3 signature NOMINAL_PACKAGE =
5 val add_nominal_datatype : bool -> string list -> (string list * bstring * mixfix *
6 (bstring * string list * mixfix) list) list -> theory -> theory
9 structure NominalPackage : NOMINAL_PACKAGE =
15 (** FIXME: DatatypePackage should export this function **)
19 fun dt_recs (DtTFree _) = []
20 | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)
21 | dt_recs (DtRec i) = [i];
23 fun dt_cases (descr: descr) (_, args, constrs) =
25 fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i)));
26 val bnames = map the_bname (distinct (List.concat (map dt_recs args)));
27 in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;
30 fun induct_cases descr =
31 DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));
33 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));
37 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);
39 fun mk_case_names_exhausts descr new =
40 map (RuleCases.case_names o exhaust_cases descr o #1)
41 (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);
45 (*******************************)
47 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
49 fun read_typ sign ((Ts, sorts), str) =
51 val T = Type.no_tvars (Sign.read_typ (sign, (AList.lookup op =)
52 (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg
53 in (Ts @ [T], add_typ_tfrees (T, sorts)) end;
55 (** taken from HOL/Tools/datatype_aux.ML **)
57 fun indtac indrule indnames i st =
59 val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
60 val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
61 (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
62 val getP = if can HOLogic.dest_imp (hd ts) then
63 (apfst SOME) o HOLogic.dest_imp else pair NONE;
64 fun abstr (t1, t2) = (case t1 of
65 NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false)
67 [Free (s, T)] => absfree (s, T, t2)
68 | _ => sys_error "indtac")
69 | SOME (_ $ t' $ _) => Abs ("x", fastype_of t', abstract_over (t', t2)))
70 val cert = cterm_of (Thm.sign_of_thm st);
71 val Ps = map (cert o head_of o snd o getP) ts;
72 val indrule' = cterm_instantiate (Ps ~~
73 (map (cert o abstr o getP) ts')) indrule
78 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
80 (* this theory is used just for parsing *)
84 Theory.add_types (map (fn (tvs, tname, mx, _) =>
85 (tname, length tvs, mx)) dts);
87 val sign = Theory.sign_of tmp_thy;
89 val atoms = atoms_of thy;
90 val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
91 val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
92 Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
93 Sign.base_name atom2)) atoms) atoms);
94 fun augment_sort S = S union classes;
95 val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
97 fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
98 let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
99 in (constrs @ [(cname, cargs', mx)], sorts') end
101 fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
102 let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
103 in (dts @ [(tvs, tname, mx, constrs')], sorts') end
105 val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
106 val sorts' = map (apsnd augment_sort) sorts;
107 val tyvars = map #1 dts';
109 val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
110 val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
111 map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
113 val ps = map (fn (_, n, _, _) =>
114 (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
115 val rps = map Library.swap ps;
117 fun replace_types (Type ("nominal.ABS", [T, U])) =
118 Type ("fun", [T, Type ("nominal.nOption", [replace_types U])])
119 | replace_types (Type (s, Ts)) =
120 Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
121 | replace_types T = T;
123 fun replace_types' (Type (s, Ts)) =
124 Type (getOpt (AList.lookup op = rps s, s), map replace_types' Ts)
125 | replace_types' T = T;
127 val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
128 map (fn (cname, cargs, mx) => (cname,
129 map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
131 val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
132 val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
134 val ({induction, ...},thy1) =
135 DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
137 val SOME {descr, ...} = Symtab.lookup
138 (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
139 fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
141 val dt_atomTs = filter (fn Type (s, []) => s mem atoms | _ => false)
142 (get_nonrec_types descr sorts);
144 (**** define permutation functions ****)
146 val permT = mk_permT (TFree ("'x", HOLogic.typeS));
147 val pi = Free ("pi", permT);
148 val perm_types = map (fn (i, _) =>
149 let val T = nth_dtyp i
150 in permT --> T --> T end) descr;
151 val perm_names = replicate (length new_type_names) "nominal.perm" @
152 DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
153 ("perm_" ^ name_of_typ (nth_dtyp i)))
154 (length new_type_names upto length descr - 1));
155 val perm_names_types = perm_names ~~ perm_types;
