src/HOLCF/domain/theorems.ML
author oheimb
Fri Jan 28 11:23:41 2000 +0100 (2000-01-28)
changeset 8149 941afb897532
parent 7906 0576dad973b1
child 8438 b8389b4fca9c
permissions -rw-r--r--
added tacticals DETERM_UNTIL and DETERM_UNTIL_SOLVED
     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 = #1 (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 local val trueI2 = prove_goal HOL.thy"f~=x ==> True"(fn _ => [rtac TrueI 1]) in
    31 val kill_neq_tac = dtac trueI2 end;
    32 fun case_UU_tac rews i v =      case_tac (v^"=UU") i THEN
    33                                 asm_simp_tac (HOLCF_ss addsimps rews) i;
    34 
    35 val chain_tac = REPEAT_DETERM o resolve_tac 
    36                 [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL];
    37 
    38 (* ----- general proofs ----------------------------------------------------- *)
    39 
    40 val all2E = prove_goal HOL.thy "[| !x y . P x y; P x y ==> R |] ==> R"
    41  (fn prems =>[
    42                                 resolve_tac prems 1,
    43                                 cut_facts_tac prems 1,
    44                                 fast_tac HOL_cs 1]);
    45 
    46 val dist_eqI = prove_goal Porder.thy "~(x::'a::po) << y ==> x ~= y" (fn prems => [
    47                                 rtac rev_contrapos 1,
    48                                 etac (antisym_less_inverse RS conjunct1) 1,
    49                                 resolve_tac prems 1]);
    50 (*
    51 infixr 0 y;
    52 val b = 0;
    53 fun _ y t = by t;
    54 fun g defs t = let val sg = sign_of thy;
    55                      val ct = Thm.cterm_of sg (inferT sg t);
    56                  in goalw_cterm defs ct end;
    57 *)
    58 
    59 in
    60 
    61 fun theorems (((dname,_),cons) : eq, eqs : eq list) thy =
    62 let
    63 
    64 val dummy = writeln ("Proving isomorphism properties of domain "^dname^" ...");
    65 val pg = pg' thy;
    66 
    67 
    68 (* ----- getting the axioms and definitions --------------------------------- *)
    69 
    70 local fun ga s dn = get_thm thy (dn^"."^s) in
    71 val ax_abs_iso    = ga "abs_iso"  dname;
    72 val ax_rep_iso    = ga "rep_iso"  dname;
    73 val ax_when_def   = ga "when_def" dname;
    74 val axs_con_def   = map (fn (con,_) => ga (extern_name con^"_def") dname) cons;
    75 val axs_dis_def   = map (fn (con,_) => ga (   dis_name con^"_def") dname) cons;
    76 val axs_sel_def   = flat(map (fn (_,args) => 
    77                     map (fn     arg => ga (sel_of arg     ^"_def") dname) args)
    78 									  cons);
    79 val ax_copy_def   = ga "copy_def" dname;
    80 end; (* local *)
    81 
    82 (* ----- theorems concerning the isomorphism -------------------------------- *)
    83 
    84 val dc_abs  = %%(dname^"_abs");
    85 val dc_rep  = %%(dname^"_rep");
    86 val dc_copy = %%(dname^"_copy");
    87 val x_name = "x";
    88 
    89 val (rep_strict, abs_strict) = let 
    90          val r = ax_rep_iso RS (ax_abs_iso RS (allI  RSN(2,allI RS iso_strict)))
    91                in (r RS conjunct1, r RS conjunct2) end;
    92 val abs_defin' = pg [] ((dc_abs`%x_name === UU) ==> (%x_name === UU)) [
    93                            res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
    94                                 etac ssubst 1, rtac rep_strict 1];
    95 val rep_defin' = pg [] ((dc_rep`%x_name === UU) ==> (%x_name === UU)) [
    96                            res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
    97                                 etac ssubst 1, rtac abs_strict 1];
    98 val iso_rews = [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
    99 
   100 local 
   101 val iso_swap = pg [] (dc_rep`%"x" === %"y" ==> %"x" === dc_abs`%"y") [
   102                             dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
   103                             etac (ax_rep_iso RS subst) 1];
   104 fun exh foldr1 cn quant foldr2 var = let
   105   fun one_con (con,args) = let val vns = map vname args in
   106     foldr quant (vns, foldr2 ((%x_name === con_app2 con (var vns) vns)::
   107                               map (defined o (var vns)) (nonlazy args))) end
   108   in foldr1 ((cn(%x_name===UU))::map one_con cons) end;
   109 in
   110 val casedist = let 
   111             fun common_tac thm = rtac thm 1 THEN contr_tac 1;
   112             fun unit_tac true = common_tac upE1
   113             |   unit_tac _    = all_tac;
   114             fun prod_tac []          = common_tac oneE
   115             |   prod_tac [arg]       = unit_tac (is_lazy arg)
   116             |   prod_tac (arg::args) = 
   117                                 common_tac sprodE THEN
   118                                 kill_neq_tac 1 THEN
   119                                 unit_tac (is_lazy arg) THEN
   120                                 prod_tac args;
