src/HOLCF/Tools/Domain/domain_theorems.ML
author wenzelm
Fri Jul 24 18:58:58 2009 +0200 (2009-07-24)
changeset 32172 c4e55f30d527
parent 32149 ef59550a55d3
child 32740 9dd0a2f83429
permissions -rw-r--r--
renamed functor ProjectRuleFun to Project_Rule;
renamed structure ProjectRule to Project_Rule;
     1 (*  Title:      HOLCF/Tools/Domain/domain_theorems.ML
     2     Author:     David von Oheimb
     3                 New proofs/tactics by 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 ref;
    15   val trace_domain: bool ref;
    16 end;
    17 
    18 structure Domain_Theorems :> DOMAIN_THEOREMS =
    19 struct
    20 
    21 val quiet_mode = ref false;
    22 val trace_domain = 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_Rep_CFunL = @{thm ch2ch_Rep_CFunL};
    37 val ch2ch_Rep_CFunR = @{thm ch2ch_Rep_CFunR};
    38 val chain_iterate = @{thm chain_iterate};
    39 val compact_ONE = @{thm compact_ONE};
    40 val compact_sinl = @{thm compact_sinl};
    41 val compact_sinr = @{thm compact_sinr};
    42 val compact_spair = @{thm compact_spair};
    43 val compact_up = @{thm compact_up};
    44 val contlub_cfun_arg = @{thm contlub_cfun_arg};
    45 val contlub_cfun_fun = @{thm contlub_cfun_fun};
    46 val fix_def2 = @{thm fix_def2};
    47 val injection_eq = @{thm injection_eq};
    48 val injection_less = @{thm injection_below};
    49 val lub_equal = @{thm lub_equal};
    50 val monofun_cfun_arg = @{thm monofun_cfun_arg};
    51 val retraction_strict = @{thm retraction_strict};
    52 val spair_eq = @{thm spair_eq};
    53 val spair_less = @{thm spair_below};
    54 val sscase1 = @{thm sscase1};
    55 val ssplit1 = @{thm ssplit1};
    56 val strictify1 = @{thm strictify1};
    57 val wfix_ind = @{thm wfix_ind};
    58 
    59 val iso_intro       = @{thm iso.intro};
    60 val iso_abs_iso     = @{thm iso.abs_iso};
    61 val iso_rep_iso     = @{thm iso.rep_iso};
    62 val iso_abs_strict  = @{thm iso.abs_strict};
    63 val iso_rep_strict  = @{thm iso.rep_strict};
    64 val iso_abs_defin'  = @{thm iso.abs_defin'};
    65 val iso_rep_defin'  = @{thm iso.rep_defin'};
    66 val iso_abs_defined = @{thm iso.abs_defined};
    67 val iso_rep_defined = @{thm iso.rep_defined};
    68 val iso_compact_abs = @{thm iso.compact_abs};
    69 val iso_compact_rep = @{thm iso.compact_rep};
    70 val iso_iso_swap    = @{thm iso.iso_swap};
    71 
    72 val exh_start = @{thm exh_start};
    73 val ex_defined_iffs = @{thms ex_defined_iffs};
    74 val exh_casedist0 = @{thm exh_casedist0};
    75 val exh_casedists = @{thms exh_casedists};
    76 
    77 open Domain_Library;
    78 infixr 0 ===>;
    79 infixr 0 ==>;
    80 infix 0 == ; 
    81 infix 1 ===;
    82 infix 1 ~= ;
    83 infix 1 <<;
    84 infix 1 ~<<;
    85 infix 9 `   ;
    86 infix 9 `% ;
    87 infix 9 `%%;
    88 infixr 9 oo;
    89 
    90 (* ----- general proof facilities ------------------------------------------- *)
    91 
    92 fun legacy_infer_term thy t =
    93   let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init thy)
    94   in singleton (Syntax.check_terms ctxt) (Sign.intern_term thy t) end;
    95 
    96 fun pg'' thy defs t tacs =
    97   let
    98     val t' = legacy_infer_term thy t;
    99     val asms = Logic.strip_imp_prems t';
   100     val prop = Logic.strip_imp_concl t';
   101     fun tac {prems, context} =
   102       rewrite_goals_tac defs THEN
   103       EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})
   104   in Goal.prove_global thy [] asms prop tac end;
   105 
   106 fun pg' thy defs t tacsf =
   107   let
   108     fun tacs {prems, context} =
   109       if null prems then tacsf context
   110       else cut_facts_tac prems 1 :: tacsf context;
   111   in pg'' thy defs t tacs end;
   112 
   113 fun case_UU_tac ctxt rews i v =
   114   InductTacs.case_tac ctxt (v^"=UU") i THEN
   115   asm_simp_tac (HOLCF_ss addsimps rews) i;
   116 
   117 val chain_tac =
   118   REPEAT_DETERM o resolve_tac 
   119     [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL];
   120 
   121 (* ----- general proofs ----------------------------------------------------- *)
   122 
   123 val all2E = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp}
   124 
   125 val dist_eqI = @{lemma "!!x::'a::po. ~ x << y ==> x ~= y" by (blast dest!: below_antisym_inverse)}
   126 
   127 in
   128 
   129 fun theorems (((dname, _), cons) : eq, eqs : eq list) thy =
   130 let
   131 
   132 val _ = message ("Proving isomorphism properties of domain "^dname^" ...");
   133 val pg = pg' thy;
   134 
   135 (* ----- getting the axioms and definitions --------------------------------- *)
   136 
   137 local
   138   fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
   139 in
   140   val ax_abs_iso  = ga "abs_iso"  dname;
   141   val ax_rep_iso  = ga "rep_iso"  dname;
   142   val ax_when_def = ga "when_def" dname;
   143   fun get_def mk_name (con,_) = ga (mk_name con^"_def") dname;
   144   val axs_con_def = map (get_def extern_name) cons;
   145   val axs_dis_def = map (get_def dis_name) cons;
   146   val axs_mat_def = map (get_def mat_name) cons;
   147   val axs_pat_def = map (get_def pat_name) cons;
   148   val axs_sel_def =
   149     let
   150       fun def_of_sel sel = ga (sel^"_def") dname;
   151       fun def_of_arg arg = Option.map def_of_sel (sel_of arg);
   152       fun defs_of_con (_, args) = List.mapPartial def_of_arg args;
   153     in
   154       maps defs_of_con cons
   155     end;
   156   val ax_copy_def = ga "copy_def" dname;
   157 end; (* local *)
