# HG changeset patch
# User paulson
# Date 828006779 -3600
# Node ID 2b8573c1b1c10a78d030f02010851c1ba95fc0ec
# Parent  4b0608ce6150a9618c6a93233b3fb00292880d29
Ran expandshort

diff -r 4b0608ce6150 -r 2b8573c1b1c1 src/ZF/Arith.ML
--- a/src/ZF/Arith.ML	Thu Mar 28 10:45:32 1996 +0100
+++ b/src/ZF/Arith.ML	Thu Mar 28 10:52:59 1996 +0100
@@ -347,16 +347,16 @@
 \           succ(m) mod n = if(succ(m mod n) = n, 0, succ(m mod n))";
 by (etac complete_induct 1);
 by (excluded_middle_tac "succ(x)<n" 1);
-(*case x<n*)
+(* case succ(x) < n *)
 by (asm_simp_tac (arith_ss addsimps [mod_less, nat_le_refl RS lt_trans,
-				     succ_neq_self]) 2);
+                                     succ_neq_self]) 2);
 by (asm_simp_tac (arith_ss addsimps [ltD RS mem_imp_not_eq]) 2);
-(*case n le x*)
+(* case n le succ(x) *)
 by (asm_full_simp_tac
      (arith_ss addsimps [not_lt_iff_le, nat_into_Ord, mod_geq]) 1);
-be leE 1;
+by (etac leE 1);
 by (asm_simp_tac (arith_ss addsimps [div_termination RS ltD, diff_succ, 
-				     mod_geq]) 1);
+                                     mod_geq]) 1);
 by (asm_simp_tac (arith_ss addsimps [mod_less, diff_self_eq_0]) 1);
 qed "mod_succ";
 
@@ -376,7 +376,7 @@
     "!!k b. [| k: nat; b<2 |] ==> k mod 2 = b | k mod 2 = if(b=1,0,1)";
 by (subgoal_tac "k mod 2: 2" 1);
 by (asm_simp_tac (arith_ss addsimps [mod_less_divisor RS ltD]) 2);
-by (dresolve_tac [ltD] 1);
+by (dtac ltD 1);
 by (asm_simp_tac (ZF_ss setloop split_tac [expand_if]) 1);
 by (fast_tac ZF_cs 1);
 qed "mod2_cases";
@@ -388,7 +388,7 @@
 qed "mod2_succ_succ";
 
 goal Arith.thy "!!m. m:nat ==> (m#+m) mod 2 = 0";
-by (eresolve_tac [nat_induct] 1);
+by (etac nat_induct 1);
 by (simp_tac (arith_ss addsimps [mod_less]) 1);
 by (asm_simp_tac (arith_ss addsimps [mod2_succ_succ, add_succ_right]) 1);
 qed "mod2_add_self";
@@ -456,5 +456,5 @@
 
 val arith_ss0 = arith_ss
 and arith_ss = arith_ss addsimps [add_0_right, add_succ_right,
-				  mult_0_right, mult_succ_right,
-				  mod_less, mod_geq, div_less, div_geq];
+                                  mult_0_right, mult_succ_right,
+                                  mod_less, mod_geq, div_less, div_geq];
diff -r 4b0608ce6150 -r 2b8573c1b1c1 src/ZF/Cardinal.ML
--- a/src/ZF/Cardinal.ML	Thu Mar 28 10:45:32 1996 +0100
+++ b/src/ZF/Cardinal.ML	Thu Mar 28 10:52:59 1996 +0100
@@ -417,10 +417,10 @@
 
 
 goal Cardinal.thy "!!A. [| A lepoll i; Ord(i) |] ==> |A| le i";
-br le_trans 1;
-be (well_ord_Memrel RS well_ord_lepoll_imp_Card_le) 1;
-ba 1;
-be Ord_cardinal_le 1;
+by (rtac le_trans 1);
+by (etac (well_ord_Memrel RS well_ord_lepoll_imp_Card_le) 1);
+by (assume_tac 1);
+by (etac Ord_cardinal_le 1);
 qed "lepoll_cardinal_le";
 
 
@@ -675,7 +675,7 @@
 by (split_tac [expand_if] 1);
 by (fast_tac (ZF_cs addSIs [InlI, InrI]) 1);
 by (asm_full_simp_tac (ZF_ss addsimps [Inl_def, Inr_def]
-		       setloop split_tac [expand_if]) 1);
+                       setloop split_tac [expand_if]) 1);
 qed "Un_lepoll_sum";
 
 
@@ -700,7 +700,9 @@
                      rewrite_rule [Finite_def] lepoll_nat_imp_Finite]) 1);
 qed "lepoll_Finite";
 
-bind_thm ("Finite_Diff", Diff_subset RS subset_imp_lepoll RS lepoll_Finite);
+bind_thm ("subset_Finite", subset_imp_lepoll RS lepoll_Finite);
+
+bind_thm ("Finite_Diff", Diff_subset RS subset_Finite);
 
 goalw Cardinal.thy [Finite_def] "!!x. Finite(x) ==> Finite(cons(y,x))";
 by (excluded_middle_tac "y:x" 1);
@@ -726,7 +728,7 @@
 goalw Cardinal.thy [Finite_def, eqpoll_def]
     "!!A. Finite(A) ==> EX r. well_ord(A,r)";
 by (fast_tac (ZF_cs addIs [well_ord_rvimage, bij_is_inj, well_ord_Memrel, 
-			   nat_into_Ord]) 1);
+                           nat_into_Ord]) 1);
 qed "Finite_imp_well_ord";
 
 
diff -r 4b0608ce6150 -r 2b8573c1b1c1 src/ZF/ex/Limit.ML
--- a/src/ZF/ex/Limit.ML	Thu Mar 28 10:45:32 1996 +0100
+++ b/src/ZF/ex/Limit.ML	Thu Mar 28 10:52:59 1996 +0100
@@ -31,12 +31,12 @@
 
 val prems = goalw Limit.thy [set_def] 
     "x:fst(D) ==> x:set(D)";
-by(resolve_tac prems 1);
+by (resolve_tac prems 1);
 val set_I = result();
 
 val prems = goalw Limit.thy [rel_def]
     "<x,y>:snd(D) ==> rel(D,x,y)";
-by(resolve_tac prems 1);
+by (resolve_tac prems 1);
 val rel_I = result();
 
