src/HOLCF/domain/theorems.ML
author oheimb
Thu Jun 27 15:24:17 1996 +0200 (1996-06-27)
changeset 1834 c780a4f39454
parent 1829 5a3687398716
child 2033 639de962ded4
permissions -rw-r--r--
re-added when_funs to library.ML
     1  (* theorems.ML
     2    Author : David von Oheimb
     3    Created: 06-Jun-95
     4    Updated: 08-Jun-95 first proof from cterms
     5    Updated: 26-Jun-95 proofs for exhaustion thms
     6    Updated: 27-Jun-95 proofs for discriminators, constructors and selectors
     7    Updated: 06-Jul-95 proofs for distinctness, invertibility and injectivity
     8    Updated: 17-Jul-95 proofs for induction rules
     9    Updated: 19-Jul-95 proof for co-induction rule
    10    Updated: 28-Aug-95 definedness theorems for selectors (completion)
    11    Updated: 05-Sep-95 simultaneous domain equations (main part)
    12    Updated: 11-Sep-95 simultaneous domain equations (coding finished)
    13    Updated: 13-Sep-95 simultaneous domain equations (debugging)
    14    Updated: 26-Oct-95 debugging and enhancement of proofs for take_apps, ind
    15    Updated: 16-Feb-96 bug concerning  domain Triv = triv  fixed
    16    Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
    17    Copyright 1995, 1996 TU Muenchen
    18 *)
    19 
    20 structure Domain_Theorems = struct
    21 
    22 local
    23 
    24 open Domain_Library;
    25 infixr 0 ===>;infixr 0 ==>;infix 0 == ; 
    26 infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<;
    27 infix 9 `   ; infix 9 `% ; infix 9 `%%; infixr 9 oo;
    28 
    29 (* ----- general proof facilities ------------------------------------------- *)
    30 
    31 fun inferT sg pre_tm = #2 (Sign.infer_types sg (K None) (K None) [] true 
    32 			   ([pre_tm],propT));
    33 
    34 fun pg'' thy defs t = let val sg = sign_of thy;
    35 		          val ct = Thm.cterm_of sg (inferT sg t);
    36 		      in prove_goalw_cterm defs ct end;
    37 fun pg'  thy defs t tacsf=pg'' thy defs t (fn []   => tacsf 
    38 				| prems=> (cut_facts_tac prems 1)::tacsf);
    39 
    40 fun REPEAT_DETERM_UNTIL p tac = 
    41 let fun drep st = if p st then Sequence.single st
    42 			  else (case Sequence.pull(tac st) of
    43 		                  None        => Sequence.null
    44 				| Some(st',_) => drep st')
    45 in drep end;
    46 val UNTIL_SOLVED = REPEAT_DETERM_UNTIL (has_fewer_prems 1);
    47 
    48 local val trueI2 = prove_goal HOL.thy"f~=x ==> True"(fn _ => [rtac TrueI 1]) in
    49 val kill_neq_tac = dtac trueI2 end;
    50 fun case_UU_tac rews i v =	case_tac (v^"=UU") i THEN
    51 				asm_simp_tac (HOLCF_ss addsimps rews) i;
    52 
    53 val chain_tac = REPEAT_DETERM o resolve_tac 
    54 		[is_chain_iterate, ch2ch_fappR, ch2ch_fappL];
    55 
    56 (* ----- general proofs ----------------------------------------------------- *)
    57 
    58 val quant_ss = HOL_ss addsimps (map (fn s => prove_goal HOL.thy s (fn _ =>[
    59 		fast_tac HOL_cs 1]))["(!x. P x & Q)=((!x. P x) & Q)",
    60 			    	     "(!x. P & Q x) = (P & (!x. Q x))"]);
    61 
    62 val all2E = prove_goal HOL.thy "[| !x y . P x y; P x y ==> R |] ==> R"
    63  (fn prems =>[
    64 				resolve_tac prems 1,
    65 				cut_facts_tac prems 1,
    66 				fast_tac HOL_cs 1]);
    67 
    68 val swap3 = prove_goal HOL.thy "[| Q ==> P; ~P |] ==> ~Q" (fn prems => [
    69                                 cut_facts_tac prems 1,
    70                                 etac swap 1,
    71                                 dtac notnotD 1,
    72 				etac (hd prems) 1]);
    73 
