src/HOLCF/Tools/Domain/domain_theorems.ML
author huffman
Tue Nov 03 18:32:30 2009 -0800 (2009-11-03)
changeset 33427 3182812d33ed
parent 33396 45c5c3c51918
child 33504 b4210cc3ac97
permissions -rw-r--r--
domain package registers fixrec_simp lemmas
     1 (*  Title:      HOLCF/Tools/Domain/domain_theorems.ML
     2     Author:     David von Oheimb
     3     Author:     Brian Huffman
     4 
     5 Proof generator for domain command.
     6 *)
     7 
     8 val HOLCF_ss = @{simpset};
     9 
    10 signature DOMAIN_THEOREMS =
    11 sig
    12   val theorems: Domain_Library.eq * Domain_Library.eq list -> theory -> thm list * theory;
    13   val comp_theorems: bstring * Domain_Library.eq list -> theory -> thm list * theory;
    14   val quiet_mode: bool Unsynchronized.ref;
    15   val trace_domain: bool Unsynchronized.ref;
    16 end;
    17 
    18 structure Domain_Theorems :> DOMAIN_THEOREMS =
    19 struct
    20 
    21 val quiet_mode = Unsynchronized.ref false;
    22 val trace_domain = Unsynchronized.ref false;
    23 
    24 fun message s = if !quiet_mode then () else writeln s;
    25 fun trace s = if !trace_domain then tracing s else ();
    26 
    27 local
    28 
    29 val adm_impl_admw = @{thm adm_impl_admw};
    30 val adm_all = @{thm adm_all};
    31 val adm_conj = @{thm adm_conj};
    32 val adm_subst = @{thm adm_subst};
    33 val antisym_less_inverse = @{thm below_antisym_inverse};
    34 val beta_cfun = @{thm beta_cfun};
    35 val cfun_arg_cong = @{thm cfun_arg_cong};
    36 val ch2ch_fst = @{thm ch2ch_fst};
    37 val ch2ch_snd = @{thm ch2ch_snd};
    38 val ch2ch_Rep_CFunL = @{thm ch2ch_Rep_CFunL};
    39 val ch2ch_Rep_CFunR = @{thm ch2ch_Rep_CFunR};
    40 val chain_iterate = @{thm chain_iterate};
    41 val compact_ONE = @{thm compact_ONE};
    42 val compact_sinl = @{thm compact_sinl};
    43 val compact_sinr = @{thm compact_sinr};
    44 val compact_spair = @{thm compact_spair};
    45 val compact_up = @{thm compact_up};
    46 val contlub_cfun_arg = @{thm contlub_cfun_arg};
    47 val contlub_cfun_fun = @{thm contlub_cfun_fun};
    48 val contlub_fst = @{thm contlub_fst};
    49 val contlub_snd = @{thm contlub_snd};
    50 val contlubE = @{thm contlubE};
    51 val cont_const = @{thm cont_const};
    52 val cont_id = @{thm cont_id};
    53 val cont2cont_fst = @{thm cont2cont_fst};
    54 val cont2cont_snd = @{thm cont2cont_snd};
    55 val cont2cont_Rep_CFun = @{thm cont2cont_Rep_CFun};
    56 val fix_def2 = @{thm fix_def2};
    57 val injection_eq = @{thm injection_eq};
    58 val injection_less = @{thm injection_below};
    59 val lub_equal = @{thm lub_equal};
    60 val monofun_cfun_arg = @{thm monofun_cfun_arg};
    61 val retraction_strict = @{thm retraction_strict};
    62 val spair_eq = @{thm spair_eq};
    63 val spair_less = @{thm spair_below};
    64 val sscase1 = @{thm sscase1};
    65 val ssplit1 = @{thm ssplit1};
    66 val strictify1 = @{thm strictify1};
    67 val wfix_ind = @{thm wfix_ind};
    68 
    69 val iso_intro       = @{thm iso.intro};
    70 val iso_abs_iso     = @{thm iso.abs_iso};
    71 val iso_rep_iso     = @{thm iso.rep_iso};
    72 val iso_abs_strict  = @{thm iso.abs_strict};
    73 val iso_rep_strict  = @{thm iso.rep_strict};
    74 val iso_abs_defin'  = @{thm iso.abs_defin'};
    75 val iso_rep_defin'  = @{thm iso.rep_defin'};
    76 val iso_abs_defined = @{thm iso.abs_defined};
    77 val iso_rep_defined = @{thm iso.rep_defined};
    78 val iso_compact_abs = @{thm iso.compact_abs};
    79 val iso_compact_rep = @{thm iso.compact_rep};
    80 val iso_iso_swap    = @{thm iso.iso_swap};
    81 
    82 val exh_start = @{thm exh_start};
    83 val ex_defined_iffs = @{thms ex_defined_iffs};
    84 val exh_casedist0 = @{thm exh_casedist0};
    85 val exh_casedists = @{thms exh_casedists};
    86 
    87 open Domain_Library;
    88 infixr 0 ===>;
    89 infixr 0 ==>;
    90 infix 0 == ; 
    91 infix 1 ===;
    92 infix 1 ~= ;
    93 infix 1 <<;
    94 infix 1 ~<<;
    95 infix 9 `   ;
    96 infix 9 `% ;
    97 infix 9 `%%;
    98 infixr 9 oo;
    99 
   100 (* ----- general proof facilities ------------------------------------------- *)
   101 
   102 fun legacy_infer_term thy t =
   103   let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init thy)
   104   in singleton (Syntax.check_terms ctxt) (Sign.intern_term thy t) end;
   105 
   106 fun pg'' thy defs t tacs =
   107   let
   108     val t' = legacy_infer_term thy t;
   109     val asms = Logic.strip_imp_prems t';
   110     val prop = Logic.strip_imp_concl t';
   111     fun tac {prems, context} =
   112       rewrite_goals_tac defs THEN
   113       EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})
   114   in Goal.prove_global thy [] asms prop tac end;
   115 
   116 fun pg' thy defs t tacsf =
   117   let
   118     fun tacs {prems, context} =
   119       if null prems then tacsf context
   120       else cut_facts_tac prems 1 :: tacsf context;
   121   in pg'' thy defs t tacs end;
   122 
   123 fun case_UU_tac ctxt rews i v =
   124   InductTacs.case_tac ctxt (v^"=UU") i THEN
   125   asm_simp_tac (HOLCF_ss addsimps rews) i;
   126 
   127 val chain_tac =
   128   REPEAT_DETERM o resolve_tac 
   129     [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL, ch2ch_fst, ch2ch_snd];
   130 
   131 (* ----- general proofs ----------------------------------------------------- *)
   132 
   133 val all2E = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp}
   134 
   135 val dist_eqI = @{lemma "!!x::'a::po. ~ x << y ==> x ~= y" by (blast dest!: below_antisym_inverse)}
   136 
   137 in
   138 
   139 fun theorems (((dname, _), cons) : eq, eqs : eq list) thy =
   140 let
   141 
   142 val _ = message ("Proving isomorphism properties of domain "^dname^" ...");
