src/HOLCF/domain/theorems.ML
author kleing
Mon Jun 21 10:25:57 2004 +0200 (2004-06-21)
changeset 14981 e73f8140af78
parent 14820 3f80d6510ee9
child 15457 1fbd4aba46e3
permissions -rw-r--r--
Merged in license change from Isabelle2004
     1 (*  Title:      HOLCF/domain/theorems.ML
     2     ID:         $Id$
     3     Author:     David von Oheimb
     4 
     5 Proof generator for domain section.
     6 *)
     7 
     8 
     9 structure Domain_Theorems = struct
    10 
    11 local
    12 
    13 open Domain_Library;
    14 infixr 0 ===>;infixr 0 ==>;infix 0 == ; 
    15 infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<;
    16 infix 9 `   ; infix 9 `% ; infix 9 `%%; infixr 9 oo;
    17 
    18 (* ----- general proof facilities ------------------------------------------- *)
    19 
    20 fun inferT sg pre_tm =
    21   #1 (Sign.infer_types (Sign.pp sg) sg (K None) (K None) [] true ([pre_tm],propT));
    22 
    23 fun pg'' thy defs t = let val sg = sign_of thy;
    24                           val ct = Thm.cterm_of sg (inferT sg t);
    25                       in prove_goalw_cterm defs ct end;
    26 fun pg'  thy defs t tacsf=pg'' thy defs t (fn []   => tacsf 
    27                                 | prems=> (cut_facts_tac prems 1)::tacsf);
    28 
    29 local val trueI2 = prove_goal HOL.thy"f~=x ==> True"(fn _ => [rtac TrueI 1]) in
    30 val kill_neq_tac = dtac trueI2 end;
    31 fun case_UU_tac rews i v =      case_tac (v^"=UU") i THEN
    32                                 asm_simp_tac (HOLCF_ss addsimps rews) i;
    33 
    34 val chain_tac = REPEAT_DETERM o resolve_tac 
    35                 [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL];
    36 
    37 (* ----- general proofs ----------------------------------------------------- *)
    38 
    39 val all2E = prove_goal HOL.thy "[| !x y . P x y; P x y ==> R |] ==> R"
    40  (fn prems =>[
    41                                 resolve_tac prems 1,
    42                                 cut_facts_tac prems 1,
    43                                 fast_tac HOL_cs 1]);
    44 
    45 val dist_eqI = prove_goal Porder.thy "!!x::'a::po. ~ x << y ==> x ~= y" 
    46              (fn prems => [
    47                (blast_tac (claset() addDs [antisym_less_inverse]) 1)]);
    48 (*
    49 infixr 0 y;
    50 val b = 0;
    51 fun _ y t = by t;
    52 fun g defs t = let val sg = sign_of thy;
    53                      val ct = Thm.cterm_of sg (inferT sg t);
    54                  in goalw_cterm defs ct end;
    55 *)
    56 
    57 in
    58 
    59 fun theorems (((dname,_),cons) : eq, eqs : eq list) thy =
    60 let
    61 
    62 val dummy = writeln ("Proving isomorphism properties of domain "^dname^" ...");
    63 val pg = pg' thy;
    64 
    65 
    66 (* ----- getting the axioms and definitions --------------------------------- *)
    67 
    68 local fun ga s dn = get_thm thy (dn^"."^s) in
    69 val ax_abs_iso    = ga "abs_iso"  dname;
    70 val ax_rep_iso    = ga "rep_iso"  dname;
    71 val ax_when_def   = ga "when_def" dname;
    72 val axs_con_def   = map (fn (con,_) => ga (extern_name con^"_def") dname) cons;
    73 val axs_dis_def   = map (fn (con,_) => ga (   dis_name con^"_def") dname) cons;
    74 val axs_sel_def   = flat(map (fn (_,args) => 
    75                     map (fn     arg => ga (sel_of arg     ^"_def") dname) args)
    76 									  cons);
    77 val ax_copy_def   = ga "copy_def" dname;
    78 end; (* local *)
    79 
    80 (* ----- theorems concerning the isomorphism -------------------------------- *)
    81 
    82 val dc_abs  = %%:(dname^"_abs");
    83 val dc_rep  = %%:(dname^"_rep");
    84 val dc_copy = %%:(dname^"_copy");
    85 val x_name = "x";
    86 
    87 val (rep_strict, abs_strict) = let 
    88          val r = ax_rep_iso RS (ax_abs_iso RS (allI  RSN(2,allI RS iso_strict)))
    89                in (r RS conjunct1, r RS conjunct2) end;
    90 val abs_defin' = pg [] ((dc_abs`%x_name === UU) ==> (%:x_name === UU)) [
    91                            res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
    92                                 etac ssubst 1, rtac rep_strict 1];
    93 val rep_defin' = pg [] ((dc_rep`%x_name === UU) ==> (%:x_name === UU)) [
    94                            res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
    95                                 etac ssubst 1, rtac abs_strict 1];
