src/HOLCF/domain/theorems.ML
 changeset 1637 b8a8ae2e5de1 parent 1512 ce37c64244c0 child 1638 69c094639823
```     1.1 --- a/src/HOLCF/domain/theorems.ML	Wed Apr 03 19:02:04 1996 +0200
1.2 +++ b/src/HOLCF/domain/theorems.ML	Wed Apr 03 19:27:14 1996 +0200
1.3 @@ -1,5 +1,4 @@
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 @@ -12,10 +11,12 @@
1.10     Updated: 05-Sep-95 simultaneous domain equations (main part)
1.11     Updated: 11-Sep-95 simultaneous domain equations (coding finished)
1.12     Updated: 13-Sep-95 simultaneous domain equations (debugging)
1.13 -   Copyright 1995 TU Muenchen
1.14 +   Updated: 26-Oct-95 debugging and enhancement of proofs for take_apps, ind
1.15 +   Updated: 16-Feb-96 bug concerning  domain Triv = triv  fixed
1.16 +   Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
1.17 +   Copyright 1995, 1996 TU Muenchen
1.18  *)
1.19
1.20 -
1.21  structure Domain_Theorems = struct
1.22
1.23  local
1.24 @@ -25,58 +26,58 @@
1.25  infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<;
1.26  infix 9 `   ; infix 9 `% ; infix 9 `%%; infixr 9 oo;
1.27
1.28 -(* ----- general proof facilities ------------------------------------------------- *)
1.29 -
1.30 -fun inferT sg pre_tm = #2(Sign.infer_types sg (K None)(K None)[]true([pre_tm],propT));
1.31 +(* ----- general proof facilities ------------------------------------------- *)
1.32
1.33 -(*
1.34 -infix 0 y;
1.35 -val b=0;
1.36 -fun _ y t = by t;
1.37 -fun  g  defs t = let val sg = sign_of thy;
1.38 -                     val ct = Thm.cterm_of sg (inferT sg t);
1.39 -                 in goalw_cterm defs ct end;
1.40 -*)
1.41 +fun inferT sg pre_tm = #2 (Sign.infer_types sg (K None) (K None) [] true
1.42 +			   ([pre_tm],propT));
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 +		          val ct = Thm.cterm_of sg (inferT sg t);
1.48 +		      in prove_goalw_cterm defs ct end;
1.49  fun pg'  thy defs t tacsf=pg'' thy defs t (fn []   => tacsf
1.50 -                                            | prems=> (cut_facts_tac prems 1)::tacsf);
1.51 +				| prems=> (cut_facts_tac prems 1)::tacsf);
1.52
1.53  fun REPEAT_DETERM_UNTIL p tac =
1.54  let fun drep st = if p st then Sequence.single st
1.55 -                          else (case Sequence.pull(tac st) of
1.56 -                                  None        => Sequence.null
1.57 -                                | Some(st',_) => drep st')
1.58 -in drep end;
1.59 +			  else (case Sequence.pull(tapply(tac,st)) of
1.60 +		                  None        => Sequence.null
1.61 +				| Some(st',_) => drep st')
1.62 +in Tactic drep end;
1.63  val UNTIL_SOLVED = REPEAT_DETERM_UNTIL (has_fewer_prems 1);
1.64
1.65 -local val trueI2 = prove_goal HOL.thy "f~=x ==> True" (fn prems => [rtac TrueI 1]) in
1.66 +local val trueI2 = prove_goal HOL.thy"f~=x ==> True"(fn prems=>[rtac TrueI 1])in
1.67  val kill_neq_tac = dtac trueI2 end;
1.68 -fun case_UU_tac rews i v =      res_inst_tac [("Q",v^"=UU")] classical2 i THEN
1.69 -                                asm_simp_tac (HOLCF_ss addsimps rews) i;
1.70 +fun case_UU_tac rews i v =	res_inst_tac [("Q",v^"=UU")] classical2 i THEN
1.71 +				asm_simp_tac (HOLCF_ss addsimps rews) i;
1.72
1.73  val chain_tac = REPEAT_DETERM o resolve_tac
1.74 -                [is_chain_iterate, ch2ch_fappR, ch2ch_fappL];
1.75 +		[is_chain_iterate, ch2ch_fappR, ch2ch_fappL];
1.76 +
1.77 +(* ----- general proofs ----------------------------------------------------- *)
1.78
1.79 -(* ----- general proofs ----------------------------------------------------------- *)
1.80 +val quant_ss = HOL_ss addsimps (map (fn s => prove_goal HOL.thy s (fn _ =>[
1.81 +		fast_tac HOL_cs 1]))["(Âx. P x À Q)=((Âx. P x) À Q)",
1.82 +			    	     "(Âx. P À Q x) = (P À (Âx. Q x))"]);
1.83 +
1.84 +val all2E = prove_goal HOL.thy "Ë Âx y . P x y; P x y êë R Ì êë R" (fn prems =>[
1.85 +				resolve_tac prems 1,
1.86 +				cut_facts_tac prems 1,
1.87 +				fast_tac HOL_cs 1]);
1.88
1.89  val swap3 = prove_goal HOL.thy "[| Q ==> P; ~P |] ==> ~Q" (fn prems => [
1.90                                  cut_facts_tac prems 1,
1.91                                  etac swap 1,
1.92                                  dtac notnotD 1,
1.93 -                                etac (hd prems) 1]);
1.94 +				etac (hd prems) 1]);
1.95
1.96 -val dist_eqI = prove_goal Porder0.thy "~ x << y ==> x ~= y" (fn prems => [
1.97 -                                cut_facts_tac prems 1,
1.98 -                                etac swap 1,
1.99 -                                dtac notnotD 1,
1.100 -                                asm_simp_tac HOLCF_ss 1]);
1.101 -val cfst_strict  = prove_goal Cprod3.thy "cfst`UU = UU" (fn _ => [
1.102 -                                (simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
1.103 -val csnd_strict  = prove_goal Cprod3.thy "csnd`UU = UU" (fn _ => [
1.104 -                        (simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
1.105 +val dist_eqI = prove_goal Porder.thy "¿ x Ý y êë x Û y" (fn prems => [
1.106 +                                rtac swap3 1,
1.107 +				etac (antisym_less_inverse RS conjunct1) 1,
1.108 +				resolve_tac prems 1]);
1.109 +val cfst_strict  = prove_goal Cprod3.thy "cfst`Ø = Ø" (fn _ => [
1.110 +			(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
1.111 +val csnd_strict  = prove_goal Cprod3.thy "csnd`Ø = Ø" (fn _ => [
1.112 +			(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
1.113
1.114  in
1.115
1.116 @@ -86,8 +87,17 @@
1.117
1.118  val dummy = writeln ("Proving isomorphism properties of domain "^dname^"...");
1.119  val pg = pg' thy;
1.120 +(*
1.121 +infixr 0 y;
1.122 +val b = 0;
1.123 +fun _ y t = by t;
1.124 +fun  g  defs t = let val sg = sign_of thy;
1.125 +		     val ct = Thm.cterm_of sg (inferT sg t);
1.126 +		 in goalw_cterm defs ct end;
1.127 +*)
1.128
1.129 -(* ----- getting the axioms and definitions --------------------------------------- *)
1.130 +
1.131 +(* ----- getting the axioms and definitions --------------------------------- *)
1.132
1.133  local val ga = get_axiom thy in
1.134  val ax_abs_iso    = ga (dname^"_abs_iso"   );
1.135 @@ -96,11 +106,11 @@
1.136  val axs_con_def   = map (fn (con,_) => ga (extern_name con ^"_def")) cons;
1.137  val axs_dis_def   = map (fn (con,_) => ga (   dis_name con ^"_def")) cons;
1.138  val axs_sel_def   = flat(map (fn (_,args) =>
1.139 -                    map (fn     arg => ga (sel_of arg      ^"_def")) args) cons);
1.140 +		    map (fn     arg => ga (sel_of arg      ^"_def")) args)cons);
1.141  val ax_copy_def   = ga (dname^"_copy_def"  );
1.142  end; (* local *)
1.143
1.144 -(* ----- theorems concerning the isomorphism -------------------------------------- *)
1.145 +(* ----- theorems concerning the isomorphism -------------------------------- *)
1.146
1.147  val dc_abs  = %%(dname^"_abs");
1.148  val dc_rep  = %%(dname^"_rep");
1.149 @@ -108,251 +118,243 @@
1.150  val x_name = "x";
1.151
1.152  val (rep_strict, abs_strict) = let
1.153 -               val r = ax_rep_iso RS (ax_abs_iso RS (allI  RSN(2,allI RS iso_strict)))