   143 val pg = pg' thy;
   144 
   145 (* ----- getting the axioms and definitions --------------------------------- *)
   146 
   147 local
   148   fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
   149 in
   150   val ax_abs_iso  = ga "abs_iso"  dname;
   151   val ax_rep_iso  = ga "rep_iso"  dname;
   152   val ax_when_def = ga "when_def" dname;
   153   fun get_def mk_name (con,_) = ga (mk_name con^"_def") dname;
   154   val axs_con_def = map (get_def extern_name) cons;
   155   val axs_dis_def = map (get_def dis_name) cons;
   156   val axs_mat_def = map (get_def mat_name) cons;
   157   val axs_pat_def = map (get_def pat_name) cons;
   158   val axs_sel_def =
   159     let
   160       fun def_of_sel sel = ga (sel^"_def") dname;
   161       fun def_of_arg arg = Option.map def_of_sel (sel_of arg);
   162       fun defs_of_con (_, args) = map_filter def_of_arg args;
   163     in
   164       maps defs_of_con cons
   165     end;
   166   val ax_copy_def = ga "copy_def" dname;
   167 end; (* local *)
   168 
   169 (* ----- theorems concerning the isomorphism -------------------------------- *)
   170 
   171 val dc_abs  = %%:(dname^"_abs");
   172 val dc_rep  = %%:(dname^"_rep");
   173 val dc_copy = %%:(dname^"_copy");
   174 val x_name = "x";
   175 
   176 val iso_locale = iso_intro OF [ax_abs_iso, ax_rep_iso];
   177 val abs_strict = ax_rep_iso RS (allI RS retraction_strict);
   178 val rep_strict = ax_abs_iso RS (allI RS retraction_strict);
   179 val abs_defin' = iso_locale RS iso_abs_defin';
   180 val rep_defin' = iso_locale RS iso_rep_defin';
   181 val iso_rews = map Drule.standard [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
   182 
   183 (* ----- generating beta reduction rules from definitions-------------------- *)
   184 
   185 val _ = trace " Proving beta reduction rules...";
   186 
   187 local
   188   fun arglist (Const _ $ Abs (s, _, t)) =
   189     let
   190       val (vars,body) = arglist t;
   191     in (s :: vars, body) end
   192     | arglist t = ([], t);
   193   fun bind_fun vars t = Library.foldr mk_All (vars, t);
   194   fun bound_vars 0 = []
   195     | bound_vars i = Bound (i-1) :: bound_vars (i - 1);
   196 in
   197   fun appl_of_def def =
   198     let
   199       val (_ $ con $ lam) = concl_of def;
   200       val (vars, rhs) = arglist lam;
   201       val lhs = list_ccomb (con, bound_vars (length vars));
   202       val appl = bind_fun vars (lhs == rhs);
   203       val cs = ContProc.cont_thms lam;
   204       val betas = map (fn c => mk_meta_eq (c RS beta_cfun)) cs;
   205     in pg (def::betas) appl (K [rtac reflexive_thm 1]) end;
   206 end;
   207 
   208 val _ = trace "Proving when_appl...";
   209 val when_appl = appl_of_def ax_when_def;
   210 val _ = trace "Proving con_appls...";
   211 val con_appls = map appl_of_def axs_con_def;
   212 
   213 local
   214   fun arg2typ n arg =
   215     let val t = TVar (("'a", n), pcpoS)
   216     in (n + 1, if is_lazy arg then mk_uT t else t) end;
   217 
   218   fun args2typ n [] = (n, oneT)
   219     | args2typ n [arg] = arg2typ n arg
   220     | args2typ n (arg::args) =
   221     let
   222       val (n1, t1) = arg2typ n arg;
   223       val (n2, t2) = args2typ n1 args
   224     in (n2, mk_sprodT (t1, t2)) end;
   225 
   226   fun cons2typ n [] = (n,oneT)
   227     | cons2typ n [con] = args2typ n (snd con)
   228     | cons2typ n (con::cons) =
   229     let
   230       val (n1, t1) = args2typ n (snd con);
   231       val (n2, t2) = cons2typ n1 cons
   232     in (n2, mk_ssumT (t1, t2)) end;
   233 in
   234   fun cons2ctyp cons = ctyp_of thy (snd (cons2typ 1 cons));
   235 end;
   236 
   237 local 
   238   val iso_swap = iso_locale RS iso_iso_swap;
   239   fun one_con (con, args) =
   240     let
   241       val vns = map vname args;
   242       val eqn = %:x_name === con_app2 con %: vns;
   243       val conj = foldr1 mk_conj (eqn :: map (defined o %:) (nonlazy args));
   244     in Library.foldr mk_ex (vns, conj) end;
   245 
   246   val conj_assoc = @{thm conj_assoc};
   247   val exh = foldr1 mk_disj ((%:x_name === UU) :: map one_con cons);
   248   val thm1 = instantiate' [SOME (cons2ctyp cons)] [] exh_start;
   249   val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1;
   250   val thm3 = rewrite_rule [mk_meta_eq @{thm conj_assoc}] thm2;
   251 
   252   (* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *)
   253   val tacs = [
   254     rtac disjE 1,
   255     etac (rep_defin' RS disjI1) 2,
   256     etac disjI2 2,
   257     rewrite_goals_tac [mk_meta_eq iso_swap],
   258     rtac thm3 1];
   259 in
   260   val _ = trace " Proving exhaust...";
   261   val exhaust = pg con_appls (mk_trp exh) (K tacs);
   262   val _ = trace " Proving casedist...";
   263   val casedist =
   264     Drule.standard (rewrite_rule exh_casedists (exhaust RS exh_casedist0));
   265 end;
   266 
   267 local 
   268   fun bind_fun t = Library.foldr mk_All (when_funs cons, t);
   269   fun bound_fun i _ = Bound (length cons - i);
   270   val when_app = list_ccomb (%%:(dname^"_when"), mapn bound_fun 1 cons);
   271 in
   272   val _ = trace " Proving when_strict...";
   273   val when_strict =
   274     let
   275       val axs = [when_appl, mk_meta_eq rep_strict];
   276       val goal = bind_fun (mk_trp (strict when_app));
   277       val tacs = [resolve_tac [sscase1, ssplit1, strictify1] 1];
   278     in pg axs goal (K tacs) end;
   279 
   280   val _ = trace " Proving when_apps...";
   281   val when_apps =
   282     let
   283       fun one_when n (con,args) =
   284         let
   285           val axs = when_appl :: con_appls;
   286           val goal = bind_fun (lift_defined %: (nonlazy args, 