   121             fun sum_rest_tac p = SELECT_GOAL(EVERY[
   122                                 rtac p 1,
   123                                 rewrite_goals_tac axs_con_def,
   124                                 dtac iso_swap 1,
   125                                 simp_tac HOLCF_ss 1,
   126                                 DETERM_UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
   127             fun sum_tac [(_,args)]       [p]        = 
   128                                 prod_tac args THEN sum_rest_tac p
   129             |   sum_tac ((_,args)::cons') (p::prems) = DETERM(
   130                                 common_tac ssumE THEN
   131                                 kill_neq_tac 1 THEN kill_neq_tac 2 THEN
   132                                 prod_tac args THEN sum_rest_tac p) THEN
   133                                 sum_tac cons' prems
   134             |   sum_tac _ _ = Imposs "theorems:sum_tac";
   135           in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%"P")))
   136                               (fn T => T ==> %"P") mk_All
   137                               (fn l => foldr (op ===>) (map mk_trp l,
   138                                                             mk_trp(%"P")))
   139                               bound_arg)
   140                              (fn prems => [
   141                                 cut_facts_tac [excluded_middle] 1,
   142                                 etac disjE 1,
   143                                 rtac (hd prems) 2,
   144                                 etac rep_defin' 2,
   145                                 if length cons = 1 andalso 
   146                                    length (snd(hd cons)) = 1 andalso 
   147                                    not(is_lazy(hd(snd(hd cons))))
   148                                 then rtac (hd (tl prems)) 1 THEN atac 2 THEN
   149                                      rewrite_goals_tac axs_con_def THEN
   150                                      simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
   151                                 else sum_tac cons (tl prems)])end;
   152 val exhaust= pg[](mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %)))[
   153                                 rtac casedist 1,
   154                                 DETERM_UNTIL_SOLVED(fast_tac HOL_cs 1)];
   155 end;
   156 
   157 local 
   158   fun bind_fun t = foldr mk_All (when_funs cons,t);
   159   fun bound_fun i _ = Bound (length cons - i);
   160   val when_app  = foldl (op `) (%%(dname^"_when"), mapn bound_fun 1 cons);
   161   val when_appl = pg [ax_when_def] (bind_fun(mk_trp(when_app`%x_name ===
   162              when_body cons (fn (m,n)=> bound_fun (n-m) 0)`(dc_rep`%x_name))))[
   163                                 simp_tac HOLCF_ss 1];
   164 in
   165 val when_strict = pg [] (bind_fun(mk_trp(strict when_app))) [
   166                         simp_tac(HOLCF_ss addsimps [when_appl,rep_strict]) 1];
   167 val when_apps = let fun one_when n (con,args) = pg axs_con_def 
   168                 (bind_fun (lift_defined % (nonlazy args, 
   169                 mk_trp(when_app`(con_app con args) ===
   170                        mk_cRep_CFun(bound_fun n 0,map %# args)))))[
   171                 asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
   172         in mapn one_when 1 cons end;
   173 end;
   174 val when_rews = when_strict::when_apps;
   175 
   176 (* ----- theorems concerning the constructors, discriminators and selectors - *)
   177 
   178 val dis_rews = let
   179   val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
   180                              strict(%%(dis_name con)))) [
   181                                 simp_tac (HOLCF_ss addsimps when_rews) 1]) cons;
   182   val dis_apps = let fun one_dis c (con,args)= pg axs_dis_def
   183                    (lift_defined % (nonlazy args,
   184                         (mk_trp((%%(dis_name c))`(con_app con args) ===
   185                               %%(if con=c then "TT" else "FF"))))) [
   186                                 asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   187         in flat(map (fn (c,_) => map (one_dis c) cons) cons) end;
   188   val dis_defins = map (fn (con,args) => pg [] (defined(%x_name) ==> 
   189                       defined(%%(dis_name con)`%x_name)) [
   190                                 rtac casedist 1,
   191                                 contr_tac 1,
   192                                 DETERM_UNTIL_SOLVED (CHANGED(asm_simp_tac 
   193                                         (HOLCF_ss addsimps dis_apps) 1))]) cons;
   194 in dis_stricts @ dis_defins @ dis_apps end;
   195 
   196 val con_stricts = flat(map (fn (con,args) => map (fn vn =>
   197                         pg (axs_con_def) 
