164 fun clean_message s = if ! quick_and_dirty then () else message s; |
163 fun clean_message s = if ! quick_and_dirty then () else message s; |
165 |
164 |
166 fun coind_prefix true = "co" |
165 fun coind_prefix true = "co" |
167 | coind_prefix false = ""; |
166 | coind_prefix false = ""; |
168 |
167 |
169 |
168 fun log b m n = if m >= n then 0 else 1 + log b (b * m) n; |
170 (*the following code ensures that each recursive set always has the |
169 |
171 same type in all introduction rules*) |
170 fun make_bool_args f g [] i = [] |
172 fun unify_consts thy cs intr_ts = |
171 | make_bool_args f g (x :: xs) i = |
173 (let |
172 (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2); |
174 val add_term_consts_2 = fold_aterms (fn Const c => insert (op =) c | _ => I); |
173 |
175 fun varify (t, (i, ts)) = |
174 fun make_bool_args' xs = |
176 let val t' = map_types (Logic.incr_tvar (i + 1)) (#1 (Type.varify (t, []))) |
175 make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs; |
177 in (maxidx_of_term t', t'::ts) end; |
176 |
178 val (i, cs') = foldr varify (~1, []) cs; |
177 fun find_arg T x [] = sys_error "find_arg" |
179 val (i', intr_ts') = foldr varify (i, []) intr_ts; |
178 | find_arg T x ((p as (_, (SOME _, _))) :: ps) = |
180 val rec_consts = fold add_term_consts_2 cs' []; |
179 apsnd (cons p) (find_arg T x ps) |
181 val intr_consts = fold add_term_consts_2 intr_ts' []; |
180 | find_arg T x ((p as (U, (NONE, y))) :: ps) = |
182 fun unify (cname, cT) = |
181 if T = U then (y, (U, (SOME x, y)) :: ps) |
183 let val consts = map snd (List.filter (fn c => fst c = cname) intr_consts) |
182 else apsnd (cons p) (find_arg T x ps); |
184 in fold (Sign.typ_unify thy) ((replicate (length consts) cT) ~~ consts) end; |
183 |
185 val (env, _) = fold unify rec_consts (Vartab.empty, i'); |
184 fun make_args Ts xs = |
186 val subst = Type.freeze o map_types (Envir.norm_type env) |
185 map (fn (T, (NONE, ())) => Const ("arbitrary", T) | (_, (SOME t, ())) => t) |
187 |
186 (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts)); |
188 in (map subst cs', map subst intr_ts') |
187 |
189 end) handle Type.TUNIFY => |
188 fun make_args' Ts xs Us = |
190 (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts)); |
189 fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs)); |
191 |
190 |
192 |
191 fun dest_predicate cs params t = |
193 (*make injections used in mutually recursive definitions*) |
192 let |
194 fun mk_inj cs sumT c x = |
193 val k = length params; |
195 let |
194 val (c, ts) = strip_comb t; |
196 fun mk_inj' T n i = |
195 val (xs, ys) = chop k ts; |
197 if n = 1 then x else |
196 val i = find_index_eq c cs; |
198 let val n2 = n div 2; |
197 in |
199 val Type (_, [T1, T2]) = T |
198 if xs = params andalso i >= 0 then |
200 in |
199 SOME (c, i, ys, chop (length ys) |
201 if i <= n2 then |
200 (List.drop (binder_types (fastype_of c), k))) |
202 Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i) |
201 else NONE |
203 else |
|
204 Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2)) |
|
205 end |
|
206 in mk_inj' sumT (length cs) (1 + find_index_eq c cs) |
|
207 end; |
202 end; |
208 |
203 |
209 (*make "vimage" terms for selecting out components of mutually rec.def*) |
204 fun mk_names a 0 = [] |
210 fun mk_vimage cs sumT t c = if length cs < 2 then t else |
205 | mk_names a 1 = [a] |
211 let |
206 | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n); |
212 val cT = HOLogic.dest_setT (fastype_of c); |
207 |
213 val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT |
|
214 in |
|
215 Const (vimage_name, vimageT) $ |
|
216 Abs ("y", cT, mk_inj cs sumT c (Bound 0)) $ t |
|
217 end; |
|
218 |
|
219 (** proper splitting **) |
|
220 |
|
221 fun prod_factors p (Const ("Pair", _) $ t $ u) = |
|
222 p :: prod_factors (1::p) t @ prod_factors (2::p) u |
|
223 | prod_factors p _ = []; |
|
224 |
|
225 fun mg_prod_factors ts (t $ u) fs = if t mem ts then |
|
226 let val f = prod_factors [] u |
|
227 in AList.update (op =) (t, f inter (AList.lookup (op =) fs t) |> the_default f) fs end |
|
228 else mg_prod_factors ts u (mg_prod_factors ts t fs) |
|
229 | mg_prod_factors ts (Abs (_, _, t)) fs = mg_prod_factors ts t fs |
|
230 | mg_prod_factors ts _ fs = fs; |
|
231 |
|
232 fun prodT_factors p ps (T as Type ("*", [T1, T2])) = |
|
233 if p mem ps then prodT_factors (1::p) ps T1 @ prodT_factors (2::p) ps T2 |
|
234 else [T] |
|
235 | prodT_factors _ _ T = [T]; |
|
236 |
|
237 fun ap_split p ps (Type ("*", [T1, T2])) T3 u = |
|
238 if p mem ps then HOLogic.split_const (T1, T2, T3) $ |
|
239 Abs ("v", T1, ap_split (2::p) ps T2 T3 (ap_split (1::p) ps T1 |
|
240 (prodT_factors (2::p) ps T2 ---> T3) (incr_boundvars 1 u) $ Bound 0)) |
|
241 else u |
|
242 | ap_split _ _ _ _ u = u; |
|
243 |
|
244 fun mk_tuple p ps (Type ("*", [T1, T2])) (tms as t::_) = |
|
245 if p mem ps then HOLogic.mk_prod (mk_tuple (1::p) ps T1 tms, |
|
246 mk_tuple (2::p) ps T2 (Library.drop (length (prodT_factors (1::p) ps T1), tms))) |
|
247 else t |
|
248 | mk_tuple _ _ _ (t::_) = t; |
|
249 |
|
250 fun split_rule_var' ((t as Var (v, Type ("fun", [T1, T2])), ps), rl) = |
|
251 let val T' = prodT_factors [] ps T1 ---> T2 |
|
252 val newt = ap_split [] ps T1 T2 (Var (v, T')) |
|
253 val cterm = Thm.cterm_of (Thm.theory_of_thm rl) |
|
254 in |
|
255 instantiate ([], [(cterm t, cterm newt)]) rl |
|
256 end |
|
257 | split_rule_var' (_, rl) = rl; |
|
258 |
|
259 val remove_split = rewrite_rule [split_conv RS eq_reflection]; |
|
260 |
|
261 fun split_rule_vars vs rl = standard (remove_split (foldr split_rule_var' |
|
262 rl (mg_prod_factors vs (Thm.prop_of rl) []))); |
|
263 |
|
264 fun split_rule vs rl = standard (remove_split (foldr split_rule_var' |
|
265 rl (List.mapPartial (fn (t as Var ((a, _), _)) => |
|
266 Option.map (pair t) (AList.lookup (op =) vs a)) (term_vars (Thm.prop_of rl))))); |
|
267 |
208 |
268 |
209 |
269 (** process rules **) |
210 (** process rules **) |
270 |
211 |
271 local |
212 local |
276 |
217 |
277 fun err_in_prem thy name t p msg = |
218 fun err_in_prem thy name t p msg = |
278 error (cat_lines ["Ill-formed premise", Sign.string_of_term thy p, |
219 error (cat_lines ["Ill-formed premise", Sign.string_of_term thy p, |
279 "in introduction rule " ^ quote name, Sign.string_of_term thy t, msg]); |
220 "in introduction rule " ^ quote name, Sign.string_of_term thy t, msg]); |
280 |
221 |
