src/HOLCF/domain/theorems.ML

function names in when_rews now meta-quantified

 (* theorems.ML
   Author : David von Oheimb
   Created: 06-Jun-95
   Updated: 08-Jun-95 first proof from cterms
   Updated: 26-Jun-95 proofs for exhaustion thms
   Updated: 27-Jun-95 proofs for discriminators, constructors and selectors
   Updated: 06-Jul-95 proofs for distinctness, invertibility and injectivity
   Updated: 17-Jul-95 proofs for induction rules
   Updated: 19-Jul-95 proof for co-induction rule
   Updated: 28-Aug-95 definedness theorems for selectors (completion)
   Updated: 05-Sep-95 simultaneous domain equations (main part)
   Updated: 11-Sep-95 simultaneous domain equations (coding finished)
   Updated: 13-Sep-95 simultaneous domain equations (debugging)
   Updated: 26-Oct-95 debugging and enhancement of proofs for take_apps, ind
   Updated: 16-Feb-96 bug concerning  domain Triv = triv  fixed
   Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
   Copyright 1995, 1996 TU Muenchen
*)

structure Domain_Theorems = struct

local

open Domain_Library;
infixr 0 ===>;infixr 0 ==>;infix 0 == ; 
infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<;
infix 9 `   ; infix 9 `% ; infix 9 `%%; infixr 9 oo;

(* ----- general proof facilities ------------------------------------------- *)

fun inferT sg pre_tm = #2 (Sign.infer_types sg (K None) (K None) [] true 
			   ([pre_tm],propT));

fun pg'' thy defs t = let val sg = sign_of thy;
		          val ct = Thm.cterm_of sg (inferT sg t);
		      in prove_goalw_cterm defs ct end;
fun pg'  thy defs t tacsf=pg'' thy defs t (fn []   => tacsf 
				| prems=> (cut_facts_tac prems 1)::tacsf);

fun REPEAT_DETERM_UNTIL p tac = 
let fun drep st = if p st then Sequence.single st
			  else (case Sequence.pull(tac st) of
		                  None        => Sequence.null
				| Some(st',_) => drep st')
in drep end;
val UNTIL_SOLVED = REPEAT_DETERM_UNTIL (has_fewer_prems 1);

local val trueI2 = prove_goal HOL.thy"f~=x ==> True"(fn _ => [rtac TrueI 1]) in
val kill_neq_tac = dtac trueI2 end;
fun case_UU_tac rews i v =	case_tac (v^"=UU") i THEN
				asm_simp_tac (HOLCF_ss addsimps rews) i;

val chain_tac = REPEAT_DETERM o resolve_tac 
		[is_chain_iterate, ch2ch_fappR, ch2ch_fappL];

(* ----- general proofs ----------------------------------------------------- *)

val quant_ss = HOL_ss addsimps (map (fn s => prove_goal HOL.thy s (fn _ =>[
		fast_tac HOL_cs 1]))["(!x. P x & Q)=((!x. P x) & Q)",
			    	     "(!x. P & Q x) = (P & (!x. Q x))"]);

val all2E = prove_goal HOL.thy "[| !x y . P x y; P x y ==> R |] ==> R"
 (fn prems =>[
				resolve_tac prems 1,
				cut_facts_tac prems 1,
				fast_tac HOL_cs 1]);

val swap3 = prove_goal HOL.thy "[| Q ==> P; ~P |] ==> ~Q" (fn prems => [
                                cut_facts_tac prems 1,
                                etac swap 1,
                                dtac notnotD 1,
				etac (hd prems) 1]);

val dist_eqI = prove_goal Porder.thy "~ x << y ==> x ~= y" (fn prems => [
                                rtac swap3 1,
				etac (antisym_less_inverse RS conjunct1) 1,
				resolve_tac prems 1]);
val cfst_strict  = prove_goal Cprod3.thy "cfst`UU = UU" (fn _ => [
			(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
val csnd_strict  = prove_goal Cprod3.thy "csnd`UU = UU" (fn _ => [
			(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);

in


fun theorems thy (((dname,_),cons) : eq, eqs :eq list) =
let

val dummy = writeln ("Proving isomorphism properties of domain "^dname^"...");
val pg = pg' thy;
(*
infixr 0 y;
val b = 0;
fun _ y t = by t;
fun  g  defs t = let val sg = sign_of thy;
		     val ct = Thm.cterm_of sg (inferT sg t);
		 in goalw_cterm defs ct end;
*)