    96 val iso_rews = map standard [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
    97 
    98 local 
    99 val iso_swap = pg [] (dc_rep`%"x" === %:"y" ==> %:"x" === dc_abs`%"y") [
   100                             dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
   101                             etac (ax_rep_iso RS subst) 1];
   102 fun exh foldr1 cn quant foldr2 var = let
   103   fun one_con (con,args) = let val vns = map vname args in
   104     foldr quant (vns, foldr2 ((%:x_name === con_app2 con (var vns) vns)::
   105                               map (defined o (var vns)) (nonlazy args))) end
   106   in foldr1 ((cn(%:x_name===UU))::map one_con cons) end;
   107 in
   108 val casedist = let 
   109             fun common_tac thm = rtac thm 1 THEN contr_tac 1;
   110             fun unit_tac true = common_tac upE1
   111             |   unit_tac _    = all_tac;
   112             fun prod_tac []          = common_tac oneE
   113             |   prod_tac [arg]       = unit_tac (is_lazy arg)
   114             |   prod_tac (arg::args) = 
   115                                 common_tac sprodE THEN
   116                                 kill_neq_tac 1 THEN
   117                                 unit_tac (is_lazy arg) THEN
   118                                 prod_tac args;
   119             fun sum_rest_tac p = SELECT_GOAL(EVERY[
   120                                 rtac p 1,
   121                                 rewrite_goals_tac axs_con_def,
   122                                 dtac iso_swap 1,
   123                                 simp_tac HOLCF_ss 1,
   124                                 DETERM_UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
   125             fun sum_tac [(_,args)]       [p]        = 
   126                                 prod_tac args THEN sum_rest_tac p
   127             |   sum_tac ((_,args)::cons') (p::prems) = DETERM(
   128                                 common_tac ssumE THEN
   129                                 kill_neq_tac 1 THEN kill_neq_tac 2 THEN
   130                                 prod_tac args THEN sum_rest_tac p) THEN
   131                                 sum_tac cons' prems
   132             |   sum_tac _ _ = Imposs "theorems:sum_tac";
   133           in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%:"P")))
   134                               (fn T => T ==> %:"P") mk_All
   135                               (fn l => foldr (op ===>) (map mk_trp l,
   136                                                             mk_trp(%:"P")))
   137                               bound_arg)
   138                              (fn prems => [
   139                                 cut_facts_tac [excluded_middle] 1,
   140                                 etac disjE 1,
   141                                 rtac (hd prems) 2,
   142                                 etac rep_defin' 2,
   143                                 if length cons = 1 andalso 
   144                                    length (snd(hd cons)) = 1 andalso 
   145                                    not(is_lazy(hd(snd(hd cons))))
   146                                 then rtac (hd (tl prems)) 1 THEN atac 2 THEN
   147                                      rewrite_goals_tac axs_con_def THEN
   148                                      simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
   149                                 else sum_tac cons (tl prems)])end;
   150 val exhaust= pg[](mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %:)))[
   151                                 rtac casedist 1,
   152                                 DETERM_UNTIL_SOLVED(fast_tac HOL_cs 1)];
   153 end;
   154 
   155 local 
   156   fun bind_fun t = foldr mk_All (when_funs cons,t);
   157   fun bound_fun i _ = Bound (length cons - i);
   158   val when_app  = foldl (op `) (%%:(dname^"_when"), mapn bound_fun 1 cons);
   159   val when_appl = pg [ax_when_def] (bind_fun(mk_trp(when_app`%x_name ===
   160              when_body cons (fn (m,n)=> bound_fun (n-m) 0)`(dc_rep`%x_name))))[
   161                                 simp_tac HOLCF_ss 1];
   162 in
   163 val when_strict = pg [] (bind_fun(mk_trp(strict when_app))) [
   164                         simp_tac(HOLCF_ss addsimps [when_appl,rep_strict]) 1];
   165 val when_apps = let fun one_when n (con,args) = pg axs_con_def 