   158 
   159 (* ----- theorems concerning the isomorphism -------------------------------- *)
   160 
   161 val dc_abs  = %%:(dname^"_abs");
   162 val dc_rep  = %%:(dname^"_rep");
   163 val dc_copy = %%:(dname^"_copy");
   164 val x_name = "x";
   165 
   166 val iso_locale = iso_intro OF [ax_abs_iso, ax_rep_iso];
   167 val abs_strict = ax_rep_iso RS (allI RS retraction_strict);
   168 val rep_strict = ax_abs_iso RS (allI RS retraction_strict);
   169 val abs_defin' = iso_locale RS iso_abs_defin';
   170 val rep_defin' = iso_locale RS iso_rep_defin';
   171 val iso_rews = map standard [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
   172 
   173 (* ----- generating beta reduction rules from definitions-------------------- *)
   174 
   175 val _ = trace " Proving beta reduction rules...";
   176 
   177 local
   178   fun arglist (Const _ $ Abs (s, _, t)) =
   179     let
   180       val (vars,body) = arglist t;
   181     in (s :: vars, body) end
   182     | arglist t = ([], t);
   183   fun bind_fun vars t = Library.foldr mk_All (vars, t);
   184   fun bound_vars 0 = []
   185     | bound_vars i = Bound (i-1) :: bound_vars (i - 1);
   186 in
   187   fun appl_of_def def =
   188     let
   189       val (_ $ con $ lam) = concl_of def;
   190       val (vars, rhs) = arglist lam;
   191       val lhs = list_ccomb (con, bound_vars (length vars));
   192       val appl = bind_fun vars (lhs == rhs);
   193       val cs = ContProc.cont_thms lam;
   194       val betas = map (fn c => mk_meta_eq (c RS beta_cfun)) cs;
   195     in pg (def::betas) appl (K [rtac reflexive_thm 1]) end;
   196 end;
   197 
   198 val _ = trace "Proving when_appl...";
   199 val when_appl = appl_of_def ax_when_def;
   200 val _ = trace "Proving con_appls...";
   201 val con_appls = map appl_of_def axs_con_def;
   202 
   203 local
   204   fun arg2typ n arg =
   205     let val t = TVar (("'a", n), pcpoS)
   206     in (n + 1, if is_lazy arg then mk_uT t else t) end;
   207 
   208   fun args2typ n [] = (n, oneT)
   209     | args2typ n [arg] = arg2typ n arg
   210     | args2typ n (arg::args) =
   211     let
   212       val (n1, t1) = arg2typ n arg;
   213       val (n2, t2) = args2typ n1 args
   214     in (n2, mk_sprodT (t1, t2)) end;
   215 
   216   fun cons2typ n [] = (n,oneT)
   217     | cons2typ n [con] = args2typ n (snd con)
   218     | cons2typ n (con::cons) =
   219     let
   220       val (n1, t1) = args2typ n (snd con);
   221       val (n2, t2) = cons2typ n1 cons
   222     in (n2, mk_ssumT (t1, t2)) end;
   223 in
   224   fun cons2ctyp cons = ctyp_of thy (snd (cons2typ 1 cons));
   225 end;
   226 
   227 local 
   228   val iso_swap = iso_locale RS iso_iso_swap;
   229   fun one_con (con, args) =
   230     let
   231       val vns = map vname args;
   232       val eqn = %:x_name === con_app2 con %: vns;
   233       val conj = foldr1 mk_conj (eqn :: map (defined o %:) (nonlazy args));
   234     in Library.foldr mk_ex (vns, conj) end;
   235 
   236   val conj_assoc = @{thm conj_assoc};
   237   val exh = foldr1 mk_disj ((%:x_name === UU) :: map one_con cons);
   238   val thm1 = instantiate' [SOME (cons2ctyp cons)] [] exh_start;
   239   val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1;
   240   val thm3 = rewrite_rule [mk_meta_eq @{thm conj_assoc}] thm2;
   241 
   242   (* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *)
   243   val tacs = [
   244     rtac disjE 1,
   245     etac (rep_defin' RS disjI1) 2,
   246     etac disjI2 2,
   247     rewrite_goals_tac [mk_meta_eq iso_swap],
   248     rtac thm3 1];
   249 in
   250   val _ = trace " Proving exhaust...";
   251   val exhaust = pg con_appls (mk_trp exh) (K tacs);
   252   val _ = trace " Proving casedist...";
   253   val casedist =
   254     standard (rewrite_rule exh_casedists (exhaust RS exh_casedist0));
   255 end;
   256 
   257 local 
   258   fun bind_fun t = Library.foldr mk_All (when_funs cons, t);
   259   fun bound_fun i _ = Bound (length cons - i);
   260   val when_app = list_ccomb (%%:(dname^"_when"), mapn bound_fun 1 cons);
   261 in
   262   val _ = trace " Proving when_strict...";
   263   val when_strict =
   264     let
   265       val axs = [when_appl, mk_meta_eq rep_strict];
   266       val goal = bind_fun (mk_trp (strict when_app));
   267       val tacs = [resolve_tac [sscase1, ssplit1, strictify1] 1];
   268     in pg axs goal (K tacs) end;
   269 
   270   val _ = trace " Proving when_apps...";
   271   val when_apps =
   272     let
   273       fun one_when n (con,args) =
   274         let
   275           val axs = when_appl :: con_appls;
   276           val goal = bind_fun (lift_defined %: (nonlazy args, 
   277                 mk_trp (when_app`(con_app con args) ===
   278                        list_ccomb (bound_fun n 0, map %# args))));
   279           val tacs = [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1];
   280         in pg axs goal (K tacs) end;
   281     in mapn one_when 1 cons end;
   282 end;
   283 val when_rews = when_strict :: when_apps;
   284 
   285 (* ----- theorems concerning the constructors, discriminators and selectors - *)
   286 
   287 local
   288   fun dis_strict (con, _) =
   289     let
   290       val goal = mk_trp (strict (%%:(dis_name con)));
   291     in pg axs_dis_def goal (K [rtac when_strict 1]) end;
   292 
   293   fun dis_app c (con, args) =
   294     let
   295       val lhs = %%:(dis_name c) ` con_app con args;
   296       val rhs = if con = c then TT else FF;
