src/HOLCF/domain/theorems.ML
author oheimb
Wed Oct 29 14:23:49 1997 +0100 (1997-10-29)
changeset 4030 ca44afcc259c
parent 4008 2444085532c6
child 4043 35766855f344
permissions -rw-r--r--
debugging concerning sort variables
theorems are now proved immediately after generating the syntax
     1 (*  Title:      HOLCF/domain/theorems.ML
     2     ID:         $Id$
     3     Author : David von Oheimb
     4     Copyright 1995, 1996 TU Muenchen
     5 
     6 proof generator for domain section
     7 *)
     8 
     9 
    10 structure Domain_Theorems = struct
    11 
    12 local
    13 
    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;
    18 
    19 (* ----- general proof facilities ------------------------------------------- *)
    20 
    21 fun inferT sg pre_tm = #2 (Sign.infer_types sg (K None) (K None) [] true 
    22                            ([pre_tm],propT));
    23 
    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);
    29 
    30 fun REPEAT_DETERM_UNTIL p tac = 
    31 let fun drep st = if p st then Sequence.single st
    32                           else (case Sequence.pull(tac st) of
    33                                   None        => Sequence.null
    34                                 | Some(st',_) => drep st')
    35 in drep end;
    36 val UNTIL_SOLVED = REPEAT_DETERM_UNTIL (has_fewer_prems 1);
    37 
    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;
    42 
    43 val chain_tac = REPEAT_DETERM o resolve_tac 
    44                 [is_chain_iterate, ch2ch_fappR, ch2ch_fappL];
    45 
    46 (* ----- general proofs ----------------------------------------------------- *)
    47 
    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]);
    53 
    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 in
    60 
    61 
    62 type thms = (thm list * thm * thm * thm list *
    63 	     thm list * thm list * thm list * thm list * thm  list * thm list *
    64 	     thm list * thm list);
    65 fun (theorems thy: eq list -> eq -> thms) eqs ((dname,_),cons)  =
    66 let
    67 
    68 val dummy = writeln ("Proving isomorphism properties of domain "^dname^" ...");
    69 val pg = pg' thy;
    70 (*
    71 infixr 0 y;
    72 val b = 0;
    73 fun _ y t = by t;
    74 fun g defs t = let val sg = sign_of thy;
    75                      val ct = Thm.cterm_of sg (inferT sg t);
    76                  in goalw_cterm defs ct end;
    77 *)
    78 
    79 
    80 (* ----- getting the axioms and definitions --------------------------------- *)
    81 
    82 local val ga = get_axiom thy in
    83 val ax_abs_iso    = ga (dname^"_abs_iso"   );
    84 val ax_rep_iso    = ga (dname^"_rep_iso"   );
    85 val ax_when_def   = ga (dname^"_when_def"  );
    86 val axs_con_def   = map (fn (con,_) => ga (extern_name con ^"_def")) cons;
    87 val axs_dis_def   = map (fn (con,_) => ga (   dis_name con ^"_def")) cons;
    88 val axs_sel_def   = flat(map (fn (_,args) => 
    89                     map (fn     arg => ga (sel_of arg      ^"_def")) args)cons);
    90 val ax_copy_def   = ga (dname^"_copy_def"  );
    91 end; (* local *)
    92 
    93 (* ----- theorems concerning the isomorphism -------------------------------- *)
    94 
    95 val dc_abs  = %%(dname^"_abs");
    96 val dc_rep  = %%(dname^"_rep");
    97 val dc_copy = %%(dname^"_copy");
    98 val x_name = "x";
    99 
   100 val (rep_strict, abs_strict) = let 
   101          val r = ax_rep_iso RS (ax_abs_iso RS (allI  RSN(2,allI RS iso_strict)))
   102                in (r RS conjunct1, r RS conjunct2) end;
   103 val abs_defin' = pg [] ((dc_abs`%x_name === UU) ==> (%x_name === UU)) [
   104                            res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
   105                                 etac ssubst 1, rtac rep_strict 1];
   106 val rep_defin' = pg [] ((dc_rep`%x_name === UU) ==> (%x_name === UU)) [
   107                            res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
   108                                 etac ssubst 1, rtac abs_strict 1];
   109 val iso_rews = [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
   110 
   111 local 
   112 val iso_swap = pg [] (dc_rep`%"x" === %"y" ==> %"x" === dc_abs`%"y") [
   113                             dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
   114                             etac (ax_rep_iso RS subst) 1];
   115 fun exh foldr1 cn quant foldr2 var = let
   116   fun one_con (con,args) = let val vns = map vname args in
   117     foldr quant (vns, foldr2 ((%x_name === con_app2 con (var vns) vns)::
   118                               map (defined o (var vns)) (nonlazy args))) end
   119   in foldr1 ((cn(%x_name===UU))::map one_con cons) end;
   120 in
   121 val cases = let 
   122             fun common_tac thm = rtac thm 1 THEN contr_tac 1;