   198                            (mk_trp(con_app2 con (fn arg => if vname arg = vn 
   199                                         then UU else %# arg) args === UU))[
   200                                 asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
   201                         ) (nonlazy args)) cons);
   202 val con_defins = map (fn (con,args) => pg []
   203                         (lift_defined % (nonlazy args,
   204                                 mk_trp(defined(con_app con args)))) ([
   205                           rtac rev_contrapos 1, 
   206                           eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
   207                           asm_simp_tac (HOLCF_ss addsimps dis_rews) 1] )) cons;
   208 val con_rews = con_stricts @ con_defins;
   209 
   210 val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%sel))) [
   211                                 simp_tac (HOLCF_ss addsimps when_rews) 1];
   212 in flat(map (fn (_,args) =>map (fn arg => one_sel (sel_of arg)) args) cons) end;
   213 val sel_apps = let fun one_sel c n sel = map (fn (con,args) => 
   214                 let val nlas = nonlazy args;
   215                     val vns  = map vname args;
   216                 in pg axs_sel_def (lift_defined %
   217                    (filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
   218                                 mk_trp((%%sel)`(con_app con args) === 
   219                                 (if con=c then %(nth_elem(n,vns)) else UU))))
   220                             ( (if con=c then [] 
   221                        else map(case_UU_tac(when_rews@con_stricts)1) nlas)
   222                      @(if con=c andalso ((nth_elem(n,vns)) mem nlas)
   223                                  then[case_UU_tac (when_rews @ con_stricts) 1 
   224                                                   (nth_elem(n,vns))] else [])
   225                      @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
   226 in flat(map  (fn (c,args) => 
   227      flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
   228 val sel_defins = if length cons=1 then map (fn arg => pg [](defined(%x_name)==> 
   229                         defined(%%(sel_of arg)`%x_name)) [
   230                                 rtac casedist 1,
   231                                 contr_tac 1,
   232                                 DETERM_UNTIL_SOLVED (CHANGED(asm_simp_tac 
   233                                              (HOLCF_ss addsimps sel_apps) 1))]) 
   234                  (filter_out is_lazy (snd(hd cons))) else [];
   235 val sel_rews = sel_stricts @ sel_defins @ sel_apps;
   236 
   237 val distincts_le = let
   238     fun dist (con1, args1) (con2, args2) = pg []
   239               (lift_defined % ((nonlazy args1),
   240                         (mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
   241                         rtac rev_contrapos 1,
   242                         eres_inst_tac[("fo",dis_name con1)] monofun_cfun_arg 1]
   243                       @map(case_UU_tac (con_stricts @ dis_rews)1)(nonlazy args2)
   244                       @[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
   245     fun distinct (con1,args1) (con2,args2) =
   246         let val arg1 = (con1, args1)
   247             val arg2 = (con2,
   248 			ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
   249                         (args2, variantlist(map vname args2,map vname args1)))
   250         in [dist arg1 arg2, dist arg2 arg1] end;
   251     fun distincts []      = []
   252     |   distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
   253 in distincts cons end;
   254 val dist_les = flat (flat distincts_le);
   255 val dist_eqs = let
   256     fun distinct (_,args1) ((_,args2),leqs) = let
   257         val (le1,le2) = (hd leqs, hd(tl leqs));
   258         val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
   259         if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
   260         if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
   261                                         [eq1, eq2] end;
   262     open BasisLibrary (*restore original List*)
   263     fun distincts []      = []
   264     |   distincts ((c,leqs)::cs) = List.concat
   265 	            (ListPair.map (distinct c) ((map #1 cs),leqs)) @
   266 		    distincts cs;
   267     in distincts (cons~~distincts_le) end;
   268 
   269 local 
   270   fun pgterm rel con args = let
   271                 fun append s = upd_vname(fn v => v^s);
   272                 val (largs,rargs) = (args, map (append "'") args);
   273                 in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
   274                       lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
   275                             mk_trp (foldr' mk_conj 
   276                                 (ListPair.map rel