    74 val dist_eqI = prove_goal Porder.thy "~ x << y ==> x ~= y" (fn prems => [
    75                                 rtac swap3 1,
    76 				etac (antisym_less_inverse RS conjunct1) 1,
    77 				resolve_tac prems 1]);
    78 val cfst_strict  = prove_goal Cprod3.thy "cfst`UU = UU" (fn _ => [
    79 			(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
    80 val csnd_strict  = prove_goal Cprod3.thy "csnd`UU = UU" (fn _ => [
    81 			(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
    82 
    83 in
    84 
    85 
    86 fun theorems thy (((dname,_),cons) : eq, eqs :eq list) =
    87 let
    88 
    89 val dummy = writeln ("Proving isomorphism properties of domain "^dname^"...");
    90 val pg = pg' thy;
    91 (*
    92 infixr 0 y;
    93 val b = 0;
    94 fun _ y t = by t;
    95 fun  g  defs t = let val sg = sign_of thy;
    96 		     val ct = Thm.cterm_of sg (inferT sg t);
    97 		 in goalw_cterm defs ct end;
    98 *)
    99 
   100 
   101 (* ----- getting the axioms and definitions --------------------------------- *)
   102 
   103 local val ga = get_axiom thy in
   104 val ax_abs_iso    = ga (dname^"_abs_iso"   );
   105 val ax_rep_iso    = ga (dname^"_rep_iso"   );
   106 val ax_when_def   = ga (dname^"_when_def"  );
   107 val axs_con_def   = map (fn (con,_) => ga (extern_name con ^"_def")) cons;
   108 val axs_dis_def   = map (fn (con,_) => ga (   dis_name con ^"_def")) cons;
   109 val axs_sel_def   = flat(map (fn (_,args) => 
   110 		    map (fn     arg => ga (sel_of arg      ^"_def")) args)cons);
   111 val ax_copy_def   = ga (dname^"_copy_def"  );
   112 end; (* local *)
   113 
   114 (* ----- theorems concerning the isomorphism -------------------------------- *)
   115 
   116 val dc_abs  = %%(dname^"_abs");
   117 val dc_rep  = %%(dname^"_rep");
   118 val dc_copy = %%(dname^"_copy");
   119 val x_name = "x";
   120 
   121 val (rep_strict, abs_strict) = let 
   122 	 val r = ax_rep_iso RS (ax_abs_iso RS (allI  RSN(2,allI RS iso_strict)))
   123 	       in (r RS conjunct1, r RS conjunct2) end;
   124 val abs_defin' = pg [] ((dc_abs`%x_name === UU) ==> (%x_name === UU)) [
   125 			   res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
   126 				etac ssubst 1, rtac rep_strict 1];
   127 val rep_defin' = pg [] ((dc_rep`%x_name === UU) ==> (%x_name === UU)) [
   128 			   res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
   129 				etac ssubst 1, rtac abs_strict 1];
   130 val iso_rews = [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
   131 
   132 local 
   133 val iso_swap = pg [] (dc_rep`%"x" === %"y" ==> %"x" === dc_abs`%"y") [
   134 			    dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
   135 			    etac (ax_rep_iso RS subst) 1];
   136 fun exh foldr1 cn quant foldr2 var = let
   137   fun one_con (con,args) = let val vns = map vname args in
   138     foldr quant (vns, foldr2 ((%x_name === con_app2 con (var vns) vns)::
   139 			      map (defined o (var vns)) (nonlazy args))) end
   140   in foldr1 ((cn(%x_name===UU))::map one_con cons) end;
   141 in
   142 val cases = let 
   143 	    fun common_tac thm = rtac thm 1 THEN contr_tac 1;
   144 	    fun unit_tac true = common_tac liftE1
   145 	    |   unit_tac _    = all_tac;
   146 	    fun prod_tac []          = common_tac oneE
   147 	    |   prod_tac [arg]       = unit_tac (is_lazy arg)
   148 	    |   prod_tac (arg::args) = 
   149 				common_tac sprodE THEN
   150 				kill_neq_tac 1 THEN
   151 				unit_tac (is_lazy arg) THEN
   152 				prod_tac args;
   153 	    fun sum_rest_tac p = SELECT_GOAL(EVERY[