 val prems = goalw Limit.thy [rel_def]
@@ -50,15 +50,15 @@
 
 val prems = goalw Limit.thy [po_def]
     "[|po(D); x:set(D)|] ==> rel(D,x,x)";
-br(hd prems RS conjunct1 RS bspec)1;
-by(resolve_tac prems 1);
+by (rtac (hd prems RS conjunct1 RS bspec) 1);
+by (resolve_tac prems 1);
 val po_refl = result();
 
 val [po,xy,yz,x,y,z] = goalw Limit.thy [po_def]
     "[|po(D); rel(D,x,y); rel(D,y,z); x:set(D);  \
 \      y:set(D); z:set(D)|] ==> rel(D,x,z)";
-br(po RS conjunct2 RS conjunct1 RS bspec RS bspec 
-   RS bspec RS mp RS mp)1;
+br (po RS conjunct2 RS conjunct1 RS bspec RS bspec 
+   RS bspec RS mp RS mp) 1;
 by (rtac x 1); 
 by (rtac y 1);
 by (rtac z 1);
@@ -68,8 +68,8 @@
 
 val prems = goalw Limit.thy [po_def]
     "[|po(D); rel(D,x,y); rel(D,y,x); x:set(D); y:set(D)|] ==> x = y";
-br(hd prems RS conjunct2 RS conjunct2 RS bspec RS bspec RS mp RS mp)1;
-by(REPEAT(resolve_tac prems 1));
+by (rtac (hd prems RS conjunct2 RS conjunct2 RS bspec RS bspec RS mp RS mp) 1);
+by (REPEAT(resolve_tac prems 1));
 val po_antisym = result();
 
 val prems = goalw Limit.thy [po_def]
@@ -78,42 +78,42 @@
 \                rel(D,x,z);  \
 \       !!x y. [| rel(D,x,y); rel(D,y,x); x:set(D); y:set(D)|] ==> x=y |] ==> \
 \    po(D)";
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 brr prems 1;
 val poI = result();
 
 val prems = goalw Limit.thy [cpo_def]
     "[| po(D); !!X. chain(D,X) ==> islub(D,X,x(D,X))|] ==> cpo(D)";
-by(safe_tac (lemmas_cs addSIs [exI]));
+by (safe_tac (lemmas_cs addSIs [exI]));
 brr prems 1;
 val cpoI = result();
 
 val [cpo] = goalw Limit.thy [cpo_def] "cpo(D) ==> po(D)";
-br(cpo RS conjunct1)1;
+by (rtac (cpo RS conjunct1) 1);
 val cpo_po = result();
 
 val prems = goal Limit.thy
     "[|cpo(D); x:set(D)|] ==> rel(D,x,x)";
 by (rtac po_refl 1);
-by(REPEAT(resolve_tac ((hd prems RS cpo_po)::prems) 1));
+by (REPEAT(resolve_tac ((hd prems RS cpo_po)::prems) 1));
 val cpo_refl = result();
 
 val prems = goal Limit.thy
     "[|cpo(D); rel(D,x,y); rel(D,y,z); x:set(D);  \
 \      y:set(D); z:set(D)|] ==> rel(D,x,z)";
 by (rtac po_trans 1);
-by(REPEAT(resolve_tac ((hd prems RS cpo_po)::prems) 1));
+by (REPEAT(resolve_tac ((hd prems RS cpo_po)::prems) 1));
 val cpo_trans = result();
 
 val prems = goal Limit.thy
     "[|cpo(D); rel(D,x,y); rel(D,y,x); x:set(D); y:set(D)|] ==> x = y";
 by (rtac po_antisym 1);
-by(REPEAT(resolve_tac ((hd prems RS cpo_po)::prems) 1));
+by (REPEAT(resolve_tac ((hd prems RS cpo_po)::prems) 1));
 val cpo_antisym = result();
 
 val [cpo,chain,ex] = goalw Limit.thy [cpo_def] (* cpo_islub *)
   "[|cpo(D); chain(D,X);!!x. islub(D,X,x) ==> R|] ==> R";
-br(chain RS (cpo RS conjunct2 RS spec RS mp) RS exE)1; 
+by (rtac (chain RS (cpo RS conjunct2 RS spec RS mp) RS exE) 1); 
 brr[ex]1; (* above theorem would loop *)
 val cpo_islub = result();
 
@@ -123,53 +123,53 @@
 
 val prems = goalw Limit.thy [islub_def]  (* islub_isub *)
     "islub(D,X,x) ==> isub(D,X,x)";
-by(simp_tac (ZF_ss addsimps prems) 1);
+by (simp_tac (ZF_ss addsimps prems) 1);
 val islub_isub = result();
 
 val prems = goal Limit.thy
     "islub(D,X,x) ==> x:set(D)";
-br(rewrite_rule[islub_def,isub_def](hd prems) RS conjunct1 RS conjunct1)1;
+by (rtac (rewrite_rule[islub_def,isub_def](hd prems) RS conjunct1 RS conjunct1) 1);
 val islub_in = result();
 
 val prems = goal Limit.thy
     "[|islub(D,X,x); n:nat|] ==> rel(D,X`n,x)";
-br(rewrite_rule[islub_def,isub_def](hd prems) RS conjunct1 
-   RS conjunct2 RS bspec)1;
-by(resolve_tac prems 1);
+br (rewrite_rule[islub_def,isub_def](hd prems) RS conjunct1 
+   RS conjunct2 RS bspec) 1;
+by (resolve_tac prems 1);
 val islub_ub = result();
 
 val prems = goalw Limit.thy [islub_def]
     "[|islub(D,X,x); isub(D,X,y)|] ==> rel(D,x,y)";
-br(hd prems RS conjunct2 RS spec RS mp)1;
-by(resolve_tac prems 1);
+by (rtac (hd prems RS conjunct2 RS spec RS mp) 1);
+by (resolve_tac prems 1);
 val islub_least = result();
 
 val prems = goalw Limit.thy [islub_def]  (* islubI *)
     "[|isub(D,X,x); !!y. isub(D,X,y) ==> rel(D,x,y)|] ==> islub(D,X,x)";
-by(safe_tac lemmas_cs);
-by(REPEAT(ares_tac prems 1));
+by (safe_tac lemmas_cs);
+by (REPEAT(ares_tac prems 1));
 val islubI = result();
 
