src/HOLCF/domain/theorems.ML
 changeset 1274 ea0668a1c0ba child 1461 6bcb44e4d6e5
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOLCF/domain/theorems.ML	Fri Oct 06 17:25:24 1995 +0100
1.3 @@ -0,0 +1,596 @@
1.4 +(* theorems.ML
1.5 +   ID:         \$Id\$
1.6 +   Author : David von Oheimb
1.7 +   Created: 06-Jun-95
1.8 +   Updated: 08-Jun-95 first proof from cterms
1.9 +   Updated: 26-Jun-95 proofs for exhaustion thms
1.10 +   Updated: 27-Jun-95 proofs for discriminators, constructors and selectors
1.11 +   Updated: 06-Jul-95 proofs for distinctness, invertibility and injectivity
1.12 +   Updated: 17-Jul-95 proofs for induction rules
1.13 +   Updated: 19-Jul-95 proof for co-induction rule
1.14 +   Updated: 28-Aug-95 definedness theorems for selectors (completion)
1.15 +   Updated: 05-Sep-95 simultaneous domain equations (main part)
1.16 +   Updated: 11-Sep-95 simultaneous domain equations (coding finished)
1.17 +   Updated: 13-Sep-95 simultaneous domain equations (debugging)
1.18 +   Copyright 1995 TU Muenchen
1.19 +*)
1.20 +
1.21 +
1.22 +structure Domain_Theorems = struct
1.23 +
1.24 +local
1.25 +
1.26 +open Domain_Library;
1.27 +infixr 0 ===>;infixr 0 ==>;infix 0 == ;
1.28 +infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<;
1.29 +infix 9 `   ; infix 9 `% ; infix 9 `%%; infixr 9 oo;
1.30 +
1.31 +(* ----- general proof facilities ------------------------------------------------- *)
1.32 +
1.33 +fun inferT sg pre_tm = #2(Sign.infer_types sg (K None)(K None)[]true([pre_tm],propT));
1.34 +
1.35 +(*
1.36 +infix 0 y;
1.37 +val b=0;
1.38 +fun _ y t = by t;
1.39 +fun  g  defs t = let val sg = sign_of thy;
1.40 +		     val ct = Thm.cterm_of sg (inferT sg t);
1.41 +		 in goalw_cterm defs ct end;
1.42 +*)
1.43 +
1.44 +fun pg'' thy defs t = let val sg = sign_of thy;
1.45 +		          val ct = Thm.cterm_of sg (inferT sg t);
1.46 +		      in prove_goalw_cterm defs ct end;
1.47 +fun pg'  thy defs t tacsf=pg'' thy defs t (fn []   => tacsf
1.48 +					    | prems=> (cut_facts_tac prems 1)::tacsf);
1.49 +
1.50 +fun REPEAT_DETERM_UNTIL p tac =
1.51 +let fun drep st = if p st then Sequence.single st
1.52 +			  else (case Sequence.pull(tapply(tac,st)) of
1.53 +		                  None        => Sequence.null
1.54 +				| Some(st',_) => drep st')
1.55 +in Tactic drep end;
1.56 +val UNTIL_SOLVED = REPEAT_DETERM_UNTIL (has_fewer_prems 1);
1.57 +
1.58 +local val trueI2 = prove_goal HOL.thy "f~=x ==> True" (fn prems => [rtac TrueI 1]) in
1.59 +val kill_neq_tac = dtac trueI2 end;
1.60 +fun case_UU_tac rews i v =	res_inst_tac [("Q",v^"=UU")] classical2 i THEN
1.61 +				asm_simp_tac (HOLCF_ss addsimps rews) i;
1.62 +
1.63 +val chain_tac = REPEAT_DETERM o resolve_tac
1.64 +		[is_chain_iterate, ch2ch_fappR, ch2ch_fappL];
1.65 +
1.66 +(* ----- general proofs ----------------------------------------------------------- *)
1.67 +
1.68 +val swap3 = prove_goal HOL.thy "[| Q ==> P; ~P |] ==> ~Q" (fn prems => [
1.69 +                                cut_facts_tac prems 1,
1.70 +                                etac swap 1,
1.71 +                                dtac notnotD 1,
1.72 +				etac (hd prems) 1]);
1.73 +
1.74 +val dist_eqI = prove_goal Porder0.thy "~ x << y ==> x ~= y" (fn prems => [
1.75 +				cut_facts_tac prems 1,
1.76 +				etac swap 1,
1.77 +				dtac notnotD 1,
1.78 +				asm_simp_tac HOLCF_ss 1]);
1.79 +val cfst_strict  = prove_goal Cprod3.thy "cfst`UU = UU" (fn _ => [
1.80 +				(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
1.81 +val csnd_strict  = prove_goal Cprod3.thy "csnd`UU = UU" (fn _ => [
