src/HOLCF/Tools/domain/domain_theorems.ML
author wenzelm
Sun Oct 21 14:21:53 2007 +0200 (2007-10-21)
changeset 25132 dffe405b090d
parent 24712 64ed05609568
child 25805 5df82bb5b982
permissions -rw-r--r--
removed obsolete ML bindings;
     1 (*  Title:      HOLCF/Tools/domain/domain_theorems.ML
     2     ID:         $Id$
     3     Author:     David von Oheimb
     4                 New proofs/tactics by Brian Huffman
     5 
     6 Proof generator for domain command.
     7 *)
     8 
     9 val HOLCF_ss = simpset();
    10 
    11 structure Domain_Theorems = struct
    12 
    13 local
    14 
    15 val adm_impl_admw = thm "adm_impl_admw";
    16 val antisym_less_inverse = thm "antisym_less_inverse";
    17 val beta_cfun = thm "beta_cfun";
    18 val cfun_arg_cong = thm "cfun_arg_cong";
    19 val ch2ch_Rep_CFunL = thm "ch2ch_Rep_CFunL";
    20 val ch2ch_Rep_CFunR = thm "ch2ch_Rep_CFunR";
    21 val chain_iterate = thm "chain_iterate";
    22 val compact_ONE = thm "compact_ONE";
    23 val compact_sinl = thm "compact_sinl";
    24 val compact_sinr = thm "compact_sinr";
    25 val compact_spair = thm "compact_spair";
    26 val compact_up = thm "compact_up";
    27 val contlub_cfun_arg = thm "contlub_cfun_arg";
    28 val contlub_cfun_fun = thm "contlub_cfun_fun";
    29 val fix_def2 = thm "fix_def2";
    30 val injection_eq = thm "injection_eq";
    31 val injection_less = thm "injection_less";
    32 val lub_equal = thm "lub_equal";
    33 val monofun_cfun_arg = thm "monofun_cfun_arg";
    34 val retraction_strict = thm "retraction_strict";
    35 val spair_eq = thm "spair_eq";
    36 val spair_less = thm "spair_less";
    37 val sscase1 = thm "sscase1";
    38 val ssplit1 = thm "ssplit1";
    39 val strictify1 = thm "strictify1";
    40 val wfix_ind = thm "wfix_ind";
    41 
    42 open Domain_Library;
    43 infixr 0 ===>;
    44 infixr 0 ==>;
    45 infix 0 == ; 
    46 infix 1 ===;
    47 infix 1 ~= ;
    48 infix 1 <<;
    49 infix 1 ~<<;
    50 infix 9 `   ;
    51 infix 9 `% ;
    52 infix 9 `%%;
    53 infixr 9 oo;
    54 
    55 (* ----- general proof facilities ------------------------------------------- *)
    56 
    57 fun legacy_infer_term thy t =
    58   let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init thy)
    59   in singleton (Syntax.check_terms ctxt) (Sign.intern_term thy t) end;
    60 
    61 fun pg'' thy defs t tacs =
    62   let
    63     val t' = legacy_infer_term thy t;
    64     val asms = Logic.strip_imp_prems t';
    65     val prop = Logic.strip_imp_concl t';
    66     fun tac prems =
    67       rewrite_goals_tac defs THEN
    68       EVERY (tacs (map (rewrite_rule defs) prems));
    69   in Goal.prove_global thy [] asms prop tac end;
    70 
    71 fun pg' thy defs t tacsf =
    72   let
    73     fun tacs [] = tacsf
    74       | tacs prems = cut_facts_tac prems 1 :: tacsf;
    75   in pg'' thy defs t tacs end;
    76 
    77 fun case_UU_tac rews i v =
    78   case_tac (v^"=UU") i THEN
    79   asm_simp_tac (HOLCF_ss addsimps rews) i;
    80 
    81 val chain_tac =
    82   REPEAT_DETERM o resolve_tac 
    83     [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL];
    84 
    85 (* ----- general proofs ----------------------------------------------------- *)
    86 
    87 val all2E = prove_goal HOL.thy "[| !x y . P x y; P x y ==> R |] ==> R"
    88   (fn prems =>[
    89     resolve_tac prems 1,
    90     cut_facts_tac prems 1,
    91     fast_tac HOL_cs 1]);
    92 
    93 val dist_eqI = prove_goal (the_context ()) "!!x::'a::po. ~ x << y ==> x ~= y" 
    94   (fn prems =>
    95     [blast_tac (claset () addDs [antisym_less_inverse]) 1]);
    96 
    97 in
    98 
    99 fun theorems (((dname, _), cons) : eq, eqs : eq list) thy =
   100 let
   101 
   102 val dummy = writeln ("Proving isomorphism properties of domain "^dname^" ...");
   103 val pg = pg' thy;
   104 
   105 (* ----- getting the axioms and definitions --------------------------------- *)
   106 
   107 local
   108   fun ga s dn = get_thm thy (Name (dn ^ "." ^ s));
   109 in
   110   val ax_abs_iso  = ga "abs_iso"  dname;
   111   val ax_rep_iso  = ga "rep_iso"  dname;
   112   val ax_when_def = ga "when_def" dname;
   113   fun get_def mk_name (con,_) = ga (mk_name con^"_def") dname;
   114   val axs_con_def = map (get_def extern_name) cons;
   115   val axs_dis_def = map (get_def dis_name) cons;
   116   val axs_mat_def = map (get_def mat_name) cons;
   117   val axs_pat_def = map (get_def pat_name) cons;
   118   val axs_sel_def =
   119     let
   120       fun def_of_sel sel = ga (sel^"_def") dname;
   121       fun def_of_arg arg = Option.map def_of_sel (sel_of arg);
   122       fun defs_of_con (_, args) = List.mapPartial def_of_arg args;
   123     in
   124       List.concat (map defs_of_con cons)
   125     end;
   126   val ax_copy_def = ga "copy_def" dname;
   127 end; (* local *)
   128 