   287                 mk_trp (when_app`(con_app con args) ===
   288                        list_ccomb (bound_fun n 0, map %# args))));
   289           val tacs = [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1];
   290         in pg axs goal (K tacs) end;
   291     in mapn one_when 1 cons end;
   292 end;
   293 val when_rews = when_strict :: when_apps;
   294 
   295 (* ----- theorems concerning the constructors, discriminators and selectors - *)
   296 
   297 local
   298   fun dis_strict (con, _) =
   299     let
   300       val goal = mk_trp (strict (%%:(dis_name con)));
   301     in pg axs_dis_def goal (K [rtac when_strict 1]) end;
   302 
   303   fun dis_app c (con, args) =
   304     let
   305       val lhs = %%:(dis_name c) ` con_app con args;
   306       val rhs = if con = c then TT else FF;
   307       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
   308       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   309     in pg axs_dis_def goal (K tacs) end;
   310 
   311   val _ = trace " Proving dis_apps...";
   312   val dis_apps = maps (fn (c,_) => map (dis_app c) cons) cons;
   313 
   314   fun dis_defin (con, args) =
   315     let
   316       val goal = defined (%:x_name) ==> defined (%%:(dis_name con) `% x_name);
   317       val tacs =
   318         [rtac casedist 1,
   319          contr_tac 1,
   320          DETERM_UNTIL_SOLVED (CHANGED
   321           (asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))];
   322     in pg [] goal (K tacs) end;
   323 
   324   val _ = trace " Proving dis_stricts...";
   325   val dis_stricts = map dis_strict cons;
   326   val _ = trace " Proving dis_defins...";
   327   val dis_defins = map dis_defin cons;
   328 in
   329   val dis_rews = dis_stricts @ dis_defins @ dis_apps;
   330 end;
   331 
   332 local
   333   fun mat_strict (con, _) =
   334     let
   335       val goal = mk_trp (%%:(mat_name con) ` UU ` %:"rhs" === UU);
   336       val tacs = [asm_simp_tac (HOLCF_ss addsimps [when_strict]) 1];
   337     in pg axs_mat_def goal (K tacs) end;
   338 
   339   val _ = trace " Proving mat_stricts...";
   340   val mat_stricts = map mat_strict cons;
   341 
   342   fun one_mat c (con, args) =
   343     let
   344       val lhs = %%:(mat_name c) ` con_app con args ` %:"rhs";
   345       val rhs =
   346         if con = c
   347         then list_ccomb (%:"rhs", map %# args)
   348         else mk_fail;
   349       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
   350       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   351     in pg axs_mat_def goal (K tacs) end;
   352 
   353   val _ = trace " Proving mat_apps...";
   354   val mat_apps =
   355     maps (fn (c,_) => map (one_mat c) cons) cons;
   356 in
   357   val mat_rews = mat_stricts @ mat_apps;
   358 end;
   359 
   360 local
   361   fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
   362 
   363   fun pat_lhs (con,args) = mk_branch (list_comb (%%:(pat_name con), ps args));
   364 
   365   fun pat_rhs (con,[]) = mk_return ((%:"rhs") ` HOLogic.unit)
   366     | pat_rhs (con,args) =
   367         (mk_branch (mk_ctuple_pat (ps args)))
   368           `(%:"rhs")`(mk_ctuple (map %# args));
   369 
   370   fun pat_strict c =
   371     let
   372       val axs = @{thm branch_def} :: axs_pat_def;
   373       val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));
   374       val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];
   375     in pg axs goal (K tacs) end;
   376 
   377   fun pat_app c (con, args) =
   378     let
   379       val axs = @{thm branch_def} :: axs_pat_def;
   380       val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);
   381       val rhs = if con = fst c then pat_rhs c else mk_fail;
   382       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
   383       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   384     in pg axs goal (K tacs) end;
   385 
   386   val _ = trace " Proving pat_stricts...";
   387   val pat_stricts = map pat_strict cons;
   388   val _ = trace " Proving pat_apps...";
   389   val pat_apps = maps (fn c => map (pat_app c) cons) cons;
   390 in
   391   val pat_rews = pat_stricts @ pat_apps;
   392 end;
   393 
   394 local
   395   fun con_strict (con, args) = 
   396     let
   397       val rules = abs_strict :: @{thms con_strict_rules};
   398       fun one_strict vn =
   399         let
   400           fun f arg = if vname arg = vn then UU else %# arg;
   401           val goal = mk_trp (con_app2 con f args === UU);
   402           val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1];
   403         in pg con_appls goal (K tacs) end;
   404     in map one_strict (nonlazy args) end;
   405 
   406   fun con_defin (con, args) =
   407     let
   408       fun iff_disj (t, []) = HOLogic.mk_not t
   409         | iff_disj (t, ts) = t === foldr1 HOLogic.mk_disj ts;
   410       val lhs = con_app con args === UU;
   411       val rhss = map (fn x => %:x === UU) (nonlazy args);
   412       val goal = mk_trp (iff_disj (lhs, rhss));
   413       val rule1 = iso_locale RS @{thm iso.abs_defined_iff};
   414       val rules = rule1 :: @{thms con_defined_iff_rules};
   415       val tacs = [simp_tac (HOL_ss addsimps rules) 1];
   416     in pg con_appls goal (K tacs) end;
   417 in
   418   val _ = trace " Proving con_stricts...";
   419   val con_stricts = maps con_strict cons;
   420   val _ = trace " Proving con_defins...";
   421   val con_defins = map con_defin cons;
   422   val con_rews = con_stricts @ con_defins;
   423 end;
   424 
   425 local
   426   val rules =
   427     [compact_sinl, compact_sinr, compact_spair, compact_up, compact_ONE];
   428   fun con_compact (con, args) =