1.82 +			(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
1.83 +
1.84 +in
1.85 +
1.86 +
1.87 +fun theorems thy (((dname,_),cons) : eq, eqs :eq list) =
1.88 +let
1.89 +
1.90 +val dummy = writeln ("Proving isomorphism properties of domain "^dname^"...");
1.91 +val pg = pg' thy;
1.92 +
1.93 +(* ----- getting the axioms and definitions --------------------------------------- *)
1.94 +
1.95 +local val ga = get_axiom thy in
1.96 +val ax_abs_iso    = ga (dname^"_abs_iso"   );
1.97 +val ax_rep_iso    = ga (dname^"_rep_iso"   );
1.98 +val ax_when_def   = ga (dname^"_when_def"  );
1.99 +val axs_con_def   = map (fn (con,_) => ga (extern_name con ^"_def")) cons;
1.100 +val axs_dis_def   = map (fn (con,_) => ga (   dis_name con ^"_def")) cons;
1.101 +val axs_sel_def   = flat(map (fn (_,args) =>
1.102 +		    map (fn     arg => ga (sel_of arg      ^"_def")) args) cons);
1.103 +val ax_copy_def   = ga (dname^"_copy_def"  );
1.104 +end; (* local *)
1.105 +
1.106 +(* ----- theorems concerning the isomorphism -------------------------------------- *)
1.107 +
1.108 +val dc_abs  = %%(dname^"_abs");
1.109 +val dc_rep  = %%(dname^"_rep");
1.110 +val dc_copy = %%(dname^"_copy");
1.111 +val x_name = "x";
1.112 +
1.113 +val (rep_strict, abs_strict) = let
1.114 +	       val r = ax_rep_iso RS (ax_abs_iso RS (allI  RSN(2,allI RS iso_strict)))
1.115 +	       in (r RS conjunct1, r RS conjunct2) end;
1.116 +val abs_defin' = pg [] ((dc_abs`%x_name === UU) ==> (%x_name === UU)) [
1.117 +				res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
1.118 +				etac ssubst 1,
1.119 +				rtac rep_strict 1];
1.120 +val rep_defin' = pg [] ((dc_rep`%x_name === UU) ==> (%x_name === UU)) [
1.121 +				res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
1.122 +				etac ssubst 1,
1.123 +				rtac abs_strict 1];
1.124 +val iso_rews = [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
1.125 +
1.126 +local
1.127 +val iso_swap = pg [] (dc_rep`%"x" === %"y" ==> %"x" === dc_abs`%"y") [
1.128 +				dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
1.129 +				etac (ax_rep_iso RS subst) 1];
1.130 +fun exh foldr1 cn quant foldr2 var = let
1.131 +  fun one_con (con,args) = let val vns = map vname args in
1.132 +    foldr quant (vns, foldr2 ((%x_name === con_app2 con (var vns) vns)::
1.133 +			      map (defined o (var vns)) (nonlazy args))) end
1.134 +  in foldr1 ((cn(%x_name===UU))::map one_con cons) end;
1.135 +in
1.136 +val cases = let
1.137 +	    fun common_tac thm = rtac thm 1 THEN contr_tac 1;
1.138 +	    fun unit_tac true = common_tac liftE1
1.139 +	    |   unit_tac _    = all_tac;
1.140 +	    fun prod_tac []          = common_tac oneE
1.141 +	    |   prod_tac [arg]       = unit_tac (is_lazy arg)
1.142 +	    |   prod_tac (arg::args) =
1.143 +				common_tac sprodE THEN
1.144 +				kill_neq_tac 1 THEN
1.145 +				unit_tac (is_lazy arg) THEN
1.146 +				prod_tac args;
1.147 +	    fun sum_one_tac p = SELECT_GOAL(EVERY[
1.148 +				rtac p 1,
1.149 +				rewrite_goals_tac axs_con_def,
1.150 +				dtac iso_swap 1,
1.151 +				simp_tac HOLCF_ss 1,
1.152 +				UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
1.153 +	    fun sum_tac [(_,args)]       [p]        =
1.154 +				prod_tac args THEN sum_one_tac p
1.155 +	    |   sum_tac ((_,args)::cons') (p::prems) = DETERM(
1.156 +				common_tac ssumE THEN
1.157 +				kill_neq_tac 1 THEN kill_neq_tac 2 THEN
1.158 +				prod_tac args THEN sum_one_tac p) THEN
1.159 +				sum_tac cons' prems
1.160 +	    |   sum_tac _ _ = Imposs "theorems:sum_tac";