157 val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
158 let val T = nth_dtyp i
159 in map (fn (cname, dts) =>
161 val Ts = map (typ_of_dtyp descr sorts') dts;
162 val names = DatatypeProp.make_tnames Ts;
163 val args = map Free (names ~~ Ts);
164 val c = Const (cname, Ts ---> T);
165 fun perm_arg (dt, x) =
166 let val T = type_of x
167 in if is_rec_type dt then
168 let val (Us, _) = strip_type T
169 in list_abs (map (pair "x") Us,
170 Const (List.nth (perm_names_types, body_index dt)) $ pi $
171 list_comb (x, map (fn (i, U) =>
172 Const ("nominal.perm", permT --> U --> U) $
173 (Const ("List.rev", permT --> permT) $ pi) $
174 Bound i) ((length Us - 1 downto 0) ~~ Us)))
176 else Const ("nominal.perm", permT --> T --> T) $ pi $ x
179 (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
180 (Const (List.nth (perm_names_types, i)) $
181 Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
183 list_comb (c, map perm_arg (dts ~~ args))))), [])
187 val (thy2, perm_simps) = thy1 |>
188 Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
189 (List.drop (perm_names_types, length new_type_names))) |>
190 PrimrecPackage.add_primrec_i "" perm_eqs;
192 (**** prove that permutation functions introduced by unfolding are ****)
193 (**** equivalent to already existing permutation functions ****)
195 val _ = warning ("length descr: " ^ string_of_int (length descr));
196 val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
198 val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
199 val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
201 val unfolded_perm_eq_thms =
202 if length descr = length new_type_names then []
203 else map standard (List.drop (split_conj_thm
204 (Goal.prove thy2 [] []
205 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
206 (map (fn (c as (s, T), x) =>
207 let val [T1, T2] = binder_types T
208 in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
209 Const ("nominal.perm", T) $ pi $ Free (x, T2))
211 (perm_names_types ~~ perm_indnames))))
212 (fn _ => EVERY [indtac induction perm_indnames 1,
213 ALLGOALS (asm_full_simp_tac
214 (simpset_of thy2 addsimps [perm_fun_def]))])),
215 length new_type_names));
217 (**** prove [] \<bullet> t = t ****)
219 val _ = warning "perm_empty_thms";
221 val perm_empty_thms = List.concat (map (fn a =>
222 let val permT = mk_permT (Type (a, []))
223 in map standard (List.take (split_conj_thm
224 (Goal.prove thy2 [] []
225 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
226 (map (fn ((s, T), x) => HOLogic.mk_eq
227 (Const (s, permT --> T --> T) $
228 Const ("List.list.Nil", permT) $ Free (x, T),
231 map body_type perm_types ~~ perm_indnames))))
232 (fn _ => EVERY [indtac induction perm_indnames 1,
233 ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
234 length new_type_names))
238 (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
240 val _ = warning "perm_append_thms";
242 (*FIXME: these should be looked up statically*)
243 val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
244 val pt2 = PureThy.get_thm thy2 (Name "pt2");
246 val perm_append_thms = List.concat (map (fn a =>
248 val permT = mk_permT (Type (a, []));
249 val pi1 = Free ("pi1", permT);
250 val pi2 = Free ("pi2", permT);
251 val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
252 val pt2' = pt_inst RS pt2;
253 val pt2_ax = PureThy.get_thm thy2
254 (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
255 in List.take (map standard (split_conj_thm
256 (Goal.prove thy2 [] []
257 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
258 (map (fn ((s, T), x) =>
259 let val perm = Const (s, permT --> T --> T)
261 (perm $ (Const ("List.op @", permT --> permT --> permT) $
262 pi1 $ pi2) $ Free (x, T),
263 perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
266 map body_type perm_types ~~ perm_indnames))))
267 (fn _ => EVERY [indtac induction perm_indnames 1,
268 ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
269 length new_type_names)
272 (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
274 val _ = warning "perm_eq_thms";
276 val pt3 = PureThy.get_thm thy2 (Name "pt3");
277 val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
279 val perm_eq_thms = List.concat (map (fn a =>
281 val permT = mk_permT (Type (a, []));
282 val pi1 = Free ("pi1", permT);
283 val pi2 = Free ("pi2", permT);
284 (*FIXME: not robust - better access these theorems using NominalData?*)
285 val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
286 val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
287 val pt3' = pt_inst RS pt3;
288 val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
289 val pt3_ax = PureThy.get_thm thy2
290 (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
291 in List.take (map standard (split_conj_thm
292 (Goal.prove thy2 [] [] (Logic.mk_implies
293 (HOLogic.mk_Trueprop (Const ("nominal.prm_eq",
294 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
295 HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
296 (map (fn ((s, T), x) =>
297 let val perm = Const (s, permT --> T --> T)