   154 				rtac p 1,
   155 				rewrite_goals_tac axs_con_def,
   156 				dtac iso_swap 1,
   157 				simp_tac HOLCF_ss 1,
   158 				UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
   159 	    fun sum_tac [(_,args)]       [p]        = 
   160 				prod_tac args THEN sum_rest_tac p
   161 	    |   sum_tac ((_,args)::cons') (p::prems) = DETERM(
   162 				common_tac ssumE THEN
   163 				kill_neq_tac 1 THEN kill_neq_tac 2 THEN
   164 				prod_tac args THEN sum_rest_tac p) THEN
   165 				sum_tac cons' prems
   166 	    |   sum_tac _ _ = Imposs "theorems:sum_tac";
   167 	  in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%"P")))
   168 			      (fn T => T ==> %"P") mk_All
   169 			      (fn l => foldr (op ===>) (map mk_trp l,
   170 							    mk_trp(%"P")))
   171 			      bound_arg)
   172 			     (fn prems => [
   173 				cut_facts_tac [excluded_middle] 1,
   174 				etac disjE 1,
   175 				rtac (hd prems) 2,
   176 				etac rep_defin' 2,
   177 				if length cons = 1 andalso 
   178 				   length (snd(hd cons)) = 1 andalso 
   179 				   not(is_lazy(hd(snd(hd cons))))
   180 				then rtac (hd (tl prems)) 1 THEN atac 2 THEN
   181 				     rewrite_goals_tac axs_con_def THEN
   182 				     simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
   183 				else sum_tac cons (tl prems)])end;
   184 val exhaust= pg[](mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %)))[
   185 				rtac cases 1,
   186 				UNTIL_SOLVED(fast_tac HOL_cs 1)];
   187 end;
   188 
   189 local 
   190   fun bind_fun t = foldr mk_All (when_funs cons,t);
   191   fun bound_fun i _ = Bound (length cons - i);
   192   val when_app  = foldl (op `) (%%(dname^"_when"), mapn bound_fun 1 cons);
   193   val when_appl = pg [ax_when_def] (bind_fun(mk_trp(when_app`%x_name ===
   194 	     when_body cons (fn (m,n)=> bound_fun (n-m) 0)`(dc_rep`%x_name))))[
   195 				simp_tac HOLCF_ss 1];
   196 in
   197 val when_strict = pg [] (bind_fun(mk_trp(strict when_app))) [
   198 			simp_tac(HOLCF_ss addsimps [when_appl,rep_strict]) 1];
   199 val when_apps = let fun one_when n (con,args) = pg axs_con_def 
   200 		(bind_fun (lift_defined % (nonlazy args, 
   201 		mk_trp(when_app`(con_app con args) ===
   202                        mk_cfapp(bound_fun n 0,map %# args)))))[
   203 		asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
   204 	in mapn one_when 1 cons end;
   205 end;
   206 val when_rews = when_strict::when_apps;
   207 
   208 (* ----- theorems concerning the constructors, discriminators and selectors - *)
   209 
   210 val dis_rews = let
   211   val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
   212 		      	     strict(%%(dis_name con)))) [
   213 				simp_tac (HOLCF_ss addsimps when_rews) 1]) cons;
   214   val dis_apps = let fun one_dis c (con,args)= pg axs_dis_def
   215 		   (lift_defined % (nonlazy args,
   216 			(mk_trp((%%(dis_name c))`(con_app con args) ===
   217 			      %%(if con=c then "TT" else "FF"))))) [
   218 				asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
   219 	in flat(map (fn (c,_) => map (one_dis c) cons) cons) end;
   220   val dis_defins = map (fn (con,args) => pg [] (defined(%x_name) ==> 
   221 		      defined(%%(dis_name con)`%x_name)) [
   222 				rtac cases 1,
   223 				contr_tac 1,
   224 				UNTIL_SOLVED (CHANGED(asm_simp_tac 
   225 				        (HOLCF_ss addsimps dis_apps) 1))]) cons;
   226 in dis_stricts @ dis_defins @ dis_apps end;
   227 
   228 val con_stricts = flat(map (fn (con,args) => map (fn vn =>
   229 			pg (axs_con_def) 
   230 			   (mk_trp(con_app2 con (fn arg => if vname arg = vn 