   297       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
   298       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   299     in pg axs_dis_def goal (K tacs) end;
   300 
   301   val _ = trace " Proving dis_apps...";
   302   val dis_apps = maps (fn (c,_) => map (dis_app c) cons) cons;
   303 
   304   fun dis_defin (con, args) =
   305     let
   306       val goal = defined (%:x_name) ==> defined (%%:(dis_name con) `% x_name);
   307       val tacs =
   308         [rtac casedist 1,
   309          contr_tac 1,
   310          DETERM_UNTIL_SOLVED (CHANGED
   311           (asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))];
   312     in pg [] goal (K tacs) end;
   313 
   314   val _ = trace " Proving dis_stricts...";
   315   val dis_stricts = map dis_strict cons;
   316   val _ = trace " Proving dis_defins...";
   317   val dis_defins = map dis_defin cons;
   318 in
   319   val dis_rews = dis_stricts @ dis_defins @ dis_apps;
   320 end;
   321 
   322 local
   323   fun mat_strict (con, _) =
   324     let
   325       val goal = mk_trp (%%:(mat_name con) ` UU ` %:"rhs" === UU);
   326       val tacs = [asm_simp_tac (HOLCF_ss addsimps [when_strict]) 1];
   327     in pg axs_mat_def goal (K tacs) end;
   328 
   329   val _ = trace " Proving mat_stricts...";
   330   val mat_stricts = map mat_strict cons;
   331 
   332   fun one_mat c (con, args) =
   333     let
   334       val lhs = %%:(mat_name c) ` con_app con args ` %:"rhs";
   335       val rhs =
   336         if con = c
   337         then list_ccomb (%:"rhs", map %# args)
   338         else mk_fail;
   339       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
   340       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   341     in pg axs_mat_def goal (K tacs) end;
   342 
   343   val _ = trace " Proving mat_apps...";
   344   val mat_apps =
   345     maps (fn (c,_) => map (one_mat c) cons) cons;
   346 in
   347   val mat_rews = mat_stricts @ mat_apps;
   348 end;
   349 
   350 local
   351   fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
   352 
   353   fun pat_lhs (con,args) = mk_branch (list_comb (%%:(pat_name con), ps args));
   354 
   355   fun pat_rhs (con,[]) = mk_return ((%:"rhs") ` HOLogic.unit)
   356     | pat_rhs (con,args) =
   357         (mk_branch (mk_ctuple_pat (ps args)))
   358           `(%:"rhs")`(mk_ctuple (map %# args));
   359 
   360   fun pat_strict c =
   361     let
   362       val axs = @{thm branch_def} :: axs_pat_def;
   363       val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));
   364       val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];
   365     in pg axs goal (K tacs) end;
   366 
   367   fun pat_app c (con, args) =
   368     let
   369       val axs = @{thm branch_def} :: axs_pat_def;
   370       val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);
   371       val rhs = if con = fst c then pat_rhs c else mk_fail;
   372       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
   373       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   374     in pg axs goal (K tacs) end;
   375 
   376   val _ = trace " Proving pat_stricts...";
   377   val pat_stricts = map pat_strict cons;
   378   val _ = trace " Proving pat_apps...";
   379   val pat_apps = maps (fn c => map (pat_app c) cons) cons;
   380 in
   381   val pat_rews = pat_stricts @ pat_apps;
   382 end;
   383 
   384 local
   385   fun con_strict (con, args) = 
   386     let
   387       val rules = abs_strict :: @{thms con_strict_rules};
   388       fun one_strict vn =
   389         let
   390           fun f arg = if vname arg = vn then UU else %# arg;
   391           val goal = mk_trp (con_app2 con f args === UU);
   392           val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1];
   393         in pg con_appls goal (K tacs) end;
   394     in map one_strict (nonlazy args) end;
   395 
   396   fun con_defin (con, args) =
   397     let
   398       fun iff_disj (t, []) = HOLogic.mk_not t
   399         | iff_disj (t, ts) = t === foldr1 HOLogic.mk_disj ts;
   400       val lhs = con_app con args === UU;
   401       val rhss = map (fn x => %:x === UU) (nonlazy args);
   402       val goal = mk_trp (iff_disj (lhs, rhss));
   403       val rule1 = iso_locale RS @{thm iso.abs_defined_iff};
   404       val rules = rule1 :: @{thms con_defined_iff_rules};
   405       val tacs = [simp_tac (HOL_ss addsimps rules) 1];
   406     in pg con_appls goal (K tacs) end;
   407 in
   408   val _ = trace " Proving con_stricts...";
   409   val con_stricts = maps con_strict cons;
   410   val _ = trace " Proving con_defins...";
   411   val con_defins = map con_defin cons;
   412   val con_rews = con_stricts @ con_defins;
   413 end;
   414 
   415 local
   416   val rules =
   417     [compact_sinl, compact_sinr, compact_spair, compact_up, compact_ONE];
   418   fun con_compact (con, args) =
   419     let
   420       val concl = mk_trp (mk_compact (con_app con args));
   421       val goal = lift (fn x => mk_compact (%#x)) (args, concl);
   422       val tacs = [
   423         rtac (iso_locale RS iso_compact_abs) 1,
   424         REPEAT (resolve_tac rules 1 ORELSE atac 1)];
   425     in pg con_appls goal (K tacs) end;
   426 in
   427   val _ = trace " Proving con_compacts...";
   428   val con_compacts = map con_compact cons;
   429 end;
   430 
   431 local
   432   fun one_sel sel =
   433     pg axs_sel_def (mk_trp (strict (%%:sel)))
   434       (K [simp_tac (HOLCF_ss addsimps when_rews) 1]);
   435 
   436   fun sel_strict (_, args) =