299 (perm $ pi1 $ Free (x, T),
300 perm $ pi2 $ Free (x, T))
303 map body_type perm_types ~~ perm_indnames)))))
304 (fn _ => EVERY [indtac induction perm_indnames 1,
305 ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
306 length new_type_names)
309 (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
311 val cp1 = PureThy.get_thm thy2 (Name "cp1");
312 val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
313 val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
314 val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
315 val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
317 fun composition_instance name1 name2 thy =
319 val name1' = Sign.base_name name1;
320 val name2' = Sign.base_name name2;
321 val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
322 val permT1 = mk_permT (Type (name1, []));
323 val permT2 = mk_permT (Type (name2, []));
324 val augment = map_type_tfree
325 (fn (x, S) => TFree (x, cp_class :: S));
326 val Ts = map (augment o body_type) perm_types;
327 val cp_inst = PureThy.get_thm thy
328 (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
329 val simps = simpset_of thy addsimps (perm_fun_def ::
330 (if name1 <> name2 then
331 let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
332 in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
335 val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
336 val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
338 [cp_inst RS cp1 RS sym,
339 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
340 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
342 val thms = split_conj_thm (standard (Goal.prove thy [] []
343 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
344 (map (fn ((s, T), x) =>
346 val pi1 = Free ("pi1", permT1);
347 val pi2 = Free ("pi2", permT2);
348 val perm1 = Const (s, permT1 --> T --> T);
349 val perm2 = Const (s, permT2 --> T --> T);
350 val perm3 = Const ("nominal.perm", permT1 --> permT2 --> permT2)
352 (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
353 perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
355 (perm_names ~~ Ts ~~ perm_indnames))))
356 (fn _ => EVERY [indtac induction perm_indnames 1,
357 ALLGOALS (asm_full_simp_tac simps)])))
359 foldl (fn ((s, tvs), thy) => AxClass.add_inst_arity_i
360 (s, replicate (length tvs) (cp_class :: classes), [cp_class])
361 (AxClass.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
362 thy (full_new_type_names' ~~ tyvars)
365 val (thy3, perm_thmss) = thy2 |>
366 fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
367 curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
368 AxClass.add_inst_arity_i (tyname, replicate (length args) classes, classes)
369 (AxClass.intro_classes_tac [] THEN REPEAT (EVERY
370 [resolve_tac perm_empty_thms 1,
371 resolve_tac perm_append_thms 1,
372 resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
373 (List.take (descr, length new_type_names)) |>
375 [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
376 unfolded_perm_eq_thms), [Simplifier.simp_add_global]),
377 ((space_implode "_" new_type_names ^ "_perm_empty",
378 perm_empty_thms), [Simplifier.simp_add_global]),
379 ((space_implode "_" new_type_names ^ "_perm_append",
380 perm_append_thms), [Simplifier.simp_add_global]),
381 ((space_implode "_" new_type_names ^ "_perm_eq",
382 perm_eq_thms), [Simplifier.simp_add_global])];
384 (**** Define representing sets ****)
386 val _ = warning "representing sets";
388 val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names
389 (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));
391 space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
392 (fn (i, ("nominal.nOption", _, _)) => NONE
393 | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
394 val _ = warning ("big_rep_name: " ^ big_rep_name);
396 fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
397 (case AList.lookup op = descr i of
398 SOME ("nominal.nOption", _, [(_, [dt']), _]) =>
399 apfst (cons dt) (strip_option dt')
401 | strip_option dt = ([], dt);
403 fun make_intr s T (cname, cargs) =
405 fun mk_prem (dt, (j, j', prems, ts)) =
407 val (dts, dt') = strip_option dt;
408 val (dts', dt'') = strip_dtyp dt';
409 val Ts = map (typ_of_dtyp descr sorts') dts;
410 val Us = map (typ_of_dtyp descr sorts') dts';
411 val T = typ_of_dtyp descr sorts' dt'';
412 val free = mk_Free "x" (Us ---> T) j;
413 val free' = app_bnds free (length Us);
414 fun mk_abs_fun (T, (i, t)) =
415 let val U = fastype_of t
416 in (i + 1, Const ("nominal.abs_fun", [T, U, T] --->
417 Type ("nominal.nOption", [U])) $ mk_Free "y" T i $ t)
419 in (j + 1, j' + length Ts,
421 DtRec k => list_all (map (pair "x") Us,
422 HOLogic.mk_Trueprop (HOLogic.mk_mem (free',
423 Const (List.nth (rep_set_names, k),
424 HOLogic.mk_setT T)))) :: prems
426 snd (foldr mk_abs_fun (j', free) Ts) :: ts)
429 val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
430 val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
431 (list_comb (Const (cname, map fastype_of ts ---> T), ts),
432 Const (s, HOLogic.mk_setT T)))
