src/HOLCF/domain/theorems.ML
 author wenzelm Thu Oct 21 18:46:33 1999 +0200 (1999-10-21) changeset 7906 0576dad973b1 parent 6092 d9db67970c73 child 8149 941afb897532 permissions -rw-r--r--
get_thm;
1 (*  Title:      HOLCF/domain/theorems.ML
2     ID:         \$Id\$
3     Author : David von Oheimb
4     Copyright 1995, 1996 TU Muenchen
6 proof generator for domain section
7 *)
10 structure Domain_Theorems = struct
12 local
14 open Domain_Library;
15 infixr 0 ===>;infixr 0 ==>;infix 0 == ;
16 infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<;
17 infix 9 `   ; infix 9 `% ; infix 9 `%%; infixr 9 oo;
19 (* ----- general proof facilities ------------------------------------------- *)
21 fun inferT sg pre_tm = #1 (Sign.infer_types sg (K None) (K None) [] true
22                            ([pre_tm],propT));
24 fun pg'' thy defs t = let val sg = sign_of thy;
25                           val ct = Thm.cterm_of sg (inferT sg t);
26                       in prove_goalw_cterm defs ct end;
27 fun pg'  thy defs t tacsf=pg'' thy defs t (fn []   => tacsf
28                                 | prems=> (cut_facts_tac prems 1)::tacsf);
30 fun REPEAT_DETERM_UNTIL p tac =
31 let fun drep st = if p st then Seq.single st
32                           else (case Seq.pull(tac st) of
33                                   None        => Seq.empty
34                                 | Some(st',_) => drep st')
35 in drep end;
36 val UNTIL_SOLVED = REPEAT_DETERM_UNTIL (has_fewer_prems 1);
38 local val trueI2 = prove_goal HOL.thy"f~=x ==> True"(fn _ => [rtac TrueI 1]) in
39 val kill_neq_tac = dtac trueI2 end;
40 fun case_UU_tac rews i v =      case_tac (v^"=UU") i THEN
41                                 asm_simp_tac (HOLCF_ss addsimps rews) i;
43 val chain_tac = REPEAT_DETERM o resolve_tac
44                 [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL];
46 (* ----- general proofs ----------------------------------------------------- *)
48 val all2E = prove_goal HOL.thy "[| !x y . P x y; P x y ==> R |] ==> R"
49  (fn prems =>[
50                                 resolve_tac prems 1,
51                                 cut_facts_tac prems 1,
52                                 fast_tac HOL_cs 1]);
54 val dist_eqI = prove_goal Porder.thy "~(x::'a::po) << y ==> x ~= y" (fn prems => [
55                                 rtac rev_contrapos 1,
56                                 etac (antisym_less_inverse RS conjunct1) 1,
57                                 resolve_tac prems 1]);
58 (*
59 infixr 0 y;
60 val b = 0;
61 fun _ y t = by t;
62 fun g defs t = let val sg = sign_of thy;
63                      val ct = Thm.cterm_of sg (inferT sg t);
64                  in goalw_cterm defs ct end;
65 *)
67 in
69 fun theorems (((dname,_),cons) : eq, eqs : eq list) thy =
70 let
72 val dummy = writeln ("Proving isomorphism properties of domain "^dname^" ...");
73 val pg = pg' thy;
76 (* ----- getting the axioms and definitions --------------------------------- *)
78 local fun ga s dn = get_thm thy (dn^"."^s) in
79 val ax_abs_iso    = ga "abs_iso"  dname;
80 val ax_rep_iso    = ga "rep_iso"  dname;
81 val ax_when_def   = ga "when_def" dname;
82 val axs_con_def   = map (fn (con,_) => ga (extern_name con^"_def") dname) cons;
83 val axs_dis_def   = map (fn (con,_) => ga (   dis_name con^"_def") dname) cons;
84 val axs_sel_def   = flat(map (fn (_,args) =>
85                     map (fn     arg => ga (sel_of arg     ^"_def") dname) args)
86 									  cons);
87 val ax_copy_def   = ga "copy_def" dname;
88 end; (* local *)
90 (* ----- theorems concerning the isomorphism -------------------------------- *)
92 val dc_abs  = %%(dname^"_abs");
93 val dc_rep  = %%(dname^"_rep");
94 val dc_copy = %%(dname^"_copy");
95 val x_name = "x";
97 val (rep_strict, abs_strict) = let
98          val r = ax_rep_iso RS (ax_abs_iso RS (allI  RSN(2,allI RS iso_strict)))
99                in (r RS conjunct1, r RS conjunct2) end;
100 val abs_defin' = pg [] ((dc_abs`%x_name === UU) ==> (%x_name === UU)) [
101                            res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
102                                 etac ssubst 1, rtac rep_strict 1];
103 val rep_defin' = pg [] ((dc_rep`%x_name === UU) ==> (%x_name === UU)) [
104                            res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
105                                 etac ssubst 1, rtac abs_strict 1];
106 val iso_rews = [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
108 local
109 val iso_swap = pg [] (dc_rep`%"x" === %"y" ==> %"x" === dc_abs`%"y") [