(* ----- getting the axioms and definitions --------------------------------- *)

local val ga = get_axiom thy in
val ax_abs_iso    = ga (dname^"_abs_iso"   );
val ax_rep_iso    = ga (dname^"_rep_iso"   );
val ax_when_def   = ga (dname^"_when_def"  );
val axs_con_def   = map (fn (con,_) => ga (extern_name con ^"_def")) cons;
val axs_dis_def   = map (fn (con,_) => ga (   dis_name con ^"_def")) cons;
val axs_sel_def   = flat(map (fn (_,args) => 
		    map (fn     arg => ga (sel_of arg      ^"_def")) args)cons);
val ax_copy_def   = ga (dname^"_copy_def"  );
end; (* local *)

(* ----- theorems concerning the isomorphism -------------------------------- *)

val dc_abs  = %%(dname^"_abs");
val dc_rep  = %%(dname^"_rep");
val dc_copy = %%(dname^"_copy");
val x_name = "x";

val (rep_strict, abs_strict) = let 
	 val r = ax_rep_iso RS (ax_abs_iso RS (allI  RSN(2,allI RS iso_strict)))
	       in (r RS conjunct1, r RS conjunct2) end;
val abs_defin' = pg [] ((dc_abs`%x_name === UU) ==> (%x_name === UU)) [
			   res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
				etac ssubst 1, rtac rep_strict 1];
val rep_defin' = pg [] ((dc_rep`%x_name === UU) ==> (%x_name === UU)) [
			   res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
				etac ssubst 1, rtac abs_strict 1];
val iso_rews = [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];

local 
val iso_swap = pg [] (dc_rep`%"x" === %"y" ==> %"x" === dc_abs`%"y") [
			    dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
			    etac (ax_rep_iso RS subst) 1];
fun exh foldr1 cn quant foldr2 var = let
  fun one_con (con,args) = let val vns = map vname args in
    foldr quant (vns, foldr2 ((%x_name === con_app2 con (var vns) vns)::
			      map (defined o (var vns)) (nonlazy args))) end
  in foldr1 ((cn(%x_name===UU))::map one_con cons) end;
in
val cases = let 
	    fun common_tac thm = rtac thm 1 THEN contr_tac 1;
	    fun unit_tac true = common_tac liftE1
	    |   unit_tac _    = all_tac;
	    fun prod_tac []          = common_tac oneE
	    |   prod_tac [arg]       = unit_tac (is_lazy arg)
	    |   prod_tac (arg::args) = 
				common_tac sprodE THEN
				kill_neq_tac 1 THEN
				unit_tac (is_lazy arg) THEN
				prod_tac args;
	    fun sum_rest_tac p = SELECT_GOAL(EVERY[
				rtac p 1,
				rewrite_goals_tac axs_con_def,
				dtac iso_swap 1,
				simp_tac HOLCF_ss 1,
				UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
	    fun sum_tac [(_,args)]       [p]        = 
				prod_tac args THEN sum_rest_tac p
	    |   sum_tac ((_,args)::cons') (p::prems) = DETERM(
				common_tac ssumE THEN
				kill_neq_tac 1 THEN kill_neq_tac 2 THEN
				prod_tac args THEN sum_rest_tac p) THEN
				sum_tac cons' prems
	    |   sum_tac _ _ = Imposs "theorems:sum_tac";
	  in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%"P")))
			      (fn T => T ==> %"P") mk_All
			      (fn l => foldr (op ===>) (map mk_trp l,
							    mk_trp(%"P")))
			      bound_arg)
			     (fn prems => [
				cut_facts_tac [excluded_middle] 1,
				etac disjE 1,
				rtac (hd prems) 2,
				etac rep_defin' 2,
				if length cons = 1 andalso 
				   length (snd(hd cons)) = 1 andalso 
				   not(is_lazy(hd(snd(hd cons))))
				then rtac (hd (tl prems)) 1 THEN atac 2 THEN
				     rewrite_goals_tac axs_con_def THEN
				     simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
				else sum_tac cons (tl prems)])end;
val exhaust= pg[](mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %)))[
				rtac cases 1,
				UNTIL_SOLVED(fast_tac HOL_cs 1)];
end;