   129 (* ----- theorems concerning the isomorphism -------------------------------- *)
   130 
   131 val dc_abs  = %%:(dname^"_abs");
   132 val dc_rep  = %%:(dname^"_rep");
   133 val dc_copy = %%:(dname^"_copy");
   134 val x_name = "x";
   135 
   136 val iso_locale = iso_intro OF [ax_abs_iso, ax_rep_iso];
   137 val abs_strict = ax_rep_iso RS (allI RS retraction_strict);
   138 val rep_strict = ax_abs_iso RS (allI RS retraction_strict);
   139 val abs_defin' = iso_locale RS iso_abs_defin';
   140 val rep_defin' = iso_locale RS iso_rep_defin';
   141 val iso_rews = map standard [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
   142 
   143 (* ----- generating beta reduction rules from definitions-------------------- *)
   144 
   145 local
   146   fun arglist (Const _ $ Abs (s, _, t)) =
   147     let
   148       val (vars,body) = arglist t;
   149     in (s :: vars, body) end
   150     | arglist t = ([], t);
   151   fun bind_fun vars t = Library.foldr mk_All (vars, t);
   152   fun bound_vars 0 = []
   153     | bound_vars i = Bound (i-1) :: bound_vars (i - 1);
   154 in
   155   fun appl_of_def def =
   156     let
   157       val (_ $ con $ lam) = concl_of def;
   158       val (vars, rhs) = arglist lam;
   159       val lhs = list_ccomb (con, bound_vars (length vars));
   160       val appl = bind_fun vars (lhs == rhs);
   161       val cs = ContProc.cont_thms lam;
   162       val betas = map (fn c => mk_meta_eq (c RS beta_cfun)) cs;
   163     in pg (def::betas) appl [rtac reflexive_thm 1] end;
   164 end;
   165 
   166 val when_appl = appl_of_def ax_when_def;
   167 val con_appls = map appl_of_def axs_con_def;
   168 
   169 local
   170   fun arg2typ n arg =
   171     let val t = TVar (("'a", n), pcpoS)
   172     in (n + 1, if is_lazy arg then mk_uT t else t) end;
   173 
   174   fun args2typ n [] = (n, oneT)
   175     | args2typ n [arg] = arg2typ n arg
   176     | args2typ n (arg::args) =
   177     let
   178       val (n1, t1) = arg2typ n arg;
   179       val (n2, t2) = args2typ n1 args
   180     in (n2, mk_sprodT (t1, t2)) end;
   181 
   182   fun cons2typ n [] = (n,oneT)
   183     | cons2typ n [con] = args2typ n (snd con)
   184     | cons2typ n (con::cons) =
   185     let
   186       val (n1, t1) = args2typ n (snd con);
   187       val (n2, t2) = cons2typ n1 cons
   188     in (n2, mk_ssumT (t1, t2)) end;
   189 in
   190   fun cons2ctyp cons = ctyp_of thy (snd (cons2typ 1 cons));
   191 end;
   192 
   193 local 
   194   val iso_swap = iso_locale RS iso_iso_swap;
   195   fun one_con (con, args) =
   196     let
   197       val vns = map vname args;
   198       val eqn = %:x_name === con_app2 con %: vns;
   199       val conj = foldr1 mk_conj (eqn :: map (defined o %:) (nonlazy args));
   200     in Library.foldr mk_ex (vns, conj) end;
   201 
   202   val conj_assoc = @{thm conj_assoc};
   203   val exh = foldr1 mk_disj ((%:x_name === UU) :: map one_con cons);
   204   val thm1 = instantiate' [SOME (cons2ctyp cons)] [] exh_start;
   205   val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1;
   206   val thm3 = rewrite_rule [mk_meta_eq conj_assoc] thm2;
   207 
   208   (* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *)
   209   val tacs = [
   210     rtac disjE 1,
   211     etac (rep_defin' RS disjI1) 2,
   212     etac disjI2 2,
   213     rewrite_goals_tac [mk_meta_eq iso_swap],
   214     rtac thm3 1];
   215 in
   216   val exhaust = pg con_appls (mk_trp exh) tacs;
   217   val casedist =
   218     standard (rewrite_rule exh_casedists (exhaust RS exh_casedist0));
   219 end;
   220 
   221 local 
   222   fun bind_fun t = Library.foldr mk_All (when_funs cons, t);
   223   fun bound_fun i _ = Bound (length cons - i);
   224   val when_app = list_ccomb (%%:(dname^"_when"), mapn bound_fun 1 cons);
   225 in
   226   val when_strict =
   227     let
   228       val axs = [when_appl, mk_meta_eq rep_strict];
   229       val goal = bind_fun (mk_trp (strict when_app));
   230       val tacs = [resolve_tac [sscase1, ssplit1, strictify1] 1];
   231     in pg axs goal tacs end;
   232 
   233   val when_apps =
   234     let
   235       fun one_when n (con,args) =
   236         let
   237           val axs = when_appl :: con_appls;
   238           val goal = bind_fun (lift_defined %: (nonlazy args, 
   239                 mk_trp (when_app`(con_app con args) ===
   240                        list_ccomb (bound_fun n 0, map %# args))));
   241           val tacs = [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1];
   242         in pg axs goal tacs end;
   243     in mapn one_when 1 cons end;
   244 end;
   245 val when_rews = when_strict :: when_apps;
   246 
   247 (* ----- theorems concerning the constructors, discriminators and selectors - *)
   248 
   249 local
   250   fun dis_strict (con, _) =
   251     let
   252       val goal = mk_trp (strict (%%:(dis_name con)));