   437     List.mapPartial (Option.map one_sel o sel_of) args;
   438 in
   439   val _ = trace " Proving sel_stricts...";
   440   val sel_stricts = maps sel_strict cons;
   441 end;
   442 
   443 local
   444   fun sel_app_same c n sel (con, args) =
   445     let
   446       val nlas = nonlazy args;
   447       val vns = map vname args;
   448       val vnn = List.nth (vns, n);
   449       val nlas' = List.filter (fn v => v <> vnn) nlas;
   450       val lhs = (%%:sel)`(con_app con args);
   451       val goal = lift_defined %: (nlas', mk_trp (lhs === %:vnn));
   452       fun tacs1 ctxt =
   453         if vnn mem nlas
   454         then [case_UU_tac ctxt (when_rews @ con_stricts) 1 vnn]
   455         else [];
   456       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   457     in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
   458 
   459   fun sel_app_diff c n sel (con, args) =
   460     let
   461       val nlas = nonlazy args;
   462       val goal = mk_trp (%%:sel ` con_app con args === UU);
   463       fun tacs1 ctxt = map (case_UU_tac ctxt (when_rews @ con_stricts) 1) nlas;
   464       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   465     in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
   466 
   467   fun sel_app c n sel (con, args) =
   468     if con = c
   469     then sel_app_same c n sel (con, args)
   470     else sel_app_diff c n sel (con, args);
   471 
   472   fun one_sel c n sel = map (sel_app c n sel) cons;
   473   fun one_sel' c n arg = Option.map (one_sel c n) (sel_of arg);
   474   fun one_con (c, args) =
   475     flat (map_filter I (mapn (one_sel' c) 0 args));
   476 in
   477   val _ = trace " Proving sel_apps...";
   478   val sel_apps = maps one_con cons;
   479 end;
   480 
   481 local
   482   fun sel_defin sel =
   483     let
   484       val goal = defined (%:x_name) ==> defined (%%:sel`%x_name);
   485       val tacs = [
   486         rtac casedist 1,
   487         contr_tac 1,
   488         DETERM_UNTIL_SOLVED (CHANGED
   489           (asm_simp_tac (HOLCF_ss addsimps sel_apps) 1))];
   490     in pg [] goal (K tacs) end;
   491 in
   492   val _ = trace " Proving sel_defins...";
   493   val sel_defins =
   494     if length cons = 1
   495     then List.mapPartial (fn arg => Option.map sel_defin (sel_of arg))
   496                  (filter_out is_lazy (snd (hd cons)))
   497     else [];
   498 end;
   499 
   500 val sel_rews = sel_stricts @ sel_defins @ sel_apps;
   501 
   502 val _ = trace " Proving dist_les...";
   503 val distincts_le =
   504   let
   505     fun dist (con1, args1) (con2, args2) =
   506       let
   507         val goal = lift_defined %: (nonlazy args1,
   508                         mk_trp (con_app con1 args1 ~<< con_app con2 args2));
   509         fun tacs ctxt = [
   510           rtac @{thm rev_contrapos} 1,
   511           eres_inst_tac ctxt [(("f", 0), dis_name con1)] monofun_cfun_arg 1]
   512           @ map (case_UU_tac ctxt (con_stricts @ dis_rews) 1) (nonlazy args2)
   513           @ [asm_simp_tac (HOLCF_ss addsimps dis_rews) 1];
   514       in pg [] goal tacs end;
   515 
   516     fun distinct (con1, args1) (con2, args2) =
   517         let
   518           val arg1 = (con1, args1);
   519           val arg2 =
   520             (con2, ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
   521               (args2, Name.variant_list (map vname args1) (map vname args2)));
   522         in [dist arg1 arg2, dist arg2 arg1] end;
   523     fun distincts []      = []
   524       | distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
   525   in distincts cons end;
   526 val dist_les = flat (flat distincts_le);
   527 
   528 val _ = trace " Proving dist_eqs...";
   529 val dist_eqs =
   530   let
   531     fun distinct (_,args1) ((_,args2), leqs) =
   532       let
   533         val (le1,le2) = (hd leqs, hd(tl leqs));
   534         val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI)
   535       in
   536         if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
   537         if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
   538           [eq1, eq2]
   539       end;
   540     fun distincts []      = []
   541       | distincts ((c,leqs)::cs) =
   542         flat
   543           (ListPair.map (distinct c) ((map #1 cs),leqs)) @
   544         distincts cs;
   545   in map standard (distincts (cons ~~ distincts_le)) end;
   546 
   547 local 
   548   fun pgterm rel con args =
   549     let
   550       fun append s = upd_vname (fn v => v^s);
   551       val (largs, rargs) = (args, map (append "'") args);
   552       val concl =
   553         foldr1 mk_conj (ListPair.map rel (map %# largs, map %# rargs));
   554       val prem = rel (con_app con largs, con_app con rargs);
   555       val sargs = case largs of [_] => [] | _ => nonlazy args;
   556       val prop = lift_defined %: (sargs, mk_trp (prem === concl));
   557     in pg con_appls prop end;
   558   val cons' = List.filter (fn (_,args) => args<>[]) cons;
   559 in
   560   val _ = trace " Proving inverts...";
   561   val inverts =
   562     let
   563       val abs_less = ax_abs_iso RS (allI RS injection_less);
   564       val tacs =
   565         [asm_full_simp_tac (HOLCF_ss addsimps [abs_less, spair_less]) 1];
   566     in map (fn (con, args) => pgterm (op <<) con args (K tacs)) cons' end;
   567 
   568   val _ = trace " Proving injects...";
   569   val injects =
   570     let
   571       val abs_eq = ax_abs_iso RS (allI RS injection_eq);