   231 					then UU else %# arg) args === UU))[
   232 				asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
   233 			) (nonlazy args)) cons);
   234 val con_defins = map (fn (con,args) => pg []
   235 			(lift_defined % (nonlazy args,
   236 				mk_trp(defined(con_app con args)))) ([
   237 			  rtac swap3 1, 
   238 			  eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
   239 			  asm_simp_tac (HOLCF_ss addsimps dis_rews) 1] )) cons;
   240 val con_rews = con_stricts @ con_defins;
   241 
   242 val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%sel))) [
   243 				simp_tac (HOLCF_ss addsimps when_rews) 1];
   244 in flat(map (fn (_,args) =>map (fn arg => one_sel (sel_of arg)) args) cons) end;
   245 val sel_apps = let fun one_sel c n sel = map (fn (con,args) => 
   246 		let val nlas = nonlazy args;
   247 		    val vns  = map vname args;
   248 		in pg axs_sel_def (lift_defined %
   249 		   (filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
   250 				mk_trp((%%sel)`(con_app con args) === 
   251 				(if con=c then %(nth_elem(n,vns)) else UU))))
   252 			    ( (if con=c then [] 
   253 		       else map(case_UU_tac(when_rews@con_stricts)1) nlas)
   254 		     @(if con=c andalso ((nth_elem(n,vns)) mem nlas)
   255 				 then[case_UU_tac (when_rews @ con_stricts) 1 
   256 						  (nth_elem(n,vns))] else [])
   257 		     @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
   258 in flat(map  (fn (c,args) => 
   259      flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
   260 val sel_defins = if length cons=1 then map (fn arg => pg [](defined(%x_name)==> 
   261 			defined(%%(sel_of arg)`%x_name)) [
   262 				rtac cases 1,
   263 				contr_tac 1,
   264 				UNTIL_SOLVED (CHANGED(asm_simp_tac 
   265 				             (HOLCF_ss addsimps sel_apps) 1))]) 
   266 		 (filter_out is_lazy (snd(hd cons))) else [];
   267 val sel_rews = sel_stricts @ sel_defins @ sel_apps;
   268 
   269 val distincts_le = let
   270     fun dist (con1, args1) (con2, args2) = pg []
   271 	      (lift_defined % ((nonlazy args1),
   272 			(mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
   273 			rtac swap3 1,
   274 			eres_inst_tac[("fo",dis_name con1)] monofun_cfun_arg 1]
   275 		      @map(case_UU_tac (con_stricts @ dis_rews)1)(nonlazy args2)
   276 		      @[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
   277     fun distinct (con1,args1) (con2,args2) =
   278 	let val arg1 = (con1, args1);
   279 	    val arg2 = (con2, (map (fn (arg,vn) => upd_vname (K vn) arg)
   280 			(args2~~variantlist(map vname args2,map vname args1))));
   281 	in [dist arg1 arg2, dist arg2 arg1] end;
   282     fun distincts []      = []
   283     |   distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
   284 in distincts cons end;
   285 val dists_le = flat (flat distincts_le);
   286 val dists_eq = let
   287     fun distinct (_,args1) ((_,args2),leqs) = let
   288 	val (le1,le2) = (hd leqs, hd(tl leqs));
   289 	val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
   290 	if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
   291 	if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
   292 					[eq1, eq2] end;
   293     fun distincts []      = []
   294     |   distincts ((c,leqs)::cs) = flat(map (distinct c) ((map fst cs)~~leqs)) @
   295 				   distincts cs;
   296     in distincts (cons~~distincts_le) end;
   297 
   298 local 
   299   fun pgterm rel con args = let
   300 		fun append s = upd_vname(fn v => v^s);
   301 		val (largs,rargs) = (args, map (append "'") args);