110                             dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
111                             etac (ax_rep_iso RS subst) 1];
112 fun exh foldr1 cn quant foldr2 var = let
113   fun one_con (con,args) = let val vns = map vname args in
114     foldr quant (vns, foldr2 ((%x_name === con_app2 con (var vns) vns)::
115                               map (defined o (var vns)) (nonlazy args))) end
116   in foldr1 ((cn(%x_name===UU))::map one_con cons) end;
117 in
118 val casedist = let
119             fun common_tac thm = rtac thm 1 THEN contr_tac 1;
120             fun unit_tac true = common_tac upE1
121             |   unit_tac _    = all_tac;
122             fun prod_tac []          = common_tac oneE
123             |   prod_tac [arg]       = unit_tac (is_lazy arg)
124             |   prod_tac (arg::args) =
125                                 common_tac sprodE THEN
126                                 kill_neq_tac 1 THEN
127                                 unit_tac (is_lazy arg) THEN
128                                 prod_tac args;
129             fun sum_rest_tac p = SELECT_GOAL(EVERY[
130                                 rtac p 1,
131                                 rewrite_goals_tac axs_con_def,
132                                 dtac iso_swap 1,
133                                 simp_tac HOLCF_ss 1,
134                                 UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
135             fun sum_tac [(_,args)]       [p]        =
136                                 prod_tac args THEN sum_rest_tac p
137             |   sum_tac ((_,args)::cons') (p::prems) = DETERM(
138                                 common_tac ssumE THEN
139                                 kill_neq_tac 1 THEN kill_neq_tac 2 THEN
140                                 prod_tac args THEN sum_rest_tac p) THEN
141                                 sum_tac cons' prems
142             |   sum_tac _ _ = Imposs "theorems:sum_tac";
143           in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%"P")))
144                               (fn T => T ==> %"P") mk_All
145                               (fn l => foldr (op ===>) (map mk_trp l,
146                                                             mk_trp(%"P")))
147                               bound_arg)
148                              (fn prems => [
149                                 cut_facts_tac [excluded_middle] 1,
150                                 etac disjE 1,
151                                 rtac (hd prems) 2,
152                                 etac rep_defin' 2,
153                                 if length cons = 1 andalso
154                                    length (snd(hd cons)) = 1 andalso
155                                    not(is_lazy(hd(snd(hd cons))))
156                                 then rtac (hd (tl prems)) 1 THEN atac 2 THEN
157                                      rewrite_goals_tac axs_con_def THEN
158                                      simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
159                                 else sum_tac cons (tl prems)])end;
160 val exhaust= pg[](mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %)))[
161                                 rtac casedist 1,
162                                 UNTIL_SOLVED(fast_tac HOL_cs 1)];
163 end;
165 local
166   fun bind_fun t = foldr mk_All (when_funs cons,t);
167   fun bound_fun i _ = Bound (length cons - i);
168   val when_app  = foldl (op `) (%%(dname^"_when"), mapn bound_fun 1 cons);
169   val when_appl = pg [ax_when_def] (bind_fun(mk_trp(when_app`%x_name ===
170              when_body cons (fn (m,n)=> bound_fun (n-m) 0)`(dc_rep`%x_name))))[
171                                 simp_tac HOLCF_ss 1];
172 in
173 val when_strict = pg [] (bind_fun(mk_trp(strict when_app))) [
175 val when_apps = let fun one_when n (con,args) = pg axs_con_def
176                 (bind_fun (lift_defined % (nonlazy args,
177                 mk_trp(when_app`(con_app con args) ===
178                        mk_cRep_CFun(bound_fun n 0,map %# args)))))[
179                 asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
180         in mapn one_when 1 cons end;
181 end;
182 val when_rews = when_strict::when_apps;
184 (* ----- theorems concerning the constructors, discriminators and selectors - *)
186 val dis_rews = let
187   val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
188                              strict(%%(dis_name con)))) [
189                                 simp_tac (HOLCF_ss addsimps when_rews) 1]) cons;
190   val dis_apps = let fun one_dis c (con,args)= pg axs_dis_def
