ [(cname, cargs', mx)], sorts') end fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) = let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs) in (dts @ [(tvs, tname, mx, constrs')], sorts') end val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts); val sorts' = map (apsnd augment_sort) sorts; val tyvars = map #1 dts'; val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts'; val constr_syntax = map (fn (tvs, tname, mx, constrs) => map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts'; val ps = map (fn (_, n, _, _) => (Sign.full_name tmp_thy n, Sign.full_name tmp_thy (n ^ "_Rep"))) dts; val rps = map Library.swap ps; fun replace_types (Type ("Nominal.ABS", [T, U])) = Type ("fun", [T, Type ("Nominal.noption", [replace_types U])]) | replace_types (Type (s, Ts)) = Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts) | replace_types T = T; val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn, map (fn (cname, cargs, mx) => (cname ^ "_Rep", map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts'; val new_type_names' = map (fn n => n ^ "_Rep") new_type_names; val full_new_type_names' = map (Sign.full_name thy) new_type_names'; val ({induction, ...},thy1) = DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy; val SOME {descr, ...} = Symtab.lookup (DatatypePackage.get_datatypes thy1) (hd full_new_type_names'); fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i); (**** define permutation functions ****) val permT = mk_permT (TFree ("'x", HOLogic.typeS)); val pi = Free ("pi", permT); val perm_types = map (fn (i, _) => let val T = nth_dtyp i in permT --> T --> T end) descr; val perm_names = replicate (length new_type_names) "Nominal.perm" @ DatatypeProp.indexify_names (map (fn i => Sign.full_name thy1 ("perm_" ^ name_of_typ (nth_dtyp i))) (length new_type_names upto length descr - 1)); val perm_names_types = perm_names ~~ perm_types; val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) => let val T = nth_dtyp i in map (fn (cname, dts) => let val Ts = map (typ_of_dtyp descr sorts') dts; val names = DatatypeProp.make_tnames Ts; val args = map Free (names ~~ Ts); val c = Const (cname, Ts ---> T); fun perm_arg (dt, x) = let val T = type_of x in if is_rec_type dt then let val (Us, _) = strip_type T in list_abs (map (pair "x") Us, Const (List.nth (perm_names_types, body_index dt)) $ pi $ list_comb (x, map (fn (i, U) => Const ("Nominal.perm", permT --> U --> U) $ (Const ("List.rev", permT --> permT) $ pi) $ Bound i) ((length Us - 1 downto 0) ~~ Us))) end else Const ("Nominal.perm", permT --> T --> T) $ pi $ x end; in (("", HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (List.nth (perm_names_types, i)) $ Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $ list_comb (c, args), list_comb (c, map perm_arg (dts ~~ args))))), []) end) constrs end) descr); val (perm_simps, thy2) = thy1 |> Sign.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn)) (List.drop (perm_names_types, length new_type_names))) |> PrimrecPackage.add_primrec_unchecked_i "" perm_eqs; (**** prove that permutation functions introduced by unfolding are ****) (**** equivalent to already existing permutation functions ****) val _ = warning ("length descr: " ^ string_of_int (length descr)); val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names)); val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types); val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def"); val unfolded_perm_eq_thms = if length descr = length new_type_names then [] else map standard (List.drop (split_conj_thm (Goal.prove_global thy2 [] [] (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn (c as (s, T), x) => let val [T1, T2] = binder_types T in HOLogic.mk_eq (Const c $ pi $ Free (x, T2), Const ("Nominal.perm", T) $ pi $ Free (x, T2)) end) (perm_names_types ~~ perm_indnames)))) (fn _ => EVERY [indtac induction perm_indnames 1, ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [perm_fun_def]))])), length new_type_names)); (**** prove [] • t = t ****) val _ = warning "perm_empty_thms"; val perm_empty_thms = List.concat (map (fn a => let val permT = mk_permT (Type (a, [])) in map standard (List.take (split_conj_thm (Goal.prove_global thy2 [] [] (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn ((s, T), x) => HOLogic.mk_eq (Const (s, permT --> T --> T) $ Const ("List.list.Nil", permT) $ Free (x, T), Free (x, T))) (perm_names ~~ map body_type perm_types ~~ perm_indnames)))) (fn _ => EVERY [indtac induction perm_indnames 1, ALLGOALS (asm_full_simp_tac (simpset_of thy2))])), length new_type_names)) end) atoms); (**** prove (pi1 @ pi2) • t = pi1 • (pi2 • t) ****) val _ = warning "perm_append_thms"; (*FIXME: these should be looked up statically*) val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst"); val pt2 = PureThy.get_thm thy2 (Name "pt2"); val perm_append_thms = List.concat (map (fn a => let val permT = mk_permT (Type (a, [])); val pi1 = Free ("pi1", permT); val pi2 = Free ("pi2", permT); val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst")); val pt2' = pt_inst RS pt2; val pt2_ax = PureThy.get_thm thy2 (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a)); in List.take (map standard (split_conj_thm (Goal.prove_global thy2 [] [] (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn ((s, T), x) => let val perm = Const (s, permT --> T --> T) in HOLogic.mk_eq (perm $ (Const ("List.append", permT --> permT --> permT) $ pi1 $ pi2) $ Free (x, T), perm $ pi1 $ (perm $ pi2 $ Free (x, T))) end) (perm_names ~~ map body_type perm_types ~~ perm_indnames)))) (fn _ => EVERY [indtac induction perm_indnames 1, ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))), length new_type_names) end) atoms); (**** prove pi1 ~ pi2 ==> pi1 • t = pi2 • t ****) val _ = warning "perm_eq_thms"; val pt3 = PureThy.get_thm thy2 (Name "pt3"); val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev"); val perm_eq_thms = List.concat (map (fn a => let val permT = mk_permT (Type (a, [])); val pi1 = Free ("pi1", permT); val pi2 = Free ("pi2", permT); (*FIXME: not robust - better access these theorems using NominalData?*) val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst")); val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst")); val pt3' = pt_inst RS pt3; val pt3_rev' = at_inst RS (pt_inst RS pt3_rev); val pt3_ax = PureThy.get_thm thy2 (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a)); in List.take (map standard (split_conj_thm (Goal.prove_global thy2 [] [] (Logic.mk_implies (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq", permT --> permT --> HOLogic.boolT) $ pi1 $ pi2), HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn ((s, T), x) => let val perm = Const (s, permT --> T --> T) in HOLogic.mk_eq (perm $ pi1 $ Free (x, T), perm $ pi2 $ Free (x, T)) end) (perm_names ~~ map body_type perm_types ~~ perm_indnames))))) (fn _ => EVERY [indtac induction perm_indnames 1, ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))), length new_type_names) end) atoms); (**** prove pi1 • (pi2 • t) = (pi1 • pi2) • (pi1 • t) ****) val cp1 = PureThy.get_thm thy2 (Name "cp1"); val dj_cp = PureThy.get_thm thy2 (Name "dj_cp"); val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose"); val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev"); val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget"); fun composition_instance name1 name2 thy = let val name1' = Sign.base_name name1; val name2' = Sign.base_name name2; val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2'); val permT1 = mk_permT (Type (name1, [])); val permT2 = mk_permT (Type (name2, [])); val augment = map_type_tfree (fn (x, S) => TFree (x, cp_class :: S)); val Ts = map (augment o body_type) perm_types; val cp_inst = PureThy.get_thm thy (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst")); val simps = simpset_of thy addsimps (perm_fun_def :: (if name1 <> name2 then let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1')) in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end else let val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst")); val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst")) in [cp_inst RS cp1 RS sym, at_inst RS (pt_inst RS pt_perm_compose) RS sym, at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym] end)) val thms = split_conj_thm (Goal.prove_global thy [] [] (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn ((s, T), x) => let val pi1 = Free ("pi1", permT1); val pi2 = Free ("pi2", permT2); val perm1 = Const (s, permT1 --> T --> T); val perm2 = Const (s, permT2 --> T --> T); val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2) in HOLogic.mk_eq (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)), perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T))) end) (perm_names ~~ Ts ~~ perm_indnames)))) (fn _ => EVERY [indtac induction perm_indnames 1, ALLGOALS (asm_full_simp_tac simps)])) in foldl (fn ((s, tvs), thy) => AxClass.prove_arity (s, replicate (length tvs) (cp_class :: classes), [cp_class]) (Class.