   302 		in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
   303 		      lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
   304 			    mk_trp (foldr' mk_conj 
   305 				(map rel (map %# largs ~~ map %# rargs)))))) end;
   306   val cons' = filter (fn (_,args) => args<>[]) cons;
   307 in
   308 val inverts = map (fn (con,args) => 
   309 		pgterm (op <<) con args (flat(map (fn arg => [
   310 				TRY(rtac conjI 1),
   311 				dres_inst_tac [("fo",sel_of arg)] monofun_cfun_arg 1,
   312 				asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
   313 			     			      ) args))) cons';
   314 val injects = map (fn ((con,args),inv_thm) => 
   315 			   pgterm (op ===) con args [
   316 				etac (antisym_less_inverse RS conjE) 1,
   317 				dtac inv_thm 1, REPEAT(atac 1),
   318 				dtac inv_thm 1, REPEAT(atac 1),
   319 				TRY(safe_tac HOL_cs),
   320 				REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
   321 		  (cons'~~inverts);
   322 end;
   323 
   324 (* ----- theorems concerning one induction step ----------------------------- *)
   325 
   326 val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [
   327 		   asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict,
   328 						   cfst_strict,csnd_strict]) 1];
   329 val copy_apps = map (fn (con,args) => pg [ax_copy_def]
   330 		    (lift_defined % (nonlazy_rec args,
   331 			mk_trp(dc_copy`%"f"`(con_app con args) ===
   332 		(con_app2 con (app_rec_arg (cproj (%"f") (length eqs))) args))))
   333 			(map (case_UU_tac (abs_strict::when_strict::con_stricts)
   334 				 1 o vname)
   335 			 (filter (fn a => not (is_rec a orelse is_lazy a)) args)
   336 			@[asm_simp_tac (HOLCF_ss addsimps when_apps) 1,
   337 		          simp_tac (HOLCF_ss addsimps axs_con_def) 1]))cons;
   338 val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU`
   339 					(con_app con args) ===UU))
   340      (let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
   341 			 in map (case_UU_tac rews 1) (nonlazy args) @ [
   342 			     asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
   343   		        (filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
   344 val copy_rews = copy_strict::copy_apps @ copy_stricts;
   345 
   346 in     (iso_rews, exhaust, cases, when_rews,
   347 	con_rews, sel_rews, dis_rews, dists_le, dists_eq, inverts, injects,
   348 	copy_rews)
   349 end; (* let *)
   350 
   351 
   352 fun comp_theorems thy (comp_dname, eqs: eq list, casess, con_rews, copy_rews) =
   353 let
   354 
   355 val dummy = writeln("Proving induction properties of domain "^comp_dname^"...");
   356 val pg = pg' thy;
   357 
   358 val dnames = map (fst o fst) eqs;
   359 val conss  = map  snd        eqs;
   360 
   361 (* ----- getting the composite axiom and definitions ------------------------ *)
   362 
   363 local val ga = get_axiom thy in
   364 val axs_reach      = map (fn dn => ga (dn ^  "_reach"   )) dnames;
   365 val axs_take_def   = map (fn dn => ga (dn ^  "_take_def")) dnames;
   366 val axs_finite_def = map (fn dn => ga (dn ^"_finite_def")) dnames;
   367 val ax_copy2_def   = ga (comp_dname^ "_copy_def");
   368 val ax_bisim_def   = ga (comp_dname^"_bisim_def");
   369 end; (* local *)
   370 
   371 fun dc_take dn = %%(dn^"_take");
   372 val x_name = idx_name dnames "x"; 
   373 val P_name = idx_name dnames "P";
   374 val n_eqs = length eqs;
   375 
   376 (* ----- theorems concerning finite approximation and finite induction ------ *)
   377 
   378 local
   379   val iterate_Cprod_ss = simpset_of "Fix"