281 val bad_concl = "Conclusion of introduction rule must have form \"t : S_i\""; |
222 val bad_concl = "Conclusion of introduction rule must be an inductive predicate"; |
282 |
223 |
283 val all_not_allowed = |
224 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate"; |
284 "Introduction rule must not have a leading \"!!\" quantifier"; |
225 |
|
226 val bad_app = "Inductive predicate must be applied to parameter(s) "; |
285 |
227 |
286 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize []; |
228 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize []; |
287 |
229 |
288 in |
230 in |
289 |
231 |
290 fun check_rule thy cs ((name, rule), att) = |
232 fun check_rule thy cs params ((name, att), rule) = |
291 let |
233 let |
292 val concl = Logic.strip_imp_concl rule; |
234 val params' = Term.variant_frees rule (Logic.strip_params rule); |
293 val prems = Logic.strip_imp_prems rule; |
235 val frees = rev (map Free params'); |
|
236 val concl = subst_bounds (frees, Logic.strip_assums_concl rule); |
|
237 val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule); |
294 val aprems = map (atomize_term thy) prems; |
238 val aprems = map (atomize_term thy) prems; |
295 val arule = Logic.list_implies (aprems, concl); |
239 val arule = list_all_free (params', Logic.list_implies (aprems, concl)); |
|
240 |
|
241 fun check_ind err t = case dest_predicate cs params t of |
|
242 NONE => err (bad_app ^ |
|
243 commas (map (Sign.string_of_term thy) params)) |
|
244 | SOME (_, _, ys, _) => |
|
245 if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs |
|
246 then err bad_ind_occ else (); |
|
247 |
|
248 fun check_prem' prem t = |
|
249 if head_of t mem cs then |
|
250 check_ind (err_in_prem thy name rule prem) t |
|
251 else (case t of |
|
252 Abs (_, _, t) => check_prem' prem t |
|
253 | t $ u => (check_prem' prem t; check_prem' prem u) |
|
254 | _ => ()); |
296 |
255 |
297 fun check_prem (prem, aprem) = |
256 fun check_prem (prem, aprem) = |
298 if can HOLogic.dest_Trueprop aprem then () |
257 if can HOLogic.dest_Trueprop aprem then check_prem' prem prem |
299 else err_in_prem thy name rule prem "Non-atomic premise"; |
258 else err_in_prem thy name rule prem "Non-atomic premise"; |
300 in |
259 in |
301 (case concl of |
260 (case concl of |
302 Const ("Trueprop", _) $ (Const ("op :", _) $ t $ u) => |
261 Const ("Trueprop", _) $ t => |
303 if u mem cs then |
262 if head_of t mem cs then |
304 if exists (Logic.occs o rpair t) cs then |
263 (check_ind (err_in_rule thy name rule) t; |
305 err_in_rule thy name rule "Recursion term on left of member symbol" |
264 List.app check_prem (prems ~~ aprems)) |
306 else List.app check_prem (prems ~~ aprems) |
265 else err_in_rule thy name rule bad_concl |
307 else err_in_rule thy name rule bad_concl |
266 | _ => err_in_rule thy name rule bad_concl); |
308 | Const ("all", _) $ _ => err_in_rule thy name rule all_not_allowed |
267 ((name, att), arule) |
309 | _ => err_in_rule thy name rule bad_concl); |
|
310 ((name, arule), att) |
|
311 end; |
268 end; |
312 |
269 |
313 val rulify = (* FIXME norm_hhf *) |
270 val rulify = (* FIXME norm_hhf *) |
314 hol_simplify inductive_conj |
271 hol_simplify inductive_conj |
315 #> hol_simplify inductive_rulify |
272 #> hol_simplify inductive_rulify |
316 #> hol_simplify inductive_rulify_fallback |
273 #> hol_simplify inductive_rulify_fallback |
317 #> standard; |
274 (*#> standard*); |
318 |
275 |
319 end; |
276 end; |
320 |
277 |
321 |
278 |
322 |
279 |
323 (** properties of (co)inductive sets **) |
280 (** properties of (co)inductive predicates **) |
324 |
|
325 (* elimination rules *) |
|
326 |
|
327 fun mk_elims cs cTs params intr_ts intr_names = |
|
328 let |
|
329 val used = foldr add_term_names [] intr_ts; |
|
330 val [aname, pname] = Name.variant_list used ["a", "P"]; |
|
331 val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT)); |
|
332 |
|
333 fun dest_intr r = |
|
334 let val Const ("op :", _) $ t $ u = |
|
335 HOLogic.dest_Trueprop (Logic.strip_imp_concl r) |
|
336 in (u, t, Logic.strip_imp_prems r) end; |
|
337 |
|
338 val intrs = map dest_intr intr_ts ~~ intr_names; |
|
339 |
|
340 fun mk_elim (c, T) = |
|
341 let |
|
342 val a = Free (aname, T); |
|
343 |
|
344 fun mk_elim_prem (_, t, ts) = |
|
345 list_all_free (map dest_Free ((foldr add_term_frees [] (t::ts)) \\ params), |
|
346 Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P)); |
|
347 val c_intrs = (List.filter (equal c o #1 o #1) intrs); |
|
348 in |
|
349 (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) :: |
|
350 map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs) |
|
351 end |
|
352 in |
|
353 map mk_elim (cs ~~ cTs) |
|
354 end; |
|
355 |
|
356 |
|
357 (* premises and conclusions of induction rules *) |
|
358 |
|
359 fun mk_indrule cs cTs params intr_ts = |
|
360 let |
|
361 val used = foldr add_term_names [] intr_ts; |
|
362 |
|
363 (* predicates for induction rule *) |
|
364 |
|
365 val preds = map Free (Name.variant_list used (if length cs < 2 then ["P"] else |
|
366 map (fn i => "P" ^ string_of_int i) (1 upto length cs)) ~~ |
|
367 map (fn T => T --> HOLogic.boolT) cTs); |
|
368 |
|
369 (* transform an introduction rule into a premise for induction rule *) |
|
370 |
|
371 fun mk_ind_prem r = |
|
372 let |
|
373 val frees = map dest_Free ((add_term_frees (r, [])) \\ params); |
|
374 |
|
375 val pred_of = AList.lookup (op aconv) (cs ~~ preds); |
|
376 |
|
377 fun subst (s as ((m as Const ("op :", T)) $ t $ u)) = |
|
378 (case pred_of u of |
|
379 NONE => (m $ fst (subst t) $ fst (subst u), NONE) |
|
380 | SOME P => (HOLogic.mk_binop inductive_conj_name (s, P $ t), SOME (s, P $ t))) |
|
381 | subst s = |
|
382 (case pred_of s of |
|
383 SOME P => (HOLogic.mk_binop "op Int" |
|
384 (s, HOLogic.Collect_const (HOLogic.dest_setT |
|
385 (fastype_of s)) $ P), NONE) |
|
386 | NONE => (case s of |
|
387 (t $ u) => (fst (subst t) $ fst (subst u), NONE) |
|
388 | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE) |
|
389 | _ => (s, NONE))); |
|
390 |
|
391 fun mk_prem (s, prems) = (case subst s of |
|
392 (_, SOME (t, u)) => t :: u :: prems |
|
393 | (t, _) => t :: prems); |
|
394 |
|
395 val Const ("op :", _) $ t $ u = |
|
396 HOLogic.dest_Trueprop (Logic.strip_imp_concl r) |
|
397 |
|
398 in list_all_free (frees, |
|
399 Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem |
|
400 [] (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r))), |