   166                 (bind_fun (lift_defined %: (nonlazy args, 
   167                 mk_trp(when_app`(con_app con args) ===
   168                        mk_cRep_CFun(bound_fun n 0,map %# args)))))[
   169                 asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
   170         in mapn one_when 1 cons end;
   171 end;
   172 val when_rews = when_strict::when_apps;
   173 
   174 (* ----- theorems concerning the constructors, discriminators and selectors - *)
   175 
   176 val dis_rews = let
   177   val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
   178                              strict(%%:(dis_name con)))) [
   179                                 simp_tac (HOLCF_ss addsimps when_rews) 1]) cons;
   180   val dis_apps = let fun one_dis c (con,args)= pg axs_dis_def
   181                    (lift_defined %: (nonlazy args,
   182                         (mk_trp((%%:(dis_name c))`(con_app con args) ===
   183                               %%:(if con=c then "TT" else "FF"))))) [
   184                                 asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   185         in flat(map (fn (c,_) => map (one_dis c) cons) cons) end;
   186   val dis_defins = map (fn (con,args) => pg [] (defined(%:x_name) ==> 
   187                       defined(%%:(dis_name con)`%x_name)) [
   188                                 rtac casedist 1,
   189                                 contr_tac 1,
   190                                 DETERM_UNTIL_SOLVED (CHANGED(asm_simp_tac 
   191                                         (HOLCF_ss addsimps dis_apps) 1))]) cons;
   192 in dis_stricts @ dis_defins @ dis_apps end;
   193 
   194 val con_stricts = flat(map (fn (con,args) => map (fn vn =>
   195                         pg (axs_con_def) 
   196                            (mk_trp(con_app2 con (fn arg => if vname arg = vn 
   197                                         then UU else %# arg) args === UU))[
   198                                 asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
   199                         ) (nonlazy args)) cons);
   200 val con_defins = map (fn (con,args) => pg []
   201                         (lift_defined %: (nonlazy args,
   202                                 mk_trp(defined(con_app con args)))) ([
   203                           rtac rev_contrapos 1, 
   204                           eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
   205                           asm_simp_tac (HOLCF_ss addsimps dis_rews) 1] )) cons;
   206 val con_rews = con_stricts @ con_defins;
   207 
   208 val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%:sel))) [
   209                                 simp_tac (HOLCF_ss addsimps when_rews) 1];
   210 in flat(map (fn (_,args) =>map (fn arg => one_sel (sel_of arg)) args) cons) end;
   211 val sel_apps = let fun one_sel c n sel = map (fn (con,args) => 
   212                 let val nlas = nonlazy args;
   213                     val vns  = map vname args;
   214                 in pg axs_sel_def (lift_defined %:
   215                    (filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
   216                                 mk_trp((%%:sel)`(con_app con args) === 
   217                                 (if con=c then %:(nth_elem(n,vns)) else UU))))
   218                             ( (if con=c then [] 
   219                        else map(case_UU_tac(when_rews@con_stricts)1) nlas)
   220                      @(if con=c andalso ((nth_elem(n,vns)) mem nlas)
   221                                  then[case_UU_tac (when_rews @ con_stricts) 1 
   222                                                   (nth_elem(n,vns))] else [])
   223                      @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
   224 in flat(map  (fn (c,args) => 
   225      flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
   226 val sel_defins = if length cons=1 then map (fn arg => pg [](defined(%:x_name)==> 
   227                         defined(%%:(sel_of arg)`%x_name)) [
   228                                 rtac casedist 1,
   229                                 contr_tac 1,
   230                                 DETERM_UNTIL_SOLVED (CHANGED(asm_simp_tac 
   231                                              (HOLCF_ss addsimps sel_apps) 1))]) 
   232                  (filter_out is_lazy (snd(hd cons))) else [];