   572       val tacs = [asm_full_simp_tac (HOLCF_ss addsimps [abs_eq, spair_eq]) 1];
   573     in map (fn (con, args) => pgterm (op ===) con args (K tacs)) cons' end;
   574 end;
   575 
   576 (* ----- theorems concerning one induction step ----------------------------- *)
   577 
   578 val copy_strict =
   579   let
   580     val _ = trace " Proving copy_strict...";
   581     val goal = mk_trp (strict (dc_copy `% "f"));
   582     val rules = [abs_strict, rep_strict] @ @{thms domain_fun_stricts};
   583     val tacs = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
   584   in pg [ax_copy_def] goal (K tacs) end;
   585 
   586 local
   587   fun copy_app (con, args) =
   588     let
   589       val lhs = dc_copy`%"f"`(con_app con args);
   590       fun one_rhs arg =
   591           if DatatypeAux.is_rec_type (dtyp_of arg)
   592           then Domain_Axioms.copy_of_dtyp (cproj (%:"f") eqs) (dtyp_of arg) ` (%# arg)
   593           else (%# arg);
   594       val rhs = con_app2 con one_rhs args;
   595       val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
   596       val args' = List.filter (fn a => not (is_rec a orelse is_lazy a)) args;
   597       val stricts = abs_strict :: rep_strict :: @{thms domain_fun_stricts};
   598       fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args';
   599       val rules = [ax_abs_iso] @ @{thms domain_fun_simps};
   600       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
   601     in pg (ax_copy_def::con_appls) goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
   602 in
   603   val _ = trace " Proving copy_apps...";
   604   val copy_apps = map copy_app cons;
   605 end;
   606 
   607 local
   608   fun one_strict (con, args) = 
   609     let
   610       val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);
   611       val rews = copy_strict :: copy_apps @ con_rews;
   612       fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @
   613         [asm_simp_tac (HOLCF_ss addsimps rews) 1];
   614     in pg [] goal tacs end;
   615 
   616   fun has_nonlazy_rec (_, args) = exists is_nonlazy_rec args;
   617 in
   618   val _ = trace " Proving copy_stricts...";
   619   val copy_stricts = map one_strict (List.filter has_nonlazy_rec cons);
   620 end;
   621 
   622 val copy_rews = copy_strict :: copy_apps @ copy_stricts;
   623 
   624 in
   625   thy
   626     |> Sign.add_path (Long_Name.base_name dname)
   627     |> snd o PureThy.add_thmss [
   628         ((Binding.name "iso_rews"  , iso_rews    ), [Simplifier.simp_add]),
   629         ((Binding.name "exhaust"   , [exhaust]   ), []),
   630         ((Binding.name "casedist"  , [casedist]  ), [Induct.cases_type dname]),
   631         ((Binding.name "when_rews" , when_rews   ), [Simplifier.simp_add]),
   632         ((Binding.name "compacts"  , con_compacts), [Simplifier.simp_add]),
   633         ((Binding.name "con_rews"  , con_rews    ), [Simplifier.simp_add]),
   634         ((Binding.name "sel_rews"  , sel_rews    ), [Simplifier.simp_add]),
   635         ((Binding.name "dis_rews"  , dis_rews    ), [Simplifier.simp_add]),
   636         ((Binding.name "pat_rews"  , pat_rews    ), [Simplifier.simp_add]),
   637         ((Binding.name "dist_les"  , dist_les    ), [Simplifier.simp_add]),
   638         ((Binding.name "dist_eqs"  , dist_eqs    ), [Simplifier.simp_add]),
   639         ((Binding.name "inverts"   , inverts     ), [Simplifier.simp_add]),
   640         ((Binding.name "injects"   , injects     ), [Simplifier.simp_add]),
   641         ((Binding.name "copy_rews" , copy_rews   ), [Simplifier.simp_add]),
   642         ((Binding.name "match_rews", mat_rews    ), [Simplifier.simp_add])]
   643     |> Sign.parent_path
   644     |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
   645         pat_rews @ dist_les @ dist_eqs @ copy_rews)
   646 end; (* let *)
   647 
   648 fun comp_theorems (comp_dnam, eqs: eq list) thy =
   649 let
   650 val global_ctxt = ProofContext.init thy;
   651 
   652 val dnames = map (fst o fst) eqs;
   653 val conss  = map  snd        eqs;
   654 val comp_dname = Sign.full_bname thy comp_dnam;
   655 
   656 val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
   657 val pg = pg' thy;
   658 
   659 (* ----- getting the composite axiom and definitions ------------------------ *)
   660 
   661 local
   662   fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
   663 in
   664   val axs_reach      = map (ga "reach"     ) dnames;
   665   val axs_take_def   = map (ga "take_def"  ) dnames;
   666   val axs_finite_def = map (ga "finite_def") dnames;
   667   val ax_copy2_def   =      ga "copy_def"  comp_dnam;
   668   val ax_bisim_def   =      ga "bisim_def" comp_dnam;
   669 end;
   670 
   671 local
   672   fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);
   673   fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
   674 in
   675   val cases = map (gt  "casedist" ) dnames;
   676   val con_rews  = maps (gts "con_rews" ) dnames;
   677   val copy_rews = maps (gts "copy_rews") dnames;
   678 end;
   679 
   680 fun dc_take dn = %%:(dn^"_take");
   681 val x_name = idx_name dnames "x"; 
   682 val P_name = idx_name dnames "P";
   683 val n_eqs = length eqs;
   684 
   685 (* ----- theorems concerning finite approximation and finite induction ------ *)
   686 
   687 local
   688   val iterate_Cprod_ss = global_simpset_of @{theory Fix};
   689   val copy_con_rews  = copy_rews @ con_rews;
   690   val copy_take_defs =
   691     (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