local 
  fun bind_fun t = foldr mk_All ((if length cons = 1 then ["f"] 
		  else mapn (fn n => K("f"^(string_of_int n))) 1 cons),t);
  fun bound_fun i _ = Bound (length cons - i);
  val when_app  = foldl (op `) (%%(dname^"_when"), mapn bound_fun 1 cons);
  val when_appl = pg [ax_when_def] (bind_fun(mk_trp(when_app`%x_name ===
	     when_body cons (fn (m,n)=> bound_fun (n-m) 0)`(dc_rep`%x_name))))[
				simp_tac HOLCF_ss 1];
in
val when_strict = pg [] (bind_fun(mk_trp(strict when_app))) [
			simp_tac(HOLCF_ss addsimps [when_appl,rep_strict]) 1];
val when_apps = let fun one_when n (con,args) = pg axs_con_def 
		(bind_fun (lift_defined % (nonlazy args, 
		mk_trp(when_app`(con_app con args) ===
                       mk_cfapp(bound_fun n 0,map %# args)))))[
		asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
	in mapn one_when 1 cons end;
end;
val when_rews = when_strict::when_apps;

(* ----- theorems concerning the constructors, discriminators and selectors - *)

val dis_rews = let
  val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
		      	     strict(%%(dis_name con)))) [
				simp_tac (HOLCF_ss addsimps when_rews) 1]) cons;
  val dis_apps = let fun one_dis c (con,args)= pg axs_dis_def
		   (lift_defined % (nonlazy args,
			(mk_trp((%%(dis_name c))`(con_app con args) ===
			      %%(if con=c then "TT" else "FF"))))) [
				asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
	in flat(map (fn (c,_) => map (one_dis c) cons) cons) end;
  val dis_defins = map (fn (con,args) => pg [] (defined(%x_name) ==> 
		      defined(%%(dis_name con)`%x_name)) [
				rtac cases 1,
				contr_tac 1,
				UNTIL_SOLVED (CHANGED(asm_simp_tac 
				        (HOLCF_ss addsimps dis_apps) 1))]) cons;
in dis_stricts @ dis_defins @ dis_apps end;

val con_stricts = flat(map (fn (con,args) => map (fn vn =>
			pg (axs_con_def) 
			   (mk_trp(con_app2 con (fn arg => if vname arg = vn 
					then UU else %# arg) args === UU))[
				asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
			) (nonlazy args)) cons);
val con_defins = map (fn (con,args) => pg []
			(lift_defined % (nonlazy args,
				mk_trp(defined(con_app con args)))) ([
			  rtac swap3 1, 
			  eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
			  asm_simp_tac (HOLCF_ss addsimps dis_rews) 1] )) cons;
val con_rews = con_stricts @ con_defins;