   233 val sel_rews = sel_stricts @ sel_defins @ sel_apps;
   234 
   235 val distincts_le = let
   236     fun dist (con1, args1) (con2, args2) = pg []
   237               (lift_defined %: ((nonlazy args1),
   238                         (mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
   239                         rtac rev_contrapos 1,
   240                         eres_inst_tac[("fo",dis_name con1)] monofun_cfun_arg 1]
   241                       @map(case_UU_tac (con_stricts @ dis_rews)1)(nonlazy args2)
   242                       @[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
   243     fun distinct (con1,args1) (con2,args2) =
   244         let val arg1 = (con1, args1)
   245             val arg2 = (con2,
   246 			ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
   247                         (args2, variantlist(map vname args2,map vname args1)))
   248         in [dist arg1 arg2, dist arg2 arg1] end;
   249     fun distincts []      = []
   250     |   distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
   251 in distincts cons end;
   252 val dist_les = flat (flat distincts_le);
   253 val dist_eqs = let
   254     fun distinct (_,args1) ((_,args2),leqs) = let
   255         val (le1,le2) = (hd leqs, hd(tl leqs));
   256         val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
   257         if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
   258         if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
   259                                         [eq1, eq2] end;
   260     open BasisLibrary (*restore original List*)
   261     fun distincts []      = []
   262     |   distincts ((c,leqs)::cs) = List.concat
   263 	            (ListPair.map (distinct c) ((map #1 cs),leqs)) @
   264 		    distincts cs;
   265     in map standard (distincts (cons~~distincts_le)) end;
   266 
   267 local 
   268   fun pgterm rel con args = let
   269                 fun append s = upd_vname(fn v => v^s);
   270                 val (largs,rargs) = (args, map (append "'") args);
   271                 in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
   272                       lift_defined %: ((nonlazy largs),lift_defined %: ((nonlazy rargs),
   273                             mk_trp (foldr' mk_conj 
   274                                 (ListPair.map rel
   275 				 (map %# largs, map %# rargs)))))) end;
   276   val cons' = filter (fn (_,args) => args<>[]) cons;
   277 in
   278 val inverts = map (fn (con,args) => 
   279                 pgterm (op <<) con args (flat(map (fn arg => [
   280                                 TRY(rtac conjI 1),
   281                                 dres_inst_tac [("fo",sel_of arg)] monofun_cfun_arg 1,
   282                                 asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
   283                                                       ) args))) cons';
   284 val injects = map (fn ((con,args),inv_thm) => 
   285                            pgterm (op ===) con args [
   286                                 etac (antisym_less_inverse RS conjE) 1,
   287                                 dtac inv_thm 1, REPEAT(atac 1),
   288                                 dtac inv_thm 1, REPEAT(atac 1),
   289                                 TRY(safe_tac HOL_cs),
   290                                 REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
   291                   (cons'~~inverts);
   292 end;
   293 
   294 (* ----- theorems concerning one induction step ----------------------------- *)
   295 
   296 val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [
   297                    asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict,
   298                                                    cfst_strict,csnd_strict]) 1];
   299 val copy_apps = map (fn (con,args) => pg [ax_copy_def]
   300                     (lift_defined %: (nonlazy_rec args,
   301                         mk_trp(dc_copy`%"f"`(con_app con args) ===
   302                 (con_app2 con (app_rec_arg (cproj (%:"f") eqs)) args))))
   303                         (map (case_UU_tac (abs_strict::when_strict::con_stricts)
   304                                  1 o vname)
   305                          (filter (fn a => not (is_rec a orelse is_lazy a)) args)
   306                         @[asm_simp_tac (HOLCF_ss addsimps when_apps) 1,
   307                           simp_tac (HOLCF_ss addsimps axs_con_def) 1]))cons;
