diff -r 3896169e6ff9 -r 9466169dc8e0 src/HOL/Nominal/nominal_datatype.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Nominal/nominal_datatype.ML Fri Jul 03 16:51:08 2009 +0200 @@ -0,0 +1,2094 @@ +(* Title: HOL/Nominal/nominal_datatype.ML + Author: Stefan Berghofer and Christian Urban, TU Muenchen + +Nominal datatype package for Isabelle/HOL. +*) + +signature NOMINAL_DATATYPE = +sig + val add_nominal_datatype : Datatype.config -> 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 + val fresh_const: typ -> typ -> term + val fresh_star_const: typ -> typ -> term +end + +structure NominalDatatype : NOMINAL_DATATYPE = +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: Datatype 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 = Long_Name.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 "_" (Long_Name.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); + + +(** 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 cp = cp_inst_of thy a b; + val dj = dj_thm_of thy 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 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 at_fin_set_supp = thm "at_fin_set_supp"; +val at_fin_set_fresh = thm "at_fin_set_fresh"; +val abs_fun_eq1 = thm "abs_fun_eq1"; + +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", _) $ (Const ("op =", _) $ _ $ _)) => [th, th RS not_sym] + | _ => [th]) ths; + +fun fresh_const T U = Const ("Nominal.fresh", T --> U --> HOLogic.boolT); +fun fresh_star_const T U = + Const ("Nominal.fresh_star", HOLogic.mk_setT T --> U --> HOLogic.boolT); + +fun gen_add_nominal_datatype prep_typ config 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, _) => + (Binding.name tname, length tvs, mx)) dts); + + val atoms = atoms_of thy; + + 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 tyvars = map (map (fn s => + (s, the (AList.lookup (op =) sorts s))) o #1) dts'; + + fun inter_sort thy S S' = Type.inter_sort (Sign.tsig_of thy) (S, S'); + fun augment_sort_typ thy S = + let val S = Sign.certify_sort thy S + in map_type_tfree (fn (s, S') => TFree (s, + if member (op = o apsnd fst) sorts s then inter_sort thy S S' else S')) + end; + fun augment_sort thy S = map_types (augment_sort_typ thy S); + + 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_bname tmp_thy n, Sign.full_bname 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, Binding.name (tname ^ "_Rep"), NoSyn, + map (fn (cname, cargs, mx) => (Binding.name (cname ^ "_Rep"), + map replace_types cargs, NoSyn)) constrs)) dts'; + + val new_type_names' = map (fn n => n ^ "_Rep") new_type_names; + + val (full_new_type_names',thy1) = + Datatype.add_datatype config new_type_names' dts'' thy; + + val {descr, induction, ...} = + Datatype.the_info thy1 (hd full_new_type_names'); + fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i); + + val big_name = space_implode "_" new_type_names; + + + (**** 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' = DatatypeProp.indexify_names (map (fn (i, _) => + "perm_" ^ name_of_typ (nth_dtyp i)) descr); + val perm_names = replicate (length new_type_names) "Nominal.perm" @ + map (Sign.full_bname thy1) (List.drop (perm_names', length new_type_names)); + val perm_names_types = perm_names ~~ perm_types; + val perm_names_types' = perm_names' ~~ perm_types; + + val perm_eqs = maps (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 = Name.variant_list ["pi"] (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, + Free (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 + (Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq + (Free (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) = + Primrec.add_primrec_overloaded + (map (fn (s, sT) => (s, sT, false)) + (List.take (perm_names' ~~ perm_names_types, length new_type_names))) + (map (fn s => (Binding.name s, NONE, NoSyn)) perm_names') perm_eqs thy1; + + (**** 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 "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 [] [] + (augment_sort thy2 [pt_class_of thy2 a] + (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 "at_pt_inst"; + val pt2 = PureThy.get_thm thy2 "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 = pt_inst_of thy2 a; + val pt2' = pt_inst RS pt2; + val pt2_ax = PureThy.