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy) thy (full_new_type_names' ~~ tyvars) end; val (perm_thmss,thy3) = thy2 |> fold (fn name1 => fold (composition_instance name1) atoms) atoms |> curry (Library.foldr (fn ((i, (tyname, args, _)), thy) => AxClass.prove_arity (tyname, replicate (length args) classes, classes) (Class.intro_classes_tac [] THEN REPEAT (EVERY [resolve_tac perm_empty_thms 1, resolve_tac perm_append_thms 1, resolve_tac perm_eq_thms 1, assume_tac 1])) thy)) (List.take (descr, length new_type_names)) |> PureThy.add_thmss [((space_implode "_" new_type_names ^ "_unfolded_perm_eq", unfolded_perm_eq_thms), [Simplifier.simp_add]), ((space_implode "_" new_type_names ^ "_perm_empty", perm_empty_thms), [Simplifier.simp_add]), ((space_implode "_" new_type_names ^ "_perm_append", perm_append_thms), [Simplifier.simp_add]), ((space_implode "_" new_type_names ^ "_perm_eq", perm_eq_thms), [Simplifier.simp_add])]; (**** Define representing sets ****) val _ = warning "representing sets"; val rep_set_names = DatatypeProp.indexify_names (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr); val big_rep_name = space_implode "_" (DatatypeProp.indexify_names (List.mapPartial (fn (i, ("Nominal.noption", _, _)) => NONE | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set"; val _ = warning ("big_rep_name: " ^ big_rep_name); fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) = (case AList.lookup op = descr i of SOME ("Nominal.noption", _, [(_, [dt']), _]) => apfst (cons dt) (strip_option dt') | _ => ([], dtf)) | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) = apfst (cons dt) (strip_option dt') | strip_option dt = ([], dt); val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts') (List.concat (map (fn (_, (_, _, cs)) => List.concat (map (List.concat o map (fst o strip_option) o snd) cs)) descr))); fun make_intr s T (cname, cargs) = let fun mk_prem (dt, (j, j', prems, ts)) = let val (dts, dt') = strip_option dt; val (dts', dt'') = strip_dtyp dt'; val Ts = map (typ_of_dtyp descr sorts') dts; val Us = map (typ_of_dtyp descr sorts') dts'; val T = typ_of_dtyp descr sorts' dt''; val free = mk_Free "x" (Us ---> T) j; val free' = app_bnds free (length Us); fun mk_abs_fun (T, (i, t)) = let val U = fastype_of t in (i + 1, Const ("Nominal.abs_fun", [T, U, T] ---> Type ("Nominal.noption", [U])) $ mk_Free "y" T i $ t) end in (j + 1, j' + length Ts, case dt'' of DtRec k => list_all (map (pair "x") Us, HOLogic.mk_Trueprop (Free (List.nth (rep_set_names, k), T --> HOLogic.boolT) $ free')) :: prems | _ => prems, snd (foldr mk_abs_fun (j', free) Ts) :: ts) end; val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs; val concl = HOLogic.mk_Trueprop (Free (s, T --> HOLogic.boolT) $ list_comb (Const (cname, map fastype_of ts ---> T), ts)) in Logic.list_implies (prems, concl) end; val (intr_ts, (rep_set_names', recTs')) = apfst List.concat (apsnd ListPair.unzip (ListPair.unzip (List.mapPartial (fn ((_, ("Nominal.noption", _, _)), _) => NONE | ((i, (_, _, constrs)), rep_set_name) => let val T = nth_dtyp i in SOME (map (make_intr rep_set_name T) constrs, (rep_set_name, T)) end) (descr ~~ rep_set_names)))); val rep_set_names'' = map (Sign.full_name thy3) rep_set_names'; val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy4) = setmp InductivePackage.quiet_mode false (InductivePackage.add_inductive_global {verbose = false, kind = Thm.internalK, alt_name = big_rep_name, coind = false, no_elim = true, no_ind = false} (map (fn (s, T) => ((s, T --> HOLogic.boolT), NoSyn)) (rep_set_names' ~~ recTs')) [] (map (fn x => (("", []), x)) intr_ts) []) thy3; (**** Prove that representing set is closed under permutation ****) val _ = warning "proving closure under permutation..."; val perm_indnames' = List.mapPartial (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x) (perm_indnames ~~ descr); fun mk_perm_closed name = map (fn th => standard (th RS mp)) (List.take (split_conj_thm (Goal.prove_global thy4 [] [] (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn ((s, T), x) => let val T = map_type_tfree (fn (s, cs) => TFree (s, cs union cp_classes)) T; val S = Const (s, T --> HOLogic.boolT); val permT = mk_permT (Type (name, [])) in HOLogic.mk_imp (S $ Free (x, T), S $ (Const ("Nominal.perm", permT --> T --> T) $ Free ("pi", permT) $ Free (x, T))) end) (rep_set_names'' ~~ recTs' ~~ perm_indnames')))) (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *) [indtac rep_induct [] 1, ALLGOALS (simp_tac (simpset_of thy4 addsimps (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))), ALLGOALS (resolve_tac rep_intrs THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])), length new_type_names)); (* FIXME: theorems are stored in database for testing only *) val perm_closed_thmss = map mk_perm_closed atoms; val (_, thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4; (**** typedef ****) val _ = warning "defining type..."; val (typedefs, thy6) = thy5 |> fold_map (fn ((((name, mx), tvs), (cname, U)), name') => fn thy => setmp TypedefPackage.quiet_mode true (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) (Const ("Collect", (U --> HOLogic.boolT) --> HOLogic.mk_setT U) $ Const (cname, U --> HOLogic.boolT)) NONE (rtac exI 1 THEN rtac CollectI 1 THEN QUIET_BREADTH_FIRST (has_fewer_prems 1) (resolve_tac rep_intrs 1))) thy |> (fn ((_, r), thy) => let val permT = mk_permT (TFree (Name.variant tvs "'a", HOLogic.typeS)); val pi = Free ("pi", permT); val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs; val T = Type (Sign.intern_type thy name, tvs'); in apfst (pair r o hd) (PureThy.add_defs_unchecked_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T), Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $ (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $ Free ("x", T))))), [])] thy) end)) (types_syntax ~~ tyvars ~~ List.take (rep_set_names'' ~~ recTs', length new_type_names) ~~ new_type_names); val perm_defs = map snd typedefs; val Abs_inverse_thms = map (collect_simp o #Abs_inverse o fst) typedefs; val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs; val Rep_thms = map (collect_simp o #Rep o fst) typedefs; val big_name = space_implode "_" new_type_names; (** prove that new types are in class pt_<name> **) val _ = warning "prove that new types are in class pt_<name> ..."; fun pt_instance ((class, atom), perm_closed_thms) = fold (fn ((((((Abs_inverse, Rep_inverse), Rep), perm_def), name), tvs), perm_closed) => fn thy => AxClass.prove_arity (Sign.intern_type thy name, replicate (length tvs) (classes @ cp_classes), [class]) (EVERY [Class.intro_classes_tac [], rewrite_goals_tac [perm_def], asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1, asm_full_simp_tac (simpset_of thy addsimps [Rep RS perm_closed RS Abs_inverse]) 1, asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy) (Abs_inverse_thms ~~ Rep_inverse_thms ~~ Rep_thms ~~ perm_defs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms); (** prove that new types are in class cp_<name1>_<name2> **) val _ = warning "prove that new types are in class cp_<name1>_<name2> ..."; fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy = let val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2; val class = Sign.intern_class thy name; val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1 in fold (fn ((((((Abs_inverse, Rep), perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy => AxClass.prove_arity (Sign.intern_type thy name, replicate (length tvs) (classes @ cp_classes), [class]) (EVERY [Class.intro_classes_tac [], rewrite_goals_tac [perm_def], asm_full_simp_tac (simpset_of thy addsimps ((Rep RS perm_closed1 RS Abs_inverse) :: (if atom1 = atom2 then [] else [Rep RS perm_closed2 RS Abs_inverse]))) 1, cong_tac 1, rtac