   308 val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU`
   309                                         (con_app con args) ===UU))
   310      (let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
   311                          in map (case_UU_tac rews 1) (nonlazy args) @ [
   312                              asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
   313                         (filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
   314 val copy_rews = copy_strict::copy_apps @ copy_stricts;
   315 in thy |> Theory.add_path (Sign.base_name dname)
   316        |> (#1 o (PureThy.add_thmss (map Thm.no_attributes [
   317 		("iso_rews" , iso_rews  ),
   318 		("exhaust"  , [exhaust] ),
   319 		("casedist" , [casedist]),
   320 		("when_rews", when_rews ),
   321 		("con_rews", con_rews),
   322 		("sel_rews", sel_rews),
   323 		("dis_rews", dis_rews),
   324 		("dist_les", dist_les),
   325 		("dist_eqs", dist_eqs),
   326 		("inverts" , inverts ),
   327 		("injects" , injects ),
   328 		("copy_rews", copy_rews)])))
   329        |> Theory.parent_path |> rpair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
   330                  dist_les @ dist_eqs @ copy_rews)
   331 end; (* let *)
   332 
   333 fun comp_theorems (comp_dnam, eqs: eq list) thy =
   334 let
   335 val dnames = map (fst o fst) eqs;
   336 val conss  = map  snd        eqs;
   337 val comp_dname = Sign.full_name (sign_of thy) comp_dnam;
   338 
   339 val d = writeln("Proving induction properties of domain "^comp_dname^" ...");
   340 val pg = pg' thy;
   341 
   342 (* ----- getting the composite axiom and definitions ------------------------ *)
   343 
   344 local fun ga s dn = get_thm thy (dn^"."^s) in
   345 val axs_reach      = map (ga "reach"     ) dnames;
   346 val axs_take_def   = map (ga "take_def"  ) dnames;
   347 val axs_finite_def = map (ga "finite_def") dnames;
   348 val ax_copy2_def   =      ga "copy_def"  comp_dnam;
   349 val ax_bisim_def   =      ga "bisim_def" comp_dnam;
   350 end; (* local *)
   351 
   352 local fun gt  s dn = get_thm  thy (dn^"."^s);
   353       fun gts s dn = get_thms thy (dn^"."^s) in
   354 val cases     =       map (gt  "casedist" ) dnames;
   355 val con_rews  = flat (map (gts "con_rews" ) dnames);
   356 val copy_rews = flat (map (gts "copy_rews") dnames);
   357 end; (* local *)
   358 
   359 fun dc_take dn = %%:(dn^"_take");
   360 val x_name = idx_name dnames "x"; 
   361 val P_name = idx_name dnames "P";
   362 val n_eqs = length eqs;
   363 
   364 (* ----- theorems concerning finite approximation and finite induction ------ *)
   365 
   366 local
   367   val iterate_Cprod_ss = simpset_of Fix.thy
   368                          addsimps [cfst_strict, csnd_strict]addsimps Cprod_rews;
   369   val copy_con_rews  = copy_rews @ con_rews;
   370   val copy_take_defs =(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
   371   val take_stricts=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=>
   372             strict(dc_take dn $ %:"n")) eqs))) ([
   373                         induct_tac "n" 1,
   374                          simp_tac iterate_Cprod_ss 1,
   375                         asm_simp_tac (iterate_Cprod_ss addsimps copy_rews)1]);
   376   val take_stricts' = rewrite_rule copy_take_defs take_stricts;
   377   val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn $ %%:"0")
   378                                                         `%x_name n === UU))[
   379                                 simp_tac iterate_Cprod_ss 1]) 1 dnames;
   380   val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1;
   381   val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj 
   382             (flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all 
   383         (map vname args,(dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args) ===
   384          con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))$ %:"n"))
   385                               args)) cons) eqs)))) ([
   386                                 simp_tac iterate_Cprod_ss 1,
   387                                 induct_tac "n" 1,
   388                             simp_tac(iterate_Cprod_ss addsimps copy_con_rews) 1,
   389                                 asm_full_simp_tac (HOLCF_ss addsimps 