val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%sel))) [
				simp_tac (HOLCF_ss addsimps when_rews) 1];
in flat(map (fn (_,args) =>map (fn arg => one_sel (sel_of arg)) args) cons) end;
val sel_apps = let fun one_sel c n sel = map (fn (con,args) => 
		let val nlas = nonlazy args;
		    val vns  = map vname args;
		in pg axs_sel_def (lift_defined %
		   (filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
				mk_trp((%%sel)`(con_app con args) === 
				(if con=c then %(nth_elem(n,vns)) else UU))))
			    ( (if con=c then [] 
		       else map(case_UU_tac(when_rews@con_stricts)1) nlas)
		     @(if con=c andalso ((nth_elem(n,vns)) mem nlas)
				 then[case_UU_tac (when_rews @ con_stricts) 1 
						  (nth_elem(n,vns))] else [])
		     @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
in flat(map  (fn (c,args) => 
     flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
val sel_defins = if length cons=1 then map (fn arg => pg [](defined(%x_name)==> 
			defined(%%(sel_of arg)`%x_name)) [
				rtac cases 1,
				contr_tac 1,
				UNTIL_SOLVED (CHANGED(asm_simp_tac 
				             (HOLCF_ss addsimps sel_apps) 1))]) 
		 (filter_out is_lazy (snd(hd cons))) else [];
val sel_rews = sel_stricts @ sel_defins @ sel_apps;

val distincts_le = let
    fun dist (con1, args1) (con2, args2) = pg []
	      (lift_defined % ((nonlazy args1),
			(mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
			rtac swap3 1,
			eres_inst_tac[("fo",dis_name con1)] monofun_cfun_arg 1]
		      @map(case_UU_tac (con_stricts @ dis_rews)1)(nonlazy args2)
		      @[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
    fun distinct (con1,args1) (con2,args2) =
	let val arg1 = (con1, args1);
	    val arg2 = (con2, (map (fn (arg,vn) => upd_vname (K vn) arg)
			(args2~~variantlist(map vname args2,map vname args1))));
	in [dist arg1 arg2, dist arg2 arg1] end;
    fun distincts []      = []
    |   distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
in distincts cons end;
val dists_le = flat (flat distincts_le);
val dists_eq = let
    fun distinct (_,args1) ((_,args2),leqs) = let
	val (le1,le2) = (hd leqs, hd(tl leqs));
	val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
	if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
	if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
					[eq1, eq2] end;
    fun distincts []      = []
    |   distincts ((c,leqs)::cs) = flat(map (distinct c) ((map fst cs)~~leqs)) @
				   distincts cs;
    in distincts (cons~~distincts_le) end;

local 
  fun pgterm rel con args = let
		fun append s = upd_vname(fn v => v^s);
		val (largs,rargs) = (args, map (append "'") args);
		in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
		      lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
			    mk_trp (foldr' mk_conj 
				(map rel (map %# largs ~~ map %# rargs)))))) end;
  val cons' = filter (fn (_,args) => args<>[]) cons;
in
val inverts = map (fn (con,args) => 
		pgterm (op <<) con args (flat(map (fn arg => [
				TRY(rtac conjI 1),
				dres_inst_tac [("fo",sel_of arg)] monofun_cfun_arg 1,
				asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
			     			      ) args))) cons';
val injects = map (fn ((con,args),inv_thm) => 
			   pgterm (op ===) con args [
				etac (antisym_less_inverse RS conjE) 1,
				dtac inv_thm 1, REPEAT(atac 1),
				dtac inv_thm 1, REPEAT(atac 1),
				TRY(safe_tac HOL_cs),
				REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
		  (cons'~~inverts);
end;

(* ----- theorems concerning one induction step ----------------------------- *)

val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [
		   asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict,
						   cfst_strict,csnd_strict]) 1];
val copy_apps = map (fn (con,args) => pg [ax_copy_def]
		    (lift_defined % (nonlazy_rec args,
			mk_trp(dc_copy`%"f"`(con_app con args) ===
		(con_app2 con (app_rec_arg (cproj (%"f") (length eqs))) args))))
			(map (case_UU_tac (abs_strict::when_strict::con_stricts)
				 1 o vname)
			 (filter (fn a => not (is_rec a orelse is_lazy a)) args)
			@[asm_simp_tac (HOLCF_ss addsimps when_apps) 1,
		          simp_tac (HOLCF_ss addsimps axs_con_def) 1]))cons;
val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU`
					(con_app con args) ===UU))
     (let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
			 in map (case_UU_tac rews 1) (nonlazy args) @ [
			     asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
  		        (filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
val copy_rews = copy_strict::copy_apps @ copy_stricts;

in     (iso_rews, exhaust, cases, when_rews,
	con_rews, sel_rews, dis_rews, dists_le, dists_eq, inverts, injects,
	copy_rews)
end; (* let *)


fun comp_theorems thy (comp_dname, eqs: eq list, casess, con_rews, copy_rews) =
let

val dummy = writeln("Proving induction properties of domain "^comp_dname^"...");
val pg = pg' thy;

val dnames = map (fst o fst) eqs;
val conss  = map  snd        eqs;