1.161 +	  in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%"P")))
1.162 +			      (fn T => T ==> %"P") mk_All
1.163 +			      (fn l => foldr (op ===>) (map mk_trp l,mk_trp(%"P")))
1.164 +			      bound_arg)
1.165 +			     (fn prems => [
1.166 +				cut_facts_tac [excluded_middle] 1,
1.167 +				etac disjE 1,
1.168 +				rtac (hd prems) 2,
1.169 +				etac rep_defin' 2,
1.170 +				if is_one_con_one_arg (not o is_lazy) cons
1.171 +				then rtac (hd (tl prems)) 1 THEN atac 2 THEN
1.172 +				     rewrite_goals_tac axs_con_def THEN
1.173 +				     simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
1.174 +				else sum_tac cons (tl prems)])end;
1.175 +val exhaust = pg [] (mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %))) [
1.176 +				rtac cases 1,
1.177 +				UNTIL_SOLVED(fast_tac HOL_cs 1)];
1.178 +end;
1.179 +
1.180 +local
1.181 +val when_app = foldl (op `) (%%(dname^"_when"), map % (when_funs cons));
1.182 +val when_appl = pg [ax_when_def] (mk_trp(when_app`%x_name===when_body cons
1.183 +		(fn (_,n) => %(nth_elem(n-1,when_funs cons)))`(dc_rep`%x_name))) [
1.184 +				simp_tac HOLCF_ss 1];
1.185 +in
1.186 +val when_strict = pg [] ((if is_one_con_one_arg (K true) cons
1.187 +	then fn t => mk_trp(strict(%"f")) ===> t else Id)(mk_trp(strict when_app))) [
1.188 +				simp_tac(HOLCF_ss addsimps [when_appl,rep_strict]) 1];
1.189 +val when_apps = let fun one_when n (con,args) = pg axs_con_def
1.190 +		(lift_defined % (nonlazy args, mk_trp(when_app`(con_app con args) ===
1.191 +		 mk_cfapp(%(nth_elem(n,when_funs cons)),map %# args))))[
1.192 +			asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
1.193 +		in mapn one_when 0 cons end;
1.194 +end;
1.195 +val when_rews = when_strict::when_apps;
1.196 +
1.197 +(* ----- theorems concerning the constructors, discriminators and selectors ------- *)
1.198 +
1.199 +val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
1.200 +			(if is_one_con_one_arg (K true) cons then mk_not else Id)
1.201 +		         (strict(%%(dis_name con))))) [
1.202 +		simp_tac (HOLCF_ss addsimps (if is_one_con_one_arg (K true) cons
1.203 +					then [ax_when_def] else when_rews)) 1]) cons;
1.204 +val dis_apps = let fun one_dis c (con,args)= pg (axs_dis_def)
1.205 +		   (lift_defined % (nonlazy args, (*(if is_one_con_one_arg is_lazy cons
1.206 +			then curry (lift_defined %#) args else Id)
1.207 +#################*)
1.208 +			(mk_trp((%%(dis_name c))`(con_app con args) ===
1.209 +			      %%(if con=c then "TT" else "FF"))))) [
1.210 +				asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
1.211 +	in flat(map (fn (c,_) => map (one_dis c) cons) cons) end;
1.212 +val dis_defins = map (fn (con,args) => pg [] (defined(%x_name)==>
1.213 +		      defined(%%(dis_name con)`%x_name)) [
1.214 +				rtac cases 1,
1.215 +				contr_tac 1,
1.216 +				UNTIL_SOLVED (CHANGED(asm_simp_tac
1.217 +				              (HOLCF_ss addsimps dis_apps) 1))]) cons;
1.218 +val dis_rews = dis_stricts @ dis_defins @ dis_apps;
1.219 +
1.220 +val con_stricts = flat(map (fn (con,args) => map (fn vn =>
1.221 +			pg (axs_con_def)
1.222 +			   (mk_trp(con_app2 con (fn arg => if vname arg = vn
1.223 +					then UU else %# arg) args === UU))[
1.224 +				asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
1.225 +			) (nonlazy args)) cons);
1.226 +val con_defins = map (fn (con,args) => pg []
1.227 +			(lift_defined % (nonlazy args,
1.228 +				mk_trp(defined(con_app con args)))) ([
1.229 +				rtac swap3 1] @ (if is_one_con_one_arg (K true) cons