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "2") a); + in List.take (map standard (split_conj_thm + (Goal.prove_global thy2 [] [] + (augment_sort thy2 [pt_class_of thy2 a] + (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 "pt3"; + val pt3_rev = PureThy.get_thm thy2 "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); + val at_inst = at_inst_of thy2 a; + val pt_inst = pt_inst_of thy2 a; + val pt3' = pt_inst RS pt3; + val pt3_rev' = at_inst RS (pt_inst RS pt3_rev); + val pt3_ax = PureThy.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "3") a); + in List.take (map standard (split_conj_thm + (Goal.prove_global thy2 [] [] + (augment_sort thy2 [pt_class_of thy2 a] (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 "cp1"; + val dj_cp = PureThy.get_thm thy2 "dj_cp"; + val pt_perm_compose = PureThy.get_thm thy2 "pt_perm_compose"; + val pt_perm_compose_rev = PureThy.get_thm thy2 "pt_perm_compose_rev"; + val dj_perm_perm_forget = PureThy.get_thm thy2 "dj_perm_perm_forget"; + + fun composition_instance name1 name2 thy = + let + val cp_class = cp_class_of thy name1 name2; + val pt_class = + if name1 = name2 then [pt_class_of thy name1] + else []; + val permT1 = mk_permT (Type (name1, [])); + val permT2 = mk_permT (Type (name2, [])); + val Ts = map body_type perm_types; + val cp_inst = cp_inst_of thy name1 name2; + val simps = simpset_of thy addsimps (perm_fun_def :: + (if name1 <> name2 then + let val dj = dj_thm_of thy name2 name1 + in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end + else + let + val at_inst = at_inst_of thy name1; + val pt_inst = pt_inst_of thy name1; + 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 sort = Sign.certify_sort thy (cp_class :: pt_class); + val thms = split_conj_thm (Goal.prove_global thy [] [] + (augment_sort thy sort + (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 + fold (fn (s, tvs) => fn thy => AxClass.prove_arity + (s, map (inter_sort thy sort o snd) tvs, [cp_class]) + (Class.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy) + (full_new_type_names' ~~ tyvars) thy + end; + + val (perm_thmss,thy3) = thy2 |> + fold (fn name1 => fold (composition_instance name1) atoms) atoms |> + fold (fn atom => fn thy => + let val pt_name = pt_class_of thy atom + in + fold (fn (s, tvs) => fn thy => AxClass.prove_arity + (s, map (inter_sort thy [pt_name] o snd) tvs, [pt_name]) + (EVERY + [Class.intro_classes_tac [], + resolve_tac perm_empty_thms 1, + resolve_tac perm_append_thms 1, + resolve_tac perm_eq_thms 1, assume_tac 1]) thy) + (full_new_type_names' ~~ tyvars) thy + end) atoms |> + PureThy.add_thmss + [((Binding.name (space_implode "_" new_type_names ^ "_unfolded_perm_eq"), + unfolded_perm_eq_thms), [Simplifier.simp_add]), + ((Binding.name (space_implode "_" new_type_names ^ "_perm_empty"), + perm_empty_thms), [Simplifier.simp_add]), + ((Binding.name (space_implode "_" new_type_names ^ "_perm_append"), + perm_append_thms), [Simplifier.simp_add]), + ((Binding.name (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))); + val dt_atoms = map (fst o dest_Type) dt_atomTs; + + 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 (List.foldr mk_abs_fun (j', free) Ts) :: ts) + end; + + val (_, _, prems, ts) = List.