   692   val _ = trace " Proving take_stricts...";
   693   val take_stricts =
   694     let
   695       fun one_eq ((dn, args), _) = strict (dc_take dn $ %:"n");
   696       val goal = mk_trp (foldr1 mk_conj (map one_eq eqs));
   697       fun tacs ctxt = [
   698         InductTacs.induct_tac ctxt [[SOME "n"]] 1,
   699         simp_tac iterate_Cprod_ss 1,
   700         asm_simp_tac (iterate_Cprod_ss addsimps copy_rews) 1];
   701     in pg copy_take_defs goal tacs end;
   702 
   703   val take_stricts' = rewrite_rule copy_take_defs take_stricts;
   704   fun take_0 n dn =
   705     let
   706       val goal = mk_trp ((dc_take dn $ %%:"HOL.zero") `% x_name n === UU);
   707     in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end;
   708   val take_0s = mapn take_0 1 dnames;
   709   fun c_UU_tac ctxt = case_UU_tac ctxt (take_stricts'::copy_con_rews) 1;
   710   val _ = trace " Proving take_apps...";
   711   val take_apps =
   712     let
   713       fun mk_eqn dn (con, args) =
   714         let
   715           fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
   716           fun one_rhs arg =
   717               if DatatypeAux.is_rec_type (dtyp_of arg)
   718               then Domain_Axioms.copy_of_dtyp mk_take (dtyp_of arg) ` (%# arg)
   719               else (%# arg);
   720           val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);
   721           val rhs = con_app2 con one_rhs args;
   722         in Library.foldr mk_all (map vname args, lhs === rhs) end;
   723       fun mk_eqns ((dn, _), cons) = map (mk_eqn dn) cons;
   724       val goal = mk_trp (foldr1 mk_conj (maps mk_eqns eqs));
   725       val simps = List.filter (has_fewer_prems 1) copy_rews;
   726       fun con_tac ctxt (con, args) =
   727         if nonlazy_rec args = []
   728         then all_tac
   729         else EVERY (map (c_UU_tac ctxt) (nonlazy_rec args)) THEN
   730           asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1;
   731       fun eq_tacs ctxt ((dn, _), cons) = map (con_tac ctxt) cons;
   732       fun tacs ctxt =
   733         simp_tac iterate_Cprod_ss 1 ::
   734         InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
   735         simp_tac (iterate_Cprod_ss addsimps copy_con_rews) 1 ::
   736         asm_full_simp_tac (HOLCF_ss addsimps simps) 1 ::
   737         TRY (safe_tac HOL_cs) ::
   738         maps (eq_tacs ctxt) eqs;
   739     in pg copy_take_defs goal tacs end;
   740 in
   741   val take_rews = map standard
   742     (atomize global_ctxt take_stricts @ take_0s @ atomize global_ctxt take_apps);
   743 end; (* local *)
   744 
   745 local
   746   fun one_con p (con,args) =
   747     let
   748       fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
   749       val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
   750       val t2 = lift ind_hyp (List.filter is_rec args, t1);
   751       val t3 = lift_defined (bound_arg (map vname args)) (nonlazy args, t2);
   752     in Library.foldr mk_All (map vname args, t3) end;
   753 
   754   fun one_eq ((p, cons), concl) =
   755     mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
   756 
   757   fun ind_term concf = Library.foldr one_eq
   758     (mapn (fn n => fn x => (P_name n, x)) 1 conss,
   759      mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
   760   val take_ss = HOL_ss addsimps take_rews;
   761   fun quant_tac ctxt i = EVERY
   762     (mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);
   763 
   764   fun ind_prems_tac prems = EVERY
   765     (maps (fn cons =>
   766       (resolve_tac prems 1 ::
   767         maps (fn (_,args) => 
   768           resolve_tac prems 1 ::
   769           map (K(atac 1)) (nonlazy args) @
   770           map (K(atac 1)) (List.filter is_rec args))
   771         cons))
   772       conss);
   773   local 
   774     (* check whether every/exists constructor of the n-th part of the equation:
   775        it has a possibly indirectly recursive argument that isn't/is possibly 
   776        indirectly lazy *)
   777     fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
   778           is_rec arg andalso not(rec_of arg mem ns) andalso
   779           ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
   780             rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
   781               (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
   782           ) o snd) cons;
   783     fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
   784     fun warn (n,cons) =
   785       if all_rec_to [] false (n,cons)
   786       then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
   787       else false;
   788     fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;
   789 
   790   in
   791     val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
   792     val is_emptys = map warn n__eqs;
   793     val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
   794   end;
   795 in (* local *)
   796   val _ = trace " Proving finite_ind...";
   797   val finite_ind =
   798     let
   799       fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
   800       val goal = ind_term concf;
   801 
   802       fun tacf {prems, context} =
   803         let
   804           val tacs1 = [
   805             quant_tac context 1,
   806             simp_tac HOL_ss 1,
   807             InductTacs.induct_tac context [[SOME "n"]] 1,
   808             simp_tac (take_ss addsimps prems) 1,
   809             TRY (safe_tac HOL_cs)];
   810           fun arg_tac arg =