1.230 +				then [
1.231 +				  if is_lazy (hd args) then rtac defined_up 2
1.232 +						       else atac 2,
1.233 +				  rtac abs_defin' 1,
1.234 +				  asm_full_simp_tac (HOLCF_ss addsimps axs_con_def) 1]
1.235 +				else [
1.236 +				  eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
1.237 +				  asm_simp_tac (HOLCF_ss addsimps dis_rews) 1])))cons;
1.238 +val con_rews = con_stricts @ con_defins;
1.239 +
1.240 +val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%sel))) [
1.241 +				simp_tac (HOLCF_ss addsimps when_rews) 1];
1.242 +in flat(map (fn (_,args) => map (fn arg => one_sel (sel_of arg)) args) cons) end;
1.243 +val sel_apps = let fun one_sel c n sel = map (fn (con,args) =>
1.244 +		let val nlas = nonlazy args;
1.245 +		    val vns  = map vname args;
1.246 +		in pg axs_sel_def (lift_defined %
1.247 +		   (filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
1.248 +   mk_trp((%%sel)`(con_app con args) === (if con=c then %(nth_elem(n,vns)) else UU))))
1.249 +			    ( (if con=c then []
1.250 +			       else map(case_UU_tac(when_rews@con_stricts)1) nlas)
1.251 +			     @(if con=c andalso ((nth_elem(n,vns)) mem nlas)
1.252 +					 then[case_UU_tac (when_rews @ con_stricts) 1
1.253 +							  (nth_elem(n,vns))] else [])
1.254 +			     @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
1.255 +in flat(map  (fn (c,args) =>
1.256 +	flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
1.257 +val sel_defins = if length cons = 1 then map (fn arg => pg [] (defined(%x_name) ==>
1.258 +			defined(%%(sel_of arg)`%x_name)) [
1.259 +				rtac cases 1,
1.260 +				contr_tac 1,
1.261 +				UNTIL_SOLVED (CHANGED(asm_simp_tac
1.262 +				              (HOLCF_ss addsimps sel_apps) 1))])
1.263 +		 (filter_out is_lazy (snd(hd cons))) else [];
1.264 +val sel_rews = sel_stricts @ sel_defins @ sel_apps;
1.265 +
1.266 +val distincts_le = let
1.267 +    fun dist (con1, args1) (con2, args2) = pg []
1.268 +	      (lift_defined % ((nonlazy args1),
1.269 +			     (mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
1.270 +			rtac swap3 1,
1.271 +			eres_inst_tac [("fo5",dis_name con1)] monofun_cfun_arg 1]
1.272 +		      @ map (case_UU_tac (con_stricts @ dis_rews) 1) (nonlazy args2)
1.273 +		      @[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
1.274 +    fun distinct (con1,args1) (con2,args2) =
1.275 +	let val arg1 = (con1, args1);
1.276 +	    val arg2 = (con2, (map (fn (arg,vn) => upd_vname (K vn) arg)
1.277 +			      (args2~~variantlist(map vname args2,map vname args1))));
1.278 +	in [dist arg1 arg2, dist arg2 arg1] end;
1.279 +    fun distincts []      = []
1.280 +    |   distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
1.281 +in distincts cons end;
1.282 +val dists_le = flat (flat distincts_le);
1.283 +val dists_eq = let
1.284 +    fun distinct (_,args1) ((_,args2),leqs) = let
1.285 +	val (le1,le2) = (hd leqs, hd(tl leqs));
1.286 +	val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
1.287 +	if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
1.288 +	if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
1.289 +					[eq1, eq2] end;
1.290 +    fun distincts []      = []
1.291 +    |   distincts ((c,leqs)::cs) = flat(map (distinct c) ((map fst cs)~~leqs)) @
1.292 +				   distincts cs;
1.293 +    in distincts (cons~~distincts_le) end;
1.294 +
1.295 +local
1.296 +  fun pgterm rel con args = let
1.297 +		fun append s = upd_vname(fn v => v^s);
1.298 +		val (largs,rargs) = (args, map (append "'") args);
1.299 +		in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