(* ----- getting the composite axiom and definitions ------------------------ *)

local val ga = get_axiom thy in
val axs_reach      = map (fn dn => ga (dn ^  "_reach"   )) dnames;
val axs_take_def   = map (fn dn => ga (dn ^  "_take_def")) dnames;
val axs_finite_def = map (fn dn => ga (dn ^"_finite_def")) dnames;
val ax_copy2_def   = ga (comp_dname^ "_copy_def");
val ax_bisim_def   = ga (comp_dname^"_bisim_def");
end; (* local *)

fun dc_take dn = %%(dn^"_take");
val x_name = idx_name dnames "x"; 
val P_name = idx_name dnames "P";
val n_eqs = length eqs;

(* ----- theorems concerning finite approximation and finite induction ------ *)

local
  val iterate_Cprod_ss = simpset_of "Fix"
			 addsimps [cfst_strict, csnd_strict]addsimps Cprod_rews;
  val copy_con_rews  = copy_rews @ con_rews;
  val copy_take_defs =(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
  val take_stricts=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=>
	    (dc_take dn $ %"n")`UU === mk_constrain(Type(dn,args),UU)) eqs)))([
			nat_ind_tac "n" 1,
			simp_tac iterate_Cprod_ss 1,
			asm_simp_tac (iterate_Cprod_ss addsimps copy_rews)1]);
  val take_stricts' = rewrite_rule copy_take_defs take_stricts;
  val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn $ %%"0")
							`%x_name n === UU))[
				simp_tac iterate_Cprod_ss 1]) 1 dnames;
  val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1;
  val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj 
	    (flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all 
	(map vname args,(dc_take dn $ (%%"Suc" $ %"n"))`(con_app con args) ===
  	 con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))$ %"n"))
			      args)) cons) eqs)))) ([
				simp_tac iterate_Cprod_ss 1,
				nat_ind_tac "n" 1,
			    simp_tac(iterate_Cprod_ss addsimps copy_con_rews) 1,
				asm_full_simp_tac (HOLCF_ss addsimps 
				      (filter (has_fewer_prems 1) copy_rews)) 1,
				TRY(safe_tac HOL_cs)] @
			(flat(map (fn ((dn,_),cons) => map (fn (con,args) => 
				if nonlazy_rec args = [] then all_tac else
				EVERY(map c_UU_tac (nonlazy_rec args)) THEN
				asm_full_simp_tac (HOLCF_ss addsimps copy_rews)1
		 					   ) cons) eqs)));
in
val take_rews = atomize take_stricts @ take_0s @ atomize take_apps;
end; (* local *)

local
  fun one_con p (con,args) = foldr mk_All (map vname args,
	lift_defined (bound_arg (map vname args)) (nonlazy args,
	lift (fn arg => %(P_name (1+rec_of arg)) $ bound_arg args arg)
         (filter is_rec args,mk_trp(%p $ con_app2 con (bound_arg args) args))));
  fun one_eq ((p,cons),concl) = (mk_trp(%p $ UU) ===> 
			   foldr (op ===>) (map (one_con p) cons,concl));
  fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss,
			mk_trp(foldr' mk_conj (mapn concf 1 dnames)));
  val take_ss = HOL_ss addsimps take_rews;
  fun quant_tac i = EVERY(mapn(fn n=> fn _=> res_inst_tac[("x",x_name n)]spec i)
			       1 dnames);
  fun ind_prems_tac prems = EVERY(flat (map (fn cons => (
				     resolve_tac prems 1 ::
				     flat (map (fn (_,args) => 
				       resolve_tac prems 1 ::
				       map (K(atac 1)) (nonlazy args) @
				       map (K(atac 1)) (filter is_rec args))
				     cons))) conss));
  local 
    (* check whether every/exists constructor of the n-th part of the equation:
       it has a possibly indirectly recursive argument that isn't/is possibly 
       indirectly lazy *)
    fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
	  is_rec arg andalso not(rec_of arg mem ns) andalso
	  ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
	    rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
	      (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
	  ) o snd) cons;
    fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
    fun warn (n,cons)  = if all_rec_to [] false (n,cons) then (writeln 
        ("WARNING: domain "^nth_elem(n,dnames)^" is empty!"); true) else false;
    fun lazy_rec_to ns = rec_to exists Id  lazy_rec_to ns;