   390                                       (filter (has_fewer_prems 1) copy_rews)) 1,
   391                                 TRY(safe_tac HOL_cs)] @
   392                         (flat(map (fn ((dn,_),cons) => map (fn (con,args) => 
   393                                 if nonlazy_rec args = [] then all_tac else
   394                                 EVERY(map c_UU_tac (nonlazy_rec args)) THEN
   395                                 asm_full_simp_tac (HOLCF_ss addsimps copy_rews)1
   396                                                            ) cons) eqs)));
   397 in
   398 val take_rews = map standard (atomize take_stricts @ take_0s @ atomize take_apps);
   399 end; (* local *)
   400 
   401 local
   402   fun one_con p (con,args) = foldr mk_All (map vname args,
   403         lift_defined (bound_arg (map vname args)) (nonlazy args,
   404         lift (fn arg => %:(P_name (1+rec_of arg)) $ bound_arg args arg)
   405          (filter is_rec args,mk_trp(%:p $ con_app2 con (bound_arg args) args))));
   406   fun one_eq ((p,cons),concl) = (mk_trp(%:p $ UU) ===> 
   407                            foldr (op ===>) (map (one_con p) cons,concl));
   408   fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss,
   409                         mk_trp(foldr' mk_conj (mapn concf 1 dnames)));
   410   val take_ss = HOL_ss addsimps take_rews;
   411   fun quant_tac i = EVERY(mapn(fn n=> fn _=> res_inst_tac[("x",x_name n)]spec i)
   412                                1 dnames);
   413   fun ind_prems_tac prems = EVERY(flat (map (fn cons => (
   414                                      resolve_tac prems 1 ::
   415                                      flat (map (fn (_,args) => 
   416                                        resolve_tac prems 1 ::
   417                                        map (K(atac 1)) (nonlazy args) @
   418                                        map (K(atac 1)) (filter is_rec args))
   419                                      cons))) conss));
   420   local 
   421     (* check whether every/exists constructor of the n-th part of the equation:
   422        it has a possibly indirectly recursive argument that isn't/is possibly 
   423        indirectly lazy *)
   424     fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
   425           is_rec arg andalso not(rec_of arg mem ns) andalso
   426           ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
   427             rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
   428               (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
   429           ) o snd) cons;
   430     fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
   431     fun warn (n,cons)  = if all_rec_to [] false (n,cons) then (warning
   432         ("domain "^nth_elem(n,dnames)^" is empty!"); true) else false;
   433     fun lazy_rec_to ns = rec_to exists Id  lazy_rec_to ns;
   434 
   435   in val n__eqs     = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
   436      val is_emptys = map warn n__eqs;
   437      val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
   438   end;
   439 in (* local *)
   440 val finite_ind = pg'' thy [] (ind_term (fn n => fn dn => %:(P_name n)$
   441                              (dc_take dn $ %:"n" `%(x_name n)))) (fn prems => [
   442                                 quant_tac 1,
   443                                 simp_tac HOL_ss 1,
   444                                 induct_tac "n" 1,
   445                                 simp_tac (take_ss addsimps prems) 1,
   446                                 TRY(safe_tac HOL_cs)]
   447                                 @ flat(map (fn (cons,cases) => [
   448                                  res_inst_tac [("x","x")] cases 1,
   449                                  asm_simp_tac (take_ss addsimps prems) 1]
   450                                  @ flat(map (fn (con,args) => 
   451                                   asm_simp_tac take_ss 1 ::
   452                                   map (fn arg =>
   453                                    case_UU_tac (prems@con_rews) 1 (
   454                            nth_elem(rec_of arg,dnames)^"_take n$"^vname arg))
   455                                   (filter is_nonlazy_rec args) @ [
   456                                   resolve_tac prems 1] @
   457                                   map (K (atac 1))      (nonlazy args) @