1.300 +		      lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
1.301 +			    mk_trp (foldr' mk_conj
1.302 +				(map rel (map %# largs ~~ map %# rargs)))))) end;
1.303 +  val cons' = filter (fn (_,args) => args<>[]) cons;
1.304 +in
1.305 +val inverts = map (fn (con,args) =>
1.306 +		pgterm (op <<) con args (flat(map (fn arg => [
1.307 +				TRY(rtac conjI 1),
1.308 +				dres_inst_tac [("fo5",sel_of arg)] monofun_cfun_arg 1,
1.309 +				asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
1.310 +			     			      ) args))) cons';
1.311 +val injects = map (fn ((con,args),inv_thm) =>
1.312 +			   pgterm (op ===) con args [
1.313 +				etac (antisym_less_inverse RS conjE) 1,
1.314 +				dtac inv_thm 1, REPEAT(atac 1),
1.315 +				dtac inv_thm 1, REPEAT(atac 1),
1.316 +				TRY(safe_tac HOL_cs),
1.317 +				REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
1.318 +		  (cons'~~inverts);
1.319 +end;
1.320 +
1.321 +(* ----- theorems concerning one induction step ----------------------------------- *)
1.322 +
1.323 +val copy_strict = pg [ax_copy_def] ((if is_one_con_one_arg (K true) cons then fn t =>
1.324 +	 mk_trp(strict(cproj (%"f") eqs (rec_of (hd(snd(hd cons)))))) ===> t
1.325 +	else Id) (mk_trp(strict(dc_copy`%"f")))) [
1.327 +							cfst_strict,csnd_strict]) 1];
1.328 +val copy_apps = map (fn (con,args) => pg (ax_copy_def::axs_con_def)
1.329 +		    (lift_defined %# (filter is_nonlazy_rec args,
1.330 +			mk_trp(dc_copy`%"f"`(con_app con args) ===
1.331 +			   (con_app2 con (app_rec_arg (cproj (%"f") eqs)) args))))
1.332 +				 (map (case_UU_tac [ax_abs_iso] 1 o vname)
1.333 +				   (filter(fn a=>not(is_rec a orelse is_lazy a))args)@
1.334 +				 [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1])
1.335 +		)cons;
1.336 +val copy_stricts = map(fn(con,args)=>pg[](mk_trp(dc_copy`UU`(con_app con args) ===UU))
1.337 +	     (let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
1.338 +			 in map (case_UU_tac rews 1) (nonlazy args) @ [
1.339 +			     asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
1.340 +		   (filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
1.341 +val copy_rews = copy_strict::copy_apps @ copy_stricts;
1.342 +
1.343 +in     (iso_rews, exhaust, cases, when_rews,
1.344 +	con_rews, sel_rews, dis_rews, dists_eq, dists_le, inverts, injects,
1.345 +	copy_rews)
1.346 +end; (* let *)
1.347 +
1.348 +
1.349 +fun comp_theorems thy (comp_dname, eqs: eq list, casess, con_rews, copy_rews) =
1.350 +let
1.351 +
1.352 +val dummy = writeln ("Proving induction properties of domain "^comp_dname^"...");
1.353 +val pg = pg' thy;
1.354 +
1.355 +val dnames = map (fst o fst) eqs;
1.356 +val conss  = map  snd        eqs;
1.357 +
1.358 +(* ----- getting the composite axiom and definitions ------------------------------ *)
1.359 +
1.360 +local val ga = get_axiom thy in
1.361 +val axs_reach      = map (fn dn => ga (dn ^  "_reach"   )) dnames;
1.362 +val axs_take_def   = map (fn dn => ga (dn ^  "_take_def")) dnames;
1.363 +val axs_finite_def = map (fn dn => ga (dn ^"_finite_def")) dnames;
1.364 +val ax_copy2_def   = ga (comp_dname^ "_copy_def");
1.365 +val ax_bisim_def   = ga (comp_dname^"_bisim_def");
1.366 +end; (* local *)
1.367 +
1.368 +(* ----- theorems concerning finiteness and induction ----------------------------- *)
1.369 +
1.370 +fun dc_take dn = %%(dn^"_take");
1.371 +val x_name = idx_name dnames "x";
1.372 +val P_name = idx_name dnames "P";
1.373 +
1.374 +local
1.375 +  val iterate_ss = simpset_of "Fix";