  in val n__eqs     = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
     val is_emptys = map warn n__eqs;
     val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
  end;
in (* local *)
val finite_ind = pg'' thy [] (ind_term (fn n => fn dn => %(P_name n)$
			     (dc_take dn $ %"n" `%(x_name n)))) (fn prems => [
				quant_tac 1,
				simp_tac quant_ss 1,
				nat_ind_tac "n" 1,
				simp_tac (take_ss addsimps prems) 1,
				TRY(safe_tac HOL_cs)]
				@ flat(map (fn (cons,cases) => [
				 res_inst_tac [("x","x")] cases 1,
				 asm_simp_tac (take_ss addsimps prems) 1]
				 @ flat(map (fn (con,args) => 
				  asm_simp_tac take_ss 1 ::
				  map (fn arg =>
				   case_UU_tac (prems@con_rews) 1 (
			   nth_elem(rec_of arg,dnames)^"_take n1`"^vname arg))
				  (filter is_nonlazy_rec args) @ [
				  resolve_tac prems 1] @
				  map (K (atac 1))      (nonlazy args) @
				  map (K (etac spec 1)) (filter is_rec args)) 
				 cons))
				(conss~~casess)));

val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n",
		mk_trp(dc_take dn $ Bound 0 `%(x_name n) === 
		       dc_take dn $ Bound 0 `%(x_name n^"'")))
	   ===> mk_trp(%(x_name n) === %(x_name n^"'"))) (fn prems => [
			res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
			res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
				rtac (fix_def2 RS ssubst) 1,
				REPEAT(CHANGED(rtac(contlub_cfun_arg RS ssubst)1
					       THEN chain_tac 1)),
				rtac (contlub_cfun_fun RS ssubst) 1,
				rtac (contlub_cfun_fun RS ssubst) 2,
				rtac lub_equal 3,
				chain_tac 1,
				rtac allI 1,
				resolve_tac prems 1])) 1 (dnames~~axs_reach);

(* ----- theorems concerning finiteness and induction ----------------------- *)