1.154 -               in (r RS conjunct1, r RS conjunct2) end;
1.155 +	 val r = ax_rep_iso RS (ax_abs_iso RS (allI  RSN(2,allI RS iso_strict)))
1.156 +	       in (r RS conjunct1, r RS conjunct2) end;
1.157  val abs_defin' = pg [] ((dc_abs`%x_name === UU) ==> (%x_name === UU)) [
1.158 -                                res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
1.159 -                                etac ssubst 1,
1.160 -                                rtac rep_strict 1];
1.161 +			   res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
1.162 +				etac ssubst 1, rtac rep_strict 1];
1.163  val rep_defin' = pg [] ((dc_rep`%x_name === UU) ==> (%x_name === UU)) [
1.164 -                                res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
1.165 -                                etac ssubst 1,
1.166 -                                rtac abs_strict 1];
1.167 +			   res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
1.168 +				etac ssubst 1, rtac abs_strict 1];
1.169  val iso_rews = [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
1.170
1.171  local
1.172  val iso_swap = pg [] (dc_rep`%"x" === %"y" ==> %"x" === dc_abs`%"y") [
1.173 -                                dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
1.174 -                                etac (ax_rep_iso RS subst) 1];
1.175 +			    dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
1.176 +			    etac (ax_rep_iso RS subst) 1];
1.177  fun exh foldr1 cn quant foldr2 var = let
1.178    fun one_con (con,args) = let val vns = map vname args in
1.179      foldr quant (vns, foldr2 ((%x_name === con_app2 con (var vns) vns)::
1.180 -                              map (defined o (var vns)) (nonlazy args))) end
1.181 +			      map (defined o (var vns)) (nonlazy args))) end
1.182    in foldr1 ((cn(%x_name===UU))::map one_con cons) end;
1.183  in
1.184  val cases = let
1.185 -            fun common_tac thm = rtac thm 1 THEN contr_tac 1;
1.186 -            fun unit_tac true = common_tac liftE1
1.187 -            |   unit_tac _    = all_tac;
1.188 -            fun prod_tac []          = common_tac oneE
1.189 -            |   prod_tac [arg]       = unit_tac (is_lazy arg)
1.190 -            |   prod_tac (arg::args) =
1.191 -                                common_tac sprodE THEN
1.192 -                                kill_neq_tac 1 THEN
1.193 -                                unit_tac (is_lazy arg) THEN
1.194 -                                prod_tac args;
1.195 -            fun sum_one_tac p = SELECT_GOAL(EVERY[
1.196 -                                rtac p 1,
1.197 -                                rewrite_goals_tac axs_con_def,
1.198 -                                dtac iso_swap 1,
1.199 -                                simp_tac HOLCF_ss 1,
1.200 -                                UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
1.201 -            fun sum_tac [(_,args)]       [p]        =
1.202 -                                prod_tac args THEN sum_one_tac p
1.203 -            |   sum_tac ((_,args)::cons') (p::prems) = DETERM(
1.204 -                                common_tac ssumE THEN
1.205 -                                kill_neq_tac 1 THEN kill_neq_tac 2 THEN
1.206 -                                prod_tac args THEN sum_one_tac p) THEN
1.207 -                                sum_tac cons' prems
1.208 -            |   sum_tac _ _ = Imposs "theorems:sum_tac";
1.209 -          in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%"P")))
1.210 -                              (fn T => T ==> %"P") mk_All
1.211 -                              (fn l => foldr (op ===>) (map mk_trp l,mk_trp(%"P")))
1.212 -                              bound_arg)
1.213 -                             (fn prems => [
1.214 -                                cut_facts_tac [excluded_middle] 1,
1.215 -                                etac disjE 1,
1.216 -                                rtac (hd prems) 2,
1.217 -                                etac rep_defin' 2,
1.218 -                                if is_one_con_one_arg (not o is_lazy) cons
1.219 -                                then rtac (hd (tl prems)) 1 THEN atac 2 THEN
1.220 -                                     rewrite_goals_tac axs_con_def THEN
1.221 -                                     simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
1.222 -                                else sum_tac cons (tl prems)])end;
1.223 -val exhaust = pg [] (mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %))) [
1.224 -                                rtac cases 1,
1.225 -                                UNTIL_SOLVED(fast_tac HOL_cs 1)];
1.226 +	    fun common_tac thm = rtac thm 1 THEN contr_tac 1;
1.227 +	    fun unit_tac true = common_tac liftE1
1.228 +	    |   unit_tac _    = all_tac;
1.229 +	    fun prod_tac []          = common_tac oneE
1.230 +	    |   prod_tac [arg]       = unit_tac (is_lazy arg)
1.231 +	    |   prod_tac (arg::args) =
1.232 +				common_tac sprodE THEN
1.233 +				kill_neq_tac 1 THEN
1.234 +				unit_tac (is_lazy arg) THEN
1.235 +				prod_tac args;
1.236 +	    fun sum_rest_tac p = SELECT_GOAL(EVERY[
1.237 +				rtac p 1,
1.238 +				rewrite_goals_tac axs_con_def,
1.239 +				dtac iso_swap 1,
1.240 +				simp_tac HOLCF_ss 1,
1.241 +				UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
1.242 +	    fun sum_tac [(_,args)]       [p]        =
1.243 +				prod_tac args THEN sum_rest_tac p
1.244 +	    |   sum_tac ((_,args)::cons') (p::prems) = DETERM(
1.245 +				common_tac ssumE THEN
1.246 +				kill_neq_tac 1 THEN kill_neq_tac 2 THEN
1.247 +				prod_tac args THEN sum_rest_tac p) THEN
1.248 +				sum_tac cons' prems
1.249 +	    |   sum_tac _ _ = Imposs "theorems:sum_tac";
1.250 +	  in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%"P")))
1.251 +			      (fn T => T ==> %"P") mk_All
1.252 +			      (fn l => foldr (op ===>) (map mk_trp l,
1.253 +							    mk_trp(%"P")))
1.254 +			      bound_arg)
1.255 +			     (fn prems => [
1.256 +				cut_facts_tac [excluded_middle] 1,
1.257 +				etac disjE 1,
1.258 +				rtac (hd prems) 2,
1.259 +				etac rep_defin' 2,
1.260 +				if length cons = 1 andalso
1.261 +				   length (snd(hd cons)) = 1 andalso
1.262 +				   not(is_lazy(hd(snd(hd cons))))
1.263 +				then rtac (hd (tl prems)) 1 THEN atac 2 THEN
1.264 +				     rewrite_goals_tac axs_con_def THEN
1.265 +				     simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
1.266 +				else sum_tac cons (tl prems)])end;
1.267 +val exhaust= pg[](mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %)))[
1.268 +				rtac cases 1,
1.269 +				UNTIL_SOLVED(fast_tac HOL_cs 1)];
1.270  end;
1.271
1.272  local
1.273 -val when_app = foldl (op `) (%%(dname^"_when"), map % (when_funs cons));
1.274 -val when_appl = pg [ax_when_def] (mk_trp(when_app`%x_name===when_body cons
1.275 -                (fn (_,n) => %(nth_elem(n-1,when_funs cons)))`(dc_rep`%x_name))) [
1.276 -                                simp_tac HOLCF_ss 1];
1.277 +  val when_app  = foldl (op `) (%%(dname^"_when"), map % (when_funs cons));