1.376 +  val iterate_Cprod_strict_ss = iterate_ss addsimps [cfst_strict, csnd_strict];
1.377 +  val iterate_Cprod_ss = iterate_ss addsimps [cfst2,csnd2,csplit2];
1.378 +  val copy_con_rews  = copy_rews @ con_rews;
1.379 +  val copy_take_defs = (if length dnames=1 then [] else [ax_copy2_def]) @axs_take_def;
1.380 +  val take_stricts = pg copy_take_defs (mk_trp(foldr' mk_conj (map (fn ((dn,args),_)=>
1.381 +		  (dc_take dn \$ %"n")`UU === mk_constrain(Type(dn,args),UU)) eqs)))([
1.382 +				nat_ind_tac "n" 1,
1.383 +				simp_tac iterate_ss 1,
1.384 +				simp_tac iterate_Cprod_strict_ss 1,
1.385 +				asm_simp_tac iterate_Cprod_ss 1,
1.386 +				TRY(safe_tac HOL_cs)] @
1.387 +			map(K(asm_simp_tac (HOL_ss addsimps copy_rews)1))dnames);
1.388 +  val take_stricts' = rewrite_rule copy_take_defs take_stricts;
1.389 +  val take_0s = mapn (fn n => fn dn => pg axs_take_def(mk_trp((dc_take dn \$ %%"0")
1.390 +								`%x_name n === UU))[
1.391 +				simp_tac iterate_Cprod_strict_ss 1]) 1 dnames;
1.392 +  val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj
1.393 +	    (flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all
1.394 +		(map vname args,(dc_take dn \$ (%%"Suc" \$ %"n"))`(con_app con args) ===
1.395 +  		 con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))\$ %"n"))
1.396 +			      args)) cons) eqs)))) ([
1.397 +				nat_ind_tac "n" 1,
1.398 +				simp_tac iterate_Cprod_strict_ss 1,
1.399 +				simp_tac (HOLCF_ss addsimps copy_con_rews) 1,
1.400 +				TRY(safe_tac HOL_cs)] @
1.401 +			(flat(map (fn ((dn,_),cons) => map (fn (con,args) => EVERY (
1.402 +				asm_full_simp_tac iterate_Cprod_ss 1::
1.403 +				map (case_UU_tac (take_stricts'::copy_con_rews) 1)
1.404 +				    (nonlazy args) @[
1.405 +				asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1])
1.406 +		 	) cons) eqs)));
1.407 +in
1.408 +val take_rews = atomize take_stricts @ take_0s @ atomize take_apps;
1.409 +end; (* local *)
1.410 +
1.411 +val take_lemmas = mapn (fn n => fn(dn,ax_reach) => pg'' thy axs_take_def (mk_All("n",
1.412 +		mk_trp(dc_take dn \$ Bound 0 `%(x_name n) ===
1.413 +		       dc_take dn \$ Bound 0 `%(x_name n^"'")))
1.414 +	   ===> mk_trp(%(x_name n) === %(x_name n^"'"))) (fn prems => [
1.415 +				res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
1.416 +				res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
1.417 +				rtac (fix_def2 RS ssubst) 1,
1.418 +				REPEAT(CHANGED(rtac (contlub_cfun_arg RS ssubst) 1
1.419 +					       THEN chain_tac 1)),
1.420 +				rtac (contlub_cfun_fun RS ssubst) 1,
1.421 +				rtac (contlub_cfun_fun RS ssubst) 2,
1.422 +				rtac lub_equal 3,
1.423 +				chain_tac 1,
1.424 +				rtac allI 1,
1.425 +				resolve_tac prems 1])) 1 (dnames~~axs_reach);
1.426 +
1.427 +local
1.428 +  fun one_con p (con,args) = foldr mk_All (map vname args,
1.429 +	lift_defined (bound_arg (map vname args)) (nonlazy args,
1.430 +	lift (fn arg => %(P_name (1+rec_of arg)) \$ bound_arg args arg)
1.431 +	     (filter is_rec args,mk_trp(%p \$ con_app2 con (bound_arg args) args))));
1.432 +  fun one_eq ((p,cons),concl) = (mk_trp(%p \$ UU) ===>
1.433 +			   foldr (op ===>) (map (one_con p) cons,concl));
1.434 +  fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x)) 1 conss,
1.435 +	mk_trp(foldr' mk_conj (mapn (fn n => concf (P_name n,x_name n)) 1 dnames)));
1.436 +  val take_ss = HOL_ss addsimps take_rews;
1.437 +  fun ind_tacs tacsf thms1 thms2 prems = TRY(safe_tac HOL_cs)::
1.438 +				flat (mapn (fn n => fn (thm1,thm2) =>
1.439 +				  tacsf (n,prems) (thm1,thm2) @