   429     let
   430       val concl = mk_trp (mk_compact (con_app con args));
   431       val goal = lift (fn x => mk_compact (%#x)) (args, concl);
   432       val tacs = [
   433         rtac (iso_locale RS iso_compact_abs) 1,
   434         REPEAT (resolve_tac rules 1 ORELSE atac 1)];
   435     in pg con_appls goal (K tacs) end;
   436 in
   437   val _ = trace " Proving con_compacts...";
   438   val con_compacts = map con_compact cons;
   439 end;
   440 
   441 local
   442   fun one_sel sel =
   443     pg axs_sel_def (mk_trp (strict (%%:sel)))
   444       (K [simp_tac (HOLCF_ss addsimps when_rews) 1]);
   445 
   446   fun sel_strict (_, args) =
   447     map_filter (Option.map one_sel o sel_of) args;
   448 in
   449   val _ = trace " Proving sel_stricts...";
   450   val sel_stricts = maps sel_strict cons;
   451 end;
   452 
   453 local
   454   fun sel_app_same c n sel (con, args) =
   455     let
   456       val nlas = nonlazy args;
   457       val vns = map vname args;
   458       val vnn = List.nth (vns, n);
   459       val nlas' = filter (fn v => v <> vnn) nlas;
   460       val lhs = (%%:sel)`(con_app con args);
   461       val goal = lift_defined %: (nlas', mk_trp (lhs === %:vnn));
   462       fun tacs1 ctxt =
   463         if vnn mem nlas
   464         then [case_UU_tac ctxt (when_rews @ con_stricts) 1 vnn]
   465         else [];
   466       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   467     in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
   468 
   469   fun sel_app_diff c n sel (con, args) =
   470     let
   471       val nlas = nonlazy args;
   472       val goal = mk_trp (%%:sel ` con_app con args === UU);
   473       fun tacs1 ctxt = map (case_UU_tac ctxt (when_rews @ con_stricts) 1) nlas;
   474       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   475     in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
   476 
   477   fun sel_app c n sel (con, args) =
   478     if con = c
   479     then sel_app_same c n sel (con, args)
   480     else sel_app_diff c n sel (con, args);
   481 
   482   fun one_sel c n sel = map (sel_app c n sel) cons;
   483   fun one_sel' c n arg = Option.map (one_sel c n) (sel_of arg);
   484   fun one_con (c, args) =
   485     flat (map_filter I (mapn (one_sel' c) 0 args));
   486 in
   487   val _ = trace " Proving sel_apps...";
   488   val sel_apps = maps one_con cons;
   489 end;
   490 
   491 local
   492   fun sel_defin sel =
   493     let
   494       val goal = defined (%:x_name) ==> defined (%%:sel`%x_name);
   495       val tacs = [
   496         rtac casedist 1,
   497         contr_tac 1,
   498         DETERM_UNTIL_SOLVED (CHANGED
   499           (asm_simp_tac (HOLCF_ss addsimps sel_apps) 1))];
   500     in pg [] goal (K tacs) end;
   501 in
   502   val _ = trace " Proving sel_defins...";
   503   val sel_defins =
   504     if length cons = 1
   505     then map_filter (fn arg => Option.map sel_defin (sel_of arg))
   506                  (filter_out is_lazy (snd (hd cons)))
   507     else [];
   508 end;
   509 
   510 val sel_rews = sel_stricts @ sel_defins @ sel_apps;
   511 
   512 val _ = trace " Proving dist_les...";
   513 val distincts_le =
   514   let
   515     fun dist (con1, args1) (con2, args2) =
   516       let
   517         val goal = lift_defined %: (nonlazy args1,
   518                         mk_trp (con_app con1 args1 ~<< con_app con2 args2));
   519         fun tacs ctxt = [
   520           rtac @{thm rev_contrapos} 1,
   521           eres_inst_tac ctxt [(("f", 0), dis_name con1)] monofun_cfun_arg 1]
   522           @ map (case_UU_tac ctxt (con_stricts @ dis_rews) 1) (nonlazy args2)
   523           @ [asm_simp_tac (HOLCF_ss addsimps dis_rews) 1];
   524       in pg [] goal tacs end;
   525 
   526     fun distinct (con1, args1) (con2, args2) =
   527         let
   528           val arg1 = (con1, args1);
   529           val arg2 =
   530             (con2, ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
   531               (args2, Name.variant_list (map vname args1) (map vname args2)));
   532         in [dist arg1 arg2, dist arg2 arg1] end;
   533     fun distincts []      = []
   534       | distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
   535   in distincts cons end;
   536 val dist_les = flat (flat distincts_le);
   537 
   538 val _ = trace " Proving dist_eqs...";
   539 val dist_eqs =
   540   let
   541     fun distinct (_,args1) ((_,args2), leqs) =
   542       let
   543         val (le1,le2) = (hd leqs, hd(tl leqs));
   544         val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI)
   545       in
   546         if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
   547         if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
   548           [eq1, eq2]
   549       end;
   550     fun distincts []      = []
   551       | distincts ((c,leqs)::cs) =
   552         flat
   553           (ListPair.map (distinct c) ((map #1 cs),leqs)) @
   554         distincts cs;
   555   in map Drule.standard (distincts (cons ~~ distincts_le)) end;
   556 
   557 local 
   558   fun pgterm rel con args =
   559     let
   560       fun append s = upd_vname (fn v => v^s);
   561       val (largs, rargs) = (args, map (append "'") args);
   562       val concl =
   563         foldr1 mk_conj (ListPair.map rel (map %# largs, map %# rargs));
   564       val prem = rel (con_app con largs, con_app con rargs);
   565       val sargs = case largs of [_] => [] | _ => nonlazy args;
   566       val prop = lift_defined %: (sargs, mk_trp (prem === concl));
   567     in pg con_appls prop end;
   568   val cons' = filter (fn (_,args) => args<>[]) cons;
   569 in
   570   val _ = trace " Proving inverts...";