1.278 +  val when_appl = pg [ax_when_def] (mk_trp(when_app`%x_name===when_body cons
1.279 +		(fn (_,n)=> %(nth_elem(n-1,when_funs cons)))`(dc_rep`%x_name)))[
1.280 +				simp_tac HOLCF_ss 1];
1.281  in
1.282 -val when_strict = pg [] ((if is_one_con_one_arg (K true) cons
1.283 -        then fn t => mk_trp(strict(%"f")) ===> t else Id)(mk_trp(strict when_app))) [
1.284 -                                simp_tac(HOLCF_ss addsimps [when_appl,rep_strict]) 1];
1.285 -val when_apps = let fun one_when n (con,args) = pg axs_con_def
1.286 -                (lift_defined % (nonlazy args, mk_trp(when_app`(con_app con args) ===
1.287 -                 mk_cfapp(%(nth_elem(n,when_funs cons)),map %# args))))[
1.288 -                        asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
1.289 -                in mapn one_when 0 cons end;
1.290 +val when_strict = pg [] (mk_trp(strict when_app)) [
1.291 +			simp_tac(HOLCF_ss addsimps [when_appl,rep_strict]) 1];
1.292 +val when_apps = let fun one_when n (con,args) = pg axs_con_def (lift_defined %
1.293 +   (nonlazy args, mk_trp(when_app`(con_app con args) ===
1.294 +	 mk_cfapp(%(nth_elem(n,when_funs cons)),map %# args))))[
1.295 +		asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
1.296 +	in mapn one_when 0 cons end;
1.297  end;
1.298  val when_rews = when_strict::when_apps;
1.299
1.300 -(* ----- theorems concerning the constructors, discriminators and selectors ------- *)
1.301 +(* ----- theorems concerning the constructors, discriminators and selectors - *)
1.302
1.303 -val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
1.304 -                        (if is_one_con_one_arg (K true) cons then mk_not else Id)
1.305 -                         (strict(%%(dis_name con))))) [
1.306 -                simp_tac (HOLCF_ss addsimps (if is_one_con_one_arg (K true) cons
1.307 -                                        then [ax_when_def] else when_rews)) 1]) cons;
1.308 -val dis_apps = let fun one_dis c (con,args)= pg (axs_dis_def)
1.309 -                   (lift_defined % (nonlazy args, (*(if is_one_con_one_arg is_lazy cons
1.310 -                        then curry (lift_defined %#) args else Id)
1.311 -#################*)
1.312 -                        (mk_trp((%%(dis_name c))`(con_app con args) ===
1.313 -                              %%(if con=c then "TT" else "FF"))))) [
1.314 -                                asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
1.315 -        in flat(map (fn (c,_) => map (one_dis c) cons) cons) end;
1.316 -val dis_defins = map (fn (con,args) => pg [] (defined(%x_name)==>
1.317 -                      defined(%%(dis_name con)`%x_name)) [
1.318 -                                rtac cases 1,
1.319 -                                contr_tac 1,
1.320 -                                UNTIL_SOLVED (CHANGED(asm_simp_tac
1.321 -                                              (HOLCF_ss addsimps dis_apps) 1))]) cons;
1.322 -val dis_rews = dis_stricts @ dis_defins @ dis_apps;
1.323 +val dis_rews = let
1.324 +  val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
1.325 +		      	     strict(%%(dis_name con)))) [
1.326 +				simp_tac (HOLCF_ss addsimps when_rews) 1]) cons;
1.327 +  val dis_apps = let fun one_dis c (con,args)= pg axs_dis_def
1.328 +		   (lift_defined % (nonlazy args,
1.329 +			(mk_trp((%%(dis_name c))`(con_app con args) ===
1.330 +			      %%(if con=c then "TT" else "FF"))))) [
1.331 +				asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
1.332 +	in flat(map (fn (c,_) => map (one_dis c) cons) cons) end;
1.333 +  val dis_defins = map (fn (con,args) => pg [] (defined(%x_name) ==>
1.334 +		      defined(%%(dis_name con)`%x_name)) [
1.335 +				rtac cases 1,
1.336 +				contr_tac 1,
1.337 +				UNTIL_SOLVED (CHANGED(asm_simp_tac
1.338 +				        (HOLCF_ss addsimps dis_apps) 1))]) cons;
1.339 +in dis_stricts @ dis_defins @ dis_apps end;
1.340
1.341  val con_stricts = flat(map (fn (con,args) => map (fn vn =>
1.342 -                        pg (axs_con_def)
1.343 -                           (mk_trp(con_app2 con (fn arg => if vname arg = vn
1.344 -                                        then UU else %# arg) args === UU))[
1.345 -                                asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
1.346 -                        ) (nonlazy args)) cons);
1.347 +			pg (axs_con_def)
1.348 +			   (mk_trp(con_app2 con (fn arg => if vname arg = vn
1.349 +					then UU else %# arg) args === UU))[
1.350 +				asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
1.351 +			) (nonlazy args)) cons);
1.352  val con_defins = map (fn (con,args) => pg []
1.353 -                        (lift_defined % (nonlazy args,
1.354 -                                mk_trp(defined(con_app con args)))) ([
1.355 -                                rtac swap3 1] @ (if is_one_con_one_arg (K true) cons
1.356 -                                then [
1.357 -                                  if is_lazy (hd args) then rtac defined_up 2
1.358 -                                                       else atac 2,
1.359 -                                  rtac abs_defin' 1,
1.360 -                                  asm_full_simp_tac (HOLCF_ss addsimps axs_con_def) 1]
1.361 -                                else [
1.362 -                                  eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
1.363 -                                  asm_simp_tac (HOLCF_ss addsimps dis_rews) 1])))cons;
1.364 +			(lift_defined % (nonlazy args,
1.365 +				mk_trp(defined(con_app con args)))) ([
1.366 +			  rtac swap3 1,
1.367 +			  eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
1.368 +			  asm_simp_tac (HOLCF_ss addsimps dis_rews) 1] )) cons;
1.369  val con_rews = con_stricts @ con_defins;
1.370
1.371  val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%sel))) [
1.372 -                                simp_tac (HOLCF_ss addsimps when_rews) 1];
1.373 -in flat(map (fn (_,args) => map (fn arg => one_sel (sel_of arg)) args) cons) end;
1.374 +				simp_tac (HOLCF_ss addsimps when_rews) 1];
1.375 +in flat(map (fn (_,args) =>map (fn arg => one_sel (sel_of arg)) args) cons) end;
1.376  val sel_apps = let fun one_sel c n sel = map (fn (con,args) =>
1.377 -                let val nlas = nonlazy args;
1.378 -                    val vns  = map vname args;
1.379 -                in pg axs_sel_def (lift_defined %
1.380 -                   (filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
1.381 -   mk_trp((%%sel)`(con_app con args) === (if con=c then %(nth_elem(n,vns)) else UU))))
1.382 -                            ( (if con=c then []
1.383 -                               else map(case_UU_tac(when_rews@con_stricts)1) nlas)
1.384 -                             @(if con=c andalso ((nth_elem(n,vns)) mem nlas)
1.385 -                                         then[case_UU_tac (when_rews @ con_stricts) 1
1.386 -                                                          (nth_elem(n,vns))] else [])
1.387 -                             @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