   380 			 addsimps [cfst_strict, csnd_strict]addsimps Cprod_rews;
   381   val copy_con_rews  = copy_rews @ con_rews;
   382   val copy_take_defs =(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
   383   val take_stricts=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=>
   384 	    (dc_take dn $ %"n")`UU === mk_constrain(Type(dn,args),UU)) eqs)))([
   385 			nat_ind_tac "n" 1,
   386 			simp_tac iterate_Cprod_ss 1,
   387 			asm_simp_tac (iterate_Cprod_ss addsimps copy_rews)1]);
   388   val take_stricts' = rewrite_rule copy_take_defs take_stricts;
   389   val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn $ %%"0")
   390 							`%x_name n === UU))[
   391 				simp_tac iterate_Cprod_ss 1]) 1 dnames;
   392   val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1;
   393   val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj 
   394 	    (flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all 
   395 	(map vname args,(dc_take dn $ (%%"Suc" $ %"n"))`(con_app con args) ===
   396   	 con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))$ %"n"))
   397 			      args)) cons) eqs)))) ([
   398 				simp_tac iterate_Cprod_ss 1,
   399 				nat_ind_tac "n" 1,
   400 			    simp_tac(iterate_Cprod_ss addsimps copy_con_rews) 1,
   401 				asm_full_simp_tac (HOLCF_ss addsimps 
   402 				      (filter (has_fewer_prems 1) copy_rews)) 1,
   403 				TRY(safe_tac HOL_cs)] @
   404 			(flat(map (fn ((dn,_),cons) => map (fn (con,args) => 
   405 				if nonlazy_rec args = [] then all_tac else
   406 				EVERY(map c_UU_tac (nonlazy_rec args)) THEN
   407 				asm_full_simp_tac (HOLCF_ss addsimps copy_rews)1
   408 		 					   ) cons) eqs)));
   409 in
   410 val take_rews = atomize take_stricts @ take_0s @ atomize take_apps;
   411 end; (* local *)
   412 
   413 local
   414   fun one_con p (con,args) = foldr mk_All (map vname args,
   415 	lift_defined (bound_arg (map vname args)) (nonlazy args,
   416 	lift (fn arg => %(P_name (1+rec_of arg)) $ bound_arg args arg)
   417          (filter is_rec args,mk_trp(%p $ con_app2 con (bound_arg args) args))));
   418   fun one_eq ((p,cons),concl) = (mk_trp(%p $ UU) ===> 
   419 			   foldr (op ===>) (map (one_con p) cons,concl));
   420   fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss,
   421 			mk_trp(foldr' mk_conj (mapn concf 1 dnames)));
   422   val take_ss = HOL_ss addsimps take_rews;
   423   fun quant_tac i = EVERY(mapn(fn n=> fn _=> res_inst_tac[("x",x_name n)]spec i)
   424 			       1 dnames);
   425   fun ind_prems_tac prems = EVERY(flat (map (fn cons => (
   426 				     resolve_tac prems 1 ::
   427 				     flat (map (fn (_,args) => 
   428 				       resolve_tac prems 1 ::
   429 				       map (K(atac 1)) (nonlazy args) @
   430 				       map (K(atac 1)) (filter is_rec args))
   431 				     cons))) conss));
   432   local 
   433     (* check whether every/exists constructor of the n-th part of the equation:
   434        it has a possibly indirectly recursive argument that isn't/is possibly 
   435        indirectly lazy *)
   436     fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
   437 	  is_rec arg andalso not(rec_of arg mem ns) andalso
   438 	  ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
   439 	    rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
   440 	      (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
   441 	  ) o snd) cons;
   442     fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
   443     fun warn (n,cons)  = if all_rec_to [] false (n,cons) then (writeln 