refl 1, rtac cp1' 1]) thy) (Abs_inverse_thms ~~ Rep_thms ~~ perm_defs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~ perm_closed_thms2) thy end; val thy7 = fold (fn x => fn thy => thy |> pt_instance x |> fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss)) (classes ~~ atoms ~~ perm_closed_thmss) thy6; (**** constructors ****) fun mk_abs_fun (x, t) = let val T = fastype_of x; val U = fastype_of t in Const ("Nominal.abs_fun", T --> U --> T --> Type ("Nominal.noption", [U])) $ x $ t end; val (ty_idxs, _) = foldl (fn ((i, ("Nominal.noption", _, _)), p) => p | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr; fun reindex (DtType (s, dts)) = DtType (s, map reindex dts) | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i)) | reindex dt = dt; fun strip_suffix i s = implode (List.take (explode s, size s - i)); (** strips the "_Rep" in type names *) fun strip_nth_name i s = let val xs = NameSpace.explode s; in NameSpace.implode (Library.nth_map (length xs - i) (strip_suffix 4) xs) end; val (descr'', ndescr) = ListPair.unzip (List.mapPartial (fn (i, ("Nominal.noption", _, _)) => NONE | (i, (s, dts, constrs)) => let val SOME index = AList.lookup op = ty_idxs i; val (constrs1, constrs2) = ListPair.unzip (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 (strip_nth_name 1 cname))) (foldl_map (fn (dts, dt) => let val (dts', dt') = strip_option dt in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end) ([], cargs))) constrs) in SOME ((index, (strip_nth_name 1 s, map reindex dts, constrs1)), (index, constrs2)) end) descr); val (descr1, descr2) = chop (length new_type_names) descr''; val descr' = [descr1, descr2]; fun partition_cargs idxs xs = map (fn (i, j) => (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs; val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts, map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs)) (constrs ~~ idxss)))) (descr'' ~~ ndescr); fun nth_dtyp' i = typ_of_dtyp descr'' sorts' (DtRec i); val rep_names = map (fn s => Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names; val abs_names = map (fn s => Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names; val recTs = get_rec_types descr'' sorts'; val newTs' = Library.take (length new_type_names, recTs'); val newTs = Library.take (length new_type_names, recTs); val full_new_type_names = map (Sign.full_name thy) new_type_names; fun make_constr_def tname T T' ((thy, defs, eqns), (((cname_rep, _), (cname, cargs)), (cname', mx))) = let fun constr_arg ((dts, dt), (j, l_args, r_args)) = let val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts' dt) i) (dts ~~ (j upto j + length dts - 1)) val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts) in (j + length dts + 1, xs @ x :: l_args, foldr mk_abs_fun (case dt of DtRec k => if k < length new_type_names then Const (List.nth (rep_names, k), typ_of_dtyp descr'' sorts' dt --> typ_of_dtyp descr sorts' dt) $ x else error "nested recursion not (yet) supported" | _ => x) xs :: r_args) end val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs; val abs_name = Sign.intern_const thy ("Abs_" ^ tname); val rep_name = Sign.intern_const thy ("Rep_" ^ tname); val constrT = map fastype_of l_args ---> T; val lhs = list_comb (Const (cname, constrT), l_args); val rhs = list_comb (Const (cname_rep, map fastype_of r_args ---> T'), r_args); val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs); val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (rep_name, T --> T') $ lhs, rhs)); val def_name = (Sign.base_name cname) ^ "_def"; val ([def_thm], thy') = thy |> Sign.add_consts_i [(cname', constrT, mx)] |> (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)] in (thy', defs @ [def_thm], eqns @ [eqn]) end; fun dt_constr_defs ((thy, defs, eqns, dist_lemmas), ((((((_, (_, _, constrs)), (_, (_, _, constrs'))), tname), T), T'), constr_syntax)) = let val rep_const = cterm_of thy (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T')); val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma); val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T') ((Sign.add_path tname thy, defs, []), constrs ~~ constrs' ~~ constr_syntax) in (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist]) end; val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs ((thy7, [], [], []), List.take (descr, length new_type_names) ~~ List.take (pdescr, length new_type_names) ~~ new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax); val abs_inject_thms = map (collect_simp o #Abs_inject o fst) typedefs val rep_inject_thms = map (#Rep_inject o fst) typedefs (* prove theorem Rep_i (Constr_j ...) = Constr'_j ... *) fun prove_constr_rep_thm eqn = let val inj_thms = map (fn r => r RS iffD1) abs_inject_thms; val rewrites = constr_defs @ map mk_meta_eq Rep_inverse_thms in Goal.prove_global thy8 [] [] eqn (fn _ => EVERY [resolve_tac inj_thms 1, rewrite_goals_tac rewrites, rtac refl 3, resolve_tac rep_intrs 2, REPEAT (resolve_tac Rep_thms 1)]) end; val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns; (* prove theorem pi • Rep_i x = Rep_i (pi • x) *) fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th => let val _ $ (_ $ (Rep $ x)) = Logic.unvarify (prop_of th); val Type ("fun", [T, U]) = fastype_of Rep; val permT = mk_permT (Type (atom, [])); val pi = Free ("pi", permT); in Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x), Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x)))) (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @ perm_closed_thms @ Rep_thms)) 1) end) Rep_thms; val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm (atoms ~~ perm_closed_thmss)); (* prove distinctness theorems *) val distinctness_limit = Config.get_thy thy8 DatatypeProp.distinctness_limit; val thy8' = Config.put_thy DatatypeProp.distinctness_limit 1000 thy8; val distinct_props = DatatypeProp.make_distincts new_type_names descr' sorts' thy8'; val thy8 = Config.put_thy DatatypeProp.distinctness_limit distinctness_limit thy8'; val dist_rewrites = map (fn (rep_thms, dist_lemma) => dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0])) (constr_rep_thmss ~~ dist_lemmas); fun prove_distinct_thms (_, []) = [] | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) = let val dist_thm = Goal.prove_global thy8 [] [] t (fn _ => simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1) in dist_thm::(standard (dist_thm RS not_sym)):: (prove_distinct_thms (p, ts)) end; val distinct_thms = map prove_distinct_thms (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props); (** prove equations for permutation functions **) val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *) val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) => let val T = nth_dtyp' i in List.concat (map (fn (atom, perm_closed_thms) => map (fn ((cname, dts), constr_rep_thm) => let val cname = Sign.intern_const thy8 (NameSpace.append tname (Sign.base_name cname)); val permT = mk_permT (Type (atom, [])); val pi = Free ("pi", permT); fun perm t = let val T = fastype_of t in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end; fun constr_arg ((dts, dt), (j, l_args, r_args)) = let val Ts = map (typ_of_dtyp descr'' sorts') dts; val xs = map (fn (T, i) => mk_Free "x" T i) (Ts ~~ (j upto j + length dts - 1)) val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts) in (j + length dts + 1, xs @ x :: l_args, map perm (xs @ [x]) @ r_args) end val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts; val c = Const (cname, map fastype_of l_args ---> T) in Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (perm (list_comb (c, l_args)), list_comb (c, r_args)))) (fn _ => EVERY [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1, simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @ constr_defs @ perm_closed_thms)) 1, TRY (simp_tac (HOL_basic_ss addsimps (symmetric perm_fun_def :: abs_perm)) 1), TRY (simp_tac (HOL_basic_ss addsimps (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @ perm_closed_thms)) 1)]) end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss)) end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss); (** prove injectivity of constructors **) val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms; val alpha = PureThy.get_thms thy8 (Name "alpha"); val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh"); val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) => let val T = nth_dtyp' i in