 val prems = goalw Limit.thy [isub_def]  (* isubI *)
     "[|x:set(D);!!n. n:nat ==> rel(D,X`n,x)|] ==> isub(D,X,x)";
-by(safe_tac lemmas_cs);
-by(REPEAT(ares_tac prems 1));
+by (safe_tac lemmas_cs);
+by (REPEAT(ares_tac prems 1));
 val isubI = result();
 
 val prems = goalw Limit.thy [isub_def]  (* isubE *)
     "!!z.[|isub(D,X,x);[|x:set(D);!!n.n:nat==>rel(D,X`n,x)|] ==> P|] ==> P";
-by(safe_tac lemmas_cs);
-by(asm_simp_tac ZF_ss 1);
+by (safe_tac lemmas_cs);
+by (asm_simp_tac ZF_ss 1);
 val isubE = result();
 
 val prems = goalw Limit.thy [isub_def]  (* isubD1 *)
     "isub(D,X,x) ==> x:set(D)";
-by(simp_tac (ZF_ss addsimps prems) 1);
+by (simp_tac (ZF_ss addsimps prems) 1);
 val isubD1 = result();
 
 val prems = goalw Limit.thy [isub_def]  (* isubD2 *)
     "[|isub(D,X,x); n:nat|]==>rel(D,X`n,x)";
-by(simp_tac (ZF_ss addsimps prems) 1);
+by (simp_tac (ZF_ss addsimps prems) 1);
 val isubD2 = result();
 
 val prems = goal Limit.thy
@@ -205,49 +205,49 @@
 
 val prems = goalw Limit.thy [chain_def]
     "chain(D,X) ==> X : nat -> set(D)";
-by(asm_simp_tac (ZF_ss addsimps prems) 1);
+by (asm_simp_tac (ZF_ss addsimps prems) 1);
 val chain_fun = result();
     
 val prems = goalw Limit.thy [chain_def]
     "[|chain(D,X); n:nat|] ==> X`n : set(D)";
-br((hd prems)RS conjunct1 RS apply_type)1;
-br(hd(tl prems))1;
+by (rtac ((hd prems)RS conjunct1 RS apply_type) 1);
+by (rtac (hd(tl prems)) 1);
 val chain_in = result();
     
 val prems = goalw Limit.thy [chain_def]
     "[|chain(D,X); n:nat|] ==> rel(D, X ` n, X ` succ(n))";
-br((hd prems)RS conjunct2 RS bspec)1;
-br(hd(tl prems))1;
+by (rtac ((hd prems)RS conjunct2 RS bspec) 1);
+by (rtac (hd(tl prems)) 1);
 val chain_rel = result();
     
 val prems = goal Limit.thy  (* chain_rel_gen_add *)
     "[|chain(D,X); cpo(D); n:nat; m:nat|] ==> rel(D,X`n,(X`(m #+ n)))";
-by(res_inst_tac [("n","m")] nat_induct 1);
-by(ALLGOALS(simp_tac arith_ss));
+by (res_inst_tac [("n","m")] nat_induct 1);
+by (ALLGOALS(simp_tac arith_ss));
 by (rtac cpo_trans 3); (* loops if repeated *)
-brr(cpo_refl::chain_in::chain_rel::nat_succI::add_type::prems)1;
+brr(cpo_refl::chain_in::chain_rel::nat_succI::add_type::prems) 1;
 val chain_rel_gen_add = result();
 
 val prems = goal Limit.thy  (* le_succ_eq *)
   "[| n le succ(x); ~ n le x; x : nat; n:nat |] ==> n = succ(x)";
 by (rtac le_anti_sym 1);
-by(resolve_tac prems 1);
-by(simp_tac arith_ss 1);
-br(not_le_iff_lt RS iffD1)1;
-by(REPEAT(resolve_tac (nat_into_Ord::prems) 1));
+by (resolve_tac prems 1);
+by (simp_tac arith_ss 1);
+by (rtac (not_le_iff_lt RS iffD1) 1);
+by (REPEAT(resolve_tac (nat_into_Ord::prems) 1));
 val le_succ_eq = result();
 
 val prems = goal Limit.thy  (* chain_rel_gen *)
     "[|n le m; chain(D,X); cpo(D); n:nat; m:nat|] ==> rel(D,X`n,X`m)";
 by (rtac impE 1);  (* The first three steps prepare for the induction proof *)
 by (assume_tac 3);
-br(hd prems)2;
-by(res_inst_tac [("n","m")] nat_induct 1);
-by(safe_tac lemmas_cs);
-by(asm_full_simp_tac (arith_ss addsimps prems) 2);
+by (rtac (hd prems) 2);
+by (res_inst_tac [("n","m")] nat_induct 1);
+by (safe_tac lemmas_cs);
+by (asm_full_simp_tac (arith_ss addsimps prems) 2);
 by (rtac cpo_trans 4);
 by (rtac (le_succ_eq RS subst) 3);
-brr(cpo_refl::chain_in::chain_rel::nat_0I::nat_succI::prems)1;
+brr(cpo_refl::chain_in::chain_rel::nat_0I::nat_succI::prems) 1;
 val chain_rel_gen = result();
 
 (*----------------------------------------------------------------------*)
@@ -265,22 +265,22 @@
 val pcpoI = result();
 
 val pcpo_cpo = prove_goalw Limit.thy [pcpo_def] "pcpo(D) ==> cpo(D)"
-    (fn [pcpo] => [rtac(pcpo RS conjunct1)1]);
+    (fn [pcpo] => [rtac(pcpo RS conjunct1) 1]);
 
 val prems = goalw Limit.thy [pcpo_def] (* pcpo_bot_ex1 *)
     "pcpo(D) ==> EX! x. x:set(D) & (ALL y:set(D). rel(D,x,y))";
-br(hd prems RS conjunct2 RS bexE)1;
+by (rtac (hd prems RS conjunct2 RS bexE) 1);
 by (rtac ex1I 1);
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 by (assume_tac 1);
 by (etac bspec 1);
 by (assume_tac 1);
 by (rtac cpo_antisym 1);
-br(hd prems RS conjunct1)1;
+by (rtac (hd prems RS conjunct1) 1);
 by (etac bspec 1);
 by (assume_tac 1);
 by (etac bspec 1);
-by(REPEAT(atac 1));
+by (REPEAT(atac 1));
 val pcpo_bot_ex1 = result();
 
 val prems = goalw Limit.thy [bot_def] (* bot_least *)
@@ -305,7 +305,7 @@
 val prems = goal Limit.thy  (* bot_unique *)
     "[| pcpo(D); x:set(D); !!y. y:set(D) ==> rel(D,x,y)|] ==> x = bot(D)";
 by (rtac cpo_antisym 1);
-brr(pcpo_cpo::bot_in::bot_least::prems)1;
+brr(pcpo_cpo::bot_in::bot_least::prems) 1;
 val bot_unique = result();
 