   458                                   map (K (etac spec 1)) (filter is_rec args)) 
   459                                  cons))
   460                                 (conss~~cases)));
   461 
   462 val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n",
   463                 mk_trp(dc_take dn $ Bound 0 `%(x_name n) === 
   464                        dc_take dn $ Bound 0 `%(x_name n^"'")))
   465            ===> mk_trp(%:(x_name n) === %:(x_name n^"'"))) (fn prems => [
   466                         res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
   467                         res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
   468                                 stac fix_def2 1,
   469                                 REPEAT(CHANGED(rtac(contlub_cfun_arg RS ssubst)1
   470                                                THEN chain_tac 1)),
   471                                 stac contlub_cfun_fun 1,
   472                                 stac contlub_cfun_fun 2,
   473                                 rtac lub_equal 3,
   474                                 chain_tac 1,
   475                                 rtac allI 1,
   476                                 resolve_tac prems 1])) 1 (dnames~~axs_reach);
   477 
   478 (* ----- theorems concerning finiteness and induction ----------------------- *)
   479 
   480 val (finites,ind) = if is_finite then
   481   let 
   482     fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
   483     val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%:"x")) ===> 
   484         mk_trp(mk_disj(mk_all("n",dc_take dn $ Bound 0 ` %:"x" === UU),
   485         take_enough dn)) ===> mk_trp(take_enough dn)) [
   486                                 etac disjE 1,
   487                                 etac notE 1,
   488                                 resolve_tac take_lemmas 1,
   489                                 asm_simp_tac take_ss 1,
   490                                 atac 1]) dnames;
   491     val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn 
   492         (fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
   493          mk_disj(dc_take dn $ Bound 1 ` Bound 0 === UU,
   494                  dc_take dn $ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
   495                                 rtac allI 1,
   496                                 induct_tac "n" 1,
   497                                 simp_tac take_ss 1,
   498                         TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @
   499                                 flat(mapn (fn n => fn (cons,cases) => [
   500                                   simp_tac take_ss 1,
   501                                   rtac allI 1,
   502                                   res_inst_tac [("x",x_name n)] cases 1,
   503                                   asm_simp_tac take_ss 1] @ 
   504                                   flat(map (fn (con,args) => 
   505                                     asm_simp_tac take_ss 1 ::
   506                                     flat(map (fn vn => [
   507                                       eres_inst_tac [("x",vn)] all_dupE 1,
   508                                       etac disjE 1,
   509                                       asm_simp_tac (HOL_ss addsimps con_rews) 1,
   510                                       asm_simp_tac take_ss 1])
   511                                     (nonlazy_rec args)))
   512                                   cons))
   513                                 1 (conss~~cases)));
   514     val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp(
   515                                                 %%:(dn^"_finite") $ %:"x"))[
   516                                 case_UU_tac take_rews 1 "x",
   517                                 eresolve_tac finite_lemmas1a 1,
   518                                 step_tac HOL_cs 1,
   519                                 step_tac HOL_cs 1,
   520                                 cut_facts_tac [l1b] 1,
   521                         fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
   522   in
   523   (finites,
   524    pg'' thy[](ind_term (fn n => fn dn => %:(P_name n) $ %:(x_name n)))(fn prems =>
   525                                 TRY(safe_tac HOL_cs) ::
   526                          flat (map (fn (finite,fin_ind) => [
   527                                rtac(rewrite_rule axs_finite_def finite RS exE)1,