|
401 HOLogic.mk_Trueprop (valOf (pred_of u) $ t))) |
|
402 end; |
|
403 |
|
404 val ind_prems = map mk_ind_prem intr_ts; |
|
405 |
|
406 val factors = Library.fold (mg_prod_factors preds) ind_prems []; |
|
407 |
|
408 (* make conclusions for induction rules *) |
|
409 |
|
410 fun mk_ind_concl ((c, P), (ts, x)) = |
|
411 let val T = HOLogic.dest_setT (fastype_of c); |
|
412 val ps = AList.lookup (op =) factors P |> the_default []; |
|
413 val Ts = prodT_factors [] ps T; |
|
414 val (frees, x') = foldr (fn (T', (fs, s)) => |
|
415 ((Free (s, T'))::fs, Symbol.bump_string s)) ([], x) Ts; |
|
416 val tuple = mk_tuple [] ps T frees; |
|
417 in ((HOLogic.mk_binop "op -->" |
|
418 (HOLogic.mk_mem (tuple, c), P $ tuple))::ts, x') |
|
419 end; |
|
420 |
|
421 val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj |
|
422 (fst (foldr mk_ind_concl ([], "xa") (cs ~~ preds)))) |
|
423 |
|
424 in (preds, ind_prems, mutual_ind_concl, |
|
425 map (apfst (fst o dest_Free)) factors) |
|
426 end; |
|
427 |
|
428 |
281 |
429 (* prepare cases and induct rules *) |
282 (* prepare cases and induct rules *) |
430 |
283 |
431 fun add_cases_induct no_elim no_induct coind names elims induct = |
284 fun add_cases_induct no_elim no_induct coind rec_name names elims induct = |
432 let |
285 let |
433 fun cases_spec name elim thy = |
286 fun cases_spec name elim = |
434 thy |
287 LocalTheory.note ((NameSpace.append (Sign.base_name name) "cases", |
435 |> Theory.parent_path |
288 [Attrib.internal (InductAttrib.cases_set name)]), [elim]) #> snd; |
436 |> Theory.add_path (Sign.base_name name) |
|
437 |> PureThy.add_thms [(("cases", elim), [InductAttrib.cases_set name])] |> snd |
|
438 |> Theory.restore_naming thy; |
|
439 val cases_specs = if no_elim then [] else map2 cases_spec names elims; |
289 val cases_specs = if no_elim then [] else map2 cases_spec names elims; |
440 |
290 |
441 val induct_att = if coind then InductAttrib.coinduct_set else InductAttrib.induct_set; |
291 val induct_att = if coind then InductAttrib.coinduct_set else InductAttrib.induct_set; |
442 fun induct_specs thy = |
292 fun induct_specs ctxt = |
443 if no_induct then thy |
293 if no_induct then ctxt |
444 else |
294 else |
445 let |
295 let |
446 val ctxt = ProofContext.init thy; |
|
447 val rules = names ~~ ProjectRule.projects ctxt (1 upto length names) induct; |
296 val rules = names ~~ ProjectRule.projects ctxt (1 upto length names) induct; |
448 val inducts = map (RuleCases.save induct o standard o #2) rules; |
297 val inducts = map (RuleCases.save induct o standard o #2) rules; |
449 in |
298 in |
450 thy |
299 ctxt |> |
451 |> PureThy.add_thms (rules |> map (fn (name, th) => |
300 LocalTheory.notes (rules |> map (fn (name, th) => |
452 (("", th), [RuleCases.consumes 1, induct_att name]))) |> snd |
301 (("", [Attrib.internal (RuleCases.consumes 1), |
453 |> PureThy.add_thmss |
302 Attrib.internal (induct_att name)]), [([th], [])]))) |> snd |> |
454 [((coind_prefix coind ^ "inducts", inducts), [RuleCases.consumes 1])] |> snd |
303 LocalTheory.note ((NameSpace.append rec_name |
|
304 (coind_prefix coind ^ "inducts"), |
|
305 [Attrib.internal (RuleCases.consumes 1)]), inducts) |> snd |
455 end; |
306 end; |
456 in Library.apply cases_specs #> induct_specs end; |
307 in Library.apply cases_specs #> induct_specs end; |
457 |
308 |
458 |
309 |
459 |
310 |
460 (** proofs for (co)inductive sets **) |
311 (** proofs for (co)inductive predicates **) |
461 |
312 |
462 (* prove monotonicity -- NOT subject to quick_and_dirty! *) |
313 (* prove monotonicity -- NOT subject to quick_and_dirty! *) |
463 |
314 |
464 fun prove_mono setT fp_fun monos thy = |
315 fun prove_mono predT fp_fun monos ctxt = |
465 (message " Proving monotonicity ..."; |
316 (message " Proving monotonicity ..."; |
466 Goal.prove_global thy [] [] (*NO quick_and_dirty here!*) |
317 Goal.prove ctxt [] [] (*NO quick_and_dirty here!*) |
467 (HOLogic.mk_Trueprop |
318 (HOLogic.mk_Trueprop |
468 (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)) |
319 (Const (mono_name, (predT --> predT) --> HOLogic.boolT) $ fp_fun)) |
469 (fn _ => EVERY [rtac monoI 1, |
320 (fn _ => EVERY [rtac monoI 1, |
470 REPEAT (ares_tac (List.concat (map mk_mono monos) @ get_monos thy) 1)])); |
321 REPEAT (resolve_tac [le_funI, le_boolI'] 1), |
|
322 REPEAT (FIRST |
|
323 [atac 1, |
|
324 resolve_tac (List.concat (map mk_mono monos) @ |
|
325 get_monos (Context.Proof ctxt)) 1, |
|
326 etac le_funE 1, dtac le_boolD 1])])); |
471 |
327 |
472 |
328 |
473 (* prove introduction rules *) |
329 (* prove introduction rules *) |
474 |
330 |
475 fun prove_intrs coind mono fp_def intr_ts rec_sets_defs ctxt = |
331 fun prove_intrs coind mono fp_def k intr_ts rec_preds_defs ctxt = |
476 let |
332 let |
477 val _ = clean_message " Proving the introduction rules ..."; |
333 val _ = clean_message " Proving the introduction rules ..."; |
478 |
334 |
479 val unfold = standard' (mono RS (fp_def RS |
335 val unfold = funpow k (fn th => th RS fun_cong) |
480 (if coind then def_gfp_unfold else def_lfp_unfold))); |
336 (mono RS (fp_def RS |
|
337 (if coind then def_gfp_unfold else def_lfp_unfold))); |
481 |
338 |
482 fun select_disj 1 1 = [] |
339 fun select_disj 1 1 = [] |
483 | select_disj _ 1 = [rtac disjI1] |
340 | select_disj _ 1 = [rtac disjI1] |
484 | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1)); |
341 | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1)); |
485 |
342 |
486 val intrs = (1 upto (length intr_ts) ~~ intr_ts) |> map (fn (i, intr) => |
343 val rules = [refl, TrueI, notFalseI, exI, conjI]; |
|
344 |
|
345 val intrs = map_index (fn (i, intr) => |
487 rulify (SkipProof.prove ctxt [] [] intr (fn _ => EVERY |
346 rulify (SkipProof.prove ctxt [] [] intr (fn _ => EVERY |
488 [rewrite_goals_tac rec_sets_defs, |
347 [rewrite_goals_tac rec_preds_defs, |
489 stac unfold 1, |
348 rtac (unfold RS iffD2) 1, |
490 REPEAT (resolve_tac [vimageI2, CollectI] 1), |
349 EVERY1 (select_disj (length intr_ts) (i + 1)), |
491 (*Now 1-2 subgoals: the disjunction, perhaps equality.*) |
|
492 EVERY1 (select_disj (length intr_ts) i), |
|
493 (*Not ares_tac, since refl must be tried before any equality assumptions; |
350 (*Not ares_tac, since refl must be tried before any equality assumptions; |
494 backtracking may occur if the premises have extra variables!*) |
351 backtracking may occur if the premises have extra variables!*) |