 (*----------------------------------------------------------------------*)
@@ -316,18 +316,18 @@
     "[|x:set(D); cpo(D)|] ==> chain(D,(lam n:nat. x))";
 by (rtac conjI 1);
 by (rtac lam_type 1);
-by(resolve_tac prems 1);
+by (resolve_tac prems 1);
 by (rtac ballI 1);
-by(asm_simp_tac (ZF_ss addsimps [nat_succI]) 1);
-brr(cpo_refl::prems)1;
+by (asm_simp_tac (ZF_ss addsimps [nat_succI]) 1);
+brr(cpo_refl::prems) 1;
 val chain_const = result();
 
 val prems = goalw Limit.thy [islub_def,isub_def] (* islub_const *)
     "[|x:set(D); cpo(D)|] ==> islub(D,(lam n:nat. x),x)";
-by(simp_tac ZF_ss 1);
-by(safe_tac lemmas_cs);
+by (simp_tac ZF_ss 1);
+by (safe_tac lemmas_cs);
 by (etac bspec 3);
-brr(cpo_refl::nat_0I::prems)1;
+brr(cpo_refl::nat_0I::prems) 1;
 val islub_const = result();
 
 val prems = goal Limit.thy  (* lub_const *)
@@ -335,9 +335,9 @@
 by (rtac islub_unique 1);
 by (rtac cpo_lub 1);
 by (rtac chain_const 1);
-by(REPEAT(resolve_tac prems 1));
+by (REPEAT(resolve_tac prems 1));
 by (rtac islub_const 1);
-by(REPEAT(resolve_tac prems 1));
+by (REPEAT(resolve_tac prems 1));
 val lub_const = result();
 
 (*----------------------------------------------------------------------*)
@@ -346,25 +346,25 @@
 
 val prems = goalw Limit.thy [isub_def,suffix_def]  (* isub_suffix *)
     "[|chain(D,X); cpo(D); n:nat|] ==> isub(D,suffix(X,n),x) <-> isub(D,X,x)";
-by(simp_tac (ZF_ss addsimps prems) 1);
-by(safe_tac lemmas_cs);
+by (simp_tac (ZF_ss addsimps prems) 1);
+by (safe_tac lemmas_cs);
 by (dtac bspec 2);
 by (assume_tac 3);      (* to instantiate unknowns properly *)
 by (rtac cpo_trans 1);
 by (rtac chain_rel_gen_add 2);
 by (dtac bspec 6);
 by (assume_tac 7);       (* to instantiate unknowns properly *)
-brr(chain_in::add_type::prems)1;
+brr(chain_in::add_type::prems) 1;
 val isub_suffix = result();
 
 val prems = goalw Limit.thy [islub_def]  (* islub_suffix *)
     "[|chain(D,X); cpo(D); n:nat|] ==> islub(D,suffix(X,n),x) <-> islub(D,X,x)";
-by(asm_simp_tac (FOL_ss addsimps isub_suffix::prems) 1);
+by (asm_simp_tac (FOL_ss addsimps isub_suffix::prems) 1);
 val islub_suffix = result();
 
 val prems = goalw Limit.thy [lub_def]  (* lub_suffix *)
     "[|chain(D,X); cpo(D); n:nat|] ==> lub(D,suffix(X,n)) = lub(D,X)";
-by(asm_simp_tac (FOL_ss addsimps islub_suffix::prems) 1);
+by (asm_simp_tac (FOL_ss addsimps islub_suffix::prems) 1);
 val lub_suffix = result();
 
 (*----------------------------------------------------------------------*)
@@ -379,20 +379,20 @@
 val [dom,isub,cpo,X,Y] = goal Limit.thy
   "[|dominate(D,X,Y); isub(D,Y,x); cpo(D);  \
 \    X:nat->set(D); Y:nat->set(D)|] ==> isub(D,X,x)";
-by(rewtac isub_def);
+by (rewtac isub_def);
 by (rtac conjI 1);
-br(rewrite_rule[isub_def]isub RS conjunct1)1;
+by (rtac (rewrite_rule[isub_def]isub RS conjunct1) 1);
 by (rtac ballI 1);
-br(rewrite_rule[dominate_def]dom RS bspec RS bexE)1;
+by (rtac (rewrite_rule[dominate_def]dom RS bspec RS bexE) 1);
 by (assume_tac 1);
 by (rtac cpo_trans 1);
 by (rtac cpo 1);
 by (assume_tac 1);
-br(rewrite_rule[isub_def]isub RS conjunct2 RS bspec)1;
+by (rtac (rewrite_rule[isub_def]isub RS conjunct2 RS bspec) 1);
 by (assume_tac 1);
-be(X RS apply_type)1;
-be(Y RS apply_type)1;
-br(rewrite_rule[isub_def]isub RS conjunct1)1;
+by (etac (X RS apply_type) 1);
+by (etac (Y RS apply_type) 1);
+by (rtac (rewrite_rule[isub_def]isub RS conjunct1) 1);
 val dominate_isub = result();
 
 val [dom,Xlub,Ylub,cpo,X,Y] = goal Limit.thy
@@ -408,37 +408,37 @@
 
 val prems = goalw Limit.thy [subchain_def]  (* subchainE *)
     "[|subchain(X,Y); n:nat;!!m. [|m:nat; X`n = Y`(n #+ m)|] ==> Q|] ==> Q";
-br(hd prems RS bspec RS bexE)1;
-by(resolve_tac prems 2);
+by (rtac (hd prems RS bspec RS bexE) 1);
+by (resolve_tac prems 2);
 by (assume_tac 3);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 val subchainE = result();
 