   277 				 (map %# largs, map %# rargs)))))) end;
   278   val cons' = filter (fn (_,args) => args<>[]) cons;
   279 in
   280 val inverts = map (fn (con,args) => 
   281                 pgterm (op <<) con args (flat(map (fn arg => [
   282                                 TRY(rtac conjI 1),
   283                                 dres_inst_tac [("fo",sel_of arg)] monofun_cfun_arg 1,
   284                                 asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
   285                                                       ) args))) cons';
   286 val injects = map (fn ((con,args),inv_thm) => 
   287                            pgterm (op ===) con args [
   288                                 etac (antisym_less_inverse RS conjE) 1,
   289                                 dtac inv_thm 1, REPEAT(atac 1),
   290                                 dtac inv_thm 1, REPEAT(atac 1),
   291                                 TRY(safe_tac HOL_cs),
   292                                 REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
   293                   (cons'~~inverts);
   294 end;
   295 
   296 (* ----- theorems concerning one induction step ----------------------------- *)
   297 
   298 val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [
   299                    asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict,
   300                                                    cfst_strict,csnd_strict]) 1];
   301 val copy_apps = map (fn (con,args) => pg [ax_copy_def]
   302                     (lift_defined % (nonlazy_rec args,
   303                         mk_trp(dc_copy`%"f"`(con_app con args) ===
   304                 (con_app2 con (app_rec_arg (cproj (%"f") eqs)) args))))
   305                         (map (case_UU_tac (abs_strict::when_strict::con_stricts)
   306                                  1 o vname)
   307                          (filter (fn a => not (is_rec a orelse is_lazy a)) args)
   308                         @[asm_simp_tac (HOLCF_ss addsimps when_apps) 1,
   309                           simp_tac (HOLCF_ss addsimps axs_con_def) 1]))cons;
   310 val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU`
   311                                         (con_app con args) ===UU))
   312      (let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
   313                          in map (case_UU_tac rews 1) (nonlazy args) @ [
   314                              asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
   315                         (filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
   316 val copy_rews = copy_strict::copy_apps @ copy_stricts;
   317 in thy |> Theory.add_path (Sign.base_name dname)
   318        |> (PureThy.add_thmss o map Thm.no_attributes) [
   319 		("iso_rews" , iso_rews  ),
   320 		("exhaust"  , [exhaust] ),
   321 		("casedist" , [casedist]),
   322 		("when_rews", when_rews ),
   323 		("con_rews", con_rews),
   324 		("sel_rews", sel_rews),
   325 		("dis_rews", dis_rews),
   326 		("dist_les", dist_les),
   327 		("dist_eqs", dist_eqs),
   328 		("inverts" , inverts ),
   329 		("injects" , injects ),
   330 		("copy_rews", copy_rews)]
   331        |> Theory.parent_path
   332 end; (* let *)
   333 
   334 fun comp_theorems (comp_dnam, eqs: eq list) thy =
   335 let
   336 val dnames = map (fst o fst) eqs;
   337 val conss  = map  snd        eqs;
   338 val comp_dname = Sign.full_name (sign_of thy) comp_dnam;
   339 
   340 val d = writeln("Proving induction   properties of domain "^comp_dname^" ...");
   341 val pg = pg' thy;
   342 
   343 (* ----- getting the composite axiom and definitions ------------------------ *)
   344 
   345 local fun ga s dn = get_thm thy (dn^"."^s) in
   346 val axs_reach      = map (ga "reach"     ) dnames;
   347 val axs_take_def   = map (ga "take_def"  ) dnames;
   348 val axs_finite_def = map (ga "finite_def") dnames;
   349 val ax_copy2_def   =      ga "copy_def"  comp_dnam;
   350 val ax_bisim_def   =      ga "bisim_def" comp_dnam;
   351 end; (* local *)
   352 
   353 local fun gt  s dn = get_thm  thy (dn^"."^s);
   354       fun gts s dn = get_thms thy (dn^"."^s) in
   355 val cases     =       map (gt  "casedist" ) dnames;
   356 val con_rews  = flat (map (gts "con_rews" ) dnames);
   357 val copy_rews = flat (map (gts "copy_rews") dnames);
   358 end; (* local *)
   359 
   360 fun dc_take dn = %%(dn^"_take");
   361 val x_name = idx_name dnames "x"; 
   362 val P_name = idx_name dnames "P";
   363 val n_eqs = length eqs;
   364 
   365 (* ----- theorems concerning finite approximation and finite induction ------ *)
   366 
   367 local
   368   val iterate_Cprod_ss = simpset_of Fix.thy
   369                          addsimps [cfst_strict, csnd_strict]addsimps Cprod_rews;
   370   val copy_con_rews  = copy_rews @ con_rews;