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_bname thy3) rep_set_names'; + + val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy4) = + Inductive.add_inductive_global (serial_string ()) + {quiet_mode = false, verbose = false, kind = Thm.internalK, + alt_name = Binding.name big_rep_name, coind = false, no_elim = true, no_ind = false, + skip_mono = true, fork_mono = false} + (map (fn (s, T) => ((Binding.name s, T --> HOLogic.boolT), NoSyn)) + (rep_set_names' ~~ recTs')) + [] (map (fn x => (Attrib.empty_binding, x)) intr_ts) [] thy3; + + (**** Prove that representing set is closed under permutation ****) + + val _ = warning "proving closure under permutation..."; + + val abs_perm = PureThy.get_thms thy4 "abs_perm"; + + 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 [] [] + (augment_sort thy4 + (pt_class_of thy4 name :: map (cp_class_of thy4 name) (dt_atoms \ name)) + (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map + (fn ((s, T), x) => + let + 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 + [indtac rep_induct [] 1, + ALLGOALS (simp_tac (simpset_of thy4 addsimps + (symmetric perm_fun_def :: abs_perm))), + ALLGOALS (resolve_tac rep_intrs THEN_ALL_NEW assume_tac)])), + length new_type_names)); + + val perm_closed_thmss = map mk_perm_closed atoms; + + (**** typedef ****) + + val _ = warning "defining type..."; + + val (typedefs, thy6) = + thy4 + |> fold_map (fn ((((name, mx), tvs), (cname, U)), name') => fn thy => + Typedef.add_typedef false (SOME (Binding.name name')) + (Binding.name name, map fst 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 (map fst tvs) "'a", HOLogic.typeS)); + val pi = Free ("pi", permT); + val T = Type (Sign.intern_type thy name, map TFree tvs); + in apfst (pair r o hd) + (PureThy.add_defs_unchecked true [((Binding.name ("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; + + + (** prove that new types are in class pt_ **) + + val _ = warning "prove that new types are in class pt_ ..."; + + fun pt_instance (atom, perm_closed_thms) = + fold (fn ((((((Abs_inverse, Rep_inverse), Rep), + perm_def), name), tvs), perm_closed) => fn thy => + let + val pt_class = pt_class_of thy atom; + val sort = Sign.certify_sort thy + (pt_class :: map (cp_class_of thy atom) (dt_atoms \ atom)) + in AxClass.prove_arity + (Sign.intern_type thy name, + map (inter_sort thy sort o snd) tvs, [pt_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 + ("pt_" ^ Long_Name.base_name atom ^ "3")]) 1]) thy + end) + (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__ **) + + val _ = warning "prove that new types are in class cp__ ..."; + + fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy = + let + val cp_class = cp_class_of thy atom1 atom2; + val sort = Sign.certify_sort thy + (pt_class_of thy atom1 :: map (cp_class_of thy atom1) (dt_atoms \ atom1) @ + (if atom1 = atom2 then [cp_class_of thy atom1 atom1] else + pt_class_of thy atom2 :: map (cp_class_of thy atom2) (dt_atoms \ atom2))); + val cp1' = cp_inst_of thy atom1 atom2 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, + map (inter_sort thy sort o snd) tvs, [cp_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 x) (atoms ~~ perm_closed_thmss)) + (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, _) = List.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 = Long_Name.explode s; + in Long_Name.implode (Library.nth_map (length xs - i) (strip_suffix 4) xs) end; + + val (descr'', ndescr) = ListPair.unzip (map_filter + (fn (i, ("Nominal.noption", _, _)) => NONE + | (i, (s, dts, constrs)) => + let + val SOME index = AList.lookup op = ty_idxs i; + val (constrs2, constrs1) = + map_split (fn (cname, cargs) => + apsnd (pair (strip_nth_name 2 (strip_nth_name 1 cname))) + (fold_map (fn dt => fn dts => + let val (dts', dt') = strip_option dt + in ((length dts, length dts'), dts @ dts' @ [reindex dt']) 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_bname 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, + List.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) = List.