 val prems = goalw Limit.thy []  (* subchain_isub *)
     "[|subchain(Y,X); isub(D,X,x)|] ==> isub(D,Y,x)";
 by (rtac isubI 1);
 val [subch,ub] = prems;
-br(ub RS isubD1)1;
-br(subch RS subchainE)1;
+by (rtac (ub RS isubD1) 1);
+by (rtac (subch RS subchainE) 1);
 by (assume_tac 1);
-by(asm_simp_tac ZF_ss 1);
+by (asm_simp_tac ZF_ss 1);
 by (rtac isubD2 1);                (* br with Destruction rule ?? *)
-by(resolve_tac prems 1);
-by(asm_simp_tac arith_ss 1);
+by (resolve_tac prems 1);
+by (asm_simp_tac arith_ss 1);
 val subchain_isub = result();
 
 val prems = goal Limit.thy  (* dominate_islub_eq *)
   "[|dominate(D,X,Y); subchain(Y,X); islub(D,X,x); islub(D,Y,y); cpo(D);  \
 \    X:nat->set(D); Y:nat->set(D)|] ==> x = y";
 by (rtac cpo_antisym 1);
-by(resolve_tac prems 1);
+by (resolve_tac prems 1);
 by (rtac dominate_islub 1);
-by(REPEAT(resolve_tac prems 1));
+by (REPEAT(resolve_tac prems 1));
 by (rtac islub_least 1);
-by(REPEAT(resolve_tac prems 1));
+by (REPEAT(resolve_tac prems 1));
 by (rtac subchain_isub 1);
 by (rtac islub_isub 2);
-by(REPEAT(resolve_tac (islub_in::prems) 1));
+by (REPEAT(resolve_tac (islub_in::prems) 1));
 val dominate_islub_eq = result();
 
 (*----------------------------------------------------------------------*)
@@ -447,34 +447,34 @@
 
 val prems = goalw Limit.thy [matrix_def]  (* matrix_fun *)
     "matrix(D,M) ==> M : nat -> (nat -> set(D))";
-by(simp_tac (ZF_ss addsimps prems) 1);
+by (simp_tac (ZF_ss addsimps prems) 1);
 val matrix_fun = result();
 
 val prems = goalw Limit.thy []  (* matrix_in_fun *)
     "[|matrix(D,M); n:nat|] ==> M`n : nat -> set(D)";
 by (rtac apply_type 1);
-by(REPEAT(resolve_tac(matrix_fun::prems)1));
+by (REPEAT(resolve_tac(matrix_fun::prems) 1));
 val matrix_in_fun = result();
 
 val prems = goalw Limit.thy []  (* matrix_in *)
     "[|matrix(D,M); n:nat; m:nat|] ==> M`n`m : set(D)";
 by (rtac apply_type 1);
-by(REPEAT(resolve_tac(matrix_in_fun::prems)1));
+by (REPEAT(resolve_tac(matrix_in_fun::prems) 1));
 val matrix_in = result();
 
 val prems = goalw Limit.thy [matrix_def]  (* matrix_rel_1_0 *)
     "[|matrix(D,M); n:nat; m:nat|] ==> rel(D,M`n`m,M`succ(n)`m)";
-by(simp_tac (ZF_ss addsimps prems) 1);
+by (simp_tac (ZF_ss addsimps prems) 1);
 val matrix_rel_1_0 = result();
 
 val prems = goalw Limit.thy [matrix_def]  (* matrix_rel_0_1 *)
     "[|matrix(D,M); n:nat; m:nat|] ==> rel(D,M`n`m,M`n`succ(m))";
-by(simp_tac (ZF_ss addsimps prems) 1);
+by (simp_tac (ZF_ss addsimps prems) 1);
 val matrix_rel_0_1 = result();
 
 val prems = goalw Limit.thy [matrix_def]  (* matrix_rel_1_1 *)
     "[|matrix(D,M); n:nat; m:nat|] ==> rel(D,M`n`m,M`succ(n)`succ(m))";
-by(simp_tac (ZF_ss addsimps prems) 1);
+by (simp_tac (ZF_ss addsimps prems) 1);
 val matrix_rel_1_1 = result();
 
 val prems = goal Limit.thy  (* fun_swap *)
@@ -483,54 +483,54 @@
 by (rtac lam_type 1);
 by (rtac apply_type 1);
 by (rtac apply_type 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 val fun_swap = result();
 
 val prems = goalw Limit.thy [matrix_def]  (* matrix_sym_axis *)
     "!!z. matrix(D,M) ==> matrix(D,lam m:nat. lam n:nat. M`n`m)";
-by(simp_tac arith_ss 1 THEN safe_tac lemmas_cs THEN 
+by (simp_tac arith_ss 1 THEN safe_tac lemmas_cs THEN 
 REPEAT(asm_simp_tac (ZF_ss addsimps [fun_swap]) 1));
 val matrix_sym_axis = result();
 
 val prems = goalw Limit.thy [chain_def]  (* matrix_chain_diag *)
     "matrix(D,M) ==> chain(D,lam n:nat. M`n`n)";
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 by (rtac lam_type 1);
 by (rtac matrix_in 1);
-by(REPEAT(ares_tac prems 1));
-by(asm_simp_tac arith_ss 1);
+by (REPEAT(ares_tac prems 1));
+by (asm_simp_tac arith_ss 1);
 by (rtac matrix_rel_1_1 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 val matrix_chain_diag = result();
 
 val prems = goalw Limit.thy [chain_def]  (* matrix_chain_left *)
     "[|matrix(D,M); n:nat|] ==> chain(D,M`n)";
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 by (rtac apply_type 1);
 by (rtac matrix_fun 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 by (rtac matrix_rel_0_1 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 val matrix_chain_left = result();
 
 val prems = goalw Limit.thy [chain_def]  (* matrix_chain_right *)
     "[|matrix(D,M); m:nat|] ==> chain(D,lam n:nat. M`n`m)";
-by(safe_tac lemmas_cs);
-by(asm_simp_tac(arith_ss addsimps prems)2);
-brr(lam_type::matrix_in::matrix_rel_1_0::prems)1;
+by (safe_tac lemmas_cs);
+by (asm_simp_tac(arith_ss addsimps prems) 2);
+brr(lam_type::matrix_in::matrix_rel_1_0::prems) 1;
 val matrix_chain_right = result();
 