   571   val inverts =
   572     let
   573       val abs_less = ax_abs_iso RS (allI RS injection_less);
   574       val tacs =
   575         [asm_full_simp_tac (HOLCF_ss addsimps [abs_less, spair_less]) 1];
   576     in map (fn (con, args) => pgterm (op <<) con args (K tacs)) cons' end;
   577 
   578   val _ = trace " Proving injects...";
   579   val injects =
   580     let
   581       val abs_eq = ax_abs_iso RS (allI RS injection_eq);
   582       val tacs = [asm_full_simp_tac (HOLCF_ss addsimps [abs_eq, spair_eq]) 1];
   583     in map (fn (con, args) => pgterm (op ===) con args (K tacs)) cons' end;
   584 end;
   585 
   586 (* ----- theorems concerning one induction step ----------------------------- *)
   587 
   588 val copy_strict =
   589   let
   590     val _ = trace " Proving copy_strict...";
   591     val goal = mk_trp (strict (dc_copy `% "f"));
   592     val rules = [abs_strict, rep_strict] @ @{thms domain_fun_stricts};
   593     val tacs = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
   594   in pg [ax_copy_def] goal (K tacs) end;
   595 
   596 local
   597   fun copy_app (con, args) =
   598     let
   599       val lhs = dc_copy`%"f"`(con_app con args);
   600       fun one_rhs arg =
   601           if DatatypeAux.is_rec_type (dtyp_of arg)
   602           then Domain_Axioms.copy_of_dtyp (proj (%:"f") eqs) (dtyp_of arg) ` (%# arg)
   603           else (%# arg);
   604       val rhs = con_app2 con one_rhs args;
   605       val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
   606       val args' = filter_out (fn a => is_rec a orelse is_lazy a) args;
   607       val stricts = abs_strict :: rep_strict :: @{thms domain_fun_stricts};
   608       fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args';
   609       val rules = [ax_abs_iso] @ @{thms domain_fun_simps};
   610       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
   611     in pg (ax_copy_def::con_appls) goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
   612 in
   613   val _ = trace " Proving copy_apps...";
   614   val copy_apps = map copy_app cons;
   615 end;
   616 
   617 local
   618   fun one_strict (con, args) = 
   619     let
   620       val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);
   621       val rews = copy_strict :: copy_apps @ con_rews;
   622       fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @
   623         [asm_simp_tac (HOLCF_ss addsimps rews) 1];
   624     in pg [] goal tacs end;
   625 
   626   fun has_nonlazy_rec (_, args) = exists is_nonlazy_rec args;
   627 in
   628   val _ = trace " Proving copy_stricts...";
   629   val copy_stricts = map one_strict (filter has_nonlazy_rec cons);
   630 end;
   631 
   632 val copy_rews = copy_strict :: copy_apps @ copy_stricts;
   633 
   634 in
   635   thy
   636     |> Sign.add_path (Long_Name.base_name dname)
   637     |> snd o PureThy.add_thmss [
   638         ((Binding.name "iso_rews"  , iso_rews    ), [Simplifier.simp_add]),
   639         ((Binding.name "exhaust"   , [exhaust]   ), []),
   640         ((Binding.name "casedist"  , [casedist]  ), [Induct.cases_type dname]),
   641         ((Binding.name "when_rews" , when_rews   ), [Simplifier.simp_add]),
   642         ((Binding.name "compacts"  , con_compacts), [Simplifier.simp_add]),
   643         ((Binding.name "con_rews"  , con_rews    ),
   644          [Simplifier.simp_add, Fixrec.fixrec_simp_add]),
   645         ((Binding.name "sel_rews"  , sel_rews    ), [Simplifier.simp_add]),
   646         ((Binding.name "dis_rews"  , dis_rews    ), [Simplifier.simp_add]),
   647         ((Binding.name "pat_rews"  , pat_rews    ), [Simplifier.simp_add]),
   648         ((Binding.name "dist_les"  , dist_les    ), [Simplifier.simp_add]),
   649         ((Binding.name "dist_eqs"  , dist_eqs    ), [Simplifier.simp_add]),
   650         ((Binding.name "inverts"   , inverts     ), [Simplifier.simp_add]),
   651         ((Binding.name "injects"   , injects     ), [Simplifier.simp_add]),
   652         ((Binding.name "copy_rews" , copy_rews   ), [Simplifier.simp_add]),
   653         ((Binding.name "match_rews", mat_rews    ),
   654          [Simplifier.simp_add, Fixrec.fixrec_simp_add])]
   655     |> Sign.parent_path
   656     |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
   657         pat_rews @ dist_les @ dist_eqs @ copy_rews)
   658 end; (* let *)
   659 
   660 fun comp_theorems (comp_dnam, eqs: eq list) thy =
   661 let
   662 val global_ctxt = ProofContext.init thy;
   663 
   664 val dnames = map (fst o fst) eqs;
   665 val conss  = map  snd        eqs;
   666 val comp_dname = Sign.full_bname thy comp_dnam;
   667 
   668 val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
   669 val pg = pg' thy;
   670 
   671 (* ----- getting the composite axiom and definitions ------------------------ *)
   672 
   673 local
   674   fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
   675 in
   676   val axs_reach      = map (ga "reach"     ) dnames;
   677   val axs_take_def   = map (ga "take_def"  ) dnames;
   678   val axs_finite_def = map (ga "finite_def") dnames;
   679   val ax_copy2_def   =      ga "copy_def"  comp_dnam;
   680   val ax_bisim_def   =      ga "bisim_def" comp_dnam;
   681 end;
   682 
   683 local
   684   fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);
   685   fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
   686 in
   687   val cases = map (gt  "casedist" ) dnames;
   688   val con_rews  = maps (gts "con_rews" ) dnames;
   689   val copy_rews = maps (gts "copy_rews") dnames;
   690 end;
   691 
   692 fun dc_take dn = %%:(dn^"_take");
   693 val x_name = idx_name dnames "x"; 
   694 val P_name = idx_name dnames "P";
   695 val n_eqs = length eqs;