   371   val copy_take_defs =(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
   372   val take_stricts=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=>
   373             strict(dc_take dn $ %"n")) eqs))) ([
   374                         nat_ind_tac "n" 1,
   375                          simp_tac iterate_Cprod_ss 1,
   376                         asm_simp_tac (iterate_Cprod_ss addsimps copy_rews)1]);
   377   val take_stricts' = rewrite_rule copy_take_defs take_stricts;
   378   val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn $ %%"0")
   379                                                         `%x_name n === UU))[
   380                                 simp_tac iterate_Cprod_ss 1]) 1 dnames;
   381   val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1;
   382   val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj 
   383             (flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all 
   384         (map vname args,(dc_take dn $ (%%"Suc" $ %"n"))`(con_app con args) ===
   385          con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))$ %"n"))
   386                               args)) cons) eqs)))) ([
   387                                 simp_tac iterate_Cprod_ss 1,
   388                                 nat_ind_tac "n" 1,
   389                             simp_tac(iterate_Cprod_ss addsimps copy_con_rews) 1,
   390                                 asm_full_simp_tac (HOLCF_ss addsimps 
   391                                       (filter (has_fewer_prems 1) copy_rews)) 1,
   392                                 TRY(safe_tac HOL_cs)] @
   393                         (flat(map (fn ((dn,_),cons) => map (fn (con,args) => 
   394                                 if nonlazy_rec args = [] then all_tac else
   395                                 EVERY(map c_UU_tac (nonlazy_rec args)) THEN
   396                                 asm_full_simp_tac (HOLCF_ss addsimps copy_rews)1
   397                                                            ) cons) eqs)));
   398 in
   399 val take_rews = atomize take_stricts @ take_0s @ atomize take_apps;
   400 end; (* local *)
   401 
   402 local
   403   fun one_con p (con,args) = foldr mk_All (map vname args,
   404         lift_defined (bound_arg (map vname args)) (nonlazy args,
   405         lift (fn arg => %(P_name (1+rec_of arg)) $ bound_arg args arg)
   406          (filter is_rec args,mk_trp(%p $ con_app2 con (bound_arg args) args))));
   407   fun one_eq ((p,cons),concl) = (mk_trp(%p $ UU) ===> 
   408                            foldr (op ===>) (map (one_con p) cons,concl));
   409   fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss,
   410                         mk_trp(foldr' mk_conj (mapn concf 1 dnames)));
   411   val take_ss = HOL_ss addsimps take_rews;
   412   fun quant_tac i = EVERY(mapn(fn n=> fn _=> res_inst_tac[("x",x_name n)]spec i)
   413                                1 dnames);
   414   fun ind_prems_tac prems = EVERY(flat (map (fn cons => (
   415                                      resolve_tac prems 1 ::
   416                                      flat (map (fn (_,args) => 
   417                                        resolve_tac prems 1 ::
   418                                        map (K(atac 1)) (nonlazy args) @
   419                                        map (K(atac 1)) (filter is_rec args))
   420                                      cons))) conss));
   421   local 
   422     (* check whether every/exists constructor of the n-th part of the equation:
   423        it has a possibly indirectly recursive argument that isn't/is possibly 
   424        indirectly lazy *)
   425     fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
   426           is_rec arg andalso not(rec_of arg mem ns) andalso
   427           ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
   428             rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
   429               (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
   430           ) o snd) cons;
   431     fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
   432     fun warn (n,cons)  = if all_rec_to [] false (n,cons) then (warning
   433         ("domain "^nth_elem(n,dnames)^" is empty!"); true) else false;
   434     fun lazy_rec_to ns = rec_to exists Id  lazy_rec_to ns;
   435 
   436   in val n__eqs     = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
   437      val is_emptys = map warn n__eqs;
   438      val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
   439   end;
   440 in (* local *)
   441 val finite_ind = pg'' thy [] (ind_term (fn n => fn dn => %(P_name n)$
   442                              (dc_take dn $ %"n" `%(x_name n)))) (fn prems => [
   443                                 quant_tac 1,
   444                                 simp_tac HOL_ss 1,
   445                                 nat_ind_tac "n" 1,