495 DEPTH_SOLVE_1 (resolve_tac [refl, exI, conjI] 1 APPEND assume_tac 1), |
352 DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts |
496 (*Now solve the equations like Inl 0 = Inl ?b2*) |
|
497 REPEAT (rtac refl 1)]))) |
|
498 |
353 |
499 in (intrs, unfold) end; |
354 in (intrs, unfold) end; |
500 |
355 |
501 |
356 |
502 (* prove elimination rules *) |
357 (* prove elimination rules *) |
503 |
358 |
504 fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs ctxt = |
359 fun prove_elims cs params intr_ts intr_names unfold rec_preds_defs ctxt = |
505 let |
360 let |
506 val _ = clean_message " Proving the elimination rules ..."; |
361 val _ = clean_message " Proving the elimination rules ..."; |
507 |
362 |
508 val rules1 = [CollectE, disjE, make_elim vimageD, exE, FalseE]; |
363 val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt; |
509 val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @ map make_elim [Inl_inject, Inr_inject]; |
364 val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT)); |
510 in |
365 |
511 mk_elims cs cTs params intr_ts intr_names |> map (fn (t, cases) => |
366 fun dest_intr r = |
512 SkipProof.prove ctxt [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t) |
367 (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))), |
513 (fn {prems, ...} => EVERY |
368 Logic.strip_assums_hyp r, Logic.strip_params r); |
514 [cut_facts_tac [hd prems] 1, |
369 |
515 rewrite_goals_tac rec_sets_defs, |
370 val intrs = map dest_intr intr_ts ~~ intr_names; |
516 dtac (unfold RS subst) 1, |
371 |
517 REPEAT (FIRSTGOAL (eresolve_tac rules1)), |
372 val rules1 = [disjE, exE, FalseE]; |
518 REPEAT (FIRSTGOAL (eresolve_tac rules2)), |
373 val rules2 = [conjE, FalseE, notTrueE]; |
519 EVERY (map (fn prem => |
374 |
520 DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_sets_defs prem, conjI] 1)) (tl prems))]) |
375 fun prove_elim c = |
521 |> rulify |
376 let |
522 |> RuleCases.name cases) |
377 val Ts = List.drop (binder_types (fastype_of c), length params); |
523 end; |
378 val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt'; |
|
379 val frees = map Free (anames ~~ Ts); |
|
380 |
|
381 fun mk_elim_prem ((_, _, us, _), ts, params') = |
|
382 list_all (params', |
|
383 Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq) |
|
384 (frees ~~ us) @ ts, P)); |
|
385 val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs); |
|
386 val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) :: |
|
387 map mk_elim_prem (map #1 c_intrs) |
|
388 in |
|
389 SkipProof.prove ctxt'' [] prems P |
|
390 (fn {prems, ...} => EVERY |
|
391 [cut_facts_tac [hd prems] 1, |
|
392 rewrite_goals_tac rec_preds_defs, |
|
393 dtac (unfold RS iffD1) 1, |
|
394 REPEAT (FIRSTGOAL (eresolve_tac rules1)), |
|
395 REPEAT (FIRSTGOAL (eresolve_tac rules2)), |
|
396 EVERY (map (fn prem => |
|
397 DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))]) |
|
398 |> rulify |
|
399 |> singleton (ProofContext.export ctxt'' ctxt) |
|
400 |> RuleCases.name (map #2 c_intrs) |
|
401 end |
|
402 |
|
403 in map prove_elim cs end; |
524 |
404 |
525 |
405 |
526 (* derivation of simplified elimination rules *) |
406 (* derivation of simplified elimination rules *) |
527 |
407 |
528 local |
408 local |
529 |
409 |
530 (*cprop should have the form t:Si where Si is an inductive set*) |
410 (*cprop should have the form "Si t" where Si is an inductive predicate*) |
531 val mk_cases_err = "mk_cases: proposition not of form \"t : S_i\""; |
411 val mk_cases_err = "mk_cases: proposition not an inductive predicate"; |
532 |
412 |
533 (*delete needless equality assumptions*) |
413 (*delete needless equality assumptions*) |
534 val refl_thin = prove_goal HOL.thy "!!P. a = a ==> P ==> P" (fn _ => [assume_tac 1]); |
414 val refl_thin = prove_goal HOL.thy "!!P. a = a ==> P ==> P" (fn _ => [assume_tac 1]); |
535 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject]; |
415 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE]; |
536 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls; |
416 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls; |
537 |
417 |
538 fun simp_case_tac solved ss i = |
418 fun simp_case_tac solved ss i = |
539 EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i |
419 EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i |
540 THEN_MAYBE (if solved then no_tac else all_tac); |
420 THEN_MAYBE (if solved then no_tac else all_tac); |
586 fun mk_cases_meth (ctxt, raw_props) = |
466 fun mk_cases_meth (ctxt, raw_props) = |
587 let |
467 let |
588 val thy = ProofContext.theory_of ctxt; |
468 val thy = ProofContext.theory_of ctxt; |
589 val ss = local_simpset_of ctxt; |
469 val ss = local_simpset_of ctxt; |
590 val cprops = map (Thm.cterm_of thy o ProofContext.read_prop ctxt) raw_props; |
470 val cprops = map (Thm.cterm_of thy o ProofContext.read_prop ctxt) raw_props; |
591 in Method.erule 0 (map (smart_mk_cases thy ss) cprops) end; |
471 in Method.erule 0 (map (smart_mk_cases (Context.Theory thy) ss) cprops) end; |
592 |
472 |
593 val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name)); |
473 val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name)); |
594 |
474 |
595 |
475 |
596 (* prove induction rule *) |
476 (* prove induction rule *) |
597 |
477 |
598 fun prove_indrule cs cTs sumT rec_const params intr_ts mono |
478 fun prove_indrule cs argTs bs xs rec_const params intr_ts mono |
599 fp_def rec_sets_defs ctxt = |
479 fp_def rec_preds_defs ctxt = |
600 let |
480 let |
601 val _ = clean_message " Proving the induction rule ..."; |
481 val _ = clean_message " Proving the induction rule ..."; |
602 val thy = ProofContext.theory_of ctxt; |
482 val thy = ProofContext.theory_of ctxt; |
603 |
483 |
604 val sum_case_rewrites = |
484 (* predicates for induction rule *) |
605 (if Context.theory_name thy = "Datatype" then |
485 |
606 PureThy.get_thms thy (Name "sum.cases") |
486 val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt; |
607 else |
487 val preds = map Free (pnames ~~ |
608 (case ThyInfo.lookup_theory "Datatype" of |
488 map (fn c => List.drop (binder_types (fastype_of c), length params) ---> |
609 NONE => [] |
489 HOLogic.boolT) cs); |
610 | SOME thy' => |
490 |
611 if Theory.subthy (thy', thy) then |
491 (* transform an introduction rule into a premise for induction rule *) |
612 PureThy.get_thms thy' (Name "sum.cases") |
492 |
613 else [])) |