191                    (lift_defined % (nonlazy args,
192                         (mk_trp((%%(dis_name c))`(con_app con args) ===
193                               %%(if con=c then "TT" else "FF"))))) [
194                                 asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
195         in flat(map (fn (c,_) => map (one_dis c) cons) cons) end;
196   val dis_defins = map (fn (con,args) => pg [] (defined(%x_name) ==>
197                       defined(%%(dis_name con)`%x_name)) [
198                                 rtac casedist 1,
199                                 contr_tac 1,
200                                 UNTIL_SOLVED (CHANGED(asm_simp_tac
201                                         (HOLCF_ss addsimps dis_apps) 1))]) cons;
202 in dis_stricts @ dis_defins @ dis_apps end;
204 val con_stricts = flat(map (fn (con,args) => map (fn vn =>
205                         pg (axs_con_def)
206                            (mk_trp(con_app2 con (fn arg => if vname arg = vn
207                                         then UU else %# arg) args === UU))[
208                                 asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
209                         ) (nonlazy args)) cons);
210 val con_defins = map (fn (con,args) => pg []
211                         (lift_defined % (nonlazy args,
212                                 mk_trp(defined(con_app con args)))) ([
213                           rtac rev_contrapos 1,
214                           eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
215                           asm_simp_tac (HOLCF_ss addsimps dis_rews) 1] )) cons;
216 val con_rews = con_stricts @ con_defins;
218 val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%sel))) [
219                                 simp_tac (HOLCF_ss addsimps when_rews) 1];
220 in flat(map (fn (_,args) =>map (fn arg => one_sel (sel_of arg)) args) cons) end;
221 val sel_apps = let fun one_sel c n sel = map (fn (con,args) =>
222                 let val nlas = nonlazy args;
223                     val vns  = map vname args;
224                 in pg axs_sel_def (lift_defined %
225                    (filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
226                                 mk_trp((%%sel)`(con_app con args) ===
227                                 (if con=c then %(nth_elem(n,vns)) else UU))))
228                             ( (if con=c then []
229                        else map(case_UU_tac(when_rews@con_stricts)1) nlas)
230                      @(if con=c andalso ((nth_elem(n,vns)) mem nlas)
231                                  then[case_UU_tac (when_rews @ con_stricts) 1
232                                                   (nth_elem(n,vns))] else [])
233                      @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
234 in flat(map  (fn (c,args) =>
235      flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
236 val sel_defins = if length cons=1 then map (fn arg => pg [](defined(%x_name)==>
237                         defined(%%(sel_of arg)`%x_name)) [
238                                 rtac casedist 1,
239                                 contr_tac 1,
240                                 UNTIL_SOLVED (CHANGED(asm_simp_tac
242                  (filter_out is_lazy (snd(hd cons))) else [];
243 val sel_rews = sel_stricts @ sel_defins @ sel_apps;
245 val distincts_le = let
246     fun dist (con1, args1) (con2, args2) = pg []
247               (lift_defined % ((nonlazy args1),
248                         (mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
249                         rtac rev_contrapos 1,
250                         eres_inst_tac[("fo",dis_name con1)] monofun_cfun_arg 1]
251                       @map(case_UU_tac (con_stricts @ dis_rews)1)(nonlazy args2)
252                       @[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
253     fun distinct (con1,args1) (con2,args2) =
254         let val arg1 = (con1, args1)
255             val arg2 = (con2,
256 			ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
257                         (args2, variantlist(map vname args2,map vname args1)))
258         in [dist arg1 arg2, dist arg2 arg1] end;
259     fun distincts []      = []
260     |   distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
261 in distincts cons end;
262 val dist_les = flat (flat distincts_le);
263 val dist_eqs = let
264     fun distinct (_,args1) ((_,args2),leqs) = let
265         val (le1,le2) = (hd leqs, hd(tl leqs));
266         val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
267         if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