   123             fun unit_tac true = common_tac upE1
   124             |   unit_tac _    = all_tac;
   125             fun prod_tac []          = common_tac oneE
   126             |   prod_tac [arg]       = unit_tac (is_lazy arg)
   127             |   prod_tac (arg::args) = 
   128                                 common_tac sprodE THEN
   129                                 kill_neq_tac 1 THEN
   130                                 unit_tac (is_lazy arg) THEN
   131                                 prod_tac args;
   132             fun sum_rest_tac p = SELECT_GOAL(EVERY[
   133                                 rtac p 1,
   134                                 rewrite_goals_tac axs_con_def,
   135                                 dtac iso_swap 1,
   136                                 simp_tac HOLCF_ss 1,
   137                                 UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
   138             fun sum_tac [(_,args)]       [p]        = 
   139                                 prod_tac args THEN sum_rest_tac p
   140             |   sum_tac ((_,args)::cons') (p::prems) = DETERM(
   141                                 common_tac ssumE THEN
   142                                 kill_neq_tac 1 THEN kill_neq_tac 2 THEN
   143                                 prod_tac args THEN sum_rest_tac p) THEN
   144                                 sum_tac cons' prems
   145             |   sum_tac _ _ = Imposs "theorems:sum_tac";
   146           in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%"P")))
   147                               (fn T => T ==> %"P") mk_All
   148                               (fn l => foldr (op ===>) (map mk_trp l,
   149                                                             mk_trp(%"P")))
   150                               bound_arg)
   151                              (fn prems => [
   152                                 cut_facts_tac [excluded_middle] 1,
   153                                 etac disjE 1,
   154                                 rtac (hd prems) 2,
   155                                 etac rep_defin' 2,
   156                                 if length cons = 1 andalso 
   157                                    length (snd(hd cons)) = 1 andalso 
   158                                    not(is_lazy(hd(snd(hd cons))))
   159                                 then rtac (hd (tl prems)) 1 THEN atac 2 THEN
   160                                      rewrite_goals_tac axs_con_def THEN
   161                                      simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
   162                                 else sum_tac cons (tl prems)])end;
   163 val exhaust= pg[](mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %)))[
   164                                 rtac cases 1,
   165                                 UNTIL_SOLVED(fast_tac HOL_cs 1)];
   166 end;
   167 
   168 local 
   169   fun bind_fun t = foldr mk_All (when_funs cons,t);
   170   fun bound_fun i _ = Bound (length cons - i);
   171   val when_app  = foldl (op `) (%%(dname^"_when"), mapn bound_fun 1 cons);
   172   val when_appl = pg [ax_when_def] (bind_fun(mk_trp(when_app`%x_name ===
   173              when_body cons (fn (m,n)=> bound_fun (n-m) 0)`(dc_rep`%x_name))))[
   174                                 simp_tac HOLCF_ss 1];
   175 in
   176 val when_strict = pg [] (bind_fun(mk_trp(strict when_app))) [
   177                         simp_tac(HOLCF_ss addsimps [when_appl,rep_strict]) 1];
   178 val when_apps = let fun one_when n (con,args) = pg axs_con_def 
   179                 (bind_fun (lift_defined % (nonlazy args, 
   180                 mk_trp(when_app`(con_app con args) ===
   181                        mk_cfapp(bound_fun n 0,map %# args)))))[
   182                 asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
   183         in mapn one_when 1 cons end;
   184 end;
   185 val when_rews = when_strict::when_apps;
   186 
   187 (* ----- theorems concerning the constructors, discriminators and selectors - *)
   188 
   189 val dis_rews = let
   190   val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
   191                              strict(%%(dis_name con)))) [
   192                                 simp_tac (HOLCF_ss addsimps when_rews) 1]) cons;
   193   val dis_apps = let fun one_dis c (con,args)= pg axs_dis_def
   194                    (lift_defined % (nonlazy args,
   195                         (mk_trp((%%(dis_name c))`(con_app con args) ===
   196                               %%(if con=c then "TT" else "FF"))))) [
   197                                 asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   198         in flat(map (fn (c,_) => map (one_dis c) cons) cons) end;
   199   val dis_defins = map (fn (con,args) => pg [] (defined(%x_name) ==> 