493 fun mk_ind_prem r = |
614 |> map mk_meta_eq; |
494 let |
615 |
495 fun subst s = (case dest_predicate cs params s of |
616 val (preds, ind_prems, mutual_ind_concl, factors) = |
496 SOME (_, i, ys, (_, Ts)) => |
617 mk_indrule cs cTs params intr_ts; |
497 let |
|
498 val k = length Ts; |
|
499 val bs = map Bound (k - 1 downto 0); |
|
500 val P = list_comb (List.nth (preds, i), ys @ bs); |
|
501 val Q = list_abs (mk_names "x" k ~~ Ts, |
|
502 HOLogic.mk_binop inductive_conj_name (list_comb (s, bs), P)) |
|
503 in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end |
|
504 | NONE => (case s of |
|
505 (t $ u) => (fst (subst t) $ fst (subst u), NONE) |
|
506 | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE) |
|
507 | _ => (s, NONE))); |
|
508 |
|
509 fun mk_prem (s, prems) = (case subst s of |
|
510 (_, SOME (t, u)) => t :: u :: prems |
|
511 | (t, _) => t :: prems); |
|
512 |
|
513 val SOME (_, i, ys, _) = dest_predicate cs params |
|
514 (HOLogic.dest_Trueprop (Logic.strip_assums_concl r)) |
|
515 |
|
516 in list_all_free (Logic.strip_params r, |
|
517 Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem |
|
518 [] (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r))), |
|
519 HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys)))) |
|
520 end; |
|
521 |
|
522 val ind_prems = map mk_ind_prem intr_ts; |
|
523 |
|
524 (* make conclusions for induction rules *) |
|
525 |
|
526 val Tss = map (binder_types o fastype_of) preds; |
|
527 val (xnames, ctxt'') = |
|
528 Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt'; |
|
529 val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj |
|
530 (map (fn (((xnames, Ts), c), P) => |
|
531 let val frees = map Free (xnames ~~ Ts) |
|
532 in HOLogic.mk_imp |
|
533 (list_comb (c, params @ frees), list_comb (P, frees)) |
|
534 end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds))); |
618 |
535 |
619 val dummy = if !trace then |
536 val dummy = if !trace then |
620 (writeln "ind_prems = "; |
537 (writeln "ind_prems = "; |
621 List.app (writeln o Sign.string_of_term thy) ind_prems) |
538 List.app (writeln o Sign.string_of_term thy) ind_prems) |
622 else (); |
539 else (); |
623 |
540 |
624 (* make predicate for instantiation of abstract induction rule *) |
541 (* make predicate for instantiation of abstract induction rule *) |
625 |
542 |
626 fun mk_ind_pred _ [P] = P |
543 val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj |
627 | mk_ind_pred T Ps = |
544 (map_index (fn (i, P) => foldr HOLogic.mk_imp |
628 let val n = (length Ps) div 2; |
545 (list_comb (P, make_args' argTs xs (binder_types (fastype_of P)))) |
629 val Type (_, [T1, T2]) = T |
546 (make_bool_args HOLogic.mk_not I bs i)) preds)); |
630 in Const ("Datatype.sum.sum_case", |
|
631 [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $ |
|
632 mk_ind_pred T1 (Library.take (n, Ps)) $ mk_ind_pred T2 (Library.drop (n, Ps)) |
|
633 end; |
|
634 |
|
635 val ind_pred = mk_ind_pred sumT preds; |
|
636 |
547 |
637 val ind_concl = HOLogic.mk_Trueprop |
548 val ind_concl = HOLogic.mk_Trueprop |
638 (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->" |
549 (HOLogic.mk_binrel "Orderings.less_eq" (rec_const, ind_pred)); |
639 (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0))); |
|
640 |
|
641 (* simplification rules for vimage and Collect *) |
|
642 |
|
643 val vimage_simps = if length cs < 2 then [] else |
|
644 map (fn c => standard (SkipProof.prove ctxt [] [] |
|
645 (HOLogic.mk_Trueprop (HOLogic.mk_eq |
|
646 (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c, |
|
647 HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $ |
|
648 List.nth (preds, find_index_eq c cs)))) |
|
649 (fn _ => EVERY |
|
650 [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites, rtac refl 1]))) cs; |
|
651 |
550 |
652 val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct)); |
551 val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct)); |
653 |
552 |
654 val dummy = if !trace then |
553 val dummy = if !trace then |
655 (writeln "raw_fp_induct = "; print_thm raw_fp_induct) |
554 (writeln "raw_fp_induct = "; print_thm raw_fp_induct) |
656 else (); |
555 else (); |
657 |
556 |
658 val induct = standard (SkipProof.prove ctxt [] ind_prems ind_concl |
557 val induct = SkipProof.prove ctxt'' [] ind_prems ind_concl |
659 (fn {prems, ...} => EVERY |
558 (fn {prems, ...} => EVERY |
660 [rewrite_goals_tac [inductive_conj_def], |
559 [rewrite_goals_tac [inductive_conj_def], |
661 rtac (impI RS allI) 1, |
560 DETERM (rtac raw_fp_induct 1), |
662 DETERM (etac raw_fp_induct 1), |
561 REPEAT (resolve_tac [le_funI, le_boolI] 1), |
663 rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)), |
562 rewrite_goals_tac (map mk_meta_eq [meet_fun_eq, meet_bool_eq] @ simp_thms'), |
664 fold_goals_tac rec_sets_defs, |
563 (*This disjE separates out the introduction rules*) |
665 (*This CollectE and disjE separates out the introduction rules*) |
564 REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])), |
666 REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE, FalseE])), |
|
667 (*Now break down the individual cases. No disjE here in case |
565 (*Now break down the individual cases. No disjE here in case |
668 some premise involves disjunction.*) |
566 some premise involves disjunction.*) |
669 REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)), |
567 REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)), |
670 rewrite_goals_tac sum_case_rewrites, |
568 REPEAT (FIRSTGOAL |
671 EVERY (map (fn prem => |
569 (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))), |
672 DEPTH_SOLVE_1 (ares_tac [rewrite_rule [inductive_conj_def] prem, conjI, refl] 1)) prems)])); |
570 EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule |
673 |
571 (inductive_conj_def :: rec_preds_defs) prem, conjI, refl] 1)) prems)]); |
674 val lemma = standard (SkipProof.prove ctxt [] [] |
572 |
|
573 val lemma = SkipProof.prove ctxt'' [] [] |
675 (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY |
574 (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY |
676 [rewrite_goals_tac rec_sets_defs, |
575 [rewrite_goals_tac rec_preds_defs, |
677 REPEAT (EVERY |
576 REPEAT (EVERY |
678 [REPEAT (resolve_tac [conjI, impI] 1), |
577 [REPEAT (resolve_tac [conjI, impI] 1), |
679 TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1, |
578 REPEAT (eresolve_tac [le_funE, le_boolE] 1), |