List.mapPartial (fn ((cname, dts), constr_rep_thm) => if null dts then NONE else SOME let val cname = Sign.intern_const thy8 (NameSpace.append tname (Sign.base_name cname)); fun make_inj ((dts, dt), (j, args1, args2, eqs)) = let val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1); val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx; val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx; val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts); val y = mk_Free "y" (typ_of_dtyp descr'' sorts' dt) (j + length dts) in (j + length dts + 1, xs @ (x :: args1), ys @ (y :: args2), HOLogic.mk_eq (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs) end; val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts; val Ts = map fastype_of args1; val c = Const (cname, Ts ---> T) in Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)), foldr1 HOLogic.mk_conj eqs))) (fn _ => EVERY [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: rep_inject_thms')) 1, TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def :: alpha @ abs_perm @ abs_fresh @ rep_inject_thms @ perm_rep_perm_thms)) 1), TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def :: expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)]) end) (constrs ~~ constr_rep_thms) end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss); (** equations for support and freshness **) val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') => let val T = nth_dtyp' i in List.concat (map (fn (cname, dts) => map (fn atom => let val cname = Sign.intern_const thy8 (NameSpace.append tname (Sign.base_name cname)); val atomT = Type (atom, []); fun process_constr ((dts, dt), (j, args1, args2)) = let val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1); val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx; val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts) in (j + length dts + 1, xs @ (x :: args1), foldr mk_abs_fun x xs :: args2) end; val (_, args1, args2) = foldr process_constr (1, [], []) dts; val Ts = map fastype_of args1; val c = list_comb (Const (cname, Ts ---> T), args1); fun supp t = Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t; fun fresh t = Const ("Nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $ Free ("a", atomT) $ t; val supp_thm = Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (supp c, if null dts then Const ("{}", HOLogic.mk_setT atomT) else foldr1 (HOLogic.mk_binop "op Un") (map supp args2)))) (fn _ => simp_tac (HOL_basic_ss addsimps (supp_def :: Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un :: symmetric empty_def :: finite_emptyI :: simp_thms @ abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1) in (supp_thm, Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (fresh c, if null dts then HOLogic.true_const else foldr1 HOLogic.mk_conj (map fresh args2)))) (fn _ => simp_tac (HOL_ss addsimps [Un_iff, empty_iff, fresh_def, supp_thm]) 1)) end) atoms) constrs) end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps'))); (**** weak induction theorem ****) fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) = let val Rep_t = Const (List.nth (rep_names, i), T --> U) $ mk_Free "x" T i; val Abs_t = Const (List.nth (abs_names, i), U --> T) in (prems @ [HOLogic.imp $ (Const (List.nth (rep_set_names'', i), U --> HOLogic.boolT) $ Rep_t) $ (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))], concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i]) end; val (indrule_lemma_prems, indrule_lemma_concls) = Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs')); val indrule_lemma = Goal.prove_global thy8 [] [] (Logic.mk_implies (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems), HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY [REPEAT (etac conjE 1), REPEAT (EVERY [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1, etac mp 1, resolve_tac Rep_thms 1])]); val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma))); val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else map (Free o apfst fst o dest_Var) Ps; val indrule_lemma' = cterm_instantiate (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma; val Abs_inverse_thms' = map (fn r => r RS subst) Abs_inverse_thms; val dt_induct_prop = DatatypeProp.make_ind descr' sorts'; val dt_induct = Goal.prove_global thy8 [] (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop) (fn prems => EVERY [rtac indrule_lemma' 1, (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1, EVERY (map (fn (prem, r) => (EVERY [REPEAT (eresolve_tac Abs_inverse_thms' 1), simp_tac (HOL_basic_ss addsimps [symmetric r]) 1, DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)])) (prems ~~ constr_defs))]); val case_names_induct = mk_case_names_induct descr''; (**** prove that new datatypes have finite support ****) val _ = warning "proving finite support for the new datatype"; val indnames = DatatypeProp.make_tnames recTs; val abs_supp = PureThy.get_thms thy8 (Name "abs_supp"); val supp_atm = PureThy.get_thms thy8 (Name "supp_atm"); val finite_supp_thms = map (fn atom => let val atomT = Type (atom, []) in map standard (List.take (split_conj_thm (Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn (s, T) => Const ("Finite_Set.finite", HOLogic.mk_setT atomT --> HOLogic.boolT) $ (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T))) (indnames ~~ recTs)))) (fn _ => indtac dt_induct indnames 1 THEN ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps (abs_supp @ supp_atm @ PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @ List.concat supp_thms))))), length new_type_names)) end) atoms; val simp_atts = replicate (length new_type_names) [Simplifier.simp_add]; (* Function to add both the simp and eqvt attributes *) (* These two attributes are duplicated on all the types in the mutual nominal datatypes *) val simp_eqvt_atts = replicate (length new_type_names) [Simplifier.simp_add, NominalThmDecls.eqvt_add]; val (_, thy9) = thy8 |> Sign.add_path big_name |> PureThy.add_thms [(("weak_induct", dt_induct), [case_names_induct])] ||>> PureThy.add_thmss [(("weak_inducts", projections dt_induct), [case_names_induct])] ||> Sign.parent_path ||>> DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>> DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>> DatatypeAux.store_thmss_atts "perm" new_type_names simp_eqvt_atts perm_simps' ||>> DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>> DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>> DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||> fold (fn (atom, ths) => fn thy => let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom) in fold (fn T => AxClass.prove_arity (fst (dest_Type T), replicate (length sorts) [class], [class]) (Class.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy end) (atoms ~~ finite_supp_thms); (**** strong induction theorem ****) val pnames = if length descr'' = 1 then ["P"] else map (fn i => "P" ^ string_of_int i) (1 upto length descr''); val ind_sort = if null dt_atomTs then HOLogic.typeS else Sign.certify_sort thy9 (map (fn T => Sign.intern_class thy9 ("fs_" ^ Sign.base_name (fst (dest_Type T)))) dt_atomTs); val fsT = TFree ("'n", ind_sort); val fsT' = TFree ("'n", HOLogic.typeS); val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T))) (DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs); fun make_pred fsT i T = Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT); fun mk_fresh1 xs [] = [] | mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop (HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x)))) (filter (fn (_, U) => T = U) (rev xs)) @ mk_fresh1 (y :: xs) ys; fun mk_fresh2 xss [] = [] | mk_fresh2 xss ((p as (ys, _)) :: yss) = List.concat (map (fn y as (_, T) => map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free x)) (rev xss @ yss)) ys) @ mk_fresh2 (p :: xss) yss; fun make_ind_prem fsT f k T ((cname, cargs), idxs) = let val recs = List.filter is_rec_type cargs; val Ts = map (typ_of_dtyp descr'' sorts') cargs; val recTs' = map (typ_of_dtyp descr'' sorts') recs; val tnames = Name.variant_list pnames (DatatypeProp.make_tnames Ts); val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs)); val frees = tnames ~~ Ts; val frees' = partition_cargs idxs frees; val z = (Name.variant tnames "z", fsT); fun mk_prem ((dt, s), T) = let val (Us, U) = strip_type T; val l = length Us