   253     in pg axs_dis_def goal [rtac when_strict 1] end;
   254 
   255   fun dis_app c (con, args) =
   256     let
   257       val lhs = %%:(dis_name c) ` con_app con args;
   258       val rhs = %%:(if con = c then TT_N else FF_N);
   259       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
   260       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   261     in pg axs_dis_def goal tacs end;
   262 
   263   val dis_apps = List.concat (map (fn (c,_) => map (dis_app c) cons) cons);
   264 
   265   fun dis_defin (con, args) =
   266     let
   267       val goal = defined (%:x_name) ==> defined (%%:(dis_name con) `% x_name);
   268       val tacs =
   269         [rtac casedist 1,
   270          contr_tac 1,
   271          DETERM_UNTIL_SOLVED (CHANGED
   272           (asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))];
   273     in pg [] goal tacs end;
   274 
   275   val dis_stricts = map dis_strict cons;
   276   val dis_defins = map dis_defin cons;
   277 in
   278   val dis_rews = dis_stricts @ dis_defins @ dis_apps;
   279 end;
   280 
   281 local
   282   fun mat_strict (con, _) =
   283     let
   284       val goal = mk_trp (strict (%%:(mat_name con)));
   285       val tacs = [rtac when_strict 1];
   286     in pg axs_mat_def goal tacs end;
   287 
   288   val mat_stricts = map mat_strict cons;
   289 
   290   fun one_mat c (con, args) =
   291     let
   292       val lhs = %%:(mat_name c) ` con_app con args;
   293       val rhs =
   294         if con = c
   295         then %%:returnN ` mk_ctuple (map %# args)
   296         else %%:failN;
   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_mat_def goal tacs end;
   300 
   301   val mat_apps =
   302     List.concat (map (fn (c,_) => map (one_mat c) cons) cons);
   303 in
   304   val mat_rews = mat_stricts @ mat_apps;
   305 end;
   306 
   307 local
   308   fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
   309 
   310   fun pat_lhs (con,args) = %%:branchN $ list_comb (%%:(pat_name con), ps args);
   311 
   312   fun pat_rhs (con,[]) = %%:returnN ` ((%:"rhs") ` HOLogic.unit)
   313     | pat_rhs (con,args) =
   314         (%%:branchN $ foldr1 cpair_pat (ps args))
   315           `(%:"rhs")`(mk_ctuple (map %# args));
   316 
   317   fun pat_strict c =
   318     let
   319       val axs = @{thm branch_def} :: axs_pat_def;
   320       val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));
   321       val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];
   322     in pg axs goal tacs end;
   323 
   324   fun pat_app c (con, args) =
   325     let
   326       val axs = @{thm branch_def} :: axs_pat_def;
   327       val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);
   328       val rhs = if con = fst c then pat_rhs c else %%:failN;
   329       val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
   330       val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   331     in pg axs goal tacs end;
   332 
   333   val pat_stricts = map pat_strict cons;
   334   val pat_apps = List.concat (map (fn c => map (pat_app c) cons) cons);
   335 in
   336   val pat_rews = pat_stricts @ pat_apps;
   337 end;
   338 
   339 local
   340   val rev_contrapos = @{thm rev_contrapos};
   341   fun con_strict (con, args) = 
   342     let
   343       fun one_strict vn =
   344         let
   345           fun f arg = if vname arg = vn then UU else %# arg;
   346           val goal = mk_trp (con_app2 con f args === UU);
   347           val tacs = [asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1];
   348         in pg con_appls goal tacs end;
   349     in map one_strict (nonlazy args) end;
   350 
   351   fun con_defin (con, args) =
   352     let
   353       val concl = mk_trp (defined (con_app con args));
   354       val goal = lift_defined %: (nonlazy args, concl);
   355       val tacs = [
   356         rtac rev_contrapos 1,
   357         eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
   358         asm_simp_tac (HOLCF_ss addsimps dis_rews) 1];
   359     in pg [] goal tacs end;
   360 in
   361   val con_stricts = List.concat (map con_strict cons);
   362   val con_defins = map con_defin cons;
   363   val con_rews = con_stricts @ con_defins;
   364 end;
   365 
   366 local
   367   val rules =
   368     [compact_sinl, compact_sinr, compact_spair, compact_up, compact_ONE];
   369   fun con_compact (con, args) =
   370     let
   371       val concl = mk_trp (%%:compactN $ con_app con args);
   372       val goal = lift (fn x => %%:compactN $ %#x) (args, concl);
   373       val tacs = [
   374         rtac (iso_locale RS iso_compact_abs) 1,
   375         REPEAT (resolve_tac rules 1 ORELSE atac 1)];
   376     in pg con_appls goal tacs end;
   377 in
   378   val con_compacts = map con_compact cons;
   379 end;
   380 
   381 local
   382   fun one_sel sel =
   383     pg axs_sel_def (mk_trp (strict (%%:sel)))