   200                       defined(%%(dis_name con)`%x_name)) [
   201                                 rtac cases 1,
   202                                 contr_tac 1,
   203                                 UNTIL_SOLVED (CHANGED(asm_simp_tac 
   204                                         (HOLCF_ss addsimps dis_apps) 1))]) cons;
   205 in dis_stricts @ dis_defins @ dis_apps end;
   206 
   207 val con_stricts = flat(map (fn (con,args) => map (fn vn =>
   208                         pg (axs_con_def) 
   209                            (mk_trp(con_app2 con (fn arg => if vname arg = vn 
   210                                         then UU else %# arg) args === UU))[
   211                                 asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
   212                         ) (nonlazy args)) cons);
   213 val con_defins = map (fn (con,args) => pg []
   214                         (lift_defined % (nonlazy args,
   215                                 mk_trp(defined(con_app con args)))) ([
   216                           rtac rev_contrapos 1, 
   217                           eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
   218                           asm_simp_tac (HOLCF_ss addsimps dis_rews) 1] )) cons;
   219 val con_rews = con_stricts @ con_defins;
   220 
   221 val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%sel))) [
   222                                 simp_tac (HOLCF_ss addsimps when_rews) 1];
   223 in flat(map (fn (_,args) =>map (fn arg => one_sel (sel_of arg)) args) cons) end;
   224 val sel_apps = let fun one_sel c n sel = map (fn (con,args) => 
   225                 let val nlas = nonlazy args;
   226                     val vns  = map vname args;
   227                 in pg axs_sel_def (lift_defined %
   228                    (filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
   229                                 mk_trp((%%sel)`(con_app con args) === 
   230                                 (if con=c then %(nth_elem(n,vns)) else UU))))
   231                             ( (if con=c then [] 
   232                        else map(case_UU_tac(when_rews@con_stricts)1) nlas)
   233                      @(if con=c andalso ((nth_elem(n,vns)) mem nlas)
   234                                  then[case_UU_tac (when_rews @ con_stricts) 1 
   235                                                   (nth_elem(n,vns))] else [])
   236                      @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
   237 in flat(map  (fn (c,args) => 
   238      flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
   239 val sel_defins = if length cons=1 then map (fn arg => pg [](defined(%x_name)==> 
   240                         defined(%%(sel_of arg)`%x_name)) [
   241                                 rtac cases 1,
   242                                 contr_tac 1,
   243                                 UNTIL_SOLVED (CHANGED(asm_simp_tac 
   244                                              (HOLCF_ss addsimps sel_apps) 1))]) 
   245                  (filter_out is_lazy (snd(hd cons))) else [];
   246 val sel_rews = sel_stricts @ sel_defins @ sel_apps;
   247 
   248 val distincts_le = let
   249     fun dist (con1, args1) (con2, args2) = pg []
   250               (lift_defined % ((nonlazy args1),
   251                         (mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
   252                         rtac rev_contrapos 1,
   253                         eres_inst_tac[("fo",dis_name con1)] monofun_cfun_arg 1]
   254                       @map(case_UU_tac (con_stricts @ dis_rews)1)(nonlazy args2)
   255                       @[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
   256     fun distinct (con1,args1) (con2,args2) =
   257         let val arg1 = (con1, args1)
   258             val arg2 = (con2,
   259 			ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
   260                         (args2, variantlist(map vname args2,map vname args1)))
   261         in [dist arg1 arg2, dist arg2 arg1] end;
   262     fun distincts []      = []
   263     |   distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
   264 in distincts cons end;
   265 val dists_le = flat (flat distincts_le);
   266 val dists_eq = let
   267     fun distinct (_,args1) ((_,args2),leqs) = let
   268         val (le1,le2) = (hd leqs, hd(tl leqs));
   269         val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
   270         if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
   271         if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
   272                                         [eq1, eq2] end;
   273     fun distincts []      = []
   274     |   distincts ((c,leqs)::cs) = List_.concat
   275 	            (ListPair.map (distinct c) ((map #1 cs),leqs)) @