268         if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
269                                         [eq1, eq2] end;
270     open BasisLibrary (*restore original List*)
271     fun distincts []      = []
272     |   distincts ((c,leqs)::cs) = List.concat
273 	            (ListPair.map (distinct c) ((map #1 cs),leqs)) @
274 		    distincts cs;
275     in distincts (cons~~distincts_le) end;
277 local
278   fun pgterm rel con args = let
279                 fun append s = upd_vname(fn v => v^s);
280                 val (largs,rargs) = (args, map (append "'") args);
281                 in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
282                       lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
283                             mk_trp (foldr' mk_conj
284                                 (ListPair.map rel
285 				 (map %# largs, map %# rargs)))))) end;
286   val cons' = filter (fn (_,args) => args<>[]) cons;
287 in
288 val inverts = map (fn (con,args) =>
289                 pgterm (op <<) con args (flat(map (fn arg => [
290                                 TRY(rtac conjI 1),
291                                 dres_inst_tac [("fo",sel_of arg)] monofun_cfun_arg 1,
292                                 asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
293                                                       ) args))) cons';
294 val injects = map (fn ((con,args),inv_thm) =>
295                            pgterm (op ===) con args [
296                                 etac (antisym_less_inverse RS conjE) 1,
297                                 dtac inv_thm 1, REPEAT(atac 1),
298                                 dtac inv_thm 1, REPEAT(atac 1),
299                                 TRY(safe_tac HOL_cs),
300                                 REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
301                   (cons'~~inverts);
302 end;
304 (* ----- theorems concerning one induction step ----------------------------- *)
306 val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [
308                                                    cfst_strict,csnd_strict]) 1];
309 val copy_apps = map (fn (con,args) => pg [ax_copy_def]
310                     (lift_defined % (nonlazy_rec args,
311                         mk_trp(dc_copy`%"f"`(con_app con args) ===
312                 (con_app2 con (app_rec_arg (cproj (%"f") eqs)) args))))
313                         (map (case_UU_tac (abs_strict::when_strict::con_stricts)
314                                  1 o vname)
315                          (filter (fn a => not (is_rec a orelse is_lazy a)) args)
316                         @[asm_simp_tac (HOLCF_ss addsimps when_apps) 1,
317                           simp_tac (HOLCF_ss addsimps axs_con_def) 1]))cons;
318 val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU`
319                                         (con_app con args) ===UU))
320      (let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
321                          in map (case_UU_tac rews 1) (nonlazy args) @ [
322                              asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
323                         (filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
324 val copy_rews = copy_strict::copy_apps @ copy_stricts;
325 in thy |> Theory.add_path (Sign.base_name dname)
326        |> (PureThy.add_thmss o map Thm.no_attributes) [
327 		("iso_rews" , iso_rews  ),
328 		("exhaust"  , [exhaust] ),
329 		("casedist" , [casedist]),
330 		("when_rews", when_rews ),
331 		("con_rews", con_rews),
332 		("sel_rews", sel_rews),
333 		("dis_rews", dis_rews),
334 		("dist_les", dist_les),
335 		("dist_eqs", dist_eqs),
336 		("inverts" , inverts ),
337 		("injects" , injects ),
338 		("copy_rews", copy_rews)]
339        |> Theory.parent_path
340 end; (* let *)
342 fun comp_theorems (comp_dnam, eqs: eq list) thy =
343 let
344 val dnames = map (fst o fst) eqs;
345 val conss  = map  snd        eqs;
346 val comp_dname = Sign.full_name (sign_of thy) comp_dnam;
348 val d = writeln("Proving induction   properties of domain "^comp_dname^" ...");
349 val pg = pg' thy;
351 (* ----- getting the composite axiom and definitions ------------------------ *)
353 local fun ga s dn = get_thm thy (dn^"."^s) in
354 val axs_reach      = map (ga "reach"     ) dnames;
355 val axs_take_def   = map (ga "take_def"  ) dnames;
356 val axs_finite_def = map (ga "finite_def") dnames;
357 val ax_copy2_def   =      ga "copy_def"  comp_dnam;
358 val ax_bisim_def   =      ga "bisim_def" comp_dnam;
359 end; (* local *)
361 local fun gt  s dn = get_thm  thy (dn^"."^s);