   384       [simp_tac (HOLCF_ss addsimps when_rews) 1];
   385 
   386   fun sel_strict (_, args) =
   387     List.mapPartial (Option.map one_sel o sel_of) args;
   388 in
   389   val sel_stricts = List.concat (map sel_strict cons);
   390 end;
   391 
   392 local
   393   fun sel_app_same c n sel (con, args) =
   394     let
   395       val nlas = nonlazy args;
   396       val vns = map vname args;
   397       val vnn = List.nth (vns, n);
   398       val nlas' = List.filter (fn v => v <> vnn) nlas;
   399       val lhs = (%%:sel)`(con_app con args);
   400       val goal = lift_defined %: (nlas', mk_trp (lhs === %:vnn));
   401       val tacs1 =
   402         if vnn mem nlas
   403         then [case_UU_tac (when_rews @ con_stricts) 1 vnn]
   404         else [];
   405       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   406     in pg axs_sel_def goal (tacs1 @ tacs2) end;
   407 
   408   fun sel_app_diff c n sel (con, args) =
   409     let
   410       val nlas = nonlazy args;
   411       val goal = mk_trp (%%:sel ` con_app con args === UU);
   412       val tacs1 = map (case_UU_tac (when_rews @ con_stricts) 1) nlas;
   413       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   414     in pg axs_sel_def goal (tacs1 @ tacs2) end;
   415 
   416   fun sel_app c n sel (con, args) =
   417     if con = c
   418     then sel_app_same c n sel (con, args)
   419     else sel_app_diff c n sel (con, args);
   420 
   421   fun one_sel c n sel = map (sel_app c n sel) cons;
   422   fun one_sel' c n arg = Option.map (one_sel c n) (sel_of arg);
   423   fun one_con (c, args) =
   424     List.concat (List.mapPartial I (mapn (one_sel' c) 0 args));
   425 in
   426   val sel_apps = List.concat (map one_con cons);
   427 end;
   428 
   429 local
   430   fun sel_defin sel =
   431     let
   432       val goal = defined (%:x_name) ==> defined (%%:sel`%x_name);
   433       val tacs = [
   434         rtac casedist 1,
   435         contr_tac 1,
   436         DETERM_UNTIL_SOLVED (CHANGED
   437           (asm_simp_tac (HOLCF_ss addsimps sel_apps) 1))];
   438     in pg [] goal tacs end;
   439 in
   440   val sel_defins =
   441     if length cons = 1
   442     then List.mapPartial (fn arg => Option.map sel_defin (sel_of arg))
   443                  (filter_out is_lazy (snd (hd cons)))
   444     else [];
   445 end;
   446 
   447 val sel_rews = sel_stricts @ sel_defins @ sel_apps;
   448 val rev_contrapos = @{thm rev_contrapos};
   449 
   450 val distincts_le =
   451   let
   452     fun dist (con1, args1) (con2, args2) =
   453       let
   454         val goal = lift_defined %: (nonlazy args1,
   455                         mk_trp (con_app con1 args1 ~<< con_app con2 args2));
   456         val tacs = [
   457           rtac rev_contrapos 1,
   458           eres_inst_tac [("f", dis_name con1)] monofun_cfun_arg 1]
   459           @ map (case_UU_tac (con_stricts @ dis_rews) 1) (nonlazy args2)
   460           @ [asm_simp_tac (HOLCF_ss addsimps dis_rews) 1];
   461       in pg [] goal tacs end;
   462 
   463     fun distinct (con1, args1) (con2, args2) =
   464         let
   465           val arg1 = (con1, args1);
   466           val arg2 =
   467             (con2, ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
   468               (args2, Name.variant_list (map vname args1) (map vname args2)));
   469         in [dist arg1 arg2, dist arg2 arg1] end;
   470     fun distincts []      = []
   471       | distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
   472   in distincts cons end;
   473 val dist_les = List.concat (List.concat distincts_le);
   474 val dist_eqs =
   475   let
   476     fun distinct (_,args1) ((_,args2), leqs) =
   477       let
   478         val (le1,le2) = (hd leqs, hd(tl leqs));
   479         val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI)
   480       in
   481         if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
   482         if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
   483           [eq1, eq2]
   484       end;
   485     fun distincts []      = []
   486       | distincts ((c,leqs)::cs) = List.concat
   487 	            (ListPair.map (distinct c) ((map #1 cs),leqs)) @
   488 		    distincts cs;
   489   in map standard (distincts (cons ~~ distincts_le)) end;
   490 
   491 local 
   492   fun pgterm rel con args =
   493     let
   494       fun append s = upd_vname (fn v => v^s);
   495       val (largs, rargs) = (args, map (append "'") args);
   496       val concl =
   497         foldr1 mk_conj (ListPair.map rel (map %# largs, map %# rargs));
   498       val prem = rel (con_app con largs, con_app con rargs);
   499       val sargs = case largs of [_] => [] | _ => nonlazy args;
   500       val prop = lift_defined %: (sargs, mk_trp (prem === concl));
   501     in pg con_appls prop end;
   502   val cons' = List.filter (fn (_,args) => args<>[]) cons;