   276 		    distincts cs;
   277     in distincts (cons~~distincts_le) end;
   278 
   279 local 
   280   fun pgterm rel con args = let
   281                 fun append s = upd_vname(fn v => v^s);
   282                 val (largs,rargs) = (args, map (append "'") args);
   283                 in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
   284                       lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
   285                             mk_trp (foldr' mk_conj 
   286                                 (ListPair.map rel
   287 				 (map %# largs, map %# rargs)))))) end;
   288   val cons' = filter (fn (_,args) => args<>[]) cons;
   289 in
   290 val inverts = map (fn (con,args) => 
   291                 pgterm (op <<) con args (flat(map (fn arg => [
   292                                 TRY(rtac conjI 1),
   293                                 dres_inst_tac [("fo",sel_of arg)] monofun_cfun_arg 1,
   294                                 asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
   295                                                       ) args))) cons';
   296 val injects = map (fn ((con,args),inv_thm) => 
   297                            pgterm (op ===) con args [
   298                                 etac (antisym_less_inverse RS conjE) 1,
   299                                 dtac inv_thm 1, REPEAT(atac 1),
   300                                 dtac inv_thm 1, REPEAT(atac 1),
   301                                 TRY(safe_tac HOL_cs),
   302                                 REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
   303                   (cons'~~inverts);
   304 end;
   305 
   306 (* ----- theorems concerning one induction step ----------------------------- *)
   307 
   308 val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [
   309                    asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict,
   310                                                    cfst_strict,csnd_strict]) 1];
   311 val copy_apps = map (fn (con,args) => pg [ax_copy_def]
   312                     (lift_defined % (nonlazy_rec args,
   313                         mk_trp(dc_copy`%"f"`(con_app con args) ===
   314                 (con_app2 con (app_rec_arg (cproj (%"f") (length eqs))) args))))
   315                         (map (case_UU_tac (abs_strict::when_strict::con_stricts)
   316                                  1 o vname)
   317                          (filter (fn a => not (is_rec a orelse is_lazy a)) args)
   318                         @[asm_simp_tac (HOLCF_ss addsimps when_apps) 1,
   319                           simp_tac (HOLCF_ss addsimps axs_con_def) 1]))cons;
   320 val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU`
   321                                         (con_app con args) ===UU))
   322      (let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
   323                          in map (case_UU_tac rews 1) (nonlazy args) @ [
   324                              asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
   325                         (filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
   326 val copy_rews = copy_strict::copy_apps @ copy_stricts;
   327 
   328 in     (iso_rews, exhaust, cases, when_rews,
   329         con_rews, sel_rews, dis_rews, dists_le, dists_eq, inverts, injects,
   330         copy_rews)
   331 end; (* let *)
   332 
   333 
   334 fun comp_theorems thy (comp_dnam, eqs: eq list, thmss: thms list) =
   335 let
   336 val casess    =       map #3  thmss;
   337 val con_rews  = flat (map #5  thmss);
   338 val copy_rews = flat (map #12 thmss);
   339 val dnames = map (fst o fst) eqs;
   340 val conss  = map  snd        eqs;
   341 val comp_dname = Sign.full_name (sign_of thy) comp_dnam;
   342 
   343 val d = writeln("Proving induction   properties of domain "^comp_dname^" ...");
   344 val pg = pg' thy;
   345 
   346 (* ----- getting the composite axiom and definitions ------------------------ *)
   347 
   348 local val ga = get_axiom thy in
   349 val axs_reach      = map (fn dn => ga (dn ^  "_reach"   )) dnames;
   350 val axs_take_def   = map (fn dn => ga (dn ^  "_take_def")) dnames;
   351 val axs_finite_def = map (fn dn => ga (dn ^"_finite_def")) dnames;
   352 val ax_copy2_def   = ga (comp_dname^ "_copy_def");
   353 val ax_bisim_def   = ga (comp_dname^"_bisim_def");
   354 end; (* local *)
   355 
   356 fun dc_take dn = %%(dn^"_take");
   357 val x_name = idx_name dnames "x"; 
   358 val P_name = idx_name dnames "P";
   359 val n_eqs = length eqs;
   360 
   361 (* ----- theorems concerning finite approximation and finite induction ------ *)
   362 
   363 local
   364   val iterate_Cprod_ss = simpset_of "Fix"
   365                          addsimps [cfst_strict, csnd_strict]addsimps Cprod_rews;