   696 
   697 (* ----- theorems concerning finite approximation and finite induction ------ *)
   698 
   699 local
   700   val iterate_Cprod_ss = global_simpset_of @{theory Fix};
   701   val copy_con_rews  = copy_rews @ con_rews;
   702   val copy_take_defs =
   703     (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
   704   val _ = trace " Proving take_stricts...";
   705   val take_stricts =
   706     let
   707       fun one_eq ((dn, args), _) = strict (dc_take dn $ %:"n");
   708       val goal = mk_trp (foldr1 mk_conj (map one_eq eqs));
   709       fun tacs ctxt = [
   710         InductTacs.induct_tac ctxt [[SOME "n"]] 1,
   711         simp_tac iterate_Cprod_ss 1,
   712         asm_simp_tac (iterate_Cprod_ss addsimps copy_rews) 1];
   713     in pg copy_take_defs goal tacs end;
   714 
   715   val take_stricts' = rewrite_rule copy_take_defs take_stricts;
   716   fun take_0 n dn =
   717     let
   718       val goal = mk_trp ((dc_take dn $ %%:"HOL.zero") `% x_name n === UU);
   719     in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end;
   720   val take_0s = mapn take_0 1 dnames;
   721   fun c_UU_tac ctxt = case_UU_tac ctxt (take_stricts'::copy_con_rews) 1;
   722   val _ = trace " Proving take_apps...";
   723   val take_apps =
   724     let
   725       fun mk_eqn dn (con, args) =
   726         let
   727           fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
   728           fun one_rhs arg =
   729               if DatatypeAux.is_rec_type (dtyp_of arg)
   730               then Domain_Axioms.copy_of_dtyp mk_take (dtyp_of arg) ` (%# arg)
   731               else (%# arg);
   732           val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);
   733           val rhs = con_app2 con one_rhs args;
   734         in Library.foldr mk_all (map vname args, lhs === rhs) end;
   735       fun mk_eqns ((dn, _), cons) = map (mk_eqn dn) cons;
   736       val goal = mk_trp (foldr1 mk_conj (maps mk_eqns eqs));
   737       val simps = filter (has_fewer_prems 1) copy_rews;
   738       fun con_tac ctxt (con, args) =
   739         if nonlazy_rec args = []
   740         then all_tac
   741         else EVERY (map (c_UU_tac ctxt) (nonlazy_rec args)) THEN
   742           asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1;
   743       fun eq_tacs ctxt ((dn, _), cons) = map (con_tac ctxt) cons;
   744       fun tacs ctxt =
   745         simp_tac iterate_Cprod_ss 1 ::
   746         InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
   747         simp_tac (iterate_Cprod_ss addsimps copy_con_rews) 1 ::
   748         asm_full_simp_tac (HOLCF_ss addsimps simps) 1 ::
   749         TRY (safe_tac HOL_cs) ::
   750         maps (eq_tacs ctxt) eqs;
   751     in pg copy_take_defs goal tacs end;
   752 in
   753   val take_rews = map Drule.standard
   754     (atomize global_ctxt take_stricts @ take_0s @ atomize global_ctxt take_apps);
   755 end; (* local *)
   756 
   757 local
   758   fun one_con p (con,args) =
   759     let
   760       fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
   761       val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
   762       val t2 = lift ind_hyp (filter is_rec args, t1);
   763       val t3 = lift_defined (bound_arg (map vname args)) (nonlazy args, t2);
   764     in Library.foldr mk_All (map vname args, t3) end;
   765 
   766   fun one_eq ((p, cons), concl) =
   767     mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
   768 
   769   fun ind_term concf = Library.foldr one_eq
   770     (mapn (fn n => fn x => (P_name n, x)) 1 conss,
   771      mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
   772   val take_ss = HOL_ss addsimps take_rews;
   773   fun quant_tac ctxt i = EVERY
   774     (mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);
   775 
   776   fun ind_prems_tac prems = EVERY
   777     (maps (fn cons =>
   778       (resolve_tac prems 1 ::
   779         maps (fn (_,args) => 
   780           resolve_tac prems 1 ::
   781           map (K(atac 1)) (nonlazy args) @
   782           map (K(atac 1)) (filter is_rec args))
   783         cons))
   784       conss);
   785   local 
   786     (* check whether every/exists constructor of the n-th part of the equation:
   787        it has a possibly indirectly recursive argument that isn't/is possibly 
   788        indirectly lazy *)
   789     fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
   790           is_rec arg andalso not(rec_of arg mem ns) andalso
   791           ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
   792             rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
   793               (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
   794           ) o snd) cons;
   795     fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
   796     fun warn (n,cons) =
   797       if all_rec_to [] false (n,cons)
   798       then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
   799       else false;
   800     fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;
   801 
   802   in
   803     val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
   804     val is_emptys = map warn n__eqs;
   805     val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
   806   end;
   807 in (* local *)
   808   val _ = trace " Proving finite_ind...";
   809   val finite_ind =
   810     let
   811       fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
   812       val goal = ind_term concf;
   813 
   814       fun tacf {prems, context} =
   815         let
   816           val tacs1 = [
   817             quant_tac context 1,
   818             simp_tac HOL_ss 1,
   819             InductTacs.induct_tac context [[SOME "n"]] 1,