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 = (Long_Name.base_name cname) ^ "_def"; + val ([def_thm], thy') = thy |> + Sign.add_consts_i [(Binding.name cname', constrT, mx)] |> + (PureThy.add_defs false o map Thm.no_attributes) [(Binding.name 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 config) 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 [] [] + (augment_sort thy8 + (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (dt_atoms \ atom)) + (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 distinct_props = DatatypeProp.make_distincts descr' sorts; + val dist_rewrites = map2 (fn rep_thms => fn 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)) (k, 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 (k, ts) + end; + + val distinct_thms = map2 prove_distinct_thms + (constr_rep_thmss ~~ dist_lemmas) distinct_props; + + (** prove equations for permutation functions **) + + 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 + (Long_Name.append tname (Long_Name.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) = List.foldr constr_arg (1, [], []) dts; + val c = Const (cname, map fastype_of l_args ---> T) + in + Goal.prove_global thy8 [] [] + (augment_sort thy8 + (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (dt_atoms \ atom)) + (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 "alpha"; + val abs_fresh = PureThy.get_thms thy8 "abs_fresh"; + + val pt_cp_sort = + map (pt_class_of thy8) dt_atoms @ + maps (fn s => map (cp_class_of thy8 s) (dt_atoms \ s)) dt_atoms; + + 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 + (Long_Name.append tname (Long_Name.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 + (List.foldr mk_abs_fun x xs, List.foldr mk_abs_fun y ys) :: eqs) + end; + + val (_, args1, args2, eqs) = List.foldr make_inj (1, [], [], []) dts; + val Ts = map fastype_of args1; + val c = Const (cname, Ts ---> T) + in + Goal.prove_global thy8 [] [] + (augment_sort thy8 pt_cp_sort + (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)]) + 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 + (Long_Name.append tname (Long_Name.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), List.foldr mk_abs_fun x xs :: args2) + end; + + val (_, args1, args2) = List.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 = fresh_const atomT (fastype_of t) $ Free ("a", atomT) $ t; + val supp_thm = Goal.prove_global thy8 [] [] + (augment_sort thy8 pt_cp_sort + (HOLogic.mk_Trueprop (HOLogic.mk_eq + (supp c, + if null dts then HOLogic.mk_set atomT [] + else foldr1 (HOLogic.mk_binop @{const_name 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 [] [] (augment_sort thy8 pt_cp_sort + (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, + (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 "abs_supp"; + val supp_atm = PureThy.get_thms thy8 "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 [] [] + (augment_sort thy8 (fs_class_of thy8 atom :: pt_cp_sort) + (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 ("fs_" ^ Long_Name.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 [((Binding.name "induct", dt_induct), [case_names_induct])] ||>> + PureThy.add_thmss [((Binding.name "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 = fs_class_of thy atom; + val sort = Sign.certify_sort thy (class :: pt_cp_sort) + in fold (fn Type (s, Ts) => AxClass.prove_arity + (s, map (inter_sort thy sort o snd o dest_TFree) Ts, [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 (fs_class_of thy9) dt_atoms); + 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 + (fresh_const T U $ 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 => + fresh_const T fsT $ 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", + (snd (split_last (binder_types T)) --> HOLogic.boolT) --> + 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))); + + val fin_set_supp = map (fn s => + at_inst_of thy9 s RS at_fin_set_supp) dt_atoms; + val fin_set_fresh = map (fn s => + at_inst_of thy9 s RS at_fin_set_fresh) dt_atoms; + val pt1_atoms = map (fn Type (s, _) => + PureThy.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "1")) dt_atomTs; + val pt2_atoms = map (fn Type (s, _) => + PureThy.