1.388 +		let val nlas = nonlazy args;
1.389 +		    val vns  = map vname args;
1.390 +		in pg axs_sel_def (lift_defined %
1.391 +		   (filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
1.392 +				mk_trp((%%sel)`(con_app con args) ===
1.393 +				(if con=c then %(nth_elem(n,vns)) else UU))))
1.394 +			    ( (if con=c then []
1.395 +		       else map(case_UU_tac(when_rews@con_stricts)1) nlas)
1.396 +		     @(if con=c andalso ((nth_elem(n,vns)) mem nlas)
1.397 +				 then[case_UU_tac (when_rews @ con_stricts) 1
1.398 +						  (nth_elem(n,vns))] else [])
1.399 +		     @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
1.400  in flat(map  (fn (c,args) =>
1.401 -        flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
1.402 -val sel_defins = if length cons = 1 then map (fn arg => pg [] (defined(%x_name) ==>
1.403 -                        defined(%%(sel_of arg)`%x_name)) [
1.404 -                                rtac cases 1,
1.405 -                                contr_tac 1,
1.406 -                                UNTIL_SOLVED (CHANGED(asm_simp_tac
1.407 -                                              (HOLCF_ss addsimps sel_apps) 1))])
1.408 -                 (filter_out is_lazy (snd(hd cons))) else [];
1.409 +     flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
1.410 +val sel_defins = if length cons=1 then map (fn arg => pg [](defined(%x_name)==>
1.411 +			defined(%%(sel_of arg)`%x_name)) [
1.412 +				rtac cases 1,
1.413 +				contr_tac 1,
1.414 +				UNTIL_SOLVED (CHANGED(asm_simp_tac
1.415 +				             (HOLCF_ss addsimps sel_apps) 1))])
1.416 +		 (filter_out is_lazy (snd(hd cons))) else [];
1.417  val sel_rews = sel_stricts @ sel_defins @ sel_apps;
1.418
1.419  val distincts_le = let
1.420      fun dist (con1, args1) (con2, args2) = pg []
1.421 -              (lift_defined % ((nonlazy args1),
1.422 -                             (mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
1.423 -                        rtac swap3 1,
1.424 -                        eres_inst_tac [("fo5",dis_name con1)] monofun_cfun_arg 1]
1.425 -                      @ map (case_UU_tac (con_stricts @ dis_rews) 1) (nonlazy args2)
1.426 -                      @[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
1.427 +	      (lift_defined % ((nonlazy args1),
1.428 +			(mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
1.429 +			rtac swap3 1,
1.430 +			eres_inst_tac[("fo5",dis_name con1)] monofun_cfun_arg 1]
1.431 +		      @map(case_UU_tac (con_stricts @ dis_rews)1)(nonlazy args2)
1.432 +		      @[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
1.433      fun distinct (con1,args1) (con2,args2) =
1.434 -        let val arg1 = (con1, args1);
1.435 -            val arg2 = (con2, (map (fn (arg,vn) => upd_vname (K vn) arg)
1.436 -                              (args2~~variantlist(map vname args2,map vname args1))));
1.437 -        in [dist arg1 arg2, dist arg2 arg1] end;
1.438 +	let val arg1 = (con1, args1);
1.439 +	    val arg2 = (con2, (map (fn (arg,vn) => upd_vname (K vn) arg)
1.440 +			(args2~~variantlist(map vname args2,map vname args1))));
1.441 +	in [dist arg1 arg2, dist arg2 arg1] end;
1.442      fun distincts []      = []
1.443      |   distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
1.444  in distincts cons end;
1.445  val dists_le = flat (flat distincts_le);
1.446  val dists_eq = let
1.447      fun distinct (_,args1) ((_,args2),leqs) = let
1.448 -        val (le1,le2) = (hd leqs, hd(tl leqs));
1.449 -        val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
1.450 -        if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
1.451 -        if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
1.452 -                                        [eq1, eq2] end;
1.453 +	val (le1,le2) = (hd leqs, hd(tl leqs));
1.454 +	val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
1.455 +	if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
1.456 +	if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
1.457 +					[eq1, eq2] end;
1.458      fun distincts []      = []
1.459      |   distincts ((c,leqs)::cs) = flat(map (distinct c) ((map fst cs)~~leqs)) @
1.460 -                                   distincts cs;
1.461 +				   distincts cs;
1.462      in distincts (cons~~distincts_le) end;
1.463
1.464  local
1.465    fun pgterm rel con args = let
1.466 -                fun append s = upd_vname(fn v => v^s);
1.467 -                val (largs,rargs) = (args, map (append "'") args);
1.468 -                in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
1.469 -                      lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
1.470 -                            mk_trp (foldr' mk_conj
1.471 -                                (map rel (map %# largs ~~ map %# rargs)))))) end;
1.472 +		fun append s = upd_vname(fn v => v^s);
1.473 +		val (largs,rargs) = (args, map (append "'") args);
1.474 +		in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
1.475 +		      lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
1.476 +			    mk_trp (foldr' mk_conj
1.477 +				(map rel (map %# largs ~~ map %# rargs)))))) end;
1.478    val cons' = filter (fn (_,args) => args<>[]) cons;
1.479  in
1.480  val inverts = map (fn (con,args) =>
1.481 -                pgterm (op <<) con args (flat(map (fn arg => [
1.482 -                                TRY(rtac conjI 1),
1.483 -                                dres_inst_tac [("fo5",sel_of arg)] monofun_cfun_arg 1,
1.484 -                                asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
1.485 -                                                      ) args))) cons';
1.486 +		pgterm (op <<) con args (flat(map (fn arg => [
1.487 +				TRY(rtac conjI 1),
1.488 +				dres_inst_tac [("fo5",sel_of arg)] monofun_cfun_arg 1,
1.489 +				asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
1.490 +			     			      ) args))) cons';
1.491  val injects = map (fn ((con,args),inv_thm) =>
1.492 -                           pgterm (op ===) con args [
1.493 -                                etac (antisym_less_inverse RS conjE) 1,
1.494 -                                dtac inv_thm 1, REPEAT(atac 1),
1.495 -                                dtac inv_thm 1, REPEAT(atac 1),
1.496 -                                TRY(safe_tac HOL_cs),
1.497 -                                REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
1.498 -                  (cons'~~inverts);
1.499 +			   pgterm (op ===) con args [
1.500 +				etac (antisym_less_inverse RS conjE) 1,
1.501 +				dtac inv_thm 1, REPEAT(atac 1),
1.502 +				dtac inv_thm 1, REPEAT(atac 1),
1.503 +				TRY(safe_tac HOL_cs),
1.504 +				REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
1.505 +		  (cons'~~inverts);
1.506  end;
1.507
1.508 -(* ----- theorems concerning one induction step ----------------------------------- *)
1.509 +(* ----- theorems concerning one induction step ----------------------------- *)
1.510
1.511 -val copy_strict = pg [ax_copy_def] ((if is_one_con_one_arg (K true) cons then fn t =>