   820             simp_tac (take_ss addsimps prems) 1,
   821             TRY (safe_tac HOL_cs)];
   822           fun arg_tac arg =
   823             case_UU_tac context (prems @ con_rews) 1
   824               (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
   825           fun con_tacs (con, args) = 
   826             asm_simp_tac take_ss 1 ::
   827             map arg_tac (filter is_nonlazy_rec args) @
   828             [resolve_tac prems 1] @
   829             map (K (atac 1)) (nonlazy args) @
   830             map (K (etac spec 1)) (filter is_rec args);
   831           fun cases_tacs (cons, cases) =
   832             res_inst_tac context [(("x", 0), "x")] cases 1 ::
   833             asm_simp_tac (take_ss addsimps prems) 1 ::
   834             maps con_tacs cons;
   835         in
   836           tacs1 @ maps cases_tacs (conss ~~ cases)
   837         end;
   838     in pg'' thy [] goal tacf
   839        handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
   840     end;
   841 
   842   val _ = trace " Proving take_lemmas...";
   843   val take_lemmas =
   844     let
   845       fun take_lemma n (dn, ax_reach) =
   846         let
   847           val lhs = dc_take dn $ Bound 0 `%(x_name n);
   848           val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");
   849           val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));
   850           val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;
   851           val rules = [contlub_fst RS contlubE RS ssubst,
   852                        contlub_snd RS contlubE RS ssubst];
   853           fun tacf {prems, context} = [
   854             res_inst_tac context [(("t", 0), x_name n    )] (ax_reach RS subst) 1,
   855             res_inst_tac context [(("t", 0), x_name n^"'")] (ax_reach RS subst) 1,
   856             stac fix_def2 1,
   857             REPEAT (CHANGED
   858               (resolve_tac rules 1 THEN chain_tac 1)),
   859             stac contlub_cfun_fun 1,
   860             stac contlub_cfun_fun 2,
   861             rtac lub_equal 3,
   862             chain_tac 1,
   863             rtac allI 1,
   864             resolve_tac prems 1];
   865         in pg'' thy axs_take_def goal tacf end;
   866     in mapn take_lemma 1 (dnames ~~ axs_reach) end;
   867 
   868 (* ----- theorems concerning finiteness and induction ----------------------- *)
   869 
   870   val _ = trace " Proving finites, ind...";
   871   val (finites, ind) =
   872   (
   873     if is_finite
   874     then (* finite case *)
   875       let 
   876         fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
   877         fun dname_lemma dn =
   878           let
   879             val prem1 = mk_trp (defined (%:"x"));
   880             val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);
   881             val prem2 = mk_trp (mk_disj (disj1, take_enough dn));
   882             val concl = mk_trp (take_enough dn);
   883             val goal = prem1 ===> prem2 ===> concl;
   884             val tacs = [
   885               etac disjE 1,
   886               etac notE 1,
   887               resolve_tac take_lemmas 1,
   888               asm_simp_tac take_ss 1,
   889               atac 1];
   890           in pg [] goal (K tacs) end;
   891         val _ = trace " Proving finite_lemmas1a";
   892         val finite_lemmas1a = map dname_lemma dnames;
   893  
   894         val _ = trace " Proving finite_lemma1b";
   895         val finite_lemma1b =
   896           let
   897             fun mk_eqn n ((dn, args), _) =
   898               let
   899                 val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;
   900                 val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;
   901               in
   902                 mk_constrainall
   903                   (x_name n, Type (dn,args), mk_disj (disj1, disj2))
   904               end;
   905             val goal =
   906               mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));
   907             fun arg_tacs ctxt vn = [
   908               eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1,
   909               etac disjE 1,
   910               asm_simp_tac (HOL_ss addsimps con_rews) 1,
   911               asm_simp_tac take_ss 1];
   912             fun con_tacs ctxt (con, args) =
   913               asm_simp_tac take_ss 1 ::
   914               maps (arg_tacs ctxt) (nonlazy_rec args);
   915             fun foo_tacs ctxt n (cons, cases) =
   916               simp_tac take_ss 1 ::
   917               rtac allI 1 ::
   918               res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 ::
   919               asm_simp_tac take_ss 1 ::
   920               maps (con_tacs ctxt) cons;
   921             fun tacs ctxt =
   922               rtac allI 1 ::
   923               InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
   924               simp_tac take_ss 1 ::
   925               TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::
   926               flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases));
   927           in pg [] goal tacs end;
   928 
   929         fun one_finite (dn, l1b) =
   930           let
   931             val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
   932             fun tacs ctxt = [
   933               case_UU_tac ctxt take_rews 1 "x",
   934               eresolve_tac finite_lemmas1a 1,
   935               step_tac HOL_cs 1,
   936               step_tac HOL_cs 1,
   937               cut_facts_tac [l1b] 1,
   938               fast_tac HOL_cs 1];
   939           in pg axs_finite_def goal tacs end;
   940 
   941         val _ = trace " Proving finites";
   942         val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b);