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "2") RS sym) dt_atomTs; + val exists_fresh' = PureThy.get_thms thy9 "exists_fresh'"; + val fs_atoms = PureThy.get_thms thy9 "fin_supp"; + val abs_supp = PureThy.get_thms thy9 "abs_supp"; + val perm_fresh_fresh = PureThy.get_thms thy9 "perm_fresh_fresh"; + val calc_atm = PureThy.get_thms thy9 "calc_atm"; + val fresh_atm = PureThy.get_thms thy9 "fresh_atm"; + val fresh_left = PureThy.get_thms thy9 "fresh_left"; + val perm_swap = PureThy.get_thms thy9 "perm_swap"; + + fun obtain_fresh_name' ths ts T (freshs1, freshs2, ctxt) = + let + val p = foldr1 HOLogic.mk_prod (ts @ freshs1); + val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop + (HOLogic.exists_const T $ Abs ("x", T, + fresh_const T (fastype_of p) $ + Bound 0 $ p))) + (fn _ => EVERY + [resolve_tac exists_fresh' 1, + simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms @ + fin_set_supp @ ths)) 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; + + fun fresh_fresh_inst thy a b = + let + val T = fastype_of a; + val SOME th = find_first (fn th => case prop_of th of + _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ _)) $ _ => U = T + | _ => false) perm_fresh_fresh + in + Drule.instantiate' [] + [SOME (cterm_of thy a), NONE, SOME (cterm_of thy b)] th + end; + + val fs_cp_sort = + map (fs_class_of thy9) dt_atoms @ + maps (fn s => map (cp_class_of thy9 s) (dt_atoms \ s)) dt_atoms; + + (********************************************************************** + The subgoals occurring in the proof of induct_aux have the + following parameters: + + x_1 ... x_k p_1 ... p_m z + + 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 + ***********************************************************************) + + val _ = warning "proving strong induction theorem ..."; + + val induct_aux = Goal.prove_global thy9 [] + (map (augment_sort thy9 fs_cp_sort) ind_prems') + (augment_sort thy9 fs_cp_sort ind_concl') (fn {prems, context} => + let + val (prems1, prems2) = chop (length dt_atomTs) prems; + val ind_ss2 = HOL_ss addsimps + finite_Diff :: abs_fresh @ abs_supp @ fs_atoms; + val ind_ss1 = ind_ss2 addsimps fresh_left @ calc_atm @ + fresh_atm @ rev_simps @ app_simps; + val ind_ss3 = HOL_ss addsimps abs_fun_eq1 :: + abs_perm @ calc_atm @ perm_swap; + val ind_ss4 = HOL_basic_ss addsimps fresh_left @ prems1 @ + fin_set_fresh @ calc_atm; + val ind_ss5 = HOL_basic_ss addsimps pt1_atoms; + val ind_ss6 = HOL_basic_ss addsimps flat perm_simps'; + val th = Goal.prove context [] [] + (augment_sort thy9 fs_cp_sort aux_ind_concl) + (fn {context = context1, ...} => + EVERY (indtac dt_induct tnames 1 :: + maps (fn ((_, (_, _, constrs)), (_, constrs')) => + map (fn ((cname, cargs), is) => + REPEAT (rtac allI 1) THEN + SUBPROOF (fn {prems = iprems, params, concl, + context = context2, ...} => + let + val concl' = term_of concl; + val _ $ (_ $ _ $ u) = concl'; + val U = fastype_of u; + val (xs, params') = + chop (length cargs) (map term_of params); + val Ts = map fastype_of xs; + val cnstr = Const (cname, Ts ---> U); + val (pis, z) = split_last params'; + val mk_pi = fold_rev (mk_perm []) pis; + val xs' = partition_cargs is xs; + val xs'' = map (fn (ts, u) => (map mk_pi ts, mk_pi u)) xs'; + val ts = maps (fn (ts, u) => ts @ [u]) xs''; + val (freshs1, freshs2, context3) = fold (fn t => + let val T = fastype_of t + in obtain_fresh_name' prems1 + (the (AList.lookup op = fresh_fs T) $ z :: ts) T + end) (maps fst xs') ([], [], context2); + val freshs1' = unflat (map fst xs') freshs1; + val freshs2' = map (Simplifier.simplify ind_ss4) + (mk_not_sym freshs2); + val ind_ss1' = ind_ss1 addsimps freshs2'; + val ind_ss3' = ind_ss3 addsimps freshs2'; + val rename_eq = + if forall (null o fst) xs' then [] + else [Goal.prove context3 [] [] + (HOLogic.mk_Trueprop (HOLogic.mk_eq + (list_comb (cnstr, ts), + list_comb (cnstr, maps (fn ((bs, t), cs) => + cs @ [fold_rev (mk_perm []) (map