1.512 -         mk_trp(strict(cproj (%"f") eqs (rec_of (hd(snd(hd cons)))))) ===> t
1.513 -        else Id) (mk_trp(strict(dc_copy`%"f")))) [
1.514 -                                asm_simp_tac(HOLCF_ss addsimps [abs_strict,rep_strict,
1.515 -                                                        cfst_strict,csnd_strict]) 1];
1.516 -val copy_apps = map (fn (con,args) => pg (ax_copy_def::axs_con_def)
1.517 -                    (lift_defined %# (filter is_nonlazy_rec args,
1.518 -                        mk_trp(dc_copy`%"f"`(con_app con args) ===
1.519 -                           (con_app2 con (app_rec_arg (cproj (%"f") eqs)) args))))
1.520 -                                 (map (case_UU_tac [ax_abs_iso] 1 o vname)
1.521 -                                   (filter(fn a=>not(is_rec a orelse is_lazy a))args)@
1.522 -                                 [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1])
1.523 -                )cons;
1.524 -val copy_stricts = map(fn(con,args)=>pg[](mk_trp(dc_copy`UU`(con_app con args) ===UU))
1.525 -             (let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
1.526 -                         in map (case_UU_tac rews 1) (nonlazy args) @ [
1.527 -                             asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
1.528 -                   (filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
1.529 +val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [
1.530 +		   asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict,
1.531 +						   cfst_strict,csnd_strict]) 1];
1.532 +val copy_apps = map (fn (con,args) => pg [ax_copy_def]
1.533 +		    (lift_defined % (nonlazy_rec args,
1.534 +			mk_trp(dc_copy`%"f"`(con_app con args) ===
1.535 +		(con_app2 con (app_rec_arg (cproj (%"f") (length eqs))) args))))
1.536 +			(map (case_UU_tac (abs_strict::when_strict::con_stricts)
1.537 +				 1 o vname)
1.538 +			 (filter (fn a => not (is_rec a orelse is_lazy a)) args)
1.539 +			@[asm_simp_tac (HOLCF_ss addsimps when_apps) 1,
1.540 +		          simp_tac (HOLCF_ss addsimps axs_con_def) 1]))cons;
1.541 +val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU`
1.542 +					(con_app con args) ===UU))
1.543 +     (let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
1.544 +			 in map (case_UU_tac rews 1) (nonlazy args) @ [
1.545 +			     asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
1.546 +  		        (filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
1.547  val copy_rews = copy_strict::copy_apps @ copy_stricts;
1.548
1.549  in     (iso_rews, exhaust, cases, when_rews,
1.550 -        con_rews, sel_rews, dis_rews, dists_eq, dists_le, inverts, injects,
1.551 -        copy_rews)
1.552 +	con_rews, sel_rews, dis_rews, dists_le, dists_eq, inverts, injects,
1.553 +	copy_rews)
1.554  end; (* let *)
1.555
1.556
1.557  fun comp_theorems thy (comp_dname, eqs: eq list, casess, con_rews, copy_rews) =
1.558  let
1.559
1.560 -val dummy = writeln ("Proving induction properties of domain "^comp_dname^"...");
1.561 +val dummy = writeln("Proving induction properties of domain "^comp_dname^"...");
1.562  val pg = pg' thy;
1.563
1.564  val dnames = map (fst o fst) eqs;
1.565  val conss  = map  snd        eqs;
1.566
1.567 -(* ----- getting the composite axiom and definitions ------------------------------ *)
1.568 +(* ----- getting the composite axiom and definitions ------------------------ *)
1.569
1.570  local val ga = get_axiom thy in
1.571  val axs_reach      = map (fn dn => ga (dn ^  "_reach"   )) dnames;
1.572 @@ -362,230 +364,235 @@
1.573  val ax_bisim_def   = ga (comp_dname^"_bisim_def");
1.574  end; (* local *)
1.575
1.576 -(* ----- theorems concerning finiteness and induction ----------------------------- *)
1.577 -
1.578  fun dc_take dn = %%(dn^"_take");
1.579  val x_name = idx_name dnames "x";
1.580  val P_name = idx_name dnames "P";
1.581 +val n_eqs = length eqs;
1.582 +
1.583 +(* ----- theorems concerning finite approximation and finite induction ------ *)
1.584
1.585  local
1.586 -  val iterate_ss = simpset_of "Fix";
1.587 -  val iterate_Cprod_strict_ss = iterate_ss addsimps [cfst_strict, csnd_strict];
1.588 -  val iterate_Cprod_ss = iterate_ss addsimps [cfst2,csnd2,csplit2];
1.589 +  val iterate_Cprod_ss = simpset_of "Fix"
1.591    val copy_con_rews  = copy_rews @ con_rews;
1.592 -  val copy_take_defs = (if length dnames=1 then [] else [ax_copy2_def]) @axs_take_def;
1.593 -  val take_stricts = pg copy_take_defs (mk_trp(foldr' mk_conj (map (fn ((dn,args),_)=>
1.594 -                  (dc_take dn \$ %"n")`UU === mk_constrain(Type(dn,args),UU)) eqs)))([
1.595 -                                nat_ind_tac "n" 1,
1.596 -                                simp_tac iterate_ss 1,
1.597 -                                simp_tac iterate_Cprod_strict_ss 1,
1.598 -                                asm_simp_tac iterate_Cprod_ss 1,
1.599 -                                TRY(safe_tac HOL_cs)] @
1.600 -                        map(K(asm_simp_tac (HOL_ss addsimps copy_rews)1))dnames);
1.601 +  val copy_take_defs =(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
1.602 +  val take_stricts=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=>
1.603 +	    (dc_take dn \$ %"n")`UU === mk_constrain(Type(dn,args),UU)) eqs)))([
1.604 +			nat_ind_tac "n" 1,
1.605 +			simp_tac iterate_Cprod_ss 1,
1.606 +			asm_simp_tac (iterate_Cprod_ss addsimps copy_rews)1]);
1.607    val take_stricts' = rewrite_rule copy_take_defs take_stricts;
1.608 -  val take_0s = mapn (fn n => fn dn => pg axs_take_def(mk_trp((dc_take dn \$ %%"0")
1.609 -                                                                `%x_name n === UU))[
1.610 -                                simp_tac iterate_Cprod_strict_ss 1]) 1 dnames;
1.611 +  val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn \$ %%"0")
1.612 +							`%x_name n === UU))[
1.613 +				simp_tac iterate_Cprod_ss 1]) 1 dnames;
1.614 +  val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1;
1.615    val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj
1.616 -            (flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all
1.617 -                (map vname args,(dc_take dn \$ (%%"Suc" \$ %"n"))`(con_app con args) ===
1.618 -                 con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))\$ %"n"))
1.619 -                              args)) cons) eqs)))) ([
1.620 -                                nat_ind_tac "n" 1,
1.621 -                                simp_tac iterate_Cprod_strict_ss 1,
1.622 -                                simp_tac (HOLCF_ss addsimps copy_con_rews) 1,
1.623 -                                TRY(safe_tac HOL_cs)] @
1.624 -                        (flat(map (fn ((dn,_),cons) => map (fn (con,args) => EVERY (
1.625 -                                asm_full_simp_tac iterate_Cprod_ss 1::
1.626 -                                map (case_UU_tac (take_stricts'::copy_con_rews) 1)
1.627 -                                    (nonlazy args) @[
1.628 -                                asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1])