   943         val _ = trace " Proving ind";
   944         val ind =
   945           let
   946             fun concf n dn = %:(P_name n) $ %:(x_name n);
   947             fun tacf {prems, context} =
   948               let
   949                 fun finite_tacs (finite, fin_ind) = [
   950                   rtac(rewrite_rule axs_finite_def finite RS exE)1,
   951                   etac subst 1,
   952                   rtac fin_ind 1,
   953                   ind_prems_tac prems];
   954               in
   955                 TRY (safe_tac HOL_cs) ::
   956                 maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)
   957               end;
   958           in pg'' thy [] (ind_term concf) tacf end;
   959       in (finites, ind) end (* let *)
   960 
   961     else (* infinite case *)
   962       let
   963         fun one_finite n dn =
   964           read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;
   965         val finites = mapn one_finite 1 dnames;
   966 
   967         val goal =
   968           let
   969             fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));
   970             fun concf n dn = %:(P_name n) $ %:(x_name n);
   971           in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
   972         val cont_rules =
   973             [cont_id, cont_const, cont2cont_Rep_CFun,
   974              cont2cont_fst, cont2cont_snd];
   975         fun tacf {prems, context} =
   976           map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
   977           quant_tac context 1,
   978           rtac (adm_impl_admw RS wfix_ind) 1,
   979           REPEAT_DETERM (rtac adm_all 1),
   980           REPEAT_DETERM (
   981             TRY (rtac adm_conj 1) THEN 
   982             rtac adm_subst 1 THEN 
   983             REPEAT (resolve_tac cont_rules 1) THEN
   984             resolve_tac prems 1),
   985           strip_tac 1,
   986           rtac (rewrite_rule axs_take_def finite_ind) 1,
   987           ind_prems_tac prems];
   988         val ind = (pg'' thy [] goal tacf
   989           handle ERROR _ =>
   990             (warning "Cannot prove infinite induction rule"; TrueI));
   991       in (finites, ind) end
   992   )
   993       handle THM _ =>
   994              (warning "Induction proofs failed (THM raised)."; ([], TrueI))
   995            | ERROR _ =>
   996              (warning "Induction proofs failed (ERROR raised)."; ([], TrueI));
   997 
   998 
   999 end; (* local *)
  1000 
  1001 (* ----- theorem concerning coinduction ------------------------------------- *)
  1002 
  1003 local
  1004   val xs = mapn (fn n => K (x_name n)) 1 dnames;
  1005   fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
  1006   val take_ss = HOL_ss addsimps take_rews;
  1007   val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
  1008   val _ = trace " Proving coind_lemma...";
  1009   val coind_lemma =
  1010     let
  1011       fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
  1012       fun mk_eqn n dn =
  1013         (dc_take dn $ %:"n" ` bnd_arg n 0) ===
  1014         (dc_take dn $ %:"n" ` bnd_arg n 1);
  1015       fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
  1016       val goal =
  1017         mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
  1018           Library.foldr mk_all2 (xs,
  1019             Library.foldr mk_imp (mapn mk_prj 0 dnames,
  1020               foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
  1021       fun x_tacs ctxt n x = [
  1022         rotate_tac (n+1) 1,
  1023         etac all2E 1,
  1024         eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
  1025         TRY (safe_tac HOL_cs),
  1026         REPEAT (CHANGED (asm_simp_tac take_ss 1))];
  1027       fun tacs ctxt = [
  1028         rtac impI 1,
  1029         InductTacs.induct_tac ctxt [[SOME "n"]] 1,
  1030         simp_tac take_ss 1,
  1031         safe_tac HOL_cs] @
  1032         flat (mapn (x_tacs ctxt) 0 xs);
  1033     in pg [ax_bisim_def] goal tacs end;
  1034 in
  1035   val _ = trace " Proving coind...";
  1036   val coind = 
  1037     let
  1038       fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
  1039       fun mk_eqn x = %:x === %:(x^"'");
  1040       val goal =
  1041         mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
  1042           Logic.list_implies (mapn mk_prj 0 xs,
  1043             mk_trp (foldr1 mk_conj (map mk_eqn xs)));
  1044       val tacs =
  1045         TRY (safe_tac HOL_cs) ::
  1046         maps (fn take_lemma => [
  1047           rtac take_lemma 1,
  1048           cut_facts_tac [coind_lemma] 1,
  1049           fast_tac HOL_cs 1])
  1050         take_lemmas;
  1051     in pg [] goal (K tacs) end;
  1052 end; (* local *)
  1053 
  1054 val inducts = Project_Rule.projections (ProofContext.init thy) ind;
  1055 fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);
  1056 val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
  1057 
  1058 in thy |> Sign.add_path comp_dnam
  1059        |> snd o PureThy.add_thmss [
  1060            ((Binding.name "take_rews"  , take_rews   ), [Simplifier.simp_add]),
  1061            ((Binding.name "take_lemmas", take_lemmas ), []),
  1062            ((Binding.name "finites"    , finites     ), []),
  1063            ((Binding.name "finite_ind" , [finite_ind]), []),
  1064            ((Binding.name "ind"        , [ind]       ), []),
  1065            ((Binding.name "coind"      , [coind]     ), [])]
  1066        |> (if induct_failed then I
  1067            else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
  1068        |> Sign.parent_path |> pair take_rews
  1069 end; (* let *)
  1070 end; (* local *)
  1071 end; (* struct *)