in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l)) end; val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs'); val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop (f T (Free p) (Free z))) (List.concat (map fst frees')) @ mk_fresh1 [] (List.concat (map fst frees')) @ mk_fresh2 [] frees' in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems, HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $ list_comb (Const (cname, Ts ---> T), map Free frees)))) end; val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) => map (make_ind_prem fsT (fn T => fn t => fn u => Const ("Nominal.fresh", T --> fsT --> HOLogic.boolT) $ t $ u) i T) (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs)); val tnames = DatatypeProp.make_tnames recTs; val zs = Name.variant_list tnames (replicate (length descr'') "z"); val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &") (map (fn ((((i, _), T), tname), z) => make_pred fsT i T $ Free (z, fsT) $ Free (tname, T)) (descr'' ~~ recTs ~~ tnames ~~ zs))); val induct = Logic.list_implies (ind_prems, ind_concl); val ind_prems' = map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')], HOLogic.mk_Trueprop (Const ("Finite_Set.finite", body_type T --> HOLogic.boolT) $ (f $ Free ("x", fsT'))))) fresh_fs @ List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) => map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $ HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T) (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs)); val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &") (map (fn ((((i, _), T), tname), z) => make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T)) (descr'' ~~ recTs ~~ tnames ~~ zs))); val induct' = Logic.list_implies (ind_prems', ind_concl'); val aux_ind_vars = (DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~ map mk_permT dt_atomTs) @ [("z", fsT')]; val aux_ind_Ts = rev (map snd aux_ind_vars); val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &") (map (fn (((i, _), T), tname) => HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $ fold_rev (mk_perm aux_ind_Ts) (map Bound (length dt_atomTs downto 1)) (Free (tname, T)))) (descr'' ~~ recTs ~~ tnames))); fun mk_ind_perm i k p l vs j = let val n = length vs; val Ts = map snd vs; val T = List.nth (Ts, i - j); val pT = NominalAtoms.mk_permT T in Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $ (HOLogic.pair_const T T $ Bound (l - j) $ fold_rev (mk_perm Ts) (map (mk_ind_perm i k p l vs) (j - 1 downto 0) @ map Bound (n - k - 1 downto n - k - p)) (Bound (i - j))) $ Const ("List.list.Nil", pT) end; fun mk_fresh i i' j k p l is vs _ _ = let val n = length vs; val Ts = map snd vs; val T = List.nth (Ts, n - i - 1 - j); val f = the (AList.lookup op = fresh_fs T); val U = List.nth (Ts, n - i' - 1); val S = HOLogic.mk_setT T; val prms' = map Bound (n - k downto n - k - p + 1); val prms = map (mk_ind_perm (n - i) k p (n - l) (("a", T) :: vs)) (j - 1 downto 0) @ prms'; val bs = rev (List.mapPartial (fn ((_, T'), i) => if T = T' then SOME (Bound i) else NONE) (List.take (vs, n - k - p - 1) ~~ (1 upto n - k - p - 1))); val ts = map (fn i => Const ("Nominal.supp", List.nth (Ts, n - i - 1) --> S) $ fold_rev (mk_perm (T :: Ts)) prms' (Bound (n - i))) is in HOLogic.mk_Trueprop (Const ("Ex", (T --> HOLogic.boolT) --> HOLogic.boolT) $ Abs ("a", T, HOLogic.Not $ (Const ("op :", T --> S --> HOLogic.boolT) $ Bound 0 $ (foldr (fn (t, u) => Const ("insert", T --> S --> S) $ t $ u) (foldl (fn (t, u) => Const ("op Un", S --> S --> S) $ u $ t) (f $ Bound (n - k - p)) (Const ("Nominal.supp", U --> S) $ fold_rev (mk_perm (T :: Ts)) prms (Bound (n - i')) :: ts)) (fold_rev (mk_perm (T :: Ts)) prms (Bound (n - i - j)) :: bs))))) end; fun mk_fresh_constr is p vs _ concl = let val n = length vs; val Ts = map snd vs; val _ $ (_ $ _ $ t) = concl; val c = head_of t; val T = body_type (fastype_of c); val k = foldr op + 0 (map (fn (_, i) => i + 1) is); val ps = map Bound (n - k - 1 downto n - k - p); val (_, _, ts, us) = foldl (fn ((_, i), (m, n, ts, us)) => (m - i - 1, n - i, ts @ map Bound (n downto n - i + 1) @ [fold_rev (mk_perm Ts) (map (mk_ind_perm m k p n vs) (i - 1 downto 0) @ ps) (Bound (m - i))], us @ map (fn j => fold_rev (mk_perm Ts) ps (Bound j)) (m downto m - i))) (n - 1, n - k - p - 2, [], []) is in HOLogic.mk_Trueprop (HOLogic.eq_const T $ list_comb (c, ts) $ list_comb (c, us)) end; val abs_fun_finite_supp = PureThy.get_thm thy9 (Name "abs_fun_finite_supp"); val at_finite_select = PureThy.get_thm thy9 (Name "at_finite_select"); val induct_aux_lemmas = List.concat (map (fn Type (s, _) => [PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "_inst")), PureThy.get_thm thy9 (Name ("fs_" ^ Sign.base_name s ^ "1")), PureThy.get_thm thy9 (Name ("at_" ^ Sign.base_name s ^ "_inst"))]) dt_atomTs); val induct_aux_lemmas' = map (fn Type (s, _) => PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "2")) RS sym) dt_atomTs; val fresh_aux = PureThy.get_thms thy9 (Name "fresh_aux"); (********************************************************************** The subgoals occurring in the proof of induct_aux have the following parameters: x_1 ... x_k p_1 ... p_m z b_1 ... b_n where x_i : constructor arguments (introduced by weak induction rule) p_i : permutations (one for each atom type in the data type) z : freshness context b_i : newly introduced names for binders (sufficiently fresh) ***********************************************************************) val _ = warning "proving strong induction theorem ..."; val induct_aux = Goal.prove_global thy9 [] ind_prems' ind_concl' (fn prems => EVERY ([mk_subgoal 1 (K (K (K aux_ind_concl))), indtac dt_induct tnames 1] @ List.concat (map (fn ((_, (_, _, constrs)), (_, constrs')) => List.concat (map (fn ((cname, cargs), is) => [simp_tac (HOL_basic_ss addsimps List.concat perm_simps') 1, REPEAT (rtac allI 1)] @ List.concat (map (fn ((_, 0), _) => [] | ((i, j), k) => List.concat (map (fn j' => let val DtType (tname, _) = List.nth (cargs, i + j'); val atom = Sign.base_name tname in [mk_subgoal 1 (mk_fresh i (i + j) j' (length cargs) (length dt_atomTs) (length cargs + length dt_atomTs + 1 + i - k) (List.mapPartial (fn (i', j) => if i = i' then NONE else SOME (i' + j)) is)), rtac at_finite_select 1, rtac (PureThy.get_thm thy9 (Name ("at_" ^ atom ^ "_inst"))) 1, asm_full_simp_tac (simpset_of thy9 addsimps [PureThy.get_thm thy9 (Name ("fs_" ^ atom ^ "1"))]) 1, resolve_tac prems 1, etac exE 1, asm_full_simp_tac (simpset_of thy9 addsimps [symmetric fresh_def]) 1] end) (0 upto j - 1))) (is ~~ (0 upto length is - 1))) @ (if exists (not o equal 0 o snd) is then [mk_subgoal 1 (mk_fresh_constr is (length dt_atomTs)), asm_full_simp_tac (simpset_of thy9 addsimps (List.concat inject_thms @ alpha @ abs_perm @ abs_fresh @ [abs_fun_finite_supp] @ induct_aux_lemmas)) 1, dtac sym 1, asm_full_simp_tac (simpset_of thy9) 1, REPEAT (etac conjE 1)] else []) @ [(resolve_tac prems THEN_ALL_NEW (atac ORELSE' SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of _ $ (Const ("Nominal.fresh", _) $ _ $ _) => asm_simp_tac (simpset_of thy9 addsimps fresh_aux) i | _ => asm_simp_tac (simpset_of thy9 addsimps induct_aux_lemmas' addsimprocs [perm_simproc]) i))) 1]) (constrs ~~ constrs'))) (descr'' ~~ ndescr)) @ [REPEAT (eresolve_tac [conjE, allE_Nil] 1), REPEAT (etac allE 1), REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac (simpset_of thy9) 1)])); val induct_aux' = Thm.instantiate ([], map (fn (s, T) => let val pT = TVar (("'n", 0), HOLogic.typeS) --> T --> HOLogic.boolT in (cterm_of thy9 (Var ((s, 0), pT)), cterm_of thy9 (Free (s, pT))) end) (pnames ~~ recTs) @ map (fn (_, f) => let val f' = Logic.varify f in (cterm_of thy9 f', cterm_of thy9 (Const ("Nominal.supp", fastype_of f'))) end) fresh_fs) induct_aux; val induct = Goal.prove_global thy9 [] ind_prems ind_concl (fn prems => EVERY [rtac induct_aux' 1, REPEAT (resolve_tac induct_aux_lemmas 1), REPEAT ((resolve_tac prems THEN_ALL_NEW (etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)]) val (_, thy10) = thy9 |> Sign.add_path big_name |> PureThy.add_thms [(("induct'", induct_aux), [])] ||>> PureThy.add_thms [(("induct", induct), [case_names_induct])] ||>> PureThy.add_thmss [(("inducts", projections induct), [case_names_induct])]; (**** recursion combinator ****) val _ = warning "defining recursion combinator ..."; val used = foldr add_typ_tfree_names [] recTs; val (rec_result_Ts', rec_fn_Ts') = DatatypeProp.make_primrec_Ts descr' sorts' used; val rec_sort = if null dt_atomTs then HOLogic.typeS else let val names = map (Sign.base_name o fst o dest_Type) dt_atomTs in Sign.certify_sort thy10 (map (Sign.intern_class thy10) (map (fn s => "pt_" ^ s) names @ List.concat (map (fn s => List.mapPartial (fn s' => if s = s' then NONE else SOME ("cp_" ^ s ^ "_" ^ s')) names) names))) end; val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts'; val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts'; val rec_set_Ts = map (fn (T1, T2) => rec_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts); val big_rec_name = big_name ^ "_rec_set"; val rec_set_names' = if length descr'' = 1 then [big_rec_name] else map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int) (1 upto (length descr'')); val rec_set_names = map (Sign.full_name thy10) rec_set_names'; val rec_fns = map (uncurry (mk_Free "f")) (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts))); val rec_sets' = map (fn c => list_comb (Free c, rec_fns)) (rec_set_names' ~~ rec_set_Ts); val rec_sets = map (fn c => list_comb (Const c, rec_fns)) (rec_set_names ~~ rec_set_Ts); (* introduction rules for graph of recursion function *) val rec_preds = map (fn (a, T) => Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts); fun mk_fresh3 rs [] = [] | mk_fresh3 rs ((p as (ys, z)) :: yss) = List.concat (map (fn y as (_, T) => List.mapPartial (fn ((_, (_, x)), r as (_, U)) => if z = x then NONE else SOME (HOLogic.mk_Trueprop (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free r))) rs) ys) @ mk_fresh3 rs yss; (* FIXME: avoid collisions with other variable names? *) val rec_ctxt = Free ("z", fsT'); fun make_rec_intr T p rec_set ((rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, l), ((cname, cargs), idxs)) = let val Ts = map (typ_of_dtyp descr'' sorts') cargs; val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts; val frees' = partition_cargs idxs frees; val binders = List.concat (map fst frees'); val atomTs = distinct op = (maps (map snd o fst) frees'); val recs = List.mapPartial (fn ((_, DtRec i), p) => SOME (i, p) | _ => NONE) (partition_cargs idxs cargs ~~ frees'); val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~ map (fn (i, _) => List.nth (rec_result_Ts, i)) recs; val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop (List.nth (rec_sets', i) $ Free x $ Free y)) (recs ~~ frees''); val prems2 = map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop (Const ("Nominal.fresh", T --> fastype_of f --> HOLogic.boolT) $ Free p $ f)) binders) rec_fns; val prems3 = mk_fresh1 [] binders @ mk_fresh2 [] frees'; val prems4 = map (fn ((i, _), y) => HOLogic.mk_Trueprop (List.nth (rec_preds, i) $ Free y)) (recs ~~ frees''); val prems5 = mk_fresh3 (recs ~~ frees'') frees'; val prems6 = maps (fn aT => map (fn y as (_, T) => HOLogic.mk_Trueprop (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $ (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ Free y))) frees'') atomTs; val prems7 = map (fn x as (_, T) => HOLogic.mk_Trueprop (Const ("Nominal.fresh", T --> fsT' --> HOLogic.boolT) $ Free x $ rec_ctxt)) binders; val result = list_comb (List.nth (rec_fns, l), map Free (frees @ frees'')); val result_freshs = map (fn p as (_, T) => Const ("Nominal.fresh", T --> fastype_of result --> HOLogic.boolT) $ Free p $ result) binders; val P = HOLogic.mk_Trueprop (p $ result) in (rec_intr_ts @ [Logic.list_implies (List.concat prems2 @ prems3 @ prems1, HOLogic.mk_Trueprop (rec_set $ list_comb (Const (cname, Ts ---> T), map Free frees) $ result))], rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))], rec_prems' @ map (fn fr => list_all_free (frees @ frees'', Logic.list_implies (List.nth (prems2, l) @ prems3 @ prems5 @ prems7 @ prems6 @ [P], HOLogic.mk_Trueprop fr))) result_freshs, rec_eq_prems @ [List.concat prems2 @ prems3], l + 1) end; val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) = Library.foldl (fn (x, ((((d, d'), T), p), rec_set)) => Library.foldl (make_rec_intr T p rec_set) (x, #3 (snd d) ~~ snd d')) (([], [], [], [], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets'); val ({intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}, thy11) = thy10 |> setmp InductivePackage.quiet_mode (!quiet_mode) (InductivePackage.add_inductive_global {verbose = false, kind = Thm.internalK, alt_name = big_rec_name, coind = false, no_elim = false, no_ind = false} (map (fn (s, T) => ((s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts)) (map dest_Free rec_fns) (map (fn x => (("", []), x)) rec_intr_ts) []) ||> PureThy.hide_thms true [NameSpace.append (Sign.full_name thy10 big_rec_name) "induct"]; (** equivariance **) val fresh_bij = PureThy.get_thms thy11 (Name "fresh_bij"); val perm_bij = PureThy.get_thms thy11 (Name "perm_bij"); val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT => let val permT = mk_permT aT; val pi = Free ("pi", permT); val rec_fns_pi = map (mk_perm [] pi o uncurry (mk_Free "f")) (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts))); val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi)) (rec_set_names ~~ rec_set_Ts); val ps = map (fn ((((T, U), R), R'), i) => let val x = Free ("x" ^ string_of_int i, T); val y = Free ("y" ^ string_of_int i, U) in (R $ x $ y, R' $ mk_perm [] pi x $ mk_perm [] pi y) end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs)); val ths = map (fn th => standard (th RS mp)) (split_conj_thm (Goal.prove_global thy11 [] [] (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps))) (fn _ => rtac rec_induct 1 THEN REPEAT (NominalPermeq.perm_simp_tac (HOL_basic_ss addsimps flat perm_simps') 1 THEN (resolve_tac rec_intrs THEN_ALL_NEW asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1)))) val ths' = map (fn ((P, Q), th) => Goal.prove_global thy11 [] [] (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P)) (fn _ => dtac (Thm.instantiate ([], [(cterm_of thy11 (Var (("pi", 0), permT)), cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN NominalPermeq.perm_simp_tac HOL_ss 1)) (ps ~~ ths) in (ths, ths') end) dt_atomTs); (** finite support **) val rec_fin_supp_thms = map (fn aT => let val name = Sign.base_name (fst (dest_Type aT)); val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1")); val aset = HOLogic.mk_setT aT; val finite = Const ("Finite_Set.finite", aset --> HOLogic.boolT); val fins = map (fn (f, T) => HOLogic.mk_Trueprop (finite $ (Const ("Nominal.supp", T --> aset) $ f))) (rec_fns ~~ rec_fn_Ts) in map (fn th => standard (th RS mp)) (split_conj_thm (Goal.prove_global thy11 [] fins (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn (((T, U), R), i) => let val x = Free ("x" ^ string_of_int i, T); val y = Free ("y" ^ string_of_int i, U) in HOLogic.mk_imp (R $ x $ y, finite $ (Const ("Nominal.supp", U --> aset) $ y)) end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ (1 upto length recTs))))) (fn fins => (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1)))) end) dt_atomTs; (** freshness **) val perm_fresh_fresh = PureThy.get_thms thy11 (Name "perm_fresh_fresh"); val perm_swap = PureThy.get_thms thy11 (Name "perm_swap"); val finite_premss = map (fn aT => map (fn (f, T) => HOLogic.mk_Trueprop (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $ (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ f))) (rec_fns ~~ rec_fn_Ts)) dt_atomTs; val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) => let val name = Sign.base_name (fst (dest_Type aT)); val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1")); val a = Free ("a", aT); val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop (Const ("Nominal.fresh", aT --> fT --> HOLogic.boolT) $ a $ f)) (rec_fns ~~ rec_fn_Ts) in map (fn (((T, U), R), eqvt_th) => let val x = Free ("x", T); val y = Free ("y", U); val y' = Free ("y'", U) in standard (Goal.prove (ProofContext.init thy11) [] (finite_prems @ [HOLogic.mk_Trueprop (R $ x $ y), HOLogic.mk_Trueprop (HOLogic.mk_all ("y'", U, HOLogic.mk_imp (R $ x $ y', HOLogic.mk_eq (y', y)))), HOLogic.mk_Trueprop (Const ("Nominal.fresh", aT --> T --> HOLogic.boolT) $ a $ x)] @ freshs) (HOLogic.mk_Trueprop (Const ("Nominal.fresh", aT --> U --> HOLogic.boolT) $ a $ y)) (fn {prems, context} => let val (finite_prems, rec_prem :: unique_prem :: fresh_prems) = chop (length finite_prems) prems; val unique_prem' = unique_prem RS spec RS mp; val unique = [unique_prem', unique_prem' RS sym] MRS trans; val _ $ (_ $ (_ $ S $ _)) $ _ = prop_of supports_fresh; val tuple = foldr1 HOLogic.mk_prod (x :: rec_fns) in EVERY [rtac (Drule.cterm_instantiate [(cterm_of thy11 S, cterm_of thy11 (Const ("Nominal.supp", fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))] supports_fresh) 1, simp_tac (HOL_basic_ss addsimps [supports_def, symmetric fresh_def, fresh_prod]) 1, REPEAT_DETERM (resolve_tac [allI, impI] 1), REPEAT_DETERM (etac conjE 1), rtac unique 1, SUBPROOF (fn {prems = prems', params = [a, b], ...