 val prems = goalw Limit.thy [matrix_def]  (* matrix_chainI *)
     "[|!!x.x:nat==>chain(D,M`x);!!y.y:nat==>chain(D,lam x:nat. M`x`y);   \
 \      M:nat->nat->set(D); cpo(D)|] ==> matrix(D,M)";
-by(safe_tac (lemmas_cs addSIs [ballI]));
-by(cut_inst_tac[("y1","m"),("n","n")](hd(tl prems) RS chain_rel)2);
-by(asm_full_simp_tac arith_ss 4);
+by (safe_tac (lemmas_cs addSIs [ballI]));
+by (cut_inst_tac[("y1","m"),("n","n")](hd(tl prems) RS chain_rel) 2);
+by (asm_full_simp_tac arith_ss 4);
 by (rtac cpo_trans 5);
-by(cut_inst_tac[("y1","m"),("n","n")](hd(tl prems) RS chain_rel)6);
-by(asm_full_simp_tac arith_ss 8);
-by(TRYALL(rtac (chain_fun RS apply_type)));
-brr(chain_rel::nat_succI::prems)1;
+by (cut_inst_tac[("y1","m"),("n","n")](hd(tl prems) RS chain_rel) 6);
+by (asm_full_simp_tac arith_ss 8);
+by (TRYALL(rtac (chain_fun RS apply_type)));
+brr(chain_rel::nat_succI::prems) 1;
 val matrix_chainI = result();
 
 val lemma = prove_goal Limit.thy
@@ -545,69 +545,69 @@
 val prems = goalw Limit.thy []  (* isub_lemma *)
     "[|isub(D,(lam n:nat. M`n`n),y); matrix(D,M); cpo(D)|] ==>  \
 \    isub(D,(lam n:nat. lub(D,lam m:nat. M`n`m)),y)";
-by(rewtac isub_def);
-by(safe_tac lemmas_cs);
+by (rewtac isub_def);
+by (safe_tac lemmas_cs);
 by (rtac isubD1 1);
-by(resolve_tac prems 1);
-by(asm_simp_tac ZF_ss 1);
-by(cut_inst_tac[("a","n")](hd(tl prems) RS matrix_fun RS apply_type)1);
+by (resolve_tac prems 1);
+by (asm_simp_tac ZF_ss 1);
+by (cut_inst_tac[("a","n")](hd(tl prems) RS matrix_fun RS apply_type) 1);
 by (assume_tac 1);
-by(asm_simp_tac ZF_ss 1);
+by (asm_simp_tac ZF_ss 1);
 by (rtac islub_least 1);
 by (rtac cpo_lub 1);
 by (rtac matrix_chain_left 1);
-by(resolve_tac prems 1);
+by (resolve_tac prems 1);
 by (assume_tac 1);
-by(resolve_tac prems 1);
-by(rewtac isub_def);
-by(safe_tac lemmas_cs);
+by (resolve_tac prems 1);
+by (rewtac isub_def);
+by (safe_tac lemmas_cs);
 by (rtac isubD1 1);
-by(resolve_tac prems 1);
-by(cut_inst_tac[("P","n le na")]excluded_middle 1);
-by(safe_tac lemmas_cs);
+by (resolve_tac prems 1);
+by (cut_inst_tac[("P","n le na")]excluded_middle 1);
+by (safe_tac lemmas_cs);
 by (rtac cpo_trans 1);
-by(resolve_tac prems 1);
-br(not_le_iff_lt RS iffD1 RS leI RS chain_rel_gen)1;
+by (resolve_tac prems 1);
+by (rtac (not_le_iff_lt RS iffD1 RS leI RS chain_rel_gen) 1);
 by (assume_tac 3);
-by(REPEAT(ares_tac (nat_into_Ord::matrix_chain_left::prems) 1));
+by (REPEAT(ares_tac (nat_into_Ord::matrix_chain_left::prems) 1));
 by (rtac lemma 1);
 by (rtac isubD2 2);
-by(REPEAT(ares_tac (matrix_in::isubD1::prems) 1));
+by (REPEAT(ares_tac (matrix_in::isubD1::prems) 1));
 by (rtac cpo_trans 1);
-by(resolve_tac prems 1);
+by (resolve_tac prems 1);
 by (rtac lemma2 1);
 by (rtac lemma 4);
 by (rtac isubD2 5);
-by(REPEAT(ares_tac
-  ([chain_rel_gen,matrix_chain_right,matrix_in,isubD1]@prems)1));
+by (REPEAT(ares_tac
+  ([chain_rel_gen,matrix_chain_right,matrix_in,isubD1]@prems) 1));
 val isub_lemma = result();
 
 val prems = goalw Limit.thy [chain_def]  (* matrix_chain_lub *)
     "[|matrix(D,M); cpo(D)|] ==> chain(D,lam n:nat.lub(D,lam m:nat.M`n`m))";
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 by (rtac lam_type 1);
 by (rtac islub_in 1);
 by (rtac cpo_lub 1);
-by(resolve_tac prems 2);
-by(asm_simp_tac arith_ss 2);
+by (resolve_tac prems 2);
+by (asm_simp_tac arith_ss 2);
 by (rtac chainI 1);
 by (rtac lam_type 1);
-by(REPEAT(ares_tac (matrix_in::prems) 1));
-by(asm_simp_tac arith_ss 1);
+by (REPEAT(ares_tac (matrix_in::prems) 1));
+by (asm_simp_tac arith_ss 1);
 by (rtac matrix_rel_0_1 1);
-by(REPEAT(ares_tac prems 1));
-by(asm_simp_tac (arith_ss addsimps 
+by (REPEAT(ares_tac prems 1));
+by (asm_simp_tac (arith_ss addsimps 
     [hd prems RS matrix_chain_left RS chain_fun RS eta]) 1);
 by (rtac dominate_islub 1);
 by (rtac cpo_lub 3);
 by (rtac cpo_lub 2);
-by(rewtac dominate_def);
+by (rewtac dominate_def);
 by (rtac ballI 1);
 by (rtac bexI 1);
 by (assume_tac 2);
 back();  (* Backtracking...... *)
 by (rtac matrix_rel_1_0 1);
-by(REPEAT(ares_tac (matrix_chain_left::nat_succI::chain_fun::prems) 1));
+by (REPEAT(ares_tac (matrix_chain_left::nat_succI::chain_fun::prems) 1));
 val matrix_chain_lub = result();
 