1.440 +				  flat (map (fn cons =>
1.441 +				    (resolve_tac prems 1 ::
1.442 +				     flat (map (fn (_,args) =>
1.443 +				       resolve_tac prems 1::
1.444 +				       map (K(atac 1)) (nonlazy args) @
1.445 +				       map (K(atac 1)) (filter is_rec args))
1.446 +				     cons)))
1.447 +				   conss))
1.448 +				0 (thms1~~thms2));
1.449 +  local
1.450 +    fun all_rec_to ns lazy_rec (n,cons) = forall (exists (fn arg =>
1.451 +		  is_rec arg andalso not(rec_of arg mem ns) andalso
1.452 +		  ((rec_of arg =  n andalso not(lazy_rec orelse is_lazy arg)) orelse
1.453 +		    rec_of arg <> n andalso all_rec_to (rec_of arg::ns)
1.454 +		      (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
1.455 +		  ) o snd) cons;
1.456 +    fun warn (n,cons) = if all_rec_to [] false (n,cons) then (writeln
1.457 +			   ("WARNING: domain "^nth_elem(n,dnames)^" is empty!"); true)
1.458 +			else false;
1.459 +    fun lazy_rec_to ns lazy_rec (n,cons) = exists (exists (fn arg =>
1.460 +		  is_rec arg andalso not(rec_of arg mem ns) andalso
1.461 +		  ((rec_of arg =  n andalso (lazy_rec orelse is_lazy arg)) orelse
1.462 +		    rec_of arg <> n andalso lazy_rec_to (rec_of arg::ns)
1.463 +		     (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
1.464 +		 ) o snd) cons;
1.465 +  in val is_emptys = map warn (mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs);
1.466 +     val is_finite = forall (not o lazy_rec_to [] false)
1.467 +			    (mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs)
1.468 +  end;
1.469 +in
1.470 +val finite_ind = pg'' thy [] (ind_term (fn (P,x) => fn dn =>
1.471 +			  mk_all(x,%P \$ (dc_take dn \$ %"n" `Bound 0)))) (fn prems=> [
1.472 +				nat_ind_tac "n" 1,
1.473 +				simp_tac (take_ss addsimps prems) 1,
1.474 +				TRY(safe_tac HOL_cs)]
1.475 +				@ flat(mapn (fn n => fn (cons,cases) => [
1.476 +				 res_inst_tac [("x",x_name n)] cases 1,
1.477 +				 asm_simp_tac (take_ss addsimps prems) 1]
1.478 +				 @ flat(map (fn (con,args) =>
1.479 +				  asm_simp_tac take_ss 1 ::
1.480 +				  map (fn arg =>
1.481 +				   case_UU_tac (prems@con_rews) 1 (
1.482 +				   nth_elem(rec_of arg,dnames)^"_take n1`"^vname arg))
1.483 +				  (filter is_nonlazy_rec args) @ [
1.484 +				  resolve_tac prems 1] @
1.485 +				  map (K (atac 1))      (nonlazy args) @
1.486 +				  map (K (etac spec 1)) (filter is_rec args))
1.487 +				 cons))
1.488 +				1 (conss~~casess)));
1.489 +
1.490 +val (finites,ind) = if is_finite then
1.491 +let
1.492 +  fun take_enough dn = mk_ex ("n",dc_take dn \$ Bound 0 ` %"x" === %"x");
1.493 +  val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%"x")) ===>
1.494 +	mk_trp(mk_disj(mk_all("n",dc_take dn \$ Bound 0 ` %"x" === UU),
1.495 +	take_enough dn)) ===> mk_trp(take_enough dn)) [
1.496 +				etac disjE 1,
1.497 +				etac notE 1,
1.498 +				resolve_tac take_lemmas 1,
1.499 +				asm_simp_tac take_ss 1,
1.500 +				atac 1]) dnames;
1.501 +  val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn
1.502 +	(fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
1.503 +	 mk_disj(dc_take dn \$ Bound 1 ` Bound 0 === UU,
1.504 +		 dc_take dn \$ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
1.505 +				rtac allI 1,
1.506 +				nat_ind_tac "n" 1,
1.507 +				simp_tac take_ss 1,
1.509 +				flat(mapn (fn n => fn (cons,cases) => [
1.510 +				  simp_tac take_ss 1,
1.511 +				  rtac allI 1,
1.512 +				  res_inst_tac [("x",x_name n)] cases 1,
1.513 +				  asm_simp_tac take_ss 1] @
1.514 +				  flat(map (fn (con,args) =>
1.515 +				    asm_simp_tac take_ss 1 ::