   811             case_UU_tac context (prems @ con_rews) 1
   812               (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
   813           fun con_tacs (con, args) = 
   814             asm_simp_tac take_ss 1 ::
   815             map arg_tac (List.filter is_nonlazy_rec args) @
   816             [resolve_tac prems 1] @
   817             map (K (atac 1))      (nonlazy args) @
   818             map (K (etac spec 1)) (List.filter is_rec args);
   819           fun cases_tacs (cons, cases) =
   820             res_inst_tac context [(("x", 0), "x")] cases 1 ::
   821             asm_simp_tac (take_ss addsimps prems) 1 ::
   822             maps con_tacs cons;
   823         in
   824           tacs1 @ maps cases_tacs (conss ~~ cases)
   825         end;
   826     in pg'' thy [] goal tacf
   827        handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
   828     end;
   829 
   830   val _ = trace " Proving take_lemmas...";
   831   val take_lemmas =
   832     let
   833       fun take_lemma n (dn, ax_reach) =
   834         let
   835           val lhs = dc_take dn $ Bound 0 `%(x_name n);
   836           val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");
   837           val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));
   838           val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;
   839           fun tacf {prems, context} = [
   840             res_inst_tac context [(("t", 0), x_name n    )] (ax_reach RS subst) 1,
   841             res_inst_tac context [(("t", 0), x_name n^"'")] (ax_reach RS subst) 1,
   842             stac fix_def2 1,
   843             REPEAT (CHANGED
   844               (rtac (contlub_cfun_arg RS ssubst) 1 THEN chain_tac 1)),
   845             stac contlub_cfun_fun 1,
   846             stac contlub_cfun_fun 2,
   847             rtac lub_equal 3,
   848             chain_tac 1,
   849             rtac allI 1,
   850             resolve_tac prems 1];
   851         in pg'' thy axs_take_def goal tacf end;
   852     in mapn take_lemma 1 (dnames ~~ axs_reach) end;
   853 
   854 (* ----- theorems concerning finiteness and induction ----------------------- *)
   855 
   856   val _ = trace " Proving finites, ind...";
   857   val (finites, ind) =
   858   (
   859     if is_finite
   860     then (* finite case *)
   861       let 
   862         fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
   863         fun dname_lemma dn =
   864           let
   865             val prem1 = mk_trp (defined (%:"x"));
   866             val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);
   867             val prem2 = mk_trp (mk_disj (disj1, take_enough dn));
   868             val concl = mk_trp (take_enough dn);
   869             val goal = prem1 ===> prem2 ===> concl;
   870             val tacs = [
   871               etac disjE 1,
   872               etac notE 1,
   873               resolve_tac take_lemmas 1,
   874               asm_simp_tac take_ss 1,
   875               atac 1];
   876           in pg [] goal (K tacs) end;
   877         val _ = trace " Proving finite_lemmas1a";
   878         val finite_lemmas1a = map dname_lemma dnames;
   879  
   880         val _ = trace " Proving finite_lemma1b";
   881         val finite_lemma1b =
   882           let
   883             fun mk_eqn n ((dn, args), _) =
   884               let
   885                 val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;
   886                 val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;
   887               in
   888                 mk_constrainall
   889                   (x_name n, Type (dn,args), mk_disj (disj1, disj2))
   890               end;
   891             val goal =
   892               mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));
   893             fun arg_tacs ctxt vn = [
   894               eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1,
   895               etac disjE 1,
   896               asm_simp_tac (HOL_ss addsimps con_rews) 1,
   897               asm_simp_tac take_ss 1];
   898             fun con_tacs ctxt (con, args) =
   899               asm_simp_tac take_ss 1 ::
   900               maps (arg_tacs ctxt) (nonlazy_rec args);
   901             fun foo_tacs ctxt n (cons, cases) =
   902               simp_tac take_ss 1 ::
   903               rtac allI 1 ::
   904               res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 ::
   905               asm_simp_tac take_ss 1 ::
   906               maps (con_tacs ctxt) cons;
   907             fun tacs ctxt =
   908               rtac allI 1 ::
   909               InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
   910               simp_tac take_ss 1 ::
   911               TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::
   912               flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases));
   913           in pg [] goal tacs end;
   914 
   915         fun one_finite (dn, l1b) =
   916           let
   917             val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
   918             fun tacs ctxt = [
   919               case_UU_tac ctxt take_rews 1 "x",
   920               eresolve_tac finite_lemmas1a 1,
   921               step_tac HOL_cs 1,
   922               step_tac HOL_cs 1,
   923               cut_facts_tac [l1b] 1,
   924               fast_tac HOL_cs 1];
   925           in pg axs_finite_def goal tacs end;
   926 
   927         val _ = trace " Proving finites";
   928         val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b);