362       fun gts s dn = get_thms thy (dn^"."^s) in
363 val cases     =       map (gt  "casedist" ) dnames;
364 val con_rews  = flat (map (gts "con_rews" ) dnames);
365 val copy_rews = flat (map (gts "copy_rews") dnames);
366 end; (* local *)
368 fun dc_take dn = %%(dn^"_take");
369 val x_name = idx_name dnames "x";
370 val P_name = idx_name dnames "P";
371 val n_eqs = length eqs;
373 (* ----- theorems concerning finite approximation and finite induction ------ *)
375 local
376   val iterate_Cprod_ss = simpset_of Fix.thy
378   val copy_con_rews  = copy_rews @ con_rews;
379   val copy_take_defs =(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
380   val take_stricts=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=>
381             strict(dc_take dn \$ %"n")) eqs))) ([
382                         nat_ind_tac "n" 1,
383                          simp_tac iterate_Cprod_ss 1,
385   val take_stricts' = rewrite_rule copy_take_defs take_stricts;
386   val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn \$ %%"0")
387                                                         `%x_name n === UU))[
388                                 simp_tac iterate_Cprod_ss 1]) 1 dnames;
389   val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1;
390   val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj
391             (flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all
392         (map vname args,(dc_take dn \$ (%%"Suc" \$ %"n"))`(con_app con args) ===
393          con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))\$ %"n"))
394                               args)) cons) eqs)))) ([
395                                 simp_tac iterate_Cprod_ss 1,
396                                 nat_ind_tac "n" 1,
399                                       (filter (has_fewer_prems 1) copy_rews)) 1,
400                                 TRY(safe_tac HOL_cs)] @
401                         (flat(map (fn ((dn,_),cons) => map (fn (con,args) =>
402                                 if nonlazy_rec args = [] then all_tac else
403                                 EVERY(map c_UU_tac (nonlazy_rec args)) THEN
405                                                            ) cons) eqs)));
406 in
407 val take_rews = atomize take_stricts @ take_0s @ atomize take_apps;
408 end; (* local *)
410 local
411   fun one_con p (con,args) = foldr mk_All (map vname args,
412         lift_defined (bound_arg (map vname args)) (nonlazy args,
413         lift (fn arg => %(P_name (1+rec_of arg)) \$ bound_arg args arg)
414          (filter is_rec args,mk_trp(%p \$ con_app2 con (bound_arg args) args))));
415   fun one_eq ((p,cons),concl) = (mk_trp(%p \$ UU) ===>
416                            foldr (op ===>) (map (one_con p) cons,concl));
417   fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss,
418                         mk_trp(foldr' mk_conj (mapn concf 1 dnames)));
419   val take_ss = HOL_ss addsimps take_rews;
420   fun quant_tac i = EVERY(mapn(fn n=> fn _=> res_inst_tac[("x",x_name n)]spec i)
421                                1 dnames);
422   fun ind_prems_tac prems = EVERY(flat (map (fn cons => (
423                                      resolve_tac prems 1 ::
424                                      flat (map (fn (_,args) =>
425                                        resolve_tac prems 1 ::
426                                        map (K(atac 1)) (nonlazy args) @
427                                        map (K(atac 1)) (filter is_rec args))
428                                      cons))) conss));
429   local
430     (* check whether every/exists constructor of the n-th part of the equation:
431        it has a possibly indirectly recursive argument that isn't/is possibly
432        indirectly lazy *)
433     fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg =>
434           is_rec arg andalso not(rec_of arg mem ns) andalso
435           ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse
436             rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns)
437               (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
438           ) o snd) cons;
439     fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
440     fun warn (n,cons)  = if all_rec_to [] false (n,cons) then (warning
441         ("domain "^nth_elem(n,dnames)^" is empty!"); true) else false;
442     fun lazy_rec_to ns = rec_to exists Id  lazy_rec_to ns;
444   in val n__eqs     = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
445      val is_emptys = map warn n__eqs;
446      val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