   366   val copy_con_rews  = copy_rews @ con_rews;
   367   val copy_take_defs =(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
   368   val take_stricts=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=>
   369             strict(dc_take dn $ %"n")) eqs))) ([
   370 			if n_eqs = 1 then all_tac else rewtac ax_copy2_def,
   371                         nat_ind_tac "n" 1,
   372                          simp_tac iterate_Cprod_ss 1,
   373                         asm_simp_tac (iterate_Cprod_ss addsimps copy_rews)1]);
   374   val take_stricts' = rewrite_rule copy_take_defs take_stricts;
   375   val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn $ %%"0")
   376                                                         `%x_name n === UU))[
   377                                 simp_tac iterate_Cprod_ss 1]) 1 dnames;
   378   val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1;
   379   val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj 
   380             (flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all 
   381         (map vname args,(dc_take dn $ (%%"Suc" $ %"n"))`(con_app con args) ===
   382          con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))$ %"n"))
   383                               args)) cons) eqs)))) ([
   384                                 simp_tac iterate_Cprod_ss 1,
   385                                 nat_ind_tac "n" 1,
   386                             simp_tac(iterate_Cprod_ss addsimps copy_con_rews) 1,
   387                                 asm_full_simp_tac (HOLCF_ss addsimps 
   388                                       (filter (has_fewer_prems 1) copy_rews)) 1,
   389                                 TRY(safe_tac HOL_cs)] @
   390                         (flat(map (fn ((dn,_),cons) => map (fn (con,args) => 
   391                                 if nonlazy_rec args = [] then all_tac else
   392                                 EVERY(map c_UU_tac (nonlazy_rec args)) THEN
   393                                 asm_full_simp_tac (HOLCF_ss addsimps copy_rews)1
   394                                                            ) cons) eqs)));
   395 in
   396 val take_rews = atomize take_stricts @ take_0s @ atomize take_apps;
   397 end; (* local *)
   398 
   399 local
   400   fun one_con p (con,args) = foldr mk_All (map vname args,
   401         lift_defined (bound_arg (map vname args)) (nonlazy args,
   402         lift (fn arg => %(P_name (1+rec_of arg)) $ bound_arg args arg)
   403          (filter is_rec args,mk_trp(%p $ con_app2 con (bound_arg args) args))));
   404   fun one_eq ((p,cons),concl) = (mk_trp(%p $ UU) ===> 
   405                            foldr (op ===>) (map (one_con p) cons,concl));
   406   fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss,
   407                         mk_trp(foldr' mk_conj (mapn concf 1 dnames)));
   408   val take_ss = HOL_ss addsimps take_rews;
   409   fun quant_tac i = EVERY(mapn(fn n=> fn _=> res_inst_tac[("x",x_name n)]spec i)
   410                                1 dnames);
   411   fun ind_prems_tac prems = EVERY(flat (map (fn cons => (
   412                                      resolve_tac prems 1 ::
   413                                      flat (map (fn (_,args) => 
   414                                        resolve_tac prems 1 ::
   415                                        map (K(atac 1)) (nonlazy args) @
   416                                        map (K(atac 1)) (filter is_rec args))
   417                                      cons))) conss));
   418   local 
   419     (* check whether every/exists constructor of the n-th part of the equation:
   420        it has a possibly indirectly recursive argument that isn't/is possibly 
   421        indirectly lazy *)
   422     fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
   423           is_rec arg andalso not(rec_of arg mem ns) andalso
   424           ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
   425             rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
   426               (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
   427           ) o snd) cons;
   428     fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
   429     fun warn (n,cons)  = if all_rec_to [] false (n,cons) then (warning
   430         ("domain "^nth_elem(n,dnames)^" is empty!"); true) else false;
   431     fun lazy_rec_to ns = rec_to exists Id  lazy_rec_to ns;
   432 
   433   in val n__eqs     = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
   434      val is_emptys = map warn n__eqs;
   435      val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
   436   end;
   437 in (* local *)
   438 val finite_ind = pg'' thy [] (ind_term (fn n => fn dn => %(P_name n)$