433 in Logic.list_implies (prems, concl)
436 val (intr_ts, ind_consts) =
437 apfst List.concat (ListPair.unzip (List.mapPartial
438 (fn ((_, ("nominal.nOption", _, _)), _) => NONE
439 | ((i, (_, _, constrs)), rep_set_name) =>
440 let val T = nth_dtyp i
441 in SOME (map (make_intr rep_set_name T) constrs,
442 Const (rep_set_name, HOLogic.mk_setT T))
444 (descr ~~ rep_set_names)));
446 val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
447 setmp InductivePackage.quiet_mode false
448 (InductivePackage.add_inductive_i false true big_rep_name false true false
449 ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;
451 (**** Prove that representing set is closed under permutation ****)
453 val _ = warning "proving closure under permutation...";
455 val perm_indnames' = List.mapPartial
456 (fn (x, (_, ("nominal.nOption", _, _))) => NONE | (x, _) => SOME x)
457 (perm_indnames ~~ descr);
459 fun mk_perm_closed name = map (fn th => standard (th RS mp))
460 (List.take (split_conj_thm (Goal.prove thy4 [] []
461 (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
464 val S = map_term_types (map_type_tfree
465 (fn (s, cs) => TFree (s, cs union cp_classes))) S;
466 val T = HOLogic.dest_setT (fastype_of S);
467 val permT = mk_permT (Type (name, []))
468 in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),
469 HOLogic.mk_mem (Const ("nominal.perm", permT --> T --> T) $
470 Free ("pi", permT) $ Free (x, T), S))
471 end) (ind_consts ~~ perm_indnames'))))
472 (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
473 [indtac rep_induct [] 1,
474 ALLGOALS (simp_tac (simpset_of thy4 addsimps
475 (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
476 ALLGOALS (resolve_tac rep_intrs
477 THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
478 length new_type_names));
480 (* FIXME: theorems are stored in database for testing only *)
481 val perm_closed_thmss = map mk_perm_closed atoms;
482 val (thy5, _) = PureThy.add_thmss [(("perm_closed",
483 List.concat perm_closed_thmss), [])] thy4;
487 val _ = warning "defining type...";
489 val (thy6, typedefs) =
490 foldl_map (fn (thy, ((((name, mx), tvs), c), name')) =>
491 setmp TypedefPackage.quiet_mode true
492 (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
494 QUIET_BREADTH_FIRST (has_fewer_prems 1)
495 (resolve_tac rep_intrs 1))) thy |> (fn (thy, r) =>
497 val permT = mk_permT (TFree (variant tvs "'a", HOLogic.typeS));
498 val pi = Free ("pi", permT);
499 val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
500 val T = Type (Sign.intern_type thy name, tvs');
501 val Const (_, Type (_, [U])) = c
502 in apsnd (pair r o hd)
503 (PureThy.add_defs_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
504 (Const ("nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
505 Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
506 (Const ("nominal.perm", permT --> U --> U) $ pi $
507 (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
508 Free ("x", T))))), [])] thy)
510 (thy5, types_syntax ~~ tyvars ~~
511 (List.take (ind_consts, length new_type_names)) ~~ new_type_names);
513 val perm_defs = map snd typedefs;
514 val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;
515 val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
516 val Rep_thms = map (#Rep o fst) typedefs;
518 val big_name = space_implode "_" new_type_names;
521 (** prove that new types are in class pt_<name> **)
523 val _ = warning "prove that new types are in class pt_<name> ...";
525 fun pt_instance ((class, atom), perm_closed_thms) =
526 fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},
527 perm_def), name), tvs), perm_closed) => fn thy =>
528 AxClass.add_inst_arity_i
529 (Sign.intern_type thy name,
530 replicate (length tvs) (classes @ cp_classes), [class])
531 (EVERY [AxClass.intro_classes_tac [],
532 rewrite_goals_tac [perm_def],
533 asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
534 asm_full_simp_tac (simpset_of thy addsimps
535 [Rep RS perm_closed RS Abs_inverse]) 1,
536 asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
537 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
538 (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);
541 (** prove that new types are in class cp_<name1>_<name2> **)
543 val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
545 fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
547 val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
548 val class = Sign.intern_class thy name;
549 val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
550 in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},
551 perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
552 AxClass.add_inst_arity_i
553 (Sign.intern_type thy name,
554 replicate (length tvs) (classes @ cp_classes), [class])
555 (EVERY [AxClass.intro_classes_tac [],
556 rewrite_goals_tac [perm_def],
557 asm_full_simp_tac (simpset_of thy addsimps
558 ((Rep RS perm_closed1 RS Abs_inverse) ::
559 (if atom1 = atom2 then []
560 else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