val (finites,ind) = if is_finite then
  let 
    fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %"x" === %"x");
    val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%"x")) ===> 
	mk_trp(mk_disj(mk_all("n",dc_take dn $ Bound 0 ` %"x" === UU),
	take_enough dn)) ===> mk_trp(take_enough dn)) [
				etac disjE 1,
				etac notE 1,
				resolve_tac take_lemmas 1,
				asm_simp_tac take_ss 1,
				atac 1]) dnames;
    val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn 
	(fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
	 mk_disj(dc_take dn $ Bound 1 ` Bound 0 === UU,
		 dc_take dn $ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
				rtac allI 1,
				nat_ind_tac "n" 1,
				simp_tac take_ss 1,
			TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @
				flat(mapn (fn n => fn (cons,cases) => [
				  simp_tac take_ss 1,
				  rtac allI 1,
				  res_inst_tac [("x",x_name n)] cases 1,
				  asm_simp_tac take_ss 1] @ 
				  flat(map (fn (con,args) => 
				    asm_simp_tac take_ss 1 ::
				    flat(map (fn vn => [
				      eres_inst_tac [("x",vn)] all_dupE 1,
				      etac disjE 1,
				      asm_simp_tac (HOL_ss addsimps con_rews) 1,
				      asm_simp_tac take_ss 1])
				    (nonlazy_rec args)))
				  cons))
				1 (conss~~casess))) handle ERROR => raise ERROR;
    val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp(
						%%(dn^"_finite") $ %"x"))[
				case_UU_tac take_rews 1 "x",
				eresolve_tac finite_lemmas1a 1,
				step_tac HOL_cs 1,
				step_tac HOL_cs 1,
				cut_facts_tac [l1b] 1,
			fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
  in
  (finites,
   pg'' thy[](ind_term (fn n => fn dn => %(P_name n) $ %(x_name n)))(fn prems =>
				TRY(safe_tac HOL_cs) ::
			 flat (map (fn (finite,fin_ind) => [
			       rtac(rewrite_rule axs_finite_def finite RS exE)1,
				etac subst 1,
				rtac fin_ind 1,
				ind_prems_tac prems]) 
			           (finites~~(atomize finite_ind)) ))
) end (* let *) else
  (mapn (fn n => fn dn => read_instantiate_sg (sign_of thy) 
	  	    [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
   pg'' thy [] (foldr (op ===>) (mapn (fn n => K(mk_trp(%%"adm" $ %(P_name n))))
	       1 dnames, ind_term (fn n => fn dn => %(P_name n) $ %(x_name n))))
		   (fn prems => map (fn ax_reach => rtac (ax_reach RS subst) 1) 
				    axs_reach @ [
				quant_tac 1,
				rtac (adm_impl_admw RS wfix_ind) 1,
				REPEAT_DETERM(rtac adm_all2 1),
				REPEAT_DETERM(TRY(rtac adm_conj 1) THEN 
						  rtac adm_subst 1 THEN 
					cont_tacR 1 THEN resolve_tac prems 1),
				strip_tac 1,
				rtac (rewrite_rule axs_take_def finite_ind) 1,
				ind_prems_tac prems])
)
end; (* local *)

(* ----- theorem concerning coinduction ------------------------------------- *)

local
  val xs = mapn (fn n => K (x_name n)) 1 dnames;
  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
  val take_ss = HOL_ss addsimps take_rews;
  val sproj   = prj (fn s => "fst("^s^")") (fn s => "snd("^s^")");
  val coind_lemma=pg[ax_bisim_def](mk_trp(mk_imp(%%(comp_dname^"_bisim") $ %"R",
		foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
		  foldr mk_imp (mapn (fn n => K(proj (%"R") n_eqs n $ 
				      bnd_arg n 0 $ bnd_arg n 1)) 0 dnames,
		    foldr' mk_conj (mapn (fn n => fn dn => 
				(dc_take dn $ %"n" `bnd_arg n 0 === 
				(dc_take dn $ %"n" `bnd_arg n 1)))0 dnames))))))
			     ([ rtac impI 1,
				nat_ind_tac "n" 1,
				simp_tac take_ss 1,
				safe_tac HOL_cs] @
				flat(mapn (fn n => fn x => [
				  rotate_tac (n+1) 1,
				  etac all2E 1,
				  eres_inst_tac [("P1", sproj "R" n_eqs n^
					" "^x^" "^x^"'")](mp RS disjE) 1,
				  TRY(safe_tac HOL_cs),
				  REPEAT(CHANGED(asm_simp_tac take_ss 1))]) 
				0 xs));
in
val coind = pg [] (mk_trp(%%(comp_dname^"_bisim") $ %"R") ===>
		foldr (op ===>) (mapn (fn n => fn x => 
		  mk_trp(proj (%"R") n_eqs n $ %x $ %(x^"'"))) 0 xs,
		  mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
				TRY(safe_tac HOL_cs)] @
				flat(map (fn take_lemma => [
				  rtac take_lemma 1,
				  cut_facts_tac [coind_lemma] 1,
				  fast_tac HOL_cs 1])
				take_lemmas));
end; (* local *)


in (take_rews, take_lemmas, finites, finite_ind, ind, coind)

end; (* let *)
end; (* local *)
end; (* struct *)