   503 in
   504   val inverts =
   505     let
   506       val abs_less = ax_abs_iso RS (allI RS injection_less);
   507       val tacs =
   508         [asm_full_simp_tac (HOLCF_ss addsimps [abs_less, spair_less]) 1];
   509     in map (fn (con, args) => pgterm (op <<) con args tacs) cons' end;
   510 
   511   val injects =
   512     let
   513       val abs_eq = ax_abs_iso RS (allI RS injection_eq);
   514       val tacs = [asm_full_simp_tac (HOLCF_ss addsimps [abs_eq, spair_eq]) 1];
   515     in map (fn (con, args) => pgterm (op ===) con args tacs) cons' end;
   516 end;
   517 
   518 (* ----- theorems concerning one induction step ----------------------------- *)
   519 
   520 val copy_strict =
   521   let
   522     val goal = mk_trp (strict (dc_copy `% "f"));
   523     val tacs = [asm_simp_tac (HOLCF_ss addsimps [abs_strict, when_strict]) 1];
   524   in pg [ax_copy_def] goal tacs end;
   525 
   526 local
   527   fun copy_app (con, args) =
   528     let
   529       val lhs = dc_copy`%"f"`(con_app con args);
   530       val rhs = con_app2 con (app_rec_arg (cproj (%:"f") eqs)) args;
   531       val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
   532       val args' = List.filter (fn a => not (is_rec a orelse is_lazy a)) args;
   533       val stricts = abs_strict::when_strict::con_stricts;
   534       val tacs1 = map (case_UU_tac stricts 1 o vname) args';
   535       val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_apps) 1];
   536     in pg [ax_copy_def] goal (tacs1 @ tacs2) end;
   537 in
   538   val copy_apps = map copy_app cons;
   539 end;
   540 
   541 local
   542   fun one_strict (con, args) = 
   543     let
   544       val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);
   545       val rews = copy_strict :: copy_apps @ con_rews;
   546       val tacs = map (case_UU_tac rews 1) (nonlazy args) @
   547         [asm_simp_tac (HOLCF_ss addsimps rews) 1];
   548     in pg [] goal tacs end;
   549 
   550   fun has_nonlazy_rec (_, args) = exists is_nonlazy_rec args;
   551 in
   552   val copy_stricts = map one_strict (List.filter has_nonlazy_rec cons);
   553 end;
   554 
   555 val copy_rews = copy_strict :: copy_apps @ copy_stricts;
   556 
   557 in
   558   thy
   559     |> Sign.add_path (Sign.base_name dname)
   560     |> (snd o (PureThy.add_thmss (map Thm.no_attributes [
   561         ("iso_rews" , iso_rews  ),
   562         ("exhaust"  , [exhaust] ),
   563         ("casedist" , [casedist]),
   564         ("when_rews", when_rews ),
   565         ("compacts", con_compacts),
   566         ("con_rews", con_rews),
   567         ("sel_rews", sel_rews),
   568         ("dis_rews", dis_rews),
   569         ("pat_rews", pat_rews),
   570         ("dist_les", dist_les),
   571         ("dist_eqs", dist_eqs),
   572         ("inverts" , inverts ),
   573         ("injects" , injects ),
   574         ("copy_rews", copy_rews)])))
   575     |> (snd o PureThy.add_thmss
   576         [(("match_rews", mat_rews), [Simplifier.simp_add])])
   577     |> Sign.parent_path
   578     |> rpair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
   579         pat_rews @ dist_les @ dist_eqs @ copy_rews)
   580 end; (* let *)
   581 
   582 fun comp_theorems (comp_dnam, eqs: eq list) thy =
   583 let
   584 val dnames = map (fst o fst) eqs;
   585 val conss  = map  snd        eqs;
   586 val comp_dname = Sign.full_name thy comp_dnam;
   587 
   588 val d = writeln("Proving induction properties of domain "^comp_dname^" ...");
   589 val pg = pg' thy;
   590 
   591 (* ----- getting the composite axiom and definitions ------------------------ *)
   592 
   593 local
   594   fun ga s dn = get_thm thy (Name (dn ^ "." ^ s));
   595 in
   596   val axs_reach      = map (ga "reach"     ) dnames;
   597   val axs_take_def   = map (ga "take_def"  ) dnames;
   598   val axs_finite_def = map (ga "finite_def") dnames;
   599   val ax_copy2_def   =      ga "copy_def"  comp_dnam;
   600   val ax_bisim_def   =      ga "bisim_def" comp_dnam;
   601 end;
   602 
   603 local
   604   fun gt  s dn = get_thm  thy (Name (dn ^ "." ^ s));
   605   fun gts s dn = get_thms thy (Name (dn ^ "." ^ s));
   606 in
   607   val cases = map (gt  "casedist" ) dnames;
   608   val con_rews  = List.concat (map (gts "con_rews" ) dnames);
   609   val copy_rews = List.concat (map (gts "copy_rews") dnames);
   610 end;
   611 
   612 fun dc_take dn = %%:(dn^"_take");
   613 val x_name = idx_name dnames "x"; 
   614 val P_name = idx_name dnames "P";
   615 val n_eqs = length eqs;
   616 
   617 (* ----- theorems concerning finite approximation and finite induction ------ *)
   618 
   619 local
   620   val iterate_Cprod_ss = simpset_of (theory "Fix");
   621   val copy_con_rews  = copy_rews @ con_rews;
   622   val copy_take_defs =
   623     (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
   624   val take_stricts =
   625     let