564 (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~
565 perm_closed_thms2) thy
568 val thy7 = fold (fn x => fn thy => thy |>
570 fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
571 (classes ~~ atoms ~~ perm_closed_thmss) thy6;
573 (**** constructors ****)
575 fun mk_abs_fun (x, t) =
577 val T = fastype_of x;
580 Const ("nominal.abs_fun", T --> U --> T -->
581 Type ("nominal.nOption", [U])) $ x $ t
584 val (ty_idxs, _) = foldl
585 (fn ((i, ("nominal.nOption", _, _)), p) => p
586 | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
588 fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
589 | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
592 fun strip_suffix i s = implode (List.take (explode s, size s - i));
594 (** strips the "_Rep" in type names *)
595 fun strip_nth_name i s =
596 let val xs = NameSpace.unpack s;
597 in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
599 val (descr'', ndescr) = ListPair.unzip (List.mapPartial
600 (fn (i, ("nominal.nOption", _, _)) => NONE
601 | (i, (s, dts, constrs)) =>
603 val SOME index = AList.lookup op = ty_idxs i;
604 val (constrs1, constrs2) = ListPair.unzip
605 (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 cname))
606 (foldl_map (fn (dts, dt) =>
607 let val (dts', dt') = strip_option dt
608 in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
609 ([], cargs))) constrs)
610 in SOME ((index, (strip_nth_name 1 s, map reindex dts, constrs1)),
614 val (descr1, descr2) = splitAt (length new_type_names, descr'');
615 val descr' = [descr1, descr2];
617 val typ_of_dtyp' = replace_types' o typ_of_dtyp descr sorts';
619 val rep_names = map (fn s =>
620 Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
621 val abs_names = map (fn s =>
622 Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
624 val recTs' = List.mapPartial
625 (fn ((_, ("nominal.nOption", _, _)), T) => NONE
626 | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');
627 val recTs = get_rec_types descr'' sorts';
628 val newTs' = Library.take (length new_type_names, recTs');
629 val newTs = Library.take (length new_type_names, recTs);
631 val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
633 fun make_constr_def tname T T' ((thy, defs, eqns), ((cname, cargs), (cname', mx))) =
635 fun constr_arg (dt, (j, l_args, r_args)) =
637 val x' = mk_Free "x" (typ_of_dtyp' dt) j;
638 val (dts, dt') = strip_option dt;
639 val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp' dt) i)
640 (dts ~~ (j upto j + length dts - 1))
641 val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts)
642 val (dts', dt'') = strip_dtyp dt'
644 DtRec k => if k < length new_type_names then
648 (list_abs (map (pair "z" o typ_of_dtyp') dts',
649 Const (List.nth (rep_names, k), typ_of_dtyp' dt'' -->
650 typ_of_dtyp descr sorts' dt'') $ app_bnds x (length dts')))
652 else error "nested recursion not (yet) supported"
653 | _ => (j + 1, x' :: l_args, x' :: r_args)
656 val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
657 val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
658 val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
659 val constrT = map fastype_of l_args ---> T;
660 val lhs = list_comb (Const (Sign.full_name thy (Sign.base_name cname),
662 val rhs = list_comb (Const (cname, map fastype_of r_args ---> T'), r_args);
663 val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
664 val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
665 (Const (rep_name, T --> T') $ lhs, rhs));
666 val def_name = (Sign.base_name cname) ^ "_def";
667 val (thy', [def_thm]) = thy |>
668 Theory.add_consts_i [(cname', constrT, mx)] |>
669 (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
670 in (thy', defs @ [def_thm], eqns @ [eqn]) end;
672 fun dt_constr_defs ((thy, defs, eqns, dist_lemmas),
673 (((((_, (_, _, constrs)), tname), T), T'), constr_syntax)) =
675 val rep_const = cterm_of thy
676 (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
677 val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
678 val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
679 ((Theory.add_path tname thy, defs, []), constrs ~~ constr_syntax)
681 (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
684 val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
685 ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
686 new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
688 val abs_inject_thms = map (fn tname =>
689 PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;
691 val rep_inject_thms = map (fn tname =>
692 PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;
694 val rep_thms = map (fn tname =>
695 PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;
697 val rep_inverse_thms = map (fn tname =>
698 PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;
700 (* prove theorem Rep_i (Constr_j ...) = Constr'_j ... *)
702 fun prove_constr_rep_thm eqn =
704 val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
705 val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms
706 in standard (Goal.prove thy8 [] [] eqn (fn _ => EVERY