   444         ("WARNING: domain "^nth_elem(n,dnames)^" is empty!"); true) else false;
   445     fun lazy_rec_to ns = rec_to exists Id  lazy_rec_to ns;
   446 
   447   in val n__eqs     = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
   448      val is_emptys = map warn n__eqs;
   449      val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
   450   end;
   451 in (* local *)
   452 val finite_ind = pg'' thy [] (ind_term (fn n => fn dn => %(P_name n)$
   453 			     (dc_take dn $ %"n" `%(x_name n)))) (fn prems => [
   454 				quant_tac 1,
   455 				simp_tac quant_ss 1,
   456 				nat_ind_tac "n" 1,
   457 				simp_tac (take_ss addsimps prems) 1,
   458 				TRY(safe_tac HOL_cs)]
   459 				@ flat(map (fn (cons,cases) => [
   460 				 res_inst_tac [("x","x")] cases 1,
   461 				 asm_simp_tac (take_ss addsimps prems) 1]
   462 				 @ flat(map (fn (con,args) => 
   463 				  asm_simp_tac take_ss 1 ::
   464 				  map (fn arg =>
   465 				   case_UU_tac (prems@con_rews) 1 (
   466 			   nth_elem(rec_of arg,dnames)^"_take n1`"^vname arg))
   467 				  (filter is_nonlazy_rec args) @ [
   468 				  resolve_tac prems 1] @
   469 				  map (K (atac 1))      (nonlazy args) @
   470 				  map (K (etac spec 1)) (filter is_rec args)) 
   471 				 cons))
   472 				(conss~~casess)));
   473 
   474 val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n",
   475 		mk_trp(dc_take dn $ Bound 0 `%(x_name n) === 
   476 		       dc_take dn $ Bound 0 `%(x_name n^"'")))
   477 	   ===> mk_trp(%(x_name n) === %(x_name n^"'"))) (fn prems => [
   478 			res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
   479 			res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
   480 				rtac (fix_def2 RS ssubst) 1,
   481 				REPEAT(CHANGED(rtac(contlub_cfun_arg RS ssubst)1
   482 					       THEN chain_tac 1)),
   483 				rtac (contlub_cfun_fun RS ssubst) 1,
   484 				rtac (contlub_cfun_fun RS ssubst) 2,
   485 				rtac lub_equal 3,
   486 				chain_tac 1,
   487 				rtac allI 1,
   488 				resolve_tac prems 1])) 1 (dnames~~axs_reach);
   489 
   490 (* ----- theorems concerning finiteness and induction ----------------------- *)
   491 
   492 val (finites,ind) = if is_finite then
   493   let 
   494     fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %"x" === %"x");
   495     val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%"x")) ===> 
   496 	mk_trp(mk_disj(mk_all("n",dc_take dn $ Bound 0 ` %"x" === UU),
   497 	take_enough dn)) ===> mk_trp(take_enough dn)) [
   498 				etac disjE 1,
   499 				etac notE 1,
   500 				resolve_tac take_lemmas 1,
   501 				asm_simp_tac take_ss 1,
   502 				atac 1]) dnames;
   503     val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn 
   504 	(fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
   505 	 mk_disj(dc_take dn $ Bound 1 ` Bound 0 === UU,
   506 		 dc_take dn $ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
   507 				rtac allI 1,
   508 				nat_ind_tac "n" 1,
   509 				simp_tac take_ss 1,
   510 			TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @
   511 				flat(mapn (fn n => fn (cons,cases) => [
   512 				  simp_tac take_ss 1,
   513 				  rtac allI 1,
   514 				  res_inst_tac [("x",x_name n)] cases 1,
   515 				  asm_simp_tac take_ss 1] @ 
   516 				  flat(map (fn (con,args) => 
   517 				    asm_simp_tac take_ss 1 ::
   518 				    flat(map (fn vn => [
   519 				      eres_inst_tac [("x",vn)] all_dupE 1,
   520 				      etac disjE 1,
   521 				      asm_simp_tac (HOL_ss addsimps con_rews) 1,
   522 				      asm_simp_tac take_ss 1])