 val prems = goal Limit.thy  (* isub_eq *)
@@ -617,19 +617,19 @@
 by (rtac iffI 1);
 by (rtac dominate_isub 1);
 by (assume_tac 2);
-by(rewtac dominate_def);
+by (rewtac dominate_def);
 by (rtac ballI 1);
 by (rtac bexI 1);
 by (assume_tac 2);
-by(asm_simp_tac ZF_ss 1);
-by(asm_simp_tac (arith_ss addsimps 
+by (asm_simp_tac ZF_ss 1);
+by (asm_simp_tac (arith_ss addsimps 
     [hd prems RS matrix_chain_left RS chain_fun RS eta]) 1);
 by (rtac islub_ub 1);
 by (rtac cpo_lub 1);
-by(REPEAT(ares_tac 
+by (REPEAT(ares_tac 
 (matrix_chain_left::matrix_chain_diag::chain_fun::matrix_chain_lub::prems) 1));
 by (rtac isub_lemma 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 val isub_eq = result();
 
 val lemma1 = prove_goalw Limit.thy [lub_def]  
@@ -646,17 +646,17 @@
     "[|matrix(D,M); cpo(D)|] ==>  \
 \    lub(D,(lam n:nat. lub(D,lam m:nat. M`n`m))) =  \
 \    lub(D,(lam n:nat. M`n`n))";
-by(simp_tac (arith_ss addsimps [lemma1,lemma2]) 1);
-by(rewtac islub_def);
-by(simp_tac (FOL_ss addsimps [hd(tl prems) RS (hd prems RS isub_eq)]) 1);
+by (simp_tac (arith_ss addsimps [lemma1,lemma2]) 1);
+by (rewtac islub_def);
+by (simp_tac (FOL_ss addsimps [hd(tl prems) RS (hd prems RS isub_eq)]) 1);
 val lub_matrix_diag = result();
 
 val [matrix,cpo] = goalw Limit.thy []  (* lub_matrix_diag_sym *)
     "[|matrix(D,M); cpo(D)|] ==>  \
 \    lub(D,(lam m:nat. lub(D,lam n:nat. M`n`m))) =  \
 \    lub(D,(lam n:nat. M`n`n))";
-by(cut_facts_tac[cpo RS (matrix RS matrix_sym_axis RS lub_matrix_diag)]1);
-by(asm_full_simp_tac ZF_ss 1);
+by (cut_facts_tac[cpo RS (matrix RS matrix_sym_axis RS lub_matrix_diag)]1);
+by (asm_full_simp_tac ZF_ss 1);
 val lub_matrix_diag_sym = result();
 
 (*----------------------------------------------------------------------*)
@@ -667,24 +667,24 @@
     "[|f:set(D)->set(E);   \
 \      !!x y. [|rel(D,x,y); x:set(D); y:set(D)|] ==> rel(E,f`x,f`y)|] ==>   \
 \     f:mono(D,E)";
-by(fast_tac(ZF_cs addSIs prems)1);
+by (fast_tac(ZF_cs addSIs prems) 1);
 val monoI = result();
 
 val prems = goal Limit.thy
     "f:mono(D,E) ==> f:set(D)->set(E)";
-br(rewrite_rule[mono_def](hd prems) RS CollectD1)1;
+by (rtac (rewrite_rule[mono_def](hd prems) RS CollectD1) 1);
 val mono_fun = result();
 
 val prems = goal Limit.thy
     "[|f:mono(D,E); x:set(D)|] ==> f`x:set(E)";
-br(hd prems RS mono_fun RS apply_type)1;
-by(resolve_tac prems 1);
+by (rtac (hd prems RS mono_fun RS apply_type) 1);
+by (resolve_tac prems 1);
 val mono_map = result();
 
 val prems = goal Limit.thy
     "[|f:mono(D,E); rel(D,x,y); x:set(D); y:set(D)|] ==> rel(E,f`x,f`y)";
-br(rewrite_rule[mono_def](hd prems) RS CollectD2 RS bspec RS bspec RS mp)1;
-by(REPEAT(resolve_tac prems 1));
+by (rtac (rewrite_rule[mono_def](hd prems) RS CollectD2 RS bspec RS bspec RS mp) 1);
+by (REPEAT(resolve_tac prems 1));
 val mono_mono = result();
 
 val prems = goalw Limit.thy [cont_def,mono_def]  (* contI *)
@@ -692,35 +692,35 @@
 \      !!x y. [|rel(D,x,y); x:set(D); y:set(D)|] ==> rel(E,f`x,f`y);   \
 \      !!X. chain(D,X) ==> f`lub(D,X) = lub(E,lam n:nat. f`(X`n))|] ==>   \
 \     f:cont(D,E)";
-by(fast_tac(ZF_cs addSIs prems)1);
+by (fast_tac(ZF_cs addSIs prems) 1);
 val contI = result();
 
 val prems = goal Limit.thy 
     "f:cont(D,E) ==> f:mono(D,E)";
-br(rewrite_rule[cont_def](hd prems) RS CollectD1)1;
+by (rtac (rewrite_rule[cont_def](hd prems) RS CollectD1) 1);
 val cont2mono = result();
 
 val prems = goal Limit.thy
     "f:cont(D,E) ==> f:set(D)->set(E)";
-br(rewrite_rule[cont_def](hd prems) RS CollectD1 RS mono_fun)1;
+by (rtac (rewrite_rule[cont_def](hd prems) RS CollectD1 RS mono_fun) 1);
 val cont_fun = result();
 
 val prems = goal Limit.thy
     "[|f:cont(D,E); x:set(D)|] ==> f`x:set(E)";
-br(hd prems RS cont_fun RS apply_type)1;
-by(resolve_tac prems 1);
+by (rtac (hd prems RS cont_fun RS apply_type) 1);
+by (resolve_tac prems 1);
 val cont_map = result();
 
 val prems = goal Limit.thy
     "[|f:cont(D,E); rel(D,x,y); x:set(D); y:set(D)|] ==> rel(E,f`x,f`y)";
-br(rewrite_rule[cont_def](hd prems) RS CollectD1 RS mono_mono)1;
-by(REPEAT(resolve_tac prems 1));
+by (rtac (rewrite_rule[cont_def](hd prems) RS CollectD1 RS mono_mono) 1);
+by (REPEAT(resolve_tac prems 1));
 val cont_mono = result();
 
 val prems = goal Limit.thy
     "[|f:cont(D,E); chain(D,X)|] ==> f`(lub(D,X)) = lub(E,lam n:nat. f`(X`n))";
-br(rewrite_rule[cont_def](hd prems) RS CollectD2 RS spec RS mp)1;
-by(REPEAT(resolve_tac prems 1));
+by (rtac (rewrite_rule[cont_def](hd prems) RS CollectD2 RS spec RS mp) 1);
+by (REPEAT(resolve_tac prems 1));
 val cont_lub = result();
 