707 [resolve_tac inj_thms 1,
708 rewrite_goals_tac rewrites,
710 resolve_tac rep_intrs 2,
711 REPEAT (resolve_tac rep_thms 1)]))
714 val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
716 (* prove theorem pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
718 fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
720 val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);
721 val Type ("fun", [T, U]) = fastype_of Rep;
722 val permT = mk_permT (Type (atom, []));
723 val pi = Free ("pi", permT);
725 standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
726 (Const ("nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
727 Rep $ (Const ("nominal.perm", permT --> T --> T) $ pi $ x))))
728 (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
729 perm_closed_thms @ Rep_thms)) 1))
732 val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
733 (atoms ~~ perm_closed_thmss));
735 (* prove distinctness theorems *)
737 val distinct_props = setmp DatatypeProp.dtK 1000
738 (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
740 val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
741 dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
742 (constr_rep_thmss ~~ dist_lemmas);
744 fun prove_distinct_thms (_, []) = []
745 | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
747 val dist_thm = standard (Goal.prove thy8 [] [] t (fn _ =>
748 simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1))
749 in dist_thm::(standard (dist_thm RS not_sym))::
750 (prove_distinct_thms (p, ts))
753 val distinct_thms = map prove_distinct_thms
754 (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
756 (** prove equations for permutation functions **)
758 val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
760 val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
761 let val T = replace_types' (nth_dtyp i)
762 in List.concat (map (fn (atom, perm_closed_thms) =>
763 map (fn ((cname, dts), constr_rep_thm) =>
765 val cname = Sign.intern_const thy8
766 (NameSpace.append tname (Sign.base_name cname));
767 val permT = mk_permT (Type (atom, []));
768 val pi = Free ("pi", permT);
771 let val T = fastype_of t
772 in Const ("nominal.perm", permT --> T --> T) $ pi $ t end;
774 fun constr_arg (dt, (j, l_args, r_args)) =
776 val x' = mk_Free "x" (typ_of_dtyp' dt) j;
777 val (dts, dt') = strip_option dt;
778 val Ts = map typ_of_dtyp' dts;
779 val xs = map (fn (T, i) => mk_Free "x" T i)
780 (Ts ~~ (j upto j + length dts - 1))
781 val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
782 val (dts', dt'') = strip_dtyp dt';
784 DtRec k => if k < length new_type_names then
787 map perm (xs @ [x]) @ r_args)
788 else error "nested recursion not (yet) supported"
789 | _ => (j + 1, x' :: l_args, perm x' :: r_args)
792 val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
793 val c = Const (cname, map fastype_of l_args ---> T)
795 standard (Goal.prove thy8 [] []
796 (HOLogic.mk_Trueprop (HOLogic.mk_eq
797 (perm (list_comb (c, l_args)), list_comb (c, r_args))))
799 [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
800 simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
801 constr_defs @ perm_closed_thms)) 1,
802 TRY (simp_tac (HOL_basic_ss addsimps
803 (symmetric perm_fun_def :: abs_perm)) 1),
804 TRY (simp_tac (HOL_basic_ss addsimps
805 (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
806 perm_closed_thms)) 1)]))
807 end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
808 end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
810 (** prove injectivity of constructors **)
812 val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
813 val alpha = PureThy.get_thms thy8 (Name "alpha");
814 val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
815 val fresh_def = PureThy.get_thm thy8 (Name "fresh_def");
816 val supp_def = PureThy.get_thm thy8 (Name "supp_def");
818 val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
819 let val T = replace_types' (nth_dtyp i)
820 in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
821 if null dts then NONE else SOME
823 val cname = Sign.intern_const thy8
824 (NameSpace.append tname (Sign.base_name cname));
826 fun make_inj (dt, (j, args1, args2, eqs)) =
828 val x' = mk_Free "x" (typ_of_dtyp' dt) j;
829 val y' = mk_Free "y" (typ_of_dtyp' dt) j;
830 val (dts, dt') = strip_option dt;
831 val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
832 val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
833 val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
834 val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
835 val y = mk_Free "y" (typ_of_dtyp' dt') (j + length dts);
836 val (dts', dt'') = strip_dtyp dt';
838 DtRec k => if k < length new_type_names then
840 xs @ (x :: args1), ys @ (y :: args2),
842 (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
843 else error "nested recursion not (yet) supported"
844 | _ => (j + 1, x' :: args1, y' :: args2, HOLogic.mk_eq (x', y') :: eqs)
847 val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
848 val Ts = map fastype_of args1;
849 val c = Const (cname, Ts ---> T)