1.629 -                        ) cons) eqs)));
1.630 +	    (flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all
1.631 +	(map vname args,(dc_take dn \$ (%%"Suc" \$ %"n"))`(con_app con args) ===
1.632 +  	 con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))\$ %"n"))
1.633 +			      args)) cons) eqs)))) ([
1.634 +				simp_tac iterate_Cprod_ss 1,
1.635 +				nat_ind_tac "n" 1,
1.636 +			    simp_tac(iterate_Cprod_ss addsimps copy_con_rews) 1,
1.637 +				asm_full_simp_tac (HOLCF_ss addsimps
1.638 +				      (filter (has_fewer_prems 1) copy_rews)) 1,
1.639 +				TRY(safe_tac HOL_cs)] @
1.640 +			(flat(map (fn ((dn,_),cons) => map (fn (con,args) =>
1.641 +				if nonlazy_rec args = [] then all_tac else
1.642 +				EVERY(map c_UU_tac (nonlazy_rec args)) THEN
1.643 +				asm_full_simp_tac (HOLCF_ss addsimps copy_rews)1
1.644 +		 					   ) cons) eqs)));
1.645  in
1.646  val take_rews = atomize take_stricts @ take_0s @ atomize take_apps;
1.647  end; (* local *)
1.648
1.649 -val take_lemmas = mapn (fn n => fn(dn,ax_reach) => pg'' thy axs_take_def (mk_All("n",
1.650 -                mk_trp(dc_take dn \$ Bound 0 `%(x_name n) ===
1.651 -                       dc_take dn \$ Bound 0 `%(x_name n^"'")))
1.652 -           ===> mk_trp(%(x_name n) === %(x_name n^"'"))) (fn prems => [
1.653 -                                res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
1.654 -                                res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
1.655 -                                rtac (fix_def2 RS ssubst) 1,
1.656 -                                REPEAT(CHANGED(rtac (contlub_cfun_arg RS ssubst) 1
1.657 -                                               THEN chain_tac 1)),
1.658 -                                rtac (contlub_cfun_fun RS ssubst) 1,
1.659 -                                rtac (contlub_cfun_fun RS ssubst) 2,
1.660 -                                rtac lub_equal 3,
1.661 -                                chain_tac 1,
1.662 -                                rtac allI 1,
1.663 -                                resolve_tac prems 1])) 1 (dnames~~axs_reach);
1.664 -
1.665  local
1.666    fun one_con p (con,args) = foldr mk_All (map vname args,
1.667 -        lift_defined (bound_arg (map vname args)) (nonlazy args,
1.668 -        lift (fn arg => %(P_name (1+rec_of arg)) \$ bound_arg args arg)
1.669 -             (filter is_rec args,mk_trp(%p \$ con_app2 con (bound_arg args) args))));
1.670 +	lift_defined (bound_arg (map vname args)) (nonlazy args,
1.671 +	lift (fn arg => %(P_name (1+rec_of arg)) \$ bound_arg args arg)
1.672 +         (filter is_rec args,mk_trp(%p \$ con_app2 con (bound_arg args) args))));
1.673    fun one_eq ((p,cons),concl) = (mk_trp(%p \$ UU) ===>
1.674 -                           foldr (op ===>) (map (one_con p) cons,concl));
1.675 -  fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x)) 1 conss,
1.676 -        mk_trp(foldr' mk_conj (mapn (fn n => concf (P_name n,x_name n)) 1 dnames)));
1.677 +			   foldr (op ===>) (map (one_con p) cons,concl));
1.678 +  fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss,
1.679 +			mk_trp(foldr' mk_conj (mapn concf 1 dnames)));
1.680    val take_ss = HOL_ss addsimps take_rews;
1.681 -  fun ind_tacs tacsf thms1 thms2 prems = TRY(safe_tac HOL_cs)::
1.682 -                                flat (mapn (fn n => fn (thm1,thm2) =>
1.683 -                                  tacsf (n,prems) (thm1,thm2) @
1.684 -                                  flat (map (fn cons =>
1.685 -                                    (resolve_tac prems 1 ::
1.686 -                                     flat (map (fn (_,args) =>
1.687 -                                       resolve_tac prems 1::
1.688 -                                       map (K(atac 1)) (nonlazy args) @
1.689 -                                       map (K(atac 1)) (filter is_rec args))
1.690 -                                     cons)))
1.691 -                                   conss))
1.692 -                                0 (thms1~~thms2));
1.693 +  fun quant_tac i = EVERY(mapn(fn n=> fn _=> res_inst_tac[("x",x_name n)]spec i)
1.694 +			       1 dnames);
1.695 +  fun ind_prems_tac prems = EVERY(flat (map (fn cons => (
1.696 +				     resolve_tac prems 1 ::
1.697 +				     flat (map (fn (_,args) =>
1.698 +				       resolve_tac prems 1 ::
1.699 +				       map (K(atac 1)) (nonlazy args) @
1.700 +				       map (K(atac 1)) (filter is_rec args))
1.701 +				     cons))) conss));
1.702    local
1.703 -    fun all_rec_to ns lazy_rec (n,cons) = forall (exists (fn arg =>
1.704 -                  is_rec arg andalso not(rec_of arg mem ns) andalso
1.705 -                  ((rec_of arg =  n andalso not(lazy_rec orelse is_lazy arg)) orelse
1.706 -                    rec_of arg <> n andalso all_rec_to (rec_of arg::ns)
1.707 -                      (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
1.708 -                  ) o snd) cons;
1.709 -    fun warn (n,cons) = if all_rec_to [] false (n,cons) then (writeln
1.710 -                           ("WARNING: domain "^nth_elem(n,dnames)^" is empty!"); true)
1.711 -                        else false;
1.712 -    fun lazy_rec_to ns lazy_rec (n,cons) = exists (exists (fn arg =>
1.713 -                  is_rec arg andalso not(rec_of arg mem ns) andalso
1.714 -                  ((rec_of arg =  n andalso (lazy_rec orelse is_lazy arg)) orelse
1.715 -                    rec_of arg <> n andalso lazy_rec_to (rec_of arg::ns)
1.716 -                     (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
1.717 -                 ) o snd) cons;
1.718 -  in val is_emptys = map warn (mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs);
1.719 -     val is_finite = forall (not o lazy_rec_to [] false)
1.720 -                            (mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs)
1.721 +    (* check whether every/exists constructor of the n-th part of the equation:
1.722 +       it has a possibly indirectly recursive argument that isn't/is possibly
1.723 +       indirectly lazy *)
1.724 +    fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg =>
1.725 +	  is_rec arg andalso not(rec_of arg mem ns) andalso
1.726 +	  ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse
1.727 +	    rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns)
1.728 +	      (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
1.729 +	  ) o snd) cons;
1.730 +    fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
1.731 +    fun warn (n,cons)  = if all_rec_to [] false (n,cons) then (writeln
1.732 +        ("WARNING: domain "^nth_elem(n,dnames)^" is empty!"); true) else false;
1.733 +    fun lazy_rec_to ns = rec_to exists Id  lazy_rec_to ns;
1.734 +
1.735 +  in val n__eqs     = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
1.736 +     val is_emptys = map warn n__eqs;
1.737 +     val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
1.738    end;
1.739 -in
1.740 -val finite_ind = pg'' thy [] (ind_term (fn (P,x) => fn dn =>