} => EVERY [cut_facts_tac [rec_prem] 1, rtac (Thm.instantiate ([], [(cterm_of thy11 (Var (("pi", 0), mk_permT aT)), cterm_of thy11 (perm_of_pair (term_of a, term_of b)))]) eqvt_th) 1, asm_simp_tac (HOL_ss addsimps (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1, rtac rec_prem 1, simp_tac (HOL_ss addsimps (fs_name :: supp_prod :: finite_Un :: finite_prems)) 1, simp_tac (HOL_ss addsimps (symmetric fresh_def :: fresh_prod :: fresh_prems)) 1] end)) end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ eqvt_ths) end) (dt_atomTs ~~ rec_equiv_thms' ~~ finite_premss); (** uniqueness **) val exists_fresh' = PureThy.get_thms thy11 (Name "exists_fresh'"); val fs_atoms = map (fn Type (s, _) => PureThy.get_thm thy11 (Name ("fs_" ^ Sign.base_name s ^ "1"))) dt_atomTs; val fresh_atm = PureThy.get_thms thy11 (Name "fresh_atm"); val calc_atm = PureThy.get_thms thy11 (Name "calc_atm"); val fresh_left = PureThy.get_thms thy11 (Name "fresh_left"); val fun_tuple = foldr1 HOLogic.mk_prod (rec_ctxt :: rec_fns); val fun_tupleT = fastype_of fun_tuple; val rec_unique_frees = DatatypeProp.indexify_names (replicate (length recTs) "x") ~~ recTs; val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees; val rec_unique_frees' = DatatypeProp.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts; val rec_unique_concls = map (fn ((x, U), R) => Const ("Ex1", (U --> HOLogic.boolT) --> HOLogic.boolT) $ Abs ("y", U, R $ Free x $ Bound 0)) (rec_unique_frees ~~ rec_result_Ts ~~ rec_sets); val induct_aux_rec = Drule.cterm_instantiate (map (pairself (cterm_of thy11)) (map (fn (aT, f) => (Logic.varify f, Abs ("z", HOLogic.unitT, Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple))) fresh_fs @ map (fn (((P, T), (x, U)), Q) => (Var ((P, 0), HOLogic.unitT --> Logic.varifyT T --> HOLogic.boolT), Abs ("z", HOLogic.unitT, absfree (x, U, Q)))) (pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @ map (fn (s, T) => (Var ((s, 0), Logic.varifyT T), Free (s, T))) rec_unique_frees)) induct_aux; fun obtain_fresh_name vs ths rec_fin_supp T (freshs1, freshs2, ctxt) = let val p = foldr1 HOLogic.mk_prod (vs @ freshs1); val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.exists_const T $ Abs ("x", T, Const ("Nominal.fresh", T --> fastype_of p --> HOLogic.boolT) $ Bound 0 $ p))) (fn _ => EVERY [cut_facts_tac ths 1, REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp T)) 1), resolve_tac exists_fresh' 1, asm_simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]); val (([cx], ths), ctxt') = Obtain.result (fn _ => EVERY [etac exE 1, full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1, REPEAT (etac conjE 1)]) [ex] ctxt in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end; val finite_ctxt_prems = map (fn aT => HOLogic.mk_Trueprop (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $ (Const ("Nominal.supp", fsT' --> HOLogic.mk_setT aT) $ rec_ctxt))) dt_atomTs; val rec_unique_thms = split_conj_thm (Goal.prove (ProofContext.init thy11) (map fst rec_unique_frees) (List.concat finite_premss @ finite_ctxt_prems @ rec_prems @ rec_prems') (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls)) (fn {prems, context} => let val k = length rec_fns; val (finite_thss, ths1) = fold_map (fn T => fn xs => apfst (pair T) (chop k xs)) dt_atomTs prems; val (finite_ctxt_ths, ths2) = chop (length dt_atomTs) ths1; val (P_ind_ths, fcbs) = chop k ths2; val P_ths = map (fn th => th RS mp) (split_conj_thm (Goal.prove context (map fst (rec_unique_frees'' @ rec_unique_frees')) [] (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn (((x, y), S), P) => HOLogic.mk_imp (S $ Free x $ Free y, P $ (Free y))) (rec_unique_frees'' ~~ rec_unique_frees' ~~ rec_sets ~~ rec_preds)))) (fn _ => rtac rec_induct 1 THEN REPEAT ((resolve_tac P_ind_ths THEN_ALL_NEW assume_tac) 1)))); val rec_fin_supp_thms' = map (fn (ths, (T, fin_ths)) => (T, map (curry op MRS fin_ths) ths)) (rec_fin_supp_thms ~~ finite_thss); in EVERY ([rtac induct_aux_rec 1] @ maps (fn ((_, finite_ths), finite_th) => [cut_facts_tac (finite_th :: finite_ths) 1, asm_simp_tac (HOL_ss addsimps [supp_prod, finite_Un]) 1]) (finite_thss ~~ finite_ctxt_ths) @ maps (fn ((_, idxss), elim) => maps (fn idxs => [full_simp_tac (HOL_ss addsimps [symmetric fresh_def, supp_prod, Un_iff]) 1, REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1), rtac ex1I 1, (resolve_tac rec_intrs THEN_ALL_NEW atac) 1, rotate_tac ~1 1, ((DETERM o etac elim) THEN_ALL_NEW full_simp_tac (HOL_ss addsimps List.concat distinct_thms)) 1] @ (if null idxs then [] else [hyp_subst_tac 1, SUBPROOF (fn {asms, concl, prems = prems', params, context = context', ...} => let val SOME prem = find_first (can (HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of)) prems'; val _ $ (_ $ lhs $ rhs) = prop_of prem; val _ $ (_ $ lhs' $ rhs') = term_of concl; val rT = fastype_of lhs'; val (c, cargsl) = strip_comb lhs; val cargsl' = partition_cargs idxs cargsl; val boundsl = List.concat (map fst cargsl'); val (_, cargsr) = strip_comb rhs; val cargsr' = partition_cargs idxs cargsr; val boundsr = List.concat (map fst cargsr'); val (params1, _ :: params2) = chop (length params div 2) (map term_of params); val params' = params1 @ params2; val rec_prems = filter (fn th => case prop_of th of _ $ (S $ _ $ _) => S mem rec_sets | _ => false) prems'; val fresh_prems = filter (fn th => case prop_of th of _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true | _ $ (Const ("Not", _) $ _) => true | _ => false) prems'; val Ts = map fastype_of boundsl; val _ = warning "step 1: obtaining fresh names"; val (freshs1, freshs2, context'') = fold (obtain_fresh_name (rec_ctxt :: rec_fns @ params') (List.concat (map snd finite_thss) @ finite_ctxt_ths @ rec_prems) rec_fin_supp_thms') Ts ([], [], context'); val pi1 = map perm_of_pair (boundsl ~~ freshs1); val rpi1 = rev pi1; val pi2 = map perm_of_pair (boundsr ~~ freshs1); val rpi2 = rev pi2; val fresh_prems' = mk_not_sym fresh_prems; val freshs2' = mk_not_sym freshs2; (** as, bs, cs # K as ts, K bs us **) val _ = warning "step 2: as, bs, cs # K as ts, K bs us"; val prove_fresh_ss = HOL_ss addsimps (finite_Diff :: List.concat fresh_thms @ fs_atoms @ abs_fresh @ abs_supp @ fresh_atm); (* FIXME: avoid asm_full_simp_tac ? *) fun prove_fresh ths y x = Goal.prove context'' [] [] (HOLogic.mk_Trueprop (Const ("Nominal.fresh", fastype_of x --> fastype_of y --> HOLogic.boolT) $ x $ y)) (fn _ => cut_facts_tac ths 1 THEN asm_full_simp_tac prove_fresh_ss 1); val constr_fresh_thms = map (prove_fresh fresh_prems lhs) boundsl @ map (prove_fresh fresh_prems rhs) boundsr @ map (prove_fresh freshs2 lhs) freshs1 @ map (prove_fresh freshs2 rhs) freshs1; (** pi1 o (K as ts) = pi2 o (K bs us) **) val _ = warning "step 3: pi1 o (K as ts) = pi2 o (K bs us)"; val pi1_pi2_eq = Goal.prove context'' [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (fold_rev (mk_perm []) pi1 lhs, fold_rev (mk_perm []) pi2 rhs))) (fn _ => EVERY [cut_facts_tac constr_fresh_thms 1, asm_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh) 1, rtac prem 1]); (** pi1 o ts = pi2 o us **) val _ = warning "step 4: pi1 o ts = pi2 o us"; val pi1_pi2_eqs = map (fn (t, u) => Goal.prove context'' [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (fold_rev (mk_perm []) pi1 t, fold_rev (mk_perm []) pi2 u))) (fn _ => EVERY [cut_facts_tac [pi1_pi2_eq] 1, asm_full_simp_tac (HOL_ss addsimps (calc_atm @ List.concat perm_simps' @ fresh_prems' @ freshs2' @ abs_perm @ alpha @ List.concat inject_thms)) 1])) (map snd cargsl' ~~ map snd cargsr'); (** pi1^-1 o pi2 o us = ts **) val _ = warning "step 5: pi1^-1 o pi2 o us = ts"; val rpi1_pi2_eqs = map (fn ((t, u), eq) => Goal.prove context'' [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (fold_rev (mk_perm []) (rpi1 @ pi2) u, t))) (fn _ => simp_tac (HOL_ss addsimps ((eq RS sym) :: perm_swap)) 1)) (map snd cargsl' ~~ map snd cargsr' ~~ pi1_pi2_eqs); val (rec_prems1, rec_prems2) = chop (length rec_prems div 2) rec_prems; (** (ts, pi1^-1 o pi2 o vs) in rec_set **) val _ = warning "step 6: (ts, pi1^-1 o pi2 o vs) in rec_set"; val rec_prems' = map (fn th => let val _ $ (S $ x $ y) = prop_of th; val k = find_index (equal S) rec_sets; val pi = rpi1 @ pi2; fun mk_pi z = fold_rev (mk_perm []) pi z; fun eqvt_tac p = let val U as Type (_, [Type (_, [T, _])]) = fastype_of p; val l = find_index (equal T) dt_atomTs; val th = List.nth (List.nth (rec_equiv_thms', l), k); val th' = Thm.instantiate ([], [(cterm_of thy11 (Var (("pi", 0), U)), cterm_of thy11 p)]) th; in rtac th' 1 end; val th' = Goal.prove context'' [] [] (HOLogic.mk_Trueprop (S $ mk_pi x $ mk_pi y)) (fn _ => EVERY (map eqvt_tac pi @ [simp_tac (HOL_ss addsimps (fresh_prems' @ freshs2' @ perm_swap @ perm_fresh_fresh)) 1, rtac th 1])) in Simplifier.simplify (HOL_basic_ss addsimps rpi1_pi2_eqs) th' end) rec_prems2; val ihs = filter (fn th => case prop_of th of _ $ (Const ("All", _) $ _) => true | _ => false) prems'; (** pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs **) val _ = warning "step 7: pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs"; val rec_eqns = map (fn (th, ih) => let val th' = th RS (ih RS spec RS mp) RS sym; val _ $ (_ $ lhs $ rhs) = prop_of th'; fun strip_perm (_ $ _ $ t) = strip_perm t | strip_perm t = t; in Goal.prove context'' [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (fold_rev (mk_perm []) pi1 lhs, fold_rev (mk_perm []) pi2 (strip_perm rhs)))) (fn _ => simp_tac (HOL_basic_ss addsimps (th' :: perm_swap)) 1) end) (rec_prems' ~~ ihs); (** as # rs **) val _ = warning "step 8: as # rs"; val rec_freshs = List.concat (map (fn (rec_prem, ih) => let val _ $ (S $ x $ (y as Free (_, T))) = prop_of rec_prem; val k = find_index (equal S) rec_sets; val atoms = List.concat (List.mapPartial (fn (bs, z) => if z = x then NONE else SOME bs) cargsl') in map (fn a as Free (_, aT) => let val l = find_index (equal aT) dt_atomTs; in Goal.prove context'' [] [] (HOLogic.mk_Trueprop (Const ("Nominal.fresh", aT --> T --> HOLogic.boolT) $ a $ y)) (fn _ => EVERY (rtac (List.nth (List.nth (rec_fresh_thms, l), k)) 1 :: map (fn th => rtac th 1) (snd (List.nth (finite_thss, l))) @ [rtac rec_prem 1, rtac ih 1, REPEAT_DETERM (resolve_tac fresh_prems 1)])) end) atoms end) (rec_prems1 ~~ ihs)); (** as # fK as ts rs , bs # fK bs us vs **) val _ = warning "step 9: as # fK as ts rs , bs # fK bs us vs"; fun prove_fresh_result (a as Free (_, aT)) = Goal.prove context'' [] [] (HOLogic.mk_Trueprop (Const ("Nominal.fresh", aT --> rT --> HOLogic.boolT) $ a $ rhs')) (fn _ => EVERY [resolve_tac fcbs 1, REPEAT_DETERM (resolve_tac (fresh_prems @ rec_freshs) 1), REPEAT_DETERM (resolve_tac (maps snd rec_fin_supp_thms') 1 THEN resolve_tac rec_prems 1), resolve_tac P_ind_ths 1, REPEAT_DETERM (resolve_tac (P_ths @ rec_prems) 1)]); val fresh_results'' = map prove_fresh_result boundsl; fun prove_fresh_result'' ((a as Free (_, aT), b), th) = let val th' = Goal.prove context'' [] [] (HOLogic.mk_Trueprop (Const ("Nominal.fresh", aT --> rT --> HOLogic.boolT) $ fold_rev (mk_perm []) (rpi2 @ pi1) a $ fold_rev (mk_perm []) (rpi2 @ pi1) rhs')) (fn _ => simp_tac (HOL_ss addsimps fresh_bij) 1 THEN rtac th 1) in Goal.prove context'' [] [] (HOLogic.mk_Trueprop (Const ("Nominal.fresh", aT --> rT --> HOLogic.boolT) $ b $ lhs')) (fn _ => EVERY [cut_facts_tac [th'] 1, NominalPermeq.perm_simp_tac (HOL_ss addsimps (rec_eqns @ pi1_pi2_eqs @ perm_swap)) 1, full_simp_tac (HOL_ss addsimps (calc_atm @ fresh_prems' @ freshs2' @ perm_fresh_fresh)) 1]) end; val fresh_results = fresh_results'' @ map prove_fresh_result'' (boundsl ~~ boundsr ~~ fresh_results''); (** cs # fK as ts rs , cs # fK bs us vs **) val _ = warning "step 10: cs # fK as ts rs , cs # fK bs us vs"; fun prove_fresh_result' recs t (a as Free (_, aT)) = Goal.prove context'' [] [] (HOLogic.mk_Trueprop (Const ("Nominal.fresh", aT --> rT --> HOLogic.boolT) $ a $ t)) (fn _ => EVERY [cut_facts_tac recs 1, REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp_thms' aT)) 1), NominalPermeq.fresh_guess_tac (HOL_ss addsimps (freshs2 @ fs_atoms @ fresh_atm @ List.concat (map snd finite_thss))) 1]); val fresh_results' = map (prove_fresh_result' rec_prems1 rhs') freshs1 @ map (prove_fresh_result' rec_prems2 lhs') freshs1; (** pi1 o (fK as ts rs) = pi2 o (fK bs us vs) **) val _ = warning "step 11: pi1 o (fK as ts rs) = pi2 o (fK bs us vs)"; val pi1_pi2_result = Goal.prove context'' [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (fold_rev (mk_perm []) pi1 rhs', fold_rev (mk_perm []) pi2 lhs'))) (fn _ => NominalPermeq.perm_simp_tac (HOL_ss addsimps pi1_pi2_eqs @ rec_eqns) 1 THEN TRY (simp_tac (HOL_ss addsimps (fresh_prems' @ freshs2' @ calc_atm @ perm_fresh_fresh)) 1)); val _ = warning "final result"; val final = Goal.prove context'' [] [] (term_of concl) (fn _ => cut_facts_tac [pi1_pi2_result RS sym] 1 THEN full_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh @ fresh_results @ fresh_results') 1); val final' = ProofContext.export context'' context' [final]; val _ = warning "finished!" in resolve_tac final' 1 end) context 1])) idxss) (ndescr ~~ rec_elims)) end)); val rec_total_thms = map (fn r => r RS theI') rec_unique_thms; (* define primrec combinators *) val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec"; val reccomb_names = map (Sign.full_name thy11) (if length descr'' = 1 then [big_reccomb_name] else (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int) (1 upto (length descr'')))); val reccombs = map (fn ((name, T), T') => list_comb (Const (name, rec_fn_Ts @ [T] ---> T'), rec_fns)) (reccomb_names ~~ recTs ~~ rec_result_Ts); val (reccomb_defs, thy12) = thy11 |> Sign.add_consts_i (map (fn ((name, T), T') => (Sign.base_name name, rec_fn_Ts @ [T] ---> T', NoSyn)) (reccomb_names ~~ recTs ~~ rec_result_Ts)) |> (PureThy.add_defs_i false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') => ((Sign.base_name name) ^ "_def", Logic.mk_equals (comb, absfree ("x", T, Const ("The", (T' --> HOLogic.boolT) --> T') $ absfree ("y", T', set $ Free ("x", T) $ Free ("y", T')))))) (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts)); (* prove characteristic equations for primrec combinators *) val rec_thms = map (fn (prems, concl) => let val _ $ (_ $ (_ $ x) $ _) = concl; val (_, cargs) = strip_comb x; val ps = map (fn (x as Free (_, T), i) => (Free ("x" ^ string_of_int i, T), x)) (cargs ~~ (1 upto length cargs)); val concl' = subst_atomic_types (rec_result_Ts' ~~ rec_result_Ts) concl; val prems' = List.concat finite_premss @ finite_ctxt_prems @ rec_prems @ rec_prems' @ map (subst_atomic ps) prems; fun solve rules prems = resolve_tac rules THEN_ALL_NEW (resolve_tac prems THEN_ALL_NEW atac) in Goal.prove_global thy12 [] prems' concl' (fn prems => EVERY [rewrite_goals_tac reccomb_defs, rtac the1_equality 1, solve rec_unique_thms prems 1, resolve_tac rec_intrs 1, REPEAT (solve (prems @ rec_total_thms) prems 1)]) end) (rec_eq_prems ~~ DatatypeProp.make_primrecs new_type_names descr' sorts' thy12); val dt_infos = map (make_dt_info pdescr sorts induct reccomb_names rec_thms) ((0 upto length descr1 - 1) ~~ descr1 ~~ distinct_thms ~~ inject_thms); (* FIXME: theorems are stored in database for testing only *) val (_, thy13) = thy12 |> PureThy.add_thmss [(("rec_equiv", List.concat rec_equiv_thms), []), (("rec_equiv'", List.concat rec_equiv_thms'), []), (("rec_fin_supp", List.concat rec_fin_supp_thms), []), (("rec_fresh", List.concat rec_fresh_thms), []), (("rec_unique", map standard rec_unique_thms), []), (("recs", rec_thms), [])] ||> Sign.parent_path ||> map_nominal_datatypes (fold Symtab.update dt_infos); in thy13 end; val add_nominal_datatype = gen_add_nominal_datatype read_typ true; (* FIXME: The following stuff should be exported by DatatypePackage *) local structure P = OuterParse and K = OuterKeyword in val datatype_decl = Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix -- (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix)); fun mk_datatype args = let val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args; val specs = map (fn ((((_, vs), t), mx), cons) => (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args; in add_nominal_datatype false names specs end; val _ = OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype)); end; end