   626       fun one_eq ((dn, args), _) = strict (dc_take dn $ %:"n");
   627       val goal = mk_trp (foldr1 mk_conj (map one_eq eqs));
   628       val tacs = [
   629         induct_tac "n" 1,
   630         simp_tac iterate_Cprod_ss 1,
   631         asm_simp_tac (iterate_Cprod_ss addsimps copy_rews) 1];
   632     in pg copy_take_defs goal tacs end;
   633 
   634   val take_stricts' = rewrite_rule copy_take_defs take_stricts;
   635   fun take_0 n dn =
   636     let
   637       val goal = mk_trp ((dc_take dn $ %%:"HOL.zero") `% x_name n === UU);
   638     in pg axs_take_def goal [simp_tac iterate_Cprod_ss 1] end;
   639   val take_0s = mapn take_0 1 dnames;
   640   val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1;
   641   val take_apps =
   642     let
   643       fun mk_eqn dn (con, args) =
   644         let
   645           fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
   646           val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);
   647           val rhs = con_app2 con (app_rec_arg mk_take) args;
   648         in Library.foldr mk_all (map vname args, lhs === rhs) end;
   649       fun mk_eqns ((dn, _), cons) = map (mk_eqn dn) cons;
   650       val goal = mk_trp (foldr1 mk_conj (List.concat (map mk_eqns eqs)));
   651       val simps = List.filter (has_fewer_prems 1) copy_rews;
   652       fun con_tac (con, args) =
   653         if nonlazy_rec args = []
   654         then all_tac
   655         else EVERY (map c_UU_tac (nonlazy_rec args)) THEN
   656           asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1;
   657       fun eq_tacs ((dn, _), cons) = map con_tac cons;
   658       val tacs =
   659         simp_tac iterate_Cprod_ss 1 ::
   660         induct_tac "n" 1 ::
   661         simp_tac (iterate_Cprod_ss addsimps copy_con_rews) 1 ::
   662         asm_full_simp_tac (HOLCF_ss addsimps simps) 1 ::
   663         TRY (safe_tac HOL_cs) ::
   664         List.concat (map eq_tacs eqs);
   665     in pg copy_take_defs goal tacs end;
   666 in
   667   val take_rews = map standard
   668     (atomize take_stricts @ take_0s @ atomize take_apps);
   669 end; (* local *)
   670 
   671 local
   672   fun one_con p (con,args) =
   673     let
   674       fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
   675       val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
   676       val t2 = lift ind_hyp (List.filter is_rec args, t1);
   677       val t3 = lift_defined (bound_arg (map vname args)) (nonlazy args, t2);
   678     in Library.foldr mk_All (map vname args, t3) end;
   679 
   680   fun one_eq ((p, cons), concl) =
   681     mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
   682 
   683   fun ind_term concf = Library.foldr one_eq
   684     (mapn (fn n => fn x => (P_name n, x)) 1 conss,
   685      mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
   686   val take_ss = HOL_ss addsimps take_rews;
   687   fun quant_tac i = EVERY
   688     (mapn (fn n => fn _ => res_inst_tac [("x", x_name n)] spec i) 1 dnames);
   689 
   690   fun ind_prems_tac prems = EVERY
   691     (List.concat (map (fn cons =>
   692       (resolve_tac prems 1 ::
   693         List.concat (map (fn (_,args) => 
   694           resolve_tac prems 1 ::
   695           map (K(atac 1)) (nonlazy args) @
   696           map (K(atac 1)) (List.filter is_rec args))
   697         cons)))
   698       conss));
   699   local 
   700     (* check whether every/exists constructor of the n-th part of the equation:
   701        it has a possibly indirectly recursive argument that isn't/is possibly 
   702        indirectly lazy *)
   703     fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
   704           is_rec arg andalso not(rec_of arg mem ns) andalso
   705           ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
   706             rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
   707               (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
   708           ) o snd) cons;
   709     fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
   710     fun warn (n,cons) =
   711       if all_rec_to [] false (n,cons)
   712       then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
   713       else false;
   714     fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;
   715 
   716   in
   717     val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
   718     val is_emptys = map warn n__eqs;
   719     val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
   720   end;
   721 in (* local *)
   722   val finite_ind =
   723     let
   724       fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
   725       val goal = ind_term concf;
   726 
   727       fun tacf prems =
   728         let
   729           val tacs1 = [
   730             quant_tac 1,
   731             simp_tac HOL_ss 1,
   732             induct_tac "n" 1,