447   end;
448 in (* local *)
449 val finite_ind = pg'' thy [] (ind_term (fn n => fn dn => %(P_name n)\$
450                              (dc_take dn \$ %"n" `%(x_name n)))) (fn prems => [
451                                 quant_tac 1,
452                                 simp_tac HOL_ss 1,
453                                 nat_ind_tac "n" 1,
454                                 simp_tac (take_ss addsimps prems) 1,
455                                 TRY(safe_tac HOL_cs)]
456                                 @ flat(map (fn (cons,cases) => [
457                                  res_inst_tac [("x","x")] cases 1,
458                                  asm_simp_tac (take_ss addsimps prems) 1]
459                                  @ flat(map (fn (con,args) =>
460                                   asm_simp_tac take_ss 1 ::
461                                   map (fn arg =>
462                                    case_UU_tac (prems@con_rews) 1 (
463                            nth_elem(rec_of arg,dnames)^"_take n`"^vname arg))
464                                   (filter is_nonlazy_rec args) @ [
465                                   resolve_tac prems 1] @
466                                   map (K (atac 1))      (nonlazy args) @
467                                   map (K (etac spec 1)) (filter is_rec args))
468                                  cons))
469                                 (conss~~cases)));
471 val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n",
472                 mk_trp(dc_take dn \$ Bound 0 `%(x_name n) ===
473                        dc_take dn \$ Bound 0 `%(x_name n^"'")))
474            ===> mk_trp(%(x_name n) === %(x_name n^"'"))) (fn prems => [
475                         res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
476                         res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
477                                 stac fix_def2 1,
478                                 REPEAT(CHANGED(rtac(contlub_cfun_arg RS ssubst)1
479                                                THEN chain_tac 1)),
480                                 stac contlub_cfun_fun 1,
481                                 stac contlub_cfun_fun 2,
482                                 rtac lub_equal 3,
483                                 chain_tac 1,
484                                 rtac allI 1,
485                                 resolve_tac prems 1])) 1 (dnames~~axs_reach);
487 (* ----- theorems concerning finiteness and induction ----------------------- *)
489 val (finites,ind) = if is_finite then
490   let
491     fun take_enough dn = mk_ex ("n",dc_take dn \$ Bound 0 ` %"x" === %"x");
492     val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%"x")) ===>
493         mk_trp(mk_disj(mk_all("n",dc_take dn \$ Bound 0 ` %"x" === UU),
494         take_enough dn)) ===> mk_trp(take_enough dn)) [
495                                 etac disjE 1,
496                                 etac notE 1,
497                                 resolve_tac take_lemmas 1,
498                                 asm_simp_tac take_ss 1,
499                                 atac 1]) dnames;
500     val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn
501         (fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
502          mk_disj(dc_take dn \$ Bound 1 ` Bound 0 === UU,
503                  dc_take dn \$ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
504                                 rtac allI 1,
505                                 nat_ind_tac "n" 1,
506                                 simp_tac take_ss 1,
508                                 flat(mapn (fn n => fn (cons,cases) => [
509                                   simp_tac take_ss 1,
510                                   rtac allI 1,
511                                   res_inst_tac [("x",x_name n)] cases 1,
512                                   asm_simp_tac take_ss 1] @
513                                   flat(map (fn (con,args) =>
514                                     asm_simp_tac take_ss 1 ::
515                                     flat(map (fn vn => [
516                                       eres_inst_tac [("x",vn)] all_dupE 1,
517                                       etac disjE 1,
518                                       asm_simp_tac (HOL_ss addsimps con_rews) 1,
519                                       asm_simp_tac take_ss 1])
520                                     (nonlazy_rec args)))
521                                   cons))
522                                 1 (conss~~cases)));
523     val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp(