   929         val _ = trace " Proving ind";
   930         val ind =
   931           let
   932             fun concf n dn = %:(P_name n) $ %:(x_name n);
   933             fun tacf {prems, context} =
   934               let
   935                 fun finite_tacs (finite, fin_ind) = [
   936                   rtac(rewrite_rule axs_finite_def finite RS exE)1,
   937                   etac subst 1,
   938                   rtac fin_ind 1,
   939                   ind_prems_tac prems];
   940               in
   941                 TRY (safe_tac HOL_cs) ::
   942                 maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)
   943               end;
   944           in pg'' thy [] (ind_term concf) tacf end;
   945       in (finites, ind) end (* let *)
   946 
   947     else (* infinite case *)
   948       let
   949         fun one_finite n dn =
   950           read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;
   951         val finites = mapn one_finite 1 dnames;
   952 
   953         val goal =
   954           let
   955             fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));
   956             fun concf n dn = %:(P_name n) $ %:(x_name n);
   957           in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
   958         fun tacf {prems, context} =
   959           map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
   960           quant_tac context 1,
   961           rtac (adm_impl_admw RS wfix_ind) 1,
   962           REPEAT_DETERM (rtac adm_all 1),
   963           REPEAT_DETERM (
   964             TRY (rtac adm_conj 1) THEN 
   965             rtac adm_subst 1 THEN 
   966             cont_tacR 1 THEN resolve_tac prems 1),
   967           strip_tac 1,
   968           rtac (rewrite_rule axs_take_def finite_ind) 1,
   969           ind_prems_tac prems];
   970         val ind = (pg'' thy [] goal tacf
   971           handle ERROR _ =>
   972             (warning "Cannot prove infinite induction rule"; refl));
   973       in (finites, ind) end
   974   )
   975       handle THM _ =>
   976              (warning "Induction proofs failed (THM raised)."; ([], TrueI))
   977            | ERROR _ =>
   978              (warning "Induction proofs failed (ERROR raised)."; ([], TrueI));
   979 
   980 
   981 end; (* local *)
   982 
   983 (* ----- theorem concerning coinduction ------------------------------------- *)
   984 
   985 local
   986   val xs = mapn (fn n => K (x_name n)) 1 dnames;
   987   fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
   988   val take_ss = HOL_ss addsimps take_rews;
   989   val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
   990   val _ = trace " Proving coind_lemma...";
   991   val coind_lemma =
   992     let
   993       fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
   994       fun mk_eqn n dn =
   995         (dc_take dn $ %:"n" ` bnd_arg n 0) ===
   996         (dc_take dn $ %:"n" ` bnd_arg n 1);
   997       fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
   998       val goal =
   999         mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
  1000           Library.foldr mk_all2 (xs,
  1001             Library.foldr mk_imp (mapn mk_prj 0 dnames,
  1002               foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
  1003       fun x_tacs ctxt n x = [
  1004         rotate_tac (n+1) 1,
  1005         etac all2E 1,
  1006         eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
  1007         TRY (safe_tac HOL_cs),
  1008         REPEAT (CHANGED (asm_simp_tac take_ss 1))];
  1009       fun tacs ctxt = [
  1010         rtac impI 1,
  1011         InductTacs.induct_tac ctxt [[SOME "n"]] 1,
  1012         simp_tac take_ss 1,
  1013         safe_tac HOL_cs] @
  1014         flat (mapn (x_tacs ctxt) 0 xs);
  1015     in pg [ax_bisim_def] goal tacs end;
  1016 in
  1017   val _ = trace " Proving coind...";
  1018   val coind = 
  1019     let
  1020       fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
  1021       fun mk_eqn x = %:x === %:(x^"'");
  1022       val goal =
  1023         mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
  1024           Logic.list_implies (mapn mk_prj 0 xs,
  1025             mk_trp (foldr1 mk_conj (map mk_eqn xs)));
  1026       val tacs =
  1027         TRY (safe_tac HOL_cs) ::
  1028         maps (fn take_lemma => [
  1029           rtac take_lemma 1,
  1030           cut_facts_tac [coind_lemma] 1,
  1031           fast_tac HOL_cs 1])
  1032         take_lemmas;
  1033     in pg [] goal (K tacs) end;
  1034 end; (* local *)
  1035 
  1036 val inducts = Project_Rule.projections (ProofContext.init thy) ind;
  1037 fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);
  1038 val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
  1039 
  1040 in thy |> Sign.add_path comp_dnam
  1041        |> snd o PureThy.add_thmss [
  1042            ((Binding.name "take_rews"  , take_rews   ), [Simplifier.simp_add]),
  1043            ((Binding.name "take_lemmas", take_lemmas ), []),
  1044            ((Binding.name "finites"    , finites     ), []),
  1045            ((Binding.name "finite_ind" , [finite_ind]), []),
  1046            ((Binding.name "ind"        , [ind]       ), []),
  1047            ((Binding.name "coind"      , [coind]     ), [])]
  1048        |> (if induct_failed then I
  1049            else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
  1050        |> Sign.parent_path |> pair take_rews
  1051 end; (* let *)
  1052 end; (* local *)
  1053 end; (* struct *)