   733             simp_tac (take_ss addsimps prems) 1,
   734             TRY (safe_tac HOL_cs)];
   735           fun arg_tac arg =
   736             case_UU_tac (prems @ con_rews) 1
   737               (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
   738           fun con_tacs (con, args) = 
   739             asm_simp_tac take_ss 1 ::
   740             map arg_tac (List.filter is_nonlazy_rec args) @
   741             [resolve_tac prems 1] @
   742             map (K (atac 1))      (nonlazy args) @
   743             map (K (etac spec 1)) (List.filter is_rec args);
   744           fun cases_tacs (cons, cases) =
   745             res_inst_tac [("x","x")] cases 1 ::
   746             asm_simp_tac (take_ss addsimps prems) 1 ::
   747             List.concat (map con_tacs cons);
   748         in
   749           tacs1 @ List.concat (map cases_tacs (conss ~~ cases))
   750         end;
   751     in pg'' thy [] goal tacf end;
   752 
   753   val take_lemmas =
   754     let
   755       fun take_lemma n (dn, ax_reach) =
   756         let
   757           val lhs = dc_take dn $ Bound 0 `%(x_name n);
   758           val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");
   759           val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));
   760           val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;
   761           fun tacf prems = [
   762             res_inst_tac [("t", x_name n    )] (ax_reach RS subst) 1,
   763             res_inst_tac [("t", x_name n^"'")] (ax_reach RS subst) 1,
   764             stac fix_def2 1,
   765             REPEAT (CHANGED
   766               (rtac (contlub_cfun_arg RS ssubst) 1 THEN chain_tac 1)),
   767             stac contlub_cfun_fun 1,
   768             stac contlub_cfun_fun 2,
   769             rtac lub_equal 3,
   770             chain_tac 1,
   771             rtac allI 1,
   772             resolve_tac prems 1];
   773         in pg'' thy axs_take_def goal tacf end;
   774     in mapn take_lemma 1 (dnames ~~ axs_reach) end;
   775 
   776 (* ----- theorems concerning finiteness and induction ----------------------- *)
   777 
   778   val (finites, ind) =
   779     if is_finite
   780     then (* finite case *)
   781       let 
   782         fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
   783         fun dname_lemma dn =
   784           let
   785             val prem1 = mk_trp (defined (%:"x"));
   786             val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);
   787             val prem2 = mk_trp (mk_disj (disj1, take_enough dn));
   788             val concl = mk_trp (take_enough dn);
   789             val goal = prem1 ===> prem2 ===> concl;
   790             val tacs = [
   791               etac disjE 1,
   792               etac notE 1,
   793               resolve_tac take_lemmas 1,
   794               asm_simp_tac take_ss 1,
   795               atac 1];
   796           in pg [] goal tacs end;
   797         val finite_lemmas1a = map dname_lemma dnames;
   798  
   799         val finite_lemma1b =
   800           let
   801             fun mk_eqn n ((dn, args), _) =
   802               let
   803                 val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;
   804                 val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;
   805               in
   806                 mk_constrainall
   807                   (x_name n, Type (dn,args), mk_disj (disj1, disj2))
   808               end;
   809             val goal =
   810               mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));
   811             fun arg_tacs vn = [
   812               eres_inst_tac [("x", vn)] all_dupE 1,
   813               etac disjE 1,
   814               asm_simp_tac (HOL_ss addsimps con_rews) 1,
   815               asm_simp_tac take_ss 1];
   816             fun con_tacs (con, args) =
   817               asm_simp_tac take_ss 1 ::
   818               List.concat (map arg_tacs (nonlazy_rec args));
   819             fun foo_tacs n (cons, cases) =
   820               simp_tac take_ss 1 ::
   821               rtac allI 1 ::
   822               res_inst_tac [("x",x_name n)] cases 1 ::
   823               asm_simp_tac take_ss 1 ::
   824               List.concat (map con_tacs cons);
   825             val tacs =
   826               rtac allI 1 ::
   827               induct_tac "n" 1 ::
   828               simp_tac take_ss 1 ::
   829               TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::
   830               List.concat (mapn foo_tacs 1 (conss ~~ cases));
   831           in pg [] goal tacs end;
   832 
   833         fun one_finite (dn, l1b) =
   834           let
   835             val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
   836             val tacs = [
   837               case_UU_tac take_rews 1 "x",
   838               eresolve_tac finite_lemmas1a 1,
   839               step_tac HOL_cs 1,