   446                                 simp_tac (take_ss addsimps prems) 1,
   447                                 TRY(safe_tac HOL_cs)]
   448                                 @ flat(map (fn (cons,cases) => [
   449                                  res_inst_tac [("x","x")] cases 1,
   450                                  asm_simp_tac (take_ss addsimps prems) 1]
   451                                  @ flat(map (fn (con,args) => 
   452                                   asm_simp_tac take_ss 1 ::
   453                                   map (fn arg =>
   454                                    case_UU_tac (prems@con_rews) 1 (
   455                            nth_elem(rec_of arg,dnames)^"_take n`"^vname arg))
   456                                   (filter is_nonlazy_rec args) @ [
   457                                   resolve_tac prems 1] @
   458                                   map (K (atac 1))      (nonlazy args) @
   459                                   map (K (etac spec 1)) (filter is_rec args)) 
   460                                  cons))
   461                                 (conss~~cases)));
   462 
   463 val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n",
   464                 mk_trp(dc_take dn $ Bound 0 `%(x_name n) === 
   465                        dc_take dn $ Bound 0 `%(x_name n^"'")))
   466            ===> mk_trp(%(x_name n) === %(x_name n^"'"))) (fn prems => [
   467                         res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
   468                         res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
   469                                 stac fix_def2 1,
   470                                 REPEAT(CHANGED(rtac(contlub_cfun_arg RS ssubst)1
   471                                                THEN chain_tac 1)),
   472                                 stac contlub_cfun_fun 1,
   473                                 stac contlub_cfun_fun 2,
   474                                 rtac lub_equal 3,
   475                                 chain_tac 1,
   476                                 rtac allI 1,
   477                                 resolve_tac prems 1])) 1 (dnames~~axs_reach);
   478 
   479 (* ----- theorems concerning finiteness and induction ----------------------- *)
   480 
   481 val (finites,ind) = if is_finite then
   482   let 
   483     fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %"x" === %"x");
   484     val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%"x")) ===> 
   485         mk_trp(mk_disj(mk_all("n",dc_take dn $ Bound 0 ` %"x" === UU),
   486         take_enough dn)) ===> mk_trp(take_enough dn)) [
   487                                 etac disjE 1,
   488                                 etac notE 1,
   489                                 resolve_tac take_lemmas 1,
   490                                 asm_simp_tac take_ss 1,
   491                                 atac 1]) dnames;
   492     val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn 
   493         (fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
   494          mk_disj(dc_take dn $ Bound 1 ` Bound 0 === UU,
   495                  dc_take dn $ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
   496                                 rtac allI 1,
   497                                 nat_ind_tac "n" 1,
   498                                 simp_tac take_ss 1,
   499                         TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @
   500                                 flat(mapn (fn n => fn (cons,cases) => [
   501                                   simp_tac take_ss 1,
   502                                   rtac allI 1,
   503                                   res_inst_tac [("x",x_name n)] cases 1,
   504                                   asm_simp_tac take_ss 1] @ 
   505                                   flat(map (fn (con,args) => 
   506                                     asm_simp_tac take_ss 1 ::
   507                                     flat(map (fn vn => [
   508                                       eres_inst_tac [("x",vn)] all_dupE 1,
   509                                       etac disjE 1,
   510                                       asm_simp_tac (HOL_ss addsimps con_rews) 1,
   511                                       asm_simp_tac take_ss 1])
   512                                     (nonlazy_rec args)))
   513                                   cons))
   514                                 1 (conss~~cases)));
   515     val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp(
   516                                                 %%(dn^"_finite") $ %"x"))[
   517                                 case_UU_tac take_rews 1 "x",
   518                                 eresolve_tac finite_lemmas1a 1,
   519                                 step_tac HOL_cs 1,