524                                                 %%(dn^"_finite") \$ %"x"))[
525                                 case_UU_tac take_rews 1 "x",
526                                 eresolve_tac finite_lemmas1a 1,
527                                 step_tac HOL_cs 1,
528                                 step_tac HOL_cs 1,
529                                 cut_facts_tac [l1b] 1,
530                         fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
531   in
532   (finites,
533    pg'' thy[](ind_term (fn n => fn dn => %(P_name n) \$ %(x_name n)))(fn prems =>
534                                 TRY(safe_tac HOL_cs) ::
535                          flat (map (fn (finite,fin_ind) => [
536                                rtac(rewrite_rule axs_finite_def finite RS exE)1,
537                                 etac subst 1,
538                                 rtac fin_ind 1,
539                                 ind_prems_tac prems])
540                                    (finites~~(atomize finite_ind)) ))
541 ) end (* let *) else
542   (mapn (fn n => fn dn => read_instantiate_sg (sign_of thy)
543                     [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
544    pg'' thy [] (foldr (op ===>) (mapn (fn n => K(mk_trp(%%"adm" \$ %(P_name n))))
545                1 dnames, ind_term (fn n => fn dn => %(P_name n) \$ %(x_name n))))
546                    (fn prems => map (fn ax_reach => rtac (ax_reach RS subst) 1)
547                                     axs_reach @ [
548                                 quant_tac 1,
553                                         cont_tacR 1 THEN resolve_tac prems 1),
554                                 strip_tac 1,
555                                 rtac (rewrite_rule axs_take_def finite_ind) 1,
556                                 ind_prems_tac prems])
557   handle ERROR => (warning "Cannot prove infinite induction rule"; refl))
558 end; (* local *)
560 (* ----- theorem concerning coinduction ------------------------------------- *)
562 local
563   val xs = mapn (fn n => K (x_name n)) 1 dnames;
564   fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
565   val take_ss = HOL_ss addsimps take_rews;
566   val sproj   = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
567   val coind_lemma=pg[ax_bisim_def](mk_trp(mk_imp(%%(comp_dname^"_bisim") \$ %"R",
568                 foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
569                   foldr mk_imp (mapn (fn n => K(proj (%"R") eqs n \$
570                                       bnd_arg n 0 \$ bnd_arg n 1)) 0 dnames,
571                     foldr' mk_conj (mapn (fn n => fn dn =>
572                                 (dc_take dn \$ %"n" `bnd_arg n 0 ===
573                                 (dc_take dn \$ %"n" `bnd_arg n 1)))0 dnames))))))
574                              ([ rtac impI 1,
575                                 nat_ind_tac "n" 1,
576                                 simp_tac take_ss 1,
577                                 safe_tac HOL_cs] @
578                                 flat(mapn (fn n => fn x => [
579                                   rotate_tac (n+1) 1,
580                                   etac all2E 1,
581                                   eres_inst_tac [("P1", sproj "R" eqs n^
582                                         " "^x^" "^x^"'")](mp RS disjE) 1,
583                                   TRY(safe_tac HOL_cs),
584                                   REPEAT(CHANGED(asm_simp_tac take_ss 1))])
585                                 0 xs));
586 in
587 val coind = pg [] (mk_trp(%%(comp_dname^"_bisim") \$ %"R") ===>
588                 foldr (op ===>) (mapn (fn n => fn x =>
589                   mk_trp(proj (%"R") eqs n \$ %x \$ %(x^"'"))) 0 xs,
590                   mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
591                                 TRY(safe_tac HOL_cs)] @
592                                 flat(map (fn take_lemma => [
593                                   rtac take_lemma 1,
594                                   cut_facts_tac [coind_lemma] 1,
595                                   fast_tac HOL_cs 1])
596                                 take_lemmas));
597 end; (* local *)
600 in thy |> Theory.add_path comp_dnam
601        |> (PureThy.add_thmss o map Thm.no_attributes) [
602 		("take_rews"  , take_rews  ),
603 		("take_lemmas", take_lemmas),
604 		("finites"    , finites    ),
605 		("finite_ind", [finite_ind]),
606 		("ind"       , [ind       ]),
607 		("coind"     , [coind     ])]
608        |> Theory.parent_path
609 end; (* let *)
610 end; (* local *)
611 end; (* struct *)