851 standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
852 (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
853 foldr1 HOLogic.mk_conj eqs)))
855 [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
856 rep_inject_thms')) 1,
857 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
858 alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
859 perm_rep_perm_thms)) 1),
860 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
861 expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)]))
862 end) (constrs ~~ constr_rep_thms)
863 end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
865 (** equations for support and freshness **)
867 val Un_assoc = PureThy.get_thm thy8 (Name "Un_assoc");
868 val de_Morgan_conj = PureThy.get_thm thy8 (Name "de_Morgan_conj");
869 val Collect_disj_eq = PureThy.get_thm thy8 (Name "Collect_disj_eq");
870 val finite_Un = PureThy.get_thm thy8 (Name "finite_Un");
872 val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
873 (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
874 let val T = replace_types' (nth_dtyp i)
875 in List.concat (map (fn (cname, dts) => map (fn atom =>
877 val cname = Sign.intern_const thy8
878 (NameSpace.append tname (Sign.base_name cname));
879 val atomT = Type (atom, []);
881 fun process_constr (dt, (j, args1, args2)) =
883 val x' = mk_Free "x" (typ_of_dtyp' dt) j;
884 val (dts, dt') = strip_option dt;
885 val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
886 val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
887 val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
888 val (dts', dt'') = strip_dtyp dt';
890 DtRec k => if k < length new_type_names then
892 xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
893 else error "nested recursion not (yet) supported"
894 | _ => (j + 1, x' :: args1, x' :: args2)
897 val (_, args1, args2) = foldr process_constr (1, [], []) dts;
898 val Ts = map fastype_of args1;
899 val c = list_comb (Const (cname, Ts ---> T), args1);
901 Const ("nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
903 Const ("nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
904 Free ("a", atomT) $ t;
905 val supp_thm = standard (Goal.prove thy8 [] []
906 (HOLogic.mk_Trueprop (HOLogic.mk_eq
908 if null dts then Const ("{}", HOLogic.mk_setT atomT)
909 else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
911 simp_tac (HOL_basic_ss addsimps (supp_def ::
912 Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
913 symmetric empty_def :: Finites.emptyI :: simp_thms @
914 abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1))
917 standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
919 if null dts then HOLogic.true_const
920 else foldr1 HOLogic.mk_conj (map fresh args2))))
922 simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1)))
924 end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
926 (**** weak induction theorem ****)
928 val arities = get_arities descr'';
930 fun mk_funs_inv thm =
932 val {sign, prop, ...} = rep_thm thm;
933 val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $
934 (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
935 val used = add_term_tfree_names (a, []);
939 val Ts = map (TFree o rpair HOLogic.typeS)
940 (variantlist (replicate i "'t", used));
941 val f = Free ("f", Ts ---> U)
942 in standard (Goal.prove sign [] [] (Logic.mk_implies
943 (HOLogic.mk_Trueprop (HOLogic.list_all
944 (map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),
945 HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
946 r $ (a $ app_bnds f i)), f))))
947 (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1]))
949 in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
951 fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
953 val Rep_t = Const (List.nth (rep_names, i), T --> U) $
956 val Abs_t = Const (List.nth (abs_names, i), U --> T)
958 in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
959 Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $
960 (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
961 concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
964 val (indrule_lemma_prems, indrule_lemma_concls) =
965 Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
967 val indrule_lemma = standard (Goal.prove thy8 [] []
969 (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
970 HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
971 [REPEAT (etac conjE 1),
973 [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
974 etac mp 1, resolve_tac Rep_thms 1])]));
976 val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
977 val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
978 map (Free o apfst fst o dest_Var) Ps;
979 val indrule_lemma' = cterm_instantiate
980 (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
982 val Abs_inverse_thms' = List.concat (map mk_funs_inv Abs_inverse_thms);
984 val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
985 val dt_induct = standard (Goal.prove thy8 []
986 (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