680 rewrite_goals_tac sum_case_rewrites, |
579 atac 1, |
681 atac 1])])) |
580 rewrite_goals_tac simp_thms', |
682 |
581 atac 1])]) |
683 in standard (split_rule factors (induct RS lemma)) end; |
582 |
684 |
583 in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end; |
685 |
584 |
686 |
585 |
687 (** specification of (co)inductive sets **) |
586 |
688 |
587 (** specification of (co)inductive predicates **) |
689 fun cond_declare_consts declare_consts cs paramTs cnames = |
588 |
690 if declare_consts then |
589 fun mk_ind_def alt_name coind cs intr_ts monos |
691 Theory.add_consts_i (map (fn (c, n) => (Sign.base_name n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames)) |
590 params cnames_syn ctxt = |
692 else I; |
591 let |
693 |
|
694 fun mk_ind_def declare_consts alt_name coind cs intr_ts monos thy |
|
695 params paramTs cTs cnames = |
|
696 let |
|
697 val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs; |
|
698 val setT = HOLogic.mk_setT sumT; |
|
699 |
|
700 val fp_name = if coind then gfp_name else lfp_name; |
592 val fp_name = if coind then gfp_name else lfp_name; |
701 |
593 |
702 val used = foldr add_term_names [] intr_ts; |
594 val argTs = fold (fn c => fn Ts => Ts @ |
703 val [sname, xname] = Name.variant_list used ["S", "x"]; |
595 (List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs []; |
|
596 val k = log 2 1 (length cs); |
|
597 val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT; |
|
598 val p :: xs = map Free (Variable.variant_frees ctxt intr_ts |
|
599 (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs))); |
|
600 val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts) |
|
601 (map (rpair HOLogic.boolT) (mk_names "b" k))); |
|
602 |
|
603 fun subst t = (case dest_predicate cs params t of |
|
604 SOME (_, i, ts, (Ts, Us)) => |
|
605 let val zs = map Bound (length Us - 1 downto 0) |
|
606 in |
|
607 list_abs (map (pair "z") Us, list_comb (p, |
|
608 make_bool_args' bs i @ make_args argTs ((ts ~~ Ts) @ (zs ~~ Us)))) |
|
609 end |
|
610 | NONE => (case t of |
|
611 t1 $ t2 => subst t1 $ subst t2 |
|
612 | Abs (x, T, u) => Abs (x, T, subst u) |
|
613 | _ => t)); |
704 |
614 |
705 (* transform an introduction rule into a conjunction *) |
615 (* transform an introduction rule into a conjunction *) |
706 (* [| t : ... S_i ... ; ... |] ==> u : S_j *) |
616 (* [| p_i t; ... |] ==> p_j u *) |
707 (* is transformed into *) |
617 (* is transformed into *) |
708 (* x = Inj_j u & t : ... Inj_i -`` S ... & ... *) |
618 (* b_j & x_j = u & p b_j t & ... *) |
709 |
619 |
710 fun transform_rule r = |
620 fun transform_rule r = |
711 let |
621 let |
712 val frees = map dest_Free ((add_term_frees (r, [])) \\ params); |
622 val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params |
713 val subst = subst_free |
623 (HOLogic.dest_Trueprop (Logic.strip_assums_concl r)) |
714 (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs)); |
624 |
715 val Const ("op :", _) $ t $ u = |
625 in foldr (fn ((x, T), P) => HOLogic.exists_const T $ (Abs (x, T, P))) |
716 HOLogic.dest_Trueprop (Logic.strip_imp_concl r) |
|
717 |
|
718 in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P)) |
|
719 (foldr1 HOLogic.mk_conj |
626 (foldr1 HOLogic.mk_conj |
720 (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t)):: |
627 (make_bool_args HOLogic.mk_not I bs i @ |
721 (map (subst o HOLogic.dest_Trueprop) |
628 map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @ |
722 (Logic.strip_imp_prems r)))) frees |
629 map (subst o HOLogic.dest_Trueprop) |
|
630 (Logic.strip_assums_hyp r))) (Logic.strip_params r) |
723 end |
631 end |
724 |
632 |
725 (* make a disjunction of all introduction rules *) |
633 (* make a disjunction of all introduction rules *) |
726 |
634 |
727 val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $ |
635 val fp_fun = fold_rev lambda (p :: bs @ xs) |
728 absfree (xname, sumT, if null intr_ts then HOLogic.false_const |
636 (if null intr_ts then HOLogic.false_const |
729 else foldr1 HOLogic.mk_disj (map transform_rule intr_ts))); |
637 else foldr1 HOLogic.mk_disj (map transform_rule intr_ts)); |
730 |
638 |
731 (* add definiton of recursive sets to theory *) |
639 (* add definiton of recursive predicates to theory *) |
732 |
640 |
733 val rec_name = if alt_name = "" then |
641 val rec_name = if alt_name = "" then |
734 space_implode "_" (map Sign.base_name cnames) else alt_name; |
642 space_implode "_" (map fst cnames_syn) else alt_name; |
735 val full_rec_name = if length cs < 2 then hd cnames |
643 |
736 else Sign.full_name thy rec_name; |
644 val ((rec_const, (_, fp_def)), ctxt') = ctxt |> |
737 |
645 Variable.add_fixes (map (fst o dest_Free) params) |> snd |> |
738 val rec_const = list_comb |
646 fold Variable.declare_term intr_ts |> |
739 (Const (full_rec_name, paramTs ---> setT), params); |
647 LocalTheory.def |
740 |
648 ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn), |
741 val fp_def_term = Logic.mk_equals (rec_const, |
649 (("", []), fold_rev lambda params |
742 Const (fp_name, (setT --> setT) --> setT) $ fp_fun); |
650 (Const (fp_name, (predT --> predT) --> predT) $ fp_fun))); |
743 |
651 val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def]) |
744 val def_terms = fp_def_term :: (if length cs < 2 then [] else |
652 (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params))); |
745 map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs); |
653 val specs = if length cs < 2 then [] else |
746 |
654 map_index (fn (i, (name_mx, c)) => |
747 val ([fp_def :: rec_sets_defs], thy') = |
655 let |
748 thy |
656 val Ts = List.drop (binder_types (fastype_of c), length params); |
749 |> cond_declare_consts declare_consts cs paramTs cnames |
657 val xs = map Free (Variable.variant_frees ctxt intr_ts |
750 |> (if length cs < 2 then I |
658 (mk_names "x" (length Ts) ~~ Ts)) |
751 else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)]) |
659 in |
752 |> Theory.add_path rec_name |
660 (name_mx, (("", []), fold_rev lambda (params @ xs) |
753 |> PureThy.add_defss_i false [(("defs", def_terms), [])]; |
661 (list_comb (rec_const, params @ make_bool_args' bs i @ |
754 |
662 make_args argTs (xs ~~ Ts))))) |
755 val mono = prove_mono setT fp_fun monos thy' |
663 end) (cnames_syn ~~ cs); |
756 |
664 val (consts_defs, ctxt'') = fold_map LocalTheory.def specs ctxt'; |
757 in (thy', rec_name, mono, fp_def, rec_sets_defs, rec_const, sumT) end; |