perm_of_pair + (bs ~~ cs)) t]) (xs'' ~~ freshs1'))))) + (fn _ => EVERY + (simp_tac (HOL_ss addsimps flat inject_thms) 1 :: + REPEAT (FIRSTGOAL (rtac conjI)) :: + maps (fn ((bs, t), cs) => + if null bs then [] + else rtac sym 1 :: maps (fn (b, c) => + [rtac trans 1, rtac sym 1, + rtac (fresh_fresh_inst thy9 b c) 1, + simp_tac ind_ss1' 1, + simp_tac ind_ss2 1, + simp_tac ind_ss3' 1]) (bs ~~ cs)) + (xs'' ~~ freshs1')))]; + val th = Goal.prove context3 [] [] concl' (fn _ => EVERY + [simp_tac (ind_ss6 addsimps rename_eq) 1, + cut_facts_tac iprems 1, + (resolve_tac prems THEN_ALL_NEW + SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of + _ $ (Const ("Nominal.fresh", _) $ _ $ _) => + simp_tac ind_ss1' i + | _ $ (Const ("Not", _) $ _) => + resolve_tac freshs2' i + | _ => asm_simp_tac (HOL_basic_ss addsimps + pt2_atoms addsimprocs [perm_simproc]) i)) 1]) + val final = ProofContext.export context3 context2 [th] + in + resolve_tac final 1 + end) context1 1) (constrs ~~ constrs')) (descr'' ~~ ndescr))) + in + EVERY + [cut_facts_tac [th] 1, + REPEAT (eresolve_tac [conjE, @{thm allE_Nil}] 1), + REPEAT (etac allE 1), + REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac ind_ss5 1)] + end); + + val induct_aux' = Thm.instantiate ([], + map (fn (s, v as Var (_, T)) => + (cterm_of thy9 v, cterm_of thy9 (Free (s, T)))) + (pnames ~~ map head_of (HOLogic.dest_conj + (HOLogic.dest_Trueprop (concl_of induct_aux)))) @ + 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 [] + (map (augment_sort thy9 fs_cp_sort) ind_prems) + (augment_sort thy9 fs_cp_sort ind_concl) + (fn {prems, ...} => EVERY + [rtac induct_aux' 1, + REPEAT (resolve_tac fs_atoms 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 [((Binding.name "strong_induct'", induct_aux), [])] ||>> + PureThy.add_thms [((Binding.name "strong_induct", induct), [case_names_induct])] ||>> + PureThy.add_thmss [((Binding.name "strong_inducts", projections induct), [case_names_induct])]; + + (**** recursion combinator ****) + + val _ = warning "defining recursion combinator ..."; + + val used = List.foldr OldTerm.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 + Sign.certify_sort thy10 pt_cp_sort; + + 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_bname 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 + (fresh_const T U $ 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 + (fresh_const T (fastype_of f) $ 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 + (fresh_const T fsT' $ 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) => + fresh_const T (fastype_of result) $ 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 |> + Inductive.add_inductive_global (serial_string ()) + {quiet_mode = #quiet config, verbose = false, kind = Thm.internalK, + alt_name = Binding.name big_rec_name, coind = false, no_elim = false, no_ind = false, + skip_mono = true, fork_mono = false} + (map (fn (s, T) => ((Binding.name s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts)) + (map dest_Free rec_fns) + (map (fn x => (Attrib.empty_binding, x)) rec_intr_ts) [] ||> + PureThy.hide_fact true (Long_Name.append (Sign.full_bname thy10 big_rec_name) "induct"); + + (** equivariance **) + + val fresh_bij = PureThy.get_thms thy11 "fresh_bij"; + val perm_bij = PureThy.get_thms thy11 "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 [] [] + (augment_sort thy1 pt_cp_sort + (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps)))) + (fn _ => rtac rec_induct 1 THEN REPEAT + (simp_tac (Simplifier.theory_context thy11 HOL_basic_ss + addsimps flat perm_simps' + addsimprocs [NominalPermeq.perm_simproc_app]) 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 [] [] + (augment_sort thy1 pt_cp_sort + (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 = Long_Name.base_name (fst (dest_Type aT)); + val fs_name = PureThy.get_thm thy11 ("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 [] + (map (augment_sort thy11 