988 [rtac indrule_lemma' 1,
989 (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
990 EVERY (map (fn (prem, r) => (EVERY
991 [REPEAT (eresolve_tac Abs_inverse_thms' 1),
992 simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
993 DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
994 (prems ~~ constr_defs))]));
996 val case_names_induct = mk_case_names_induct descr'';
998 (**** prove that new datatypes have finite support ****)
1000 val indnames = DatatypeProp.make_tnames recTs;
1002 val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
1003 val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
1005 val finite_supp_thms = map (fn atom =>
1006 let val atomT = Type (atom, [])
1007 in map standard (List.take
1008 (split_conj_thm (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop
1009 (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
1010 (Const ("nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
1011 Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
1012 (indnames ~~ recTs))))
1013 (fn _ => indtac dt_induct indnames 1 THEN
1014 ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
1015 (abs_supp @ supp_atm @
1016 PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
1017 List.concat supp_thms))))),
1018 length new_type_names))
1021 (**** strong induction theorem ****)
1023 val pnames = if length descr'' = 1 then ["P"]
1024 else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
1025 val ind_sort = map (fn T => Sign.intern_class thy8 ("fs_" ^
1026 Sign.base_name (fst (dest_Type T)))) dt_atomTs;
1027 val fsT = TFree ("'n", ind_sort);
1030 Free (List.nth (pnames, i), T --> fsT --> HOLogic.boolT);
1032 fun make_ind_prem k T ((cname, cargs), idxs) =
1034 val recs = List.filter is_rec_type cargs;
1035 val Ts = map (typ_of_dtyp descr'' sorts') cargs;
1036 val recTs' = map (typ_of_dtyp descr'' sorts') recs;
1037 val tnames = variantlist (DatatypeProp.make_tnames Ts, pnames);
1038 val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
1039 val frees = tnames ~~ Ts;
1040 val z = (variant tnames "z", fsT);
1042 fun mk_prem ((dt, s), T) =
1044 val (Us, U) = strip_type T;
1046 in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
1047 (make_pred (body_index dt) U $ app_bnds (Free (s, T)) l $ Bound l))
1050 val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
1051 val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
1052 (Const ("nominal.fresh", T --> fsT --> HOLogic.boolT) $
1054 (map (curry List.nth frees) (List.concat (map (fn (m, n) =>
1055 m upto m + n - 1) idxs)))
1057 in list_all_free (z :: frees, Logic.list_implies (prems' @ prems,
1058 HOLogic.mk_Trueprop (make_pred k T $
1059 list_comb (Const (cname, Ts ---> T), map Free frees) $ Free z)))
1062 val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
1063 map (make_ind_prem i T) (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
1064 val tnames = DatatypeProp.make_tnames recTs;
1065 val z = (variant tnames "z", fsT);
1066 val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
1067 (map (fn (((i, _), T), tname) => make_pred i T $ Free (tname, T) $ Free z)
1068 (descr'' ~~ recTs ~~ tnames)));
1069 val induct = Logic.list_implies (ind_prems, ind_concl);
1071 val simp_atts = replicate (length new_type_names) [Simplifier.simp_add_global];
1073 val (thy9, _) = thy8 |>
1074 Theory.add_path big_name |>
1075 PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] |>>
1076 Theory.parent_path |>>>
1077 DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms |>>>
1078 DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss |>>>
1079 DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' |>>>
1080 DatatypeAux.store_thmss "inject" new_type_names inject_thms |>>>
1081 DatatypeAux.store_thmss "supp" new_type_names supp_thms |>>>
1082 DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms |>>
1083 fold (fn (atom, ths) => fn thy =>
1084 let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
1085 in fold (fn T => AxClass.add_inst_arity_i
1087 replicate (length sorts) [class], [class])
1088 (AxClass.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
1089 end) (atoms ~~ finite_supp_thms) |>>
1090 Theory.add_path big_name |>>>
1091 PureThy.add_axioms_i [(("induct_unsafe", induct), [case_names_induct])] |>>
1098 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
1101 (* FIXME: The following stuff should be exported by DatatypePackage *)
1103 local structure P = OuterParse and K = OuterKeyword in
1106 Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
1107 (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
1109 fun mk_datatype args =
1111 val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
1112 val specs = map (fn ((((_, vs), t), mx), cons) =>
1113 (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
1114 in add_nominal_datatype false names specs end;
1116 val nominal_datatypeP =
1117 OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
1118 (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
1120 val _ = OuterSyntax.add_parsers [nominal_datatypeP];