   840               step_tac HOL_cs 1,
   841               cut_facts_tac [l1b] 1,
   842               fast_tac HOL_cs 1];
   843           in pg axs_finite_def goal tacs end;
   844 
   845         val finites = map one_finite (dnames ~~ atomize finite_lemma1b);
   846         val ind =
   847           let
   848             fun concf n dn = %:(P_name n) $ %:(x_name n);
   849             fun tacf prems =
   850               let
   851                 fun finite_tacs (finite, fin_ind) = [
   852                   rtac(rewrite_rule axs_finite_def finite RS exE)1,
   853                   etac subst 1,
   854                   rtac fin_ind 1,
   855                   ind_prems_tac prems];
   856               in
   857                 TRY (safe_tac HOL_cs) ::
   858                 List.concat (map finite_tacs (finites ~~ atomize finite_ind))
   859               end;
   860           in pg'' thy [] (ind_term concf) tacf end;
   861       in (finites, ind) end (* let *)
   862 
   863     else (* infinite case *)
   864       let
   865         fun one_finite n dn =
   866           read_instantiate_sg thy
   867             [("P",dn^"_finite "^x_name n)] excluded_middle;
   868         val finites = mapn one_finite 1 dnames;
   869 
   870         val goal =
   871           let
   872             fun one_adm n _ = mk_trp (%%:admN $ %:(P_name n));
   873             fun concf n dn = %:(P_name n) $ %:(x_name n);
   874           in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
   875         fun tacf prems =
   876           map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
   877           quant_tac 1,
   878           rtac (adm_impl_admw RS wfix_ind) 1,
   879           REPEAT_DETERM (rtac adm_all2 1),
   880           REPEAT_DETERM (
   881             TRY (rtac adm_conj 1) THEN 
   882             rtac adm_subst 1 THEN 
   883             cont_tacR 1 THEN resolve_tac prems 1),
   884           strip_tac 1,
   885           rtac (rewrite_rule axs_take_def finite_ind) 1,
   886           ind_prems_tac prems];
   887         val ind = (pg'' thy [] goal tacf
   888           handle ERROR _ =>
   889             (warning "Cannot prove infinite induction rule"; refl));
   890       in (finites, ind) end;
   891 end; (* local *)
   892 
   893 (* ----- theorem concerning coinduction ------------------------------------- *)
   894 
   895 local
   896   val xs = mapn (fn n => K (x_name n)) 1 dnames;
   897   fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
   898   val take_ss = HOL_ss addsimps take_rews;
   899   val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
   900   val coind_lemma =
   901     let
   902       fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
   903       fun mk_eqn n dn =
   904         (dc_take dn $ %:"n" ` bnd_arg n 0) ===
   905         (dc_take dn $ %:"n" ` bnd_arg n 1);
   906       fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
   907       val goal =
   908         mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
   909           Library.foldr mk_all2 (xs,
   910             Library.foldr mk_imp (mapn mk_prj 0 dnames,
   911               foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
   912       fun x_tacs n x = [
   913         rotate_tac (n+1) 1,
   914         etac all2E 1,
   915         eres_inst_tac [("P1", sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
   916         TRY (safe_tac HOL_cs),
   917         REPEAT (CHANGED (asm_simp_tac take_ss 1))];
   918       val tacs = [
   919         rtac impI 1,
   920         induct_tac "n" 1,
   921         simp_tac take_ss 1,
   922         safe_tac HOL_cs] @
   923         List.concat (mapn x_tacs 0 xs);
   924     in pg [ax_bisim_def] goal tacs end;
   925 in
   926   val coind = 
   927     let
   928       fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
   929       fun mk_eqn x = %:x === %:(x^"'");
   930       val goal =
   931         mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
   932           Logic.list_implies (mapn mk_prj 0 xs,
   933             mk_trp (foldr1 mk_conj (map mk_eqn xs)));
   934       val tacs =
   935         TRY (safe_tac HOL_cs) ::
   936         List.concat (map (fn take_lemma => [
   937           rtac take_lemma 1,
   938           cut_facts_tac [coind_lemma] 1,
   939           fast_tac HOL_cs 1])
   940         take_lemmas);
   941     in pg [] goal tacs end;
   942 end; (* local *)
   943 
   944 in thy |> Sign.add_path comp_dnam
   945        |> (snd o (PureThy.add_thmss (map Thm.no_attributes [
   946 		("take_rews"  , take_rews  ),
   947 		("take_lemmas", take_lemmas),
   948 		("finites"    , finites    ),
   949 		("finite_ind", [finite_ind]),
   950 		("ind"       , [ind       ]),
   951 		("coind"     , [coind     ])])))
   952        |> Sign.parent_path |> rpair take_rews
   953 end; (* let *)
   954 end; (* local *)
   955 end; (* struct *)