1.516 +				    flat(map (fn arg => [
1.517 +				      eres_inst_tac [("x",vname arg)] all_dupE 1,
1.518 +				      etac disjE 1,
1.519 +				      asm_simp_tac (HOL_ss addsimps con_rews) 1,
1.520 +				      asm_simp_tac take_ss 1])
1.521 +				    (filter is_nonlazy_rec args)))
1.522 +				  cons))
1.523 +				1 (conss~~casess))) handle ERROR => raise ERROR;
1.524 +  val all_finite=map (fn(dn,l1b)=>pg axs_finite_def (mk_trp(%%(dn^"_finite") \$ %"x"))[
1.525 +				case_UU_tac take_rews 1 "x",
1.526 +				eresolve_tac finite_lemmas1a 1,
1.527 +				step_tac HOL_cs 1,
1.528 +				step_tac HOL_cs 1,
1.529 +				cut_facts_tac [l1b] 1,
1.530 +				fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
1.531 +in
1.532 +(all_finite,
1.533 + pg'' thy [] (ind_term (fn (P,x) => fn dn => %P \$ %x))
1.534 +			       (ind_tacs (fn _ => fn (all_fin,finite_ind) => [
1.535 +				rtac (rewrite_rule axs_finite_def all_fin RS exE) 1,
1.536 +				etac subst 1,
1.537 +				rtac finite_ind 1]) all_finite (atomize finite_ind))
1.538 +) end (* let *) else
1.539 +(mapn (fn n => fn dn => read_instantiate_sg (sign_of thy)
1.540 +	  	    [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
1.541 + pg'' thy [] (foldr (op ===>) (mapn (fn n =>K(mk_trp(%%"adm" \$ %(P_name n))))1
1.542 +				       dnames,ind_term (fn(P,x)=>fn dn=> %P \$ %x)))
1.543 +			       (ind_tacs (fn (n,prems) => fn (ax_reach,finite_ind) =>[
1.544 +				rtac (ax_reach RS subst) 1,
1.545 +				res_inst_tac [("x",x_name n)] spec 1,
1.546 +				rtac wfix_ind 1,
1.550 +				cont_tacR 1,
1.551 +				resolve_tac prems 1,
1.552 +				strip_tac 1,
1.553 +			        rtac(rewrite_rule axs_take_def finite_ind) 1])
1.554 +				 axs_reach (atomize finite_ind))
1.555 +)
1.556 +end; (* local *)
1.557 +
1.558 +local
1.559 +  val xs = mapn (fn n => K (x_name n)) 1 dnames;
1.560 +  fun bnd_arg n i = Bound(2*(length dnames - n)-i-1);
1.561 +  val take_ss = HOL_ss addsimps take_rews;
1.562 +  val sproj   = bin_branchr (fn s => "fst("^s^")") (fn s => "snd("^s^")");
1.563 +  val coind_lemma = pg [ax_bisim_def] (mk_trp(mk_imp(%%(comp_dname^"_bisim") \$ %"R",
1.564 +		foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
1.565 +		  foldr mk_imp (mapn (fn n => K(proj (%"R") dnames n \$
1.566 +				      bnd_arg n 0 \$ bnd_arg n 1)) 0 dnames,
1.567 +		    foldr' mk_conj (mapn (fn n => fn dn =>
1.568 +				(dc_take dn \$ %"n" `bnd_arg n 0 ===
1.569 +				(dc_take dn \$ %"n" `bnd_arg n 1))) 0 dnames)))))) ([
1.570 +				rtac impI 1,
1.571 +				nat_ind_tac "n" 1,
1.572 +				simp_tac take_ss 1,
1.573 +				safe_tac HOL_cs] @
1.574 +				flat(mapn (fn n => fn x => [
1.575 +				  etac allE 1, etac allE 1,
1.576 +				  eres_inst_tac [("P1",sproj "R" dnames n^
1.577 +						  " "^x^" "^x^"'")](mp RS disjE) 1,
1.578 +				  TRY(safe_tac HOL_cs),
1.579 +				  REPEAT(CHANGED(asm_simp_tac take_ss 1))])
1.580 +				0 xs));
1.581 +in
1.582 +val coind = pg [] (mk_trp(%%(comp_dname^"_bisim") \$ %"R") ===>
1.583 +		foldr (op ===>) (mapn (fn n => fn x =>
1.584 +			mk_trp(proj (%"R") dnames n \$ %x \$ %(x^"'"))) 0 xs,
1.585 +			mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
1.586 +				TRY(safe_tac HOL_cs)] @
1.587 +				flat(map (fn take_lemma => [
1.588 +				  rtac take_lemma 1,
1.589 +				  cut_facts_tac [coind_lemma] 1,
1.590 +				  fast_tac HOL_cs 1])
1.591 +				take_lemmas));
1.592 +end; (* local *)
1.593 +
1.594 +
1.595 +in (take_rews, take_lemmas, finites, finite_ind, ind, coind)
1.596 +
1.597 +end; (* let *)
1.598 +end; (* local *)
1.599 +end; (* struct *)
```