   520                                 step_tac HOL_cs 1,
   521                                 cut_facts_tac [l1b] 1,
   522                         fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
   523   in
   524   (finites,
   525    pg'' thy[](ind_term (fn n => fn dn => %(P_name n) $ %(x_name n)))(fn prems =>
   526                                 TRY(safe_tac HOL_cs) ::
   527                          flat (map (fn (finite,fin_ind) => [
   528                                rtac(rewrite_rule axs_finite_def finite RS exE)1,
   529                                 etac subst 1,
   530                                 rtac fin_ind 1,
   531                                 ind_prems_tac prems]) 
   532                                    (finites~~(atomize finite_ind)) ))
   533 ) end (* let *) else
   534   (mapn (fn n => fn dn => read_instantiate_sg (sign_of thy) 
   535                     [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
   536    pg'' thy [] (foldr (op ===>) (mapn (fn n => K(mk_trp(%%"adm" $ %(P_name n))))
   537                1 dnames, ind_term (fn n => fn dn => %(P_name n) $ %(x_name n))))
   538                    (fn prems => map (fn ax_reach => rtac (ax_reach RS subst) 1) 
   539                                     axs_reach @ [
   540                                 quant_tac 1,
   541                                 rtac (adm_impl_admw RS wfix_ind) 1,
   542                                  REPEAT_DETERM(rtac adm_all2 1),
   543                                  REPEAT_DETERM(TRY(rtac adm_conj 1) THEN 
   544                                                    rtac adm_subst 1 THEN 
   545                                         cont_tacR 1 THEN resolve_tac prems 1),
   546                                 strip_tac 1,
   547                                 rtac (rewrite_rule axs_take_def finite_ind) 1,
   548                                 ind_prems_tac prems])
   549   handle ERROR => (warning "Cannot prove infinite induction rule"; refl))
   550 end; (* local *)
   551 
   552 (* ----- theorem concerning coinduction ------------------------------------- *)
   553 
   554 local
   555   val xs = mapn (fn n => K (x_name n)) 1 dnames;
   556   fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
   557   val take_ss = HOL_ss addsimps take_rews;
   558   val sproj   = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
   559   val coind_lemma=pg[ax_bisim_def](mk_trp(mk_imp(%%(comp_dname^"_bisim") $ %"R",
   560                 foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
   561                   foldr mk_imp (mapn (fn n => K(proj (%"R") eqs n $ 
   562                                       bnd_arg n 0 $ bnd_arg n 1)) 0 dnames,
   563                     foldr' mk_conj (mapn (fn n => fn dn => 
   564                                 (dc_take dn $ %"n" `bnd_arg n 0 === 
   565                                 (dc_take dn $ %"n" `bnd_arg n 1)))0 dnames))))))
   566                              ([ rtac impI 1,
   567                                 nat_ind_tac "n" 1,
   568                                 simp_tac take_ss 1,
   569                                 safe_tac HOL_cs] @
   570                                 flat(mapn (fn n => fn x => [
   571                                   rotate_tac (n+1) 1,
   572                                   etac all2E 1,
   573                                   eres_inst_tac [("P1", sproj "R" eqs n^
   574                                         " "^x^" "^x^"'")](mp RS disjE) 1,
   575                                   TRY(safe_tac HOL_cs),
   576                                   REPEAT(CHANGED(asm_simp_tac take_ss 1))]) 
   577                                 0 xs));
   578 in
   579 val coind = pg [] (mk_trp(%%(comp_dname^"_bisim") $ %"R") ===>
   580                 foldr (op ===>) (mapn (fn n => fn x => 
   581                   mk_trp(proj (%"R") eqs n $ %x $ %(x^"'"))) 0 xs,
   582                   mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
   583                                 TRY(safe_tac HOL_cs)] @
   584                                 flat(map (fn take_lemma => [
   585                                   rtac take_lemma 1,
   586                                   cut_facts_tac [coind_lemma] 1,
   587                                   fast_tac HOL_cs 1])
   588                                 take_lemmas));
   589 end; (* local *)
   590 
   591 
   592 in thy |> Theory.add_path comp_dnam
   593        |> (PureThy.add_thmss o map Thm.no_attributes) [
   594 		("take_rews"  , take_rews  ),
   595 		("take_lemmas", take_lemmas),
   596 		("finites"    , finites    ),
   597 		("finite_ind", [finite_ind]),
   598 		("ind"       , [ind       ]),
   599 		("coind"     , [coind     ])]
   600        |> Theory.parent_path
   601 end; (* let *)
   602 end; (* local *)
   603 end; (* struct *)