fs_cp_sort) fins) + (augment_sort thy11 fs_cp_sort + (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 {prems = 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 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_fns' = map (augment_sort thy11 fs_cp_sort) rec_fns; + + val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) => + let + val name = Long_Name.base_name (fst (dest_Type aT)); + val fs_name = PureThy.get_thm thy11 ("fs_" ^ name ^ "1"); + val a = Free ("a", aT); + val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop + (fresh_const aT fT $ a $ f)) (rec_fns ~~ rec_fn_Ts) + in + map (fn (((T, U), R), eqvt_th) => + let + val x = Free ("x", augment_sort_typ thy11 fs_cp_sort T); + val y = Free ("y", U); + val y' = Free ("y'", U) + in + standard (Goal.prove (ProofContext.init thy11) [] + (map (augment_sort thy11 fs_cp_sort) + (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 (fresh_const aT T $ a $ x)] @ + freshs)) + (HOLogic.mk_Trueprop (fresh_const aT U $ 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 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) o apsnd (augment_sort thy11 fs_cp_sort)) + (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), Logic.varifyT (fsT' --> 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, + fresh_const T (fastype_of p) $ 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) + (map (augment_sort thy11 fs_cp_sort) + (List.concat finite_premss @ finite_ctxt_prems @ rec_prems @ rec_prems')) + (augment_sort thy11 fs_cp_sort + (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')) [] + (augment_sort thy11 fs_cp_sort + (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 + _ $ p => (case head_of p of + Const (s, _) => s mem rec_set_names + | _ => false) + | _ => 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 (fresh_const + (fastype_of x) (fastype_of y) $ 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 Const (s, _) = head_of S; + val k = find_index (equal s) rec_set_names; + 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 (fresh_const aT T $ 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 (fresh_const aT rT $ 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 (fresh_const aT rT $ + 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 (fresh_const aT rT $ b $ lhs')) + (fn _ => EVERY + [cut_facts_tac [th'] 1, + full_simp_tac (Simplifier.theory_context thy11 HOL_ss + addsimps rec_eqns @ pi1_pi2_eqs @ perm_swap + addsimprocs [NominalPermeq.perm_simproc_app]) 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 (fresh_const aT rT $ 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 _ => simp_tac (Simplifier.context context'' HOL_ss + addsimps pi1_pi2_eqs @ rec_eqns + addsimprocs [NominalPermeq.perm_simproc_app]) 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_bname 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') => + (Binding.name (Long_Name.base_name name), rec_fn_Ts @ [T] ---> T', NoSyn)) + (reccomb_names ~~ recTs ~~ rec_result_Ts)) + |> (PureThy.add_defs false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') => + (Binding.name (Long_Name.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 [] + (map (augment_sort thy12 fs_cp_sort) prems') + (augment_sort thy12 fs_cp_sort 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 + [((Binding.name "rec_equiv", List.concat rec_equiv_thms), []), + ((Binding.name "rec_equiv'", List.concat rec_equiv_thms'), []), + ((Binding.name "rec_fin_supp", List.concat rec_fin_supp_thms), []), + ((Binding.name "rec_fresh", List.concat rec_fresh_thms), []), + ((Binding.name "rec_unique", map standard rec_unique_thms), []), + ((Binding.name "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 Datatype.read_typ; + + +(* FIXME: The following stuff should be exported by Datatype *) + +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 Datatype.default_config 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