   523 				    (nonlazy_rec args)))
   524 				  cons))
   525 				1 (conss~~casess))) handle ERROR => raise ERROR;
   526     val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp(
   527 						%%(dn^"_finite") $ %"x"))[
   528 				case_UU_tac take_rews 1 "x",
   529 				eresolve_tac finite_lemmas1a 1,
   530 				step_tac HOL_cs 1,
   531 				step_tac HOL_cs 1,
   532 				cut_facts_tac [l1b] 1,
   533 			fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
   534   in
   535   (finites,
   536    pg'' thy[](ind_term (fn n => fn dn => %(P_name n) $ %(x_name n)))(fn prems =>
   537 				TRY(safe_tac HOL_cs) ::
   538 			 flat (map (fn (finite,fin_ind) => [
   539 			       rtac(rewrite_rule axs_finite_def finite RS exE)1,
   540 				etac subst 1,
   541 				rtac fin_ind 1,
   542 				ind_prems_tac prems]) 
   543 			           (finites~~(atomize finite_ind)) ))
   544 ) end (* let *) else
   545   (mapn (fn n => fn dn => read_instantiate_sg (sign_of thy) 
   546 	  	    [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
   547    pg'' thy [] (foldr (op ===>) (mapn (fn n => K(mk_trp(%%"adm" $ %(P_name n))))
   548 	       1 dnames, ind_term (fn n => fn dn => %(P_name n) $ %(x_name n))))
   549 		   (fn prems => map (fn ax_reach => rtac (ax_reach RS subst) 1) 
   550 				    axs_reach @ [
   551 				quant_tac 1,
   552 				rtac (adm_impl_admw RS wfix_ind) 1,
   553 				REPEAT_DETERM(rtac adm_all2 1),
   554 				REPEAT_DETERM(TRY(rtac adm_conj 1) THEN 
   555 						  rtac adm_subst 1 THEN 
   556 					cont_tacR 1 THEN resolve_tac prems 1),
   557 				strip_tac 1,
   558 				rtac (rewrite_rule axs_take_def finite_ind) 1,
   559 				ind_prems_tac prems])
   560 )
   561 end; (* local *)
   562 
   563 (* ----- theorem concerning coinduction ------------------------------------- *)
   564 
   565 local
   566   val xs = mapn (fn n => K (x_name n)) 1 dnames;
   567   fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
   568   val take_ss = HOL_ss addsimps take_rews;
   569   val sproj   = prj (fn s => "fst("^s^")") (fn s => "snd("^s^")");
   570   val coind_lemma=pg[ax_bisim_def](mk_trp(mk_imp(%%(comp_dname^"_bisim") $ %"R",
   571 		foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
   572 		  foldr mk_imp (mapn (fn n => K(proj (%"R") n_eqs n $ 
   573 				      bnd_arg n 0 $ bnd_arg n 1)) 0 dnames,
   574 		    foldr' mk_conj (mapn (fn n => fn dn => 
   575 				(dc_take dn $ %"n" `bnd_arg n 0 === 
   576 				(dc_take dn $ %"n" `bnd_arg n 1)))0 dnames))))))
   577 			     ([ rtac impI 1,
   578 				nat_ind_tac "n" 1,
   579 				simp_tac take_ss 1,
   580 				safe_tac HOL_cs] @
   581 				flat(mapn (fn n => fn x => [
   582 				  rotate_tac (n+1) 1,
   583 				  etac all2E 1,
   584 				  eres_inst_tac [("P1", sproj "R" n_eqs n^
   585 					" "^x^" "^x^"'")](mp RS disjE) 1,
   586 				  TRY(safe_tac HOL_cs),
   587 				  REPEAT(CHANGED(asm_simp_tac take_ss 1))]) 
   588 				0 xs));
   589 in
   590 val coind = pg [] (mk_trp(%%(comp_dname^"_bisim") $ %"R") ===>
   591 		foldr (op ===>) (mapn (fn n => fn x => 
   592 		  mk_trp(proj (%"R") n_eqs n $ %x $ %(x^"'"))) 0 xs,
   593 		  mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
   594 				TRY(safe_tac HOL_cs)] @
   595 				flat(map (fn take_lemma => [
   596 				  rtac take_lemma 1,
   597 				  cut_facts_tac [coind_lemma] 1,
   598 				  fast_tac HOL_cs 1])
   599 				take_lemmas));
   600 end; (* local *)
   601 
   602 
   603 in (take_rews, take_lemmas, finites, finite_ind, ind, coind)
   604 
   605 end; (* let *)
   606 end; (* local *)
   607 end; (* struct *)