1.741 -                          mk_all(x,%P \$ (dc_take dn \$ %"n" `Bound 0)))) (fn prems=> [
1.742 -                                nat_ind_tac "n" 1,
1.743 -                                simp_tac (take_ss addsimps prems) 1,
1.744 -                                TRY(safe_tac HOL_cs)]
1.745 -                                @ flat(mapn (fn n => fn (cons,cases) => [
1.746 -                                 res_inst_tac [("x",x_name n)] cases 1,
1.747 -                                 asm_simp_tac (take_ss addsimps prems) 1]
1.748 -                                 @ flat(map (fn (con,args) =>
1.749 -                                  asm_simp_tac take_ss 1 ::
1.750 -                                  map (fn arg =>
1.751 -                                   case_UU_tac (prems@con_rews) 1 (
1.752 -                                   nth_elem(rec_of arg,dnames)^"_take n1`"^vname arg))
1.753 -                                  (filter is_nonlazy_rec args) @ [
1.754 -                                  resolve_tac prems 1] @
1.755 -                                  map (K (atac 1))      (nonlazy args) @
1.756 -                                  map (K (etac spec 1)) (filter is_rec args))
1.757 -                                 cons))
1.758 -                                1 (conss~~casess)));
1.759 +in (* local *)
1.760 +val finite_ind = pg'' thy [] (ind_term (fn n => fn dn => %(P_name n)\$
1.761 +			     (dc_take dn \$ %"n" `%(x_name n)))) (fn prems => [
1.762 +				quant_tac 1,
1.763 +				simp_tac quant_ss 1,
1.764 +				nat_ind_tac "n" 1,
1.765 +				simp_tac (take_ss addsimps prems) 1,
1.766 +				TRY(safe_tac HOL_cs)]
1.767 +				@ flat(map (fn (cons,cases) => [
1.768 +				 res_inst_tac [("x","x")] cases 1,
1.769 +				 asm_simp_tac (take_ss addsimps prems) 1]
1.770 +				 @ flat(map (fn (con,args) =>
1.771 +				  asm_simp_tac take_ss 1 ::
1.772 +				  map (fn arg =>
1.773 +				   case_UU_tac (prems@con_rews) 1 (
1.774 +			   nth_elem(rec_of arg,dnames)^"_take n1`"^vname arg))
1.775 +				  (filter is_nonlazy_rec args) @ [
1.776 +				  resolve_tac prems 1] @
1.777 +				  map (K (atac 1))      (nonlazy args) @
1.778 +				  map (K (etac spec 1)) (filter is_rec args))
1.779 +				 cons))
1.780 +				(conss~~casess)));
1.781 +
1.782 +val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n",
1.783 +		mk_trp(dc_take dn \$ Bound 0 `%(x_name n) ===
1.784 +		       dc_take dn \$ Bound 0 `%(x_name n^"'")))
1.785 +	   ===> mk_trp(%(x_name n) === %(x_name n^"'"))) (fn prems => [
1.786 +			res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
1.787 +			res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
1.788 +				rtac (fix_def2 RS ssubst) 1,
1.789 +				REPEAT(CHANGED(rtac(contlub_cfun_arg RS ssubst)1
1.790 +					       THEN chain_tac 1)),
1.791 +				rtac (contlub_cfun_fun RS ssubst) 1,
1.792 +				rtac (contlub_cfun_fun RS ssubst) 2,
1.793 +				rtac lub_equal 3,
1.794 +				chain_tac 1,
1.795 +				rtac allI 1,
1.796 +				resolve_tac prems 1])) 1 (dnames~~axs_reach);
1.797 +
1.798 +(* ----- theorems concerning finiteness and induction ----------------------- *)
1.799
1.800  val (finites,ind) = if is_finite then
1.801 -let
1.802 -  fun take_enough dn = mk_ex ("n",dc_take dn \$ Bound 0 ` %"x" === %"x");
1.803 -  val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%"x")) ===>
1.804 -        mk_trp(mk_disj(mk_all("n",dc_take dn \$ Bound 0 ` %"x" === UU),
1.805 -        take_enough dn)) ===> mk_trp(take_enough dn)) [
1.806 -                                etac disjE 1,
1.807 -                                etac notE 1,
1.808 -                                resolve_tac take_lemmas 1,
1.809 -                                asm_simp_tac take_ss 1,
1.810 -                                atac 1]) dnames;
1.811 -  val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn
1.812 -        (fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
1.813 -         mk_disj(dc_take dn \$ Bound 1 ` Bound 0 === UU,
1.814 -                 dc_take dn \$ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
1.815 -                                rtac allI 1,
1.816 -                                nat_ind_tac "n" 1,
1.817 -                                simp_tac take_ss 1,
1.819 -                                flat(mapn (fn n => fn (cons,cases) => [
1.820 -                                  simp_tac take_ss 1,
1.821 -                                  rtac allI 1,
1.822 -                                  res_inst_tac [("x",x_name n)] cases 1,
1.823 -                                  asm_simp_tac take_ss 1] @
1.824 -                                  flat(map (fn (con,args) =>
1.825 -                                    asm_simp_tac take_ss 1 ::
1.826 -                                    flat(map (fn arg => [
1.827 -                                      eres_inst_tac [("x",vname arg)] all_dupE 1,
1.828 -                                      etac disjE 1,
1.829 -                                      asm_simp_tac (HOL_ss addsimps con_rews) 1,
1.830 -                                      asm_simp_tac take_ss 1])
1.831 -                                    (filter is_nonlazy_rec args)))
1.832 -                                  cons))
1.833 -                                1 (conss~~casess))) handle ERROR => raise ERROR;
1.834 -  val all_finite=map (fn(dn,l1b)=>pg axs_finite_def (mk_trp(%%(dn^"_finite") \$ %"x"))[
1.835 -                                case_UU_tac take_rews 1 "x",
1.836 -                                eresolve_tac finite_lemmas1a 1,
1.837 -                                step_tac HOL_cs 1,
1.838 -                                step_tac HOL_cs 1,
1.839 -                                cut_facts_tac [l1b] 1,
1.840 -                                fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
1.841 -in
1.842 -(all_finite,
1.843 - pg'' thy [] (ind_term (fn (P,x) => fn dn => %P \$ %x))
1.844 -                               (ind_tacs (fn _ => fn (all_fin,finite_ind) => [
1.845 -                                rtac (rewrite_rule axs_finite_def all_fin RS exE) 1,
1.846 -                                etac subst 1,
1.847 -                                rtac finite_ind 1]) all_finite (atomize finite_ind))
1.848 +  let
1.849 +    fun take_enough dn = mk_ex ("n",dc_take dn \$ Bound 0 ` %"x" === %"x");
1.850 +    val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%"x")) ===>
1.851 +	mk_trp(mk_disj(mk_all("n",dc_take dn \$ Bound 0 ` %"x" === UU),
1.852 +	take_enough dn)) ===> mk_trp(take_enough dn)) [
1.853 +				etac disjE 1,
1.854 +				etac notE 1,
1.855 +				resolve_tac take_lemmas 1,
1.856 +				asm_simp_tac take_ss 1,
1.857 +				atac 1]) dnames;
1.858 +    val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn
1.859 +	(fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
1.860 +	 mk_disj(dc_take dn \$ Bound 1 ` Bound 0 === UU,
1.861 +		 dc_take dn \$ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
1.862 +				rtac allI 1,
1.863 +				nat_ind_tac "n" 1,
1.864 +				simp_tac take_ss 1,
1.866 +				flat(mapn (fn n => fn (cons,cases) => [