665 val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs); |
758 |
666 |
759 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs |
667 val mono = prove_mono predT fp_fun monos ctxt'' |
760 intros monos thy params paramTs cTs cnames induct_cases = |
668 |
|
669 in (ctxt'', rec_name, mono, fp_def', map (#2 o #2) consts_defs, |
|
670 list_comb (rec_const, params), preds, argTs, bs, xs) |
|
671 end; |
|
672 |
|
673 fun add_ind_def verbose alt_name coind no_elim no_ind cs |
|
674 intros monos params cnames_syn induct_cases ctxt = |
761 let |
675 let |
762 val _ = |
676 val _ = |
763 if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^ |
677 if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ |
764 commas_quote (map Sign.base_name cnames)) else (); |
678 commas_quote (map fst cnames_syn)) else (); |
765 |
679 |
766 val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros); |
680 val ((intr_names, intr_atts), intr_ts) = apfst split_list (split_list intros); |
767 |
681 |
768 val (thy1, rec_name, mono, fp_def, rec_sets_defs, rec_const, sumT) = |
682 val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds, |
769 mk_ind_def declare_consts alt_name coind cs intr_ts monos thy |
683 argTs, bs, xs) = mk_ind_def alt_name coind cs intr_ts |
770 params paramTs cTs cnames; |
684 monos params cnames_syn ctxt; |
771 val ctxt1 = ProofContext.init thy1; |
685 |
772 |
686 val (intrs, unfold) = prove_intrs coind mono fp_def (length bs + length xs) |
773 val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts rec_sets_defs ctxt1; |
687 intr_ts rec_preds_defs ctxt1; |
774 val elims = if no_elim then [] else |
688 val elims = ProofContext.export ctxt1 ctxt (if no_elim then [] else |
775 prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs ctxt1; |
689 prove_elims cs params intr_ts intr_names unfold rec_preds_defs ctxt1); |
776 val raw_induct = if no_ind then Drule.asm_rl else |
690 val raw_induct = singleton (ProofContext.export ctxt1 ctxt) |
777 if coind then standard (rule_by_tactic |
691 (if no_ind then Drule.asm_rl else |
778 (rewrite_tac [mk_meta_eq vimage_Un] THEN |
692 if coind then ObjectLogic.rulify (rule_by_tactic |
779 fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct))) |
693 (rewrite_tac [le_fun_def, le_bool_def] THEN |
780 else |
694 fold_tac rec_preds_defs) (mono RS (fp_def RS def_coinduct))) |
781 prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def |
695 else |
782 rec_sets_defs ctxt1; |
696 prove_indrule cs argTs bs xs rec_const params intr_ts mono fp_def |
|
697 rec_preds_defs ctxt1); |
783 val induct = |
698 val induct = |
784 if coind then |
699 if coind then |
785 (raw_induct, [RuleCases.case_names [rec_name], |
700 (raw_induct, [RuleCases.case_names [rec_name], |
786 RuleCases.case_conclusion (rec_name, induct_cases), |
701 RuleCases.case_conclusion (rec_name, induct_cases), |
787 RuleCases.consumes 1]) |
702 RuleCases.consumes 1]) |
788 else if no_ind orelse length cs > 1 then |
703 else if no_ind orelse length cs > 1 then |
789 (raw_induct, [RuleCases.case_names induct_cases, RuleCases.consumes 0]) |
704 (raw_induct, [RuleCases.case_names induct_cases, RuleCases.consumes 0]) |
790 else (raw_induct RSN (2, rev_mp), [RuleCases.case_names induct_cases, RuleCases.consumes 1]); |
705 else (raw_induct RSN (2, rev_mp), [RuleCases.case_names induct_cases, RuleCases.consumes 1]); |
791 |
706 |
792 val (intrs', thy2) = |
707 val (intrs', ctxt2) = |
793 thy1 |
708 ctxt1 |> |
794 |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts); |
709 LocalTheory.notes (map (NameSpace.append rec_name) intr_names ~~ intr_atts ~~ |
795 val (([_, elims'], [induct']), thy3) = |
710 map (single o rpair [] o single) (ProofContext.export ctxt1 ctxt intrs)) |>> |
796 thy2 |
711 map (hd o snd); (* FIXME? *) |
797 |> PureThy.add_thmss |
712 val (((_, (_, elims')), (_, [induct'])), ctxt3) = |
798 [(("intros", intrs'), []), |
713 ctxt2 |> |
799 (("elims", elims), [RuleCases.consumes 1])] |
714 LocalTheory.note ((NameSpace.append rec_name "intros", []), intrs') ||>> |
800 ||>> PureThy.add_thms |
715 LocalTheory.note ((NameSpace.append rec_name "elims", |
801 [((coind_prefix coind ^ "induct", rulify (#1 induct)), #2 induct)]; |
716 [Attrib.internal (RuleCases.consumes 1)]), elims) ||>> |
802 in (thy3, |
717 LocalTheory.note ((NameSpace.append rec_name (coind_prefix coind ^ "induct"), |
803 {defs = fp_def :: rec_sets_defs, |
718 map Attrib.internal (#2 induct)), [rulify (#1 induct)]) |
804 mono = mono, |
719 in (ctxt3, rec_name, |
805 unfold = unfold, |
720 {preds = preds, |
|
721 defs = fp_def :: rec_preds_defs, |
|
722 mono = singleton (ProofContext.export ctxt1 ctxt) mono, |
|
723 unfold = singleton (ProofContext.export ctxt1 ctxt) unfold, |
806 intrs = intrs', |
724 intrs = intrs', |
807 elims = elims', |
725 elims = elims', |
808 mk_cases = mk_cases elims', |
726 mk_cases = mk_cases elims', |
809 raw_induct = rulify raw_induct, |
727 raw_induct = rulify raw_induct, |
810 induct = induct'}) |
728 induct = induct'}) |
811 end; |
729 end; |
812 |
730 |
813 |
731 |
814 (* external interfaces *) |
732 (* external interfaces *) |
815 |
733 |
816 fun try_term f msg thy t = |
734 fun add_inductive_i verbose alt_name coind no_elim no_ind cnames_syn pnames pre_intros monos ctxt = |
817 (case Library.try f t of |
735 let |
818 SOME x => x |
736 val thy = ProofContext.theory_of ctxt; |
819 | NONE => error (msg ^ Sign.string_of_term thy t)); |
|
820 |
|
821 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs pre_intros monos thy = |
|
822 let |
|
823 val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions"); |
737 val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions"); |
824 |
738 |
825 (*parameters should agree for all mutually recursive components*) |
739 val frees = fold (Term.add_frees o snd) pre_intros []; |
826 val (_, params) = strip_comb (hd cs); |
740 fun type_of s = (case AList.lookup op = frees s of |
827 val paramTs = map (try_term (snd o dest_Free) "Parameter in recursive\ |
741 NONE => error ("No such variable: " ^ s) | SOME T => T); |
828 \ component is not a free variable: " thy) params; |
742 |
829 |
743 val params = map |
830 val cTs = map (try_term (HOLogic.dest_setT o fastype_of) |
744 (fn (s, SOME T) => Free (s, T) | (s, NONE) => Free (s, type_of s)) pnames; |
831 "Recursive component not of type set: " thy) cs; |
745 val cs = map |
832 |