   439                              (dc_take dn $ %"n" `%(x_name n)))) (fn prems => [
   440                                 quant_tac 1,
   441                                 simp_tac HOL_ss 1,
   442                                 nat_ind_tac "n" 1,
   443                                 simp_tac (take_ss addsimps prems) 1,
   444                                 TRY(safe_tac HOL_cs)]
   445                                 @ flat(map (fn (cons,cases) => [
   446                                  res_inst_tac [("x","x")] cases 1,
   447                                  asm_simp_tac (take_ss addsimps prems) 1]
   448                                  @ flat(map (fn (con,args) => 
   449                                   asm_simp_tac take_ss 1 ::
   450                                   map (fn arg =>
   451                                    case_UU_tac (prems@con_rews) 1 (
   452                            nth_elem(rec_of arg,dnames)^"_take n`"^vname arg))
   453                                   (filter is_nonlazy_rec args) @ [
   454                                   resolve_tac prems 1] @
   455                                   map (K (atac 1))      (nonlazy args) @
   456                                   map (K (etac spec 1)) (filter is_rec args)) 
   457                                  cons))
   458                                 (conss~~casess)));
   459 
   460 val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n",
   461                 mk_trp(dc_take dn $ Bound 0 `%(x_name n) === 
   462                        dc_take dn $ Bound 0 `%(x_name n^"'")))
   463            ===> mk_trp(%(x_name n) === %(x_name n^"'"))) (fn prems => [
   464                         res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
   465                         res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
   466                                 stac fix_def2 1,
   467                                 REPEAT(CHANGED(rtac(contlub_cfun_arg RS ssubst)1
   468                                                THEN chain_tac 1)),
   469                                 stac contlub_cfun_fun 1,
   470                                 stac contlub_cfun_fun 2,
   471                                 rtac lub_equal 3,
   472                                 chain_tac 1,
   473                                 rtac allI 1,
   474                                 resolve_tac prems 1])) 1 (dnames~~axs_reach);
   475 
   476 (* ----- theorems concerning finiteness and induction ----------------------- *)
   477 
   478 val (finites,ind) = if is_finite then
   479   let 
   480     fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %"x" === %"x");
   481     val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%"x")) ===> 
   482         mk_trp(mk_disj(mk_all("n",dc_take dn $ Bound 0 ` %"x" === UU),
   483         take_enough dn)) ===> mk_trp(take_enough dn)) [
   484                                 etac disjE 1,
   485                                 etac notE 1,
   486                                 resolve_tac take_lemmas 1,
   487                                 asm_simp_tac take_ss 1,
   488                                 atac 1]) dnames;
   489     val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn 
   490         (fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
   491          mk_disj(dc_take dn $ Bound 1 ` Bound 0 === UU,
   492                  dc_take dn $ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
   493                                 rtac allI 1,
   494                                 nat_ind_tac "n" 1,
   495                                 simp_tac take_ss 1,
   496                         TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @
   497                                 flat(mapn (fn n => fn (cons,cases) => [
   498                                   simp_tac take_ss 1,
   499                                   rtac allI 1,
   500                                   res_inst_tac [("x",x_name n)] cases 1,
   501                                   asm_simp_tac take_ss 1] @ 
   502                                   flat(map (fn (con,args) => 
   503                                     asm_simp_tac take_ss 1 ::
   504                                     flat(map (fn vn => [
   505                                       eres_inst_tac [("x",vn)] all_dupE 1,
   506                                       etac disjE 1,
   507                                       asm_simp_tac (HOL_ss addsimps con_rews) 1,
   508                                       asm_simp_tac take_ss 1])
   509                                     (nonlazy_rec args)))
   510                                   cons))
   511                                 1 (conss~~casess)));
   512     val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp(