1.867 +				  simp_tac take_ss 1,
1.868 +				  rtac allI 1,
1.869 +				  res_inst_tac [("x",x_name n)] cases 1,
1.870 +				  asm_simp_tac take_ss 1] @
1.871 +				  flat(map (fn (con,args) =>
1.872 +				    asm_simp_tac take_ss 1 ::
1.873 +				    flat(map (fn vn => [
1.874 +				      eres_inst_tac [("x",vn)] all_dupE 1,
1.875 +				      etac disjE 1,
1.876 +				      asm_simp_tac (HOL_ss addsimps con_rews) 1,
1.877 +				      asm_simp_tac take_ss 1])
1.878 +				    (nonlazy_rec args)))
1.879 +				  cons))
1.880 +				1 (conss~~casess))) handle ERROR => raise ERROR;
1.881 +    val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp(
1.882 +						%%(dn^"_finite") \$ %"x"))[
1.883 +				case_UU_tac take_rews 1 "x",
1.884 +				eresolve_tac finite_lemmas1a 1,
1.885 +				step_tac HOL_cs 1,
1.886 +				step_tac HOL_cs 1,
1.887 +				cut_facts_tac [l1b] 1,
1.888 +			fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
1.889 +  in
1.890 +  (finites,
1.891 +   pg'' thy[](ind_term (fn n => fn dn => %(P_name n) \$ %(x_name n)))(fn prems =>
1.892 +				TRY(safe_tac HOL_cs) ::
1.893 +			 flat (map (fn (finite,fin_ind) => [
1.894 +			       rtac(rewrite_rule axs_finite_def finite RS exE)1,
1.895 +				etac subst 1,
1.896 +				rtac fin_ind 1,
1.897 +				ind_prems_tac prems])
1.898 +			           (finites~~(atomize finite_ind)) ))
1.899  ) end (* let *) else
1.900 -(mapn (fn n => fn dn => read_instantiate_sg (sign_of thy)
1.901 -                    [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
1.902 - pg'' thy [] (foldr (op ===>) (mapn (fn n =>K(mk_trp(%%"adm" \$ %(P_name n))))1
1.903 -                                       dnames,ind_term (fn(P,x)=>fn dn=> %P \$ %x)))
1.904 -                               (ind_tacs (fn (n,prems) => fn (ax_reach,finite_ind) =>[
1.905 -                                rtac (ax_reach RS subst) 1,
1.906 -                                res_inst_tac [("x",x_name n)] spec 1,
1.907 -                                rtac wfix_ind 1,
1.909 -                                resolve_tac adm_thms 1,
1.910 -                                rtac adm_subst 1,
1.911 -                                cont_tacR 1,
1.912 -                                resolve_tac prems 1,
1.913 -                                strip_tac 1,
1.914 -                                rtac(rewrite_rule axs_take_def finite_ind) 1])
1.915 -                                 axs_reach (atomize finite_ind))
1.916 +  (mapn (fn n => fn dn => read_instantiate_sg (sign_of thy)
1.917 +	  	    [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
1.918 +   pg'' thy [] (foldr (op ===>) (mapn (fn n => K(mk_trp(%%"adm" \$ %(P_name n))))
1.919 +	       1 dnames, ind_term (fn n => fn dn => %(P_name n) \$ %(x_name n))))
1.920 +		   (fn prems => map (fn ax_reach => rtac (ax_reach RS subst) 1)
1.921 +				    axs_reach @ [
1.922 +				quant_tac 1,
1.923 +				rtac (adm_impl_admw RS wfix_ind) 1,
1.924 +				REPEAT_DETERM(rtac adm_all2 1),
1.925 +				REPEAT_DETERM(TRY(rtac adm_conj 1) THEN
1.926 +						  rtac adm_subst 1 THEN
1.927 +					cont_tacR 1 THEN resolve_tac prems 1),
1.928 +				strip_tac 1,
1.929 +				rtac (rewrite_rule axs_take_def finite_ind) 1,
1.930 +				ind_prems_tac prems])
1.931  )
1.932  end; (* local *)
1.933
1.934 +(* ----- theorem concerning coinduction ------------------------------------- *)
1.935 +
1.936  local
1.937    val xs = mapn (fn n => K (x_name n)) 1 dnames;
1.938 -  fun bnd_arg n i = Bound(2*(length dnames - n)-i-1);
1.939 +  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
1.940    val take_ss = HOL_ss addsimps take_rews;
1.941 -  val sproj   = bin_branchr (fn s => "fst("^s^")") (fn s => "snd("^s^")");
1.942 -  val coind_lemma = pg [ax_bisim_def] (mk_trp(mk_imp(%%(comp_dname^"_bisim") \$ %"R",
1.943 -                foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
1.944 -                  foldr mk_imp (mapn (fn n => K(proj (%"R") dnames n \$
1.945 -                                      bnd_arg n 0 \$ bnd_arg n 1)) 0 dnames,
1.946 -                    foldr' mk_conj (mapn (fn n => fn dn =>
1.947 -                                (dc_take dn \$ %"n" `bnd_arg n 0 ===
1.948 -                                (dc_take dn \$ %"n" `bnd_arg n 1))) 0 dnames)))))) ([
1.949 -                                rtac impI 1,
1.950 -                                nat_ind_tac "n" 1,
1.951 -                                simp_tac take_ss 1,
1.952 -                                safe_tac HOL_cs] @
1.953 -                                flat(mapn (fn n => fn x => [
1.954 -                                  etac allE 1, etac allE 1,
1.955 -                                  eres_inst_tac [("P1",sproj "R" dnames n^
1.956 -                                                  " "^x^" "^x^"'")](mp RS disjE) 1,
1.957 -                                  TRY(safe_tac HOL_cs),
1.958 -                                  REPEAT(CHANGED(asm_simp_tac take_ss 1))])
1.959 -                                0 xs));
1.960 +  val sproj   = prj (fn s => "fst("^s^")") (fn s => "snd("^s^")");
1.961 +  val coind_lemma=pg[ax_bisim_def](mk_trp(mk_imp(%%(comp_dname^"_bisim") \$ %"R",
1.962 +		foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
1.963 +		  foldr mk_imp (mapn (fn n => K(proj (%"R") n_eqs n \$
1.964 +				      bnd_arg n 0 \$ bnd_arg n 1)) 0 dnames,
1.965 +		    foldr' mk_conj (mapn (fn n => fn dn =>
1.966 +				(dc_take dn \$ %"n" `bnd_arg n 0 ===
1.967 +				(dc_take dn \$ %"n" `bnd_arg n 1)))0 dnames))))))
1.968 +			     ([ rtac impI 1,
1.969 +				nat_ind_tac "n" 1,
1.970 +				simp_tac take_ss 1,
1.971 +				safe_tac HOL_cs] @
1.972 +				flat(mapn (fn n => fn x => [
1.973 +				  rotate_tac (n+1) 1,
1.974 +				  etac all2E 1,
1.975 +				  eres_inst_tac [("P1", sproj "R" n_eqs n^
1.976 +					" "^x^" "^x^"'")](mp RS disjE) 1,
1.977 +				  TRY(safe_tac HOL_cs),
1.978 +				  REPEAT(CHANGED(asm_simp_tac take_ss 1))])
1.979 +				0 xs));
1.980  in
1.981  val coind = pg [] (mk_trp(%%(comp_dname^"_bisim") \$ %"R") ===>
1.982 -                foldr (op ===>) (mapn (fn n => fn x =>
1.983 -                        mk_trp(proj (%"R") dnames n \$ %x \$ %(x^"'"))) 0 xs,
1.984 -                        mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
1.985 -                                TRY(safe_tac HOL_cs)] @
1.986 -                                flat(map (fn take_lemma => [
1.987 -                                  rtac take_lemma 1,
1.988 -                                  cut_facts_tac [coind_lemma] 1,
1.989 -                                  fast_tac HOL_cs 1])
1.990 -                                take_lemmas));
1.991 +		foldr (op ===>) (mapn (fn n => fn x =>
1.992 +		  mk_trp(proj (%"R") n_eqs n \$ %x \$ %(x^"'"))) 0 xs,
1.993 +		  mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
1.994 +				TRY(safe_tac HOL_cs)] @
1.995 +				flat(map (fn take_lemma => [
1.996 +				  rtac take_lemma 1,
1.997 +				  cut_facts_tac [coind_lemma] 1,
1.998 +				  fast_tac HOL_cs 1])
1.999 +				take_lemmas));
1.1000  end; (* local *)
1.1001
1.1002
```