746 (fn (s, SOME T, _) => Free (s, T) | (s, NONE, _) => Free (s, type_of s)) cnames_syn; |
833 val cnames = map (try_term (fst o dest_Const o head_of) |
747 val cnames_syn' = map (fn (s, _, mx) => (s, mx)) cnames_syn; |
834 "Recursive set not previously declared as constant: " thy) cs; |
748 val cnames = map (Sign.full_name thy o #1) cnames_syn; |
835 |
749 |
836 val save_thy = thy |
750 fun close_rule (x, r) = (x, list_all_free (rev (fold_aterms |
837 |> Theory.copy |> cond_declare_consts declare_consts cs paramTs cnames; |
751 (fn t as Free (v as (s, _)) => |
838 val intros = map (check_rule save_thy cs) pre_intros; |
752 if Variable.is_fixed ctxt s orelse member op = cs t orelse |
|
753 member op = params t then I else insert op = v |
|
754 | _ => I) r []), r)); |
|
755 |
|
756 val intros = map (close_rule o check_rule thy cs params) pre_intros; |
839 val induct_cases = map (#1 o #1) intros; |
757 val induct_cases = map (#1 o #1) intros; |
840 |
758 |
841 val (thy1, result as {elims, induct, ...}) = |
759 val (ctxt1, rec_name, result as {elims, induct, ...}) = |
842 add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs intros monos |
760 add_ind_def verbose alt_name coind no_elim no_ind cs intros monos |
843 thy params paramTs cTs cnames induct_cases; |
761 params cnames_syn' induct_cases ctxt; |
844 val thy2 = thy1 |
762 val ctxt2 = ctxt1 |
845 |> put_inductives cnames ({names = cnames, coind = coind}, result) |
763 |> LocalTheory.declaration |
846 |> add_cases_induct no_elim no_ind coind cnames elims induct |
764 (put_inductives cnames ({names = cnames, coind = coind}, result)) |
847 |> Theory.parent_path; |
765 |> add_cases_induct no_elim no_ind coind rec_name cnames elims induct; |
848 in (thy2, result) end; |
766 in (ctxt2, result) end; |
849 |
767 |
850 fun add_inductive verbose coind c_strings intro_srcs raw_monos thy = |
768 fun add_inductive verbose coind cnames_syn pnames_syn intro_srcs raw_monos ctxt = |
851 let |
769 let |
852 val cs = map (Sign.read_term thy) c_strings; |
770 val (_, ctxt') = Specification.read_specification (cnames_syn @ pnames_syn) [] ctxt; |
853 |
771 val intrs = map (fn spec => apsnd hd (hd (snd (fst |
854 val intr_names = map (fst o fst) intro_srcs; |
772 (Specification.read_specification [] [apsnd single spec] ctxt'))))) intro_srcs; |
855 fun read_rule s = Thm.read_cterm thy (s, propT) |
773 val pnames = map (fn (s, _, _) => |
856 handle ERROR msg => cat_error msg ("The error(s) above occurred for " ^ s); |
774 (s, SOME (ProofContext.infer_type ctxt' s))) pnames_syn; |
857 val intr_ts = map (Thm.term_of o read_rule o snd o fst) intro_srcs; |
775 val cnames_syn' = map (fn (s, _, mx) => |
858 val intr_atts = map (map (Attrib.attribute thy) o snd) intro_srcs; |
776 (s, SOME (ProofContext.infer_type ctxt' s), mx)) cnames_syn; |
859 val (cs', intr_ts') = unify_consts thy cs intr_ts; |
777 val (monos, ctxt'') = LocalTheory.theory_result (IsarThy.apply_theorems raw_monos) ctxt; |
860 |
|
861 val (monos, thy') = thy |> IsarThy.apply_theorems raw_monos; |
|
862 in |
778 in |
863 add_inductive_i verbose false "" coind false false cs' |
779 add_inductive_i verbose "" coind false false cnames_syn' pnames intrs monos ctxt'' |
864 ((intr_names ~~ intr_ts') ~~ intr_atts) monos thy' |
|
865 end; |
780 end; |
866 |
781 |
867 |
782 |
868 |
783 |
869 (** package setup **) |
784 (** package setup **) |
870 |
785 |
871 (* setup theory *) |
786 (* setup theory *) |
872 |
787 |
873 val setup = |
788 val setup = |
874 InductiveData.init #> |
789 InductiveData.init #> |
875 Method.add_methods [("ind_cases", mk_cases_meth oo mk_cases_args, |
790 Method.add_methods [("ind_cases2", mk_cases_meth oo mk_cases_args, |
876 "dynamic case analysis on sets")] #> |
791 "dynamic case analysis on predicates")] #> |
877 Attrib.add_attributes [("mono", Attrib.add_del_args mono_add mono_del, |
792 Attrib.add_attributes [("mono2", Attrib.add_del_args mono_add mono_del, |
878 "declaration of monotonicity rule")]; |
793 "declaration of monotonicity rule")]; |
879 |
794 |
880 |
795 |
881 (* outer syntax *) |
796 (* outer syntax *) |
882 |
797 |
883 local structure P = OuterParse and K = OuterKeyword in |
798 local structure P = OuterParse and K = OuterKeyword in |
884 |
799 |
885 fun mk_ind coind ((sets, intrs), monos) = |
800 fun mk_ind coind ((((loc, preds), params), intrs), monos) = |
886 #1 o add_inductive true coind sets (map P.triple_swap intrs) monos; |
801 Toplevel.local_theory loc |
|
802 (#1 o add_inductive true coind preds params intrs monos); |
887 |
803 |
888 fun ind_decl coind = |
804 fun ind_decl coind = |
889 Scan.repeat1 P.term -- |
805 P.opt_locale_target -- |
|
806 P.fixes -- Scan.optional (P.$$$ "for" |-- P.fixes) [] -- |
890 (P.$$$ "intros" |-- |
807 (P.$$$ "intros" |-- |
891 P.!!! (Scan.repeat (P.opt_thm_name ":" -- P.prop))) -- |
808 P.!!! (Scan.repeat (P.opt_thm_name ":" -- P.prop))) -- |
892 Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1) [] |
809 Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1) [] |
893 >> (Toplevel.theory o mk_ind coind); |
810 >> mk_ind coind; |
894 |
811 |
895 val inductiveP = |
812 val inductiveP = |
896 OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false); |
813 OuterSyntax.command "inductive2" "define inductive predicates" K.thy_decl (ind_decl false); |
897 |
814 |
898 val coinductiveP = |
815 val coinductiveP = |
899 OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true); |
816 OuterSyntax.command "coinductive2" "define coinductive predicates" K.thy_decl (ind_decl true); |
900 |
817 |
901 |
818 |
902 val ind_cases = |
819 val ind_cases = |
903 P.and_list1 (P.opt_thm_name ":" -- Scan.repeat1 P.prop) |
820 P.and_list1 (P.opt_thm_name ":" -- Scan.repeat1 P.prop) |
904 >> (Toplevel.theory o inductive_cases); |
821 >> (Toplevel.theory o inductive_cases); |
905 |
822 |
906 val inductive_casesP = |
823 val inductive_casesP = |
907 OuterSyntax.command "inductive_cases" |
824 OuterSyntax.command "inductive_cases2" |
908 "create simplified instances of elimination rules (improper)" K.thy_script ind_cases; |
825 "create simplified instances of elimination rules (improper)" K.thy_script ind_cases; |
909 |
826 |
910 val _ = OuterSyntax.add_keywords ["intros", "monos"]; |
827 val _ = OuterSyntax.add_keywords ["intros", "monos"]; |
911 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP]; |
828 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP]; |
912 |
829 |