   513                                                 %%(dn^"_finite") $ %"x"))[
   514                                 case_UU_tac take_rews 1 "x",
   515                                 eresolve_tac finite_lemmas1a 1,
   516                                 step_tac HOL_cs 1,
   517                                 step_tac HOL_cs 1,
   518                                 cut_facts_tac [l1b] 1,
   519                         fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
   520   in
   521   (finites,
   522    pg'' thy[](ind_term (fn n => fn dn => %(P_name n) $ %(x_name n)))(fn prems =>
   523                                 TRY(safe_tac HOL_cs) ::
   524                          flat (map (fn (finite,fin_ind) => [
   525                                rtac(rewrite_rule axs_finite_def finite RS exE)1,
   526                                 etac subst 1,
   527                                 rtac fin_ind 1,
   528                                 ind_prems_tac prems]) 
   529                                    (finites~~(atomize finite_ind)) ))
   530 ) end (* let *) else
   531   (mapn (fn n => fn dn => read_instantiate_sg (sign_of thy) 
   532                     [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
   533    pg'' thy [] (foldr (op ===>) (mapn (fn n => K(mk_trp(%%"adm" $ %(P_name n))))
   534                1 dnames, ind_term (fn n => fn dn => %(P_name n) $ %(x_name n))))
   535                    (fn prems => map (fn ax_reach => rtac (ax_reach RS subst) 1) 
   536                                     axs_reach @ [
   537                                 quant_tac 1,
   538                                 rtac (adm_impl_admw RS wfix_ind) 1,
   539                                  REPEAT_DETERM(rtac adm_all2 1),
   540                                  REPEAT_DETERM(TRY(rtac adm_conj 1) THEN 
   541                                                    rtac adm_subst 1 THEN 
   542                                         cont_tacR 1 THEN resolve_tac prems 1),
   543                                 strip_tac 1,
   544                                 rtac (rewrite_rule axs_take_def finite_ind) 1,
   545                                 ind_prems_tac prems])
   546 )
   547 end; (* local *)
   548 
   549 (* ----- theorem concerning coinduction ------------------------------------- *)
   550 
   551 local
   552   val xs = mapn (fn n => K (x_name n)) 1 dnames;
   553   fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
   554   val take_ss = HOL_ss addsimps take_rews;
   555   val sproj   = prj (fn s => "fst("^s^")") (fn s => "snd("^s^")");
   556   val coind_lemma=pg[ax_bisim_def](mk_trp(mk_imp(%%(comp_dname^"_bisim") $ %"R",
   557                 foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
   558                   foldr mk_imp (mapn (fn n => K(proj (%"R") n_eqs n $ 
   559                                       bnd_arg n 0 $ bnd_arg n 1)) 0 dnames,
   560                     foldr' mk_conj (mapn (fn n => fn dn => 
   561                                 (dc_take dn $ %"n" `bnd_arg n 0 === 
   562                                 (dc_take dn $ %"n" `bnd_arg n 1)))0 dnames))))))
   563                              ([ rtac impI 1,
   564                                 nat_ind_tac "n" 1,
   565                                 simp_tac take_ss 1,
   566                                 safe_tac HOL_cs] @
   567                                 flat(mapn (fn n => fn x => [
   568                                   rotate_tac (n+1) 1,
   569                                   etac all2E 1,
   570                                   eres_inst_tac [("P1", sproj "R" n_eqs n^
   571                                         " "^x^" "^x^"'")](mp RS disjE) 1,
   572                                   TRY(safe_tac HOL_cs),
   573                                   REPEAT(CHANGED(asm_simp_tac take_ss 1))]) 
   574                                 0 xs));
   575 in
   576 val coind = pg [] (mk_trp(%%(comp_dname^"_bisim") $ %"R") ===>
   577                 foldr (op ===>) (mapn (fn n => fn x => 
   578                   mk_trp(proj (%"R") n_eqs n $ %x $ %(x^"'"))) 0 xs,
   579                   mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
   580                                 TRY(safe_tac HOL_cs)] @
   581                                 flat(map (fn take_lemma => [
   582                                   rtac take_lemma 1,
   583                                   cut_facts_tac [coind_lemma] 1,
   584                                   fast_tac HOL_cs 1])
   585                                 take_lemmas));
   586 end; (* local *)
   587 
   588 
   589 in (take_rews, take_lemmas, finites, finite_ind, ind, coind)
   590 
   591 end; (* let *)
   592 end; (* local *)
   593 end; (* struct *)