   528                                 etac subst 1,
   529                                 rtac fin_ind 1,
   530                                 ind_prems_tac prems]) 
   531                                    (finites~~(atomize finite_ind)) ))
   532 ) end (* let *) else
   533   (mapn (fn n => fn dn => read_instantiate_sg (sign_of thy) 
   534                     [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
   535    pg'' thy [] (foldr (op ===>) (mapn (fn n => K(mk_trp(%%:"adm" $ %:(P_name n))))
   536                1 dnames, ind_term (fn n => fn dn => %:(P_name n) $ %:(x_name n))))
   537                    (fn prems => map (fn ax_reach => rtac (ax_reach RS subst) 1) 
   538                                     axs_reach @ [
   539                                 quant_tac 1,
   540                                 rtac (adm_impl_admw RS wfix_ind) 1,
   541                                  REPEAT_DETERM(rtac adm_all2 1),
   542                                  REPEAT_DETERM(TRY(rtac adm_conj 1) THEN 
   543                                                    rtac adm_subst 1 THEN 
   544                                         cont_tacR 1 THEN resolve_tac prems 1),
   545                                 strip_tac 1,
   546                                 rtac (rewrite_rule axs_take_def finite_ind) 1,
   547                                 ind_prems_tac prems])
   548   handle ERROR => (warning "Cannot prove infinite induction rule"; refl))
   549 end; (* local *)
   550 
   551 (* ----- theorem concerning coinduction ------------------------------------- *)
   552 
   553 local
   554   val xs = mapn (fn n => K (x_name n)) 1 dnames;
   555   fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
   556   val take_ss = HOL_ss addsimps take_rews;
   557   val sproj   = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
   558   val coind_lemma=pg[ax_bisim_def](mk_trp(mk_imp(%%:(comp_dname^"_bisim") $ %:"R",
   559                 foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
   560                   foldr mk_imp (mapn (fn n => K(proj (%:"R") eqs n $ 
   561                                       bnd_arg n 0 $ bnd_arg n 1)) 0 dnames,
   562                     foldr' mk_conj (mapn (fn n => fn dn => 
   563                                 (dc_take dn $ %:"n" `bnd_arg n 0 === 
   564                                 (dc_take dn $ %:"n" `bnd_arg n 1)))0 dnames))))))
   565                              ([ rtac impI 1,
   566                                 induct_tac "n" 1,
   567                                 simp_tac take_ss 1,
   568                                 safe_tac HOL_cs] @
   569                                 flat(mapn (fn n => fn x => [
   570                                   rotate_tac (n+1) 1,
   571                                   etac all2E 1,
   572                                   eres_inst_tac [("P1", sproj "R" eqs n^
   573                                         " "^x^" "^x^"'")](mp RS disjE) 1,
   574                                   TRY(safe_tac HOL_cs),
   575                                   REPEAT(CHANGED(asm_simp_tac take_ss 1))]) 
   576                                 0 xs));
   577 in
   578 val coind = pg [] (mk_trp(%%:(comp_dname^"_bisim") $ %:"R") ===>
   579                 foldr (op ===>) (mapn (fn n => fn x => 
   580                   mk_trp(proj (%:"R") eqs n $ %:x $ %:(x^"'"))) 0 xs,
   581                   mk_trp(foldr' mk_conj (map (fn x => %:x === %:(x^"'")) xs)))) ([
   582                                 TRY(safe_tac HOL_cs)] @
   583                                 flat(map (fn take_lemma => [
   584                                   rtac take_lemma 1,
   585                                   cut_facts_tac [coind_lemma] 1,
   586                                   fast_tac HOL_cs 1])
   587                                 take_lemmas));
   588 end; (* local *)
   589 
   590 
   591 in thy |> Theory.add_path comp_dnam
   592        |> (#1 o (PureThy.add_thmss (map Thm.no_attributes [
   593 		("take_rews"  , take_rews  ),
   594 		("take_lemmas", take_lemmas),
   595 		("finites"    , finites    ),
   596 		("finite_ind", [finite_ind]),
   597 		("ind"       , [ind       ]),
   598 		("coind"     , [coind     ])])))
   599        |> Theory.parent_path |> rpair take_rews
   600 end; (* let *)
   601 end; (* local *)
   602 end; (* struct *)