 (*----------------------------------------------------------------------*)
@@ -729,27 +729,27 @@
 
 val prems = goalw Limit.thy []  (* mono_chain *)
     "[|f:mono(D,E); chain(D,X)|] ==> chain(E,lam n:nat. f`(X`n))";
-by(rewtac chain_def);
-by(simp_tac arith_ss 1);
-by(safe_tac lemmas_cs);
+by (rewtac chain_def);
+by (simp_tac arith_ss 1);
+by (safe_tac lemmas_cs);
 by (rtac lam_type 1);
 by (rtac mono_map 1);
-by(resolve_tac prems 1);
+by (resolve_tac prems 1);
 by (rtac chain_in 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 by (rtac mono_mono 1);
-by(resolve_tac prems 1);
+by (resolve_tac prems 1);
 by (rtac chain_rel 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 by (rtac chain_in 1);
 by (rtac chain_in 3);
-by(REPEAT(ares_tac (nat_succI::prems) 1));
+by (REPEAT(ares_tac (nat_succI::prems) 1));
 val mono_chain = result();
 
 val prems = goalw Limit.thy []  (* cont_chain *)
     "[|f:cont(D,E); chain(D,X)|] ==> chain(E,lam n:nat. f`(X`n))";
 by (rtac mono_chain 1);
-by(REPEAT(resolve_tac (cont2mono::prems) 1));
+by (REPEAT(resolve_tac (cont2mono::prems) 1));
 val cont_chain = result();
 
 (*----------------------------------------------------------------------*)
@@ -760,13 +760,13 @@
 
 val prems = goalw Limit.thy [set_def,cf_def]
     "!!z. f:set(cf(D,E)) ==> f:cont(D,E)";
-by(asm_full_simp_tac ZF_ss 1);
+by (asm_full_simp_tac ZF_ss 1);
 val in_cf = result();
 val cf_cont = result();
 
 val prems = goalw Limit.thy [set_def,cf_def]  (* Non-trivial with relation *)
     "!!z. f:cont(D,E) ==> f:set(cf(D,E))";
-by(asm_full_simp_tac ZF_ss 1);
+by (asm_full_simp_tac ZF_ss 1);
 val cont_cf = result();
 
 (* rel_cf originally an equality. Now stated as two rules. Seemed easiest. 
@@ -776,14 +776,14 @@
     "[|!!x. x:set(D) ==> rel(E,f`x,g`x); f:cont(D,E); g:cont(D,E)|] ==> \
 \    rel(cf(D,E),f,g)";
 by (rtac rel_I 1);
-by(simp_tac (ZF_ss addsimps [cf_def])1);
-by(safe_tac lemmas_cs);
+by (simp_tac (ZF_ss addsimps [cf_def]) 1);
+by (safe_tac lemmas_cs);
 brr prems 1;
 val rel_cfI = result();
 
 val prems = goalw Limit.thy [rel_def,cf_def]
     "!!z. [|rel(cf(D,E),f,g); x:set(D)|] ==> rel(E,f`x,g`x)";
-by(asm_full_simp_tac ZF_ss 1);
+by (asm_full_simp_tac ZF_ss 1);
 val rel_cf = result();
 
 (*----------------------------------------------------------------------*)
@@ -796,39 +796,39 @@
 by (rtac lam_type 1);
 by (rtac apply_type 1);
 by (resolve_tac prems 2);
-by(REPEAT(ares_tac([cont_fun,in_cf,chain_in]@prems)1));
-by(asm_simp_tac arith_ss 1);
-by(REPEAT(ares_tac([rel_cf,chain_rel]@prems)1));
+by (REPEAT(ares_tac([cont_fun,in_cf,chain_in]@prems) 1));
+by (asm_simp_tac arith_ss 1);
+by (REPEAT(ares_tac([rel_cf,chain_rel]@prems) 1));
 val chain_cf = result();
 
 val prems = goal Limit.thy  (* matrix_lemma *)
     "[|chain(cf(D,E),X); chain(D,Xa); cpo(D); cpo(E) |] ==>   \
 \    matrix(E,lam x:nat. lam xa:nat. X`x`(Xa`xa))";
 by (rtac matrix_chainI 1);
-by(asm_simp_tac ZF_ss 1);
-by(asm_simp_tac ZF_ss 2);
+by (asm_simp_tac ZF_ss 1);
+by (asm_simp_tac ZF_ss 2);
 by (rtac chainI 1);
 by (rtac lam_type 1);
 by (rtac apply_type 1);
 by (rtac (chain_in RS cf_cont RS cont_fun) 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 by (rtac chain_in 1);
-by(REPEAT(ares_tac prems 1));
-by(asm_simp_tac arith_ss 1);
+by (REPEAT(ares_tac prems 1));
+by (asm_simp_tac arith_ss 1);
 by (rtac cont_mono 1);
 by (rtac (chain_in RS cf_cont) 1);
 brr prems 1;
-brr (chain_rel::chain_in::nat_succI::prems)1;
+brr (chain_rel::chain_in::nat_succI::prems) 1;
 by (rtac chainI 1);
 by (rtac lam_type 1);
 by (rtac apply_type 1);
 by (rtac (chain_in RS cf_cont RS cont_fun) 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 by (rtac chain_in 1);
-by(REPEAT(ares_tac prems 1));
-by(asm_simp_tac arith_ss 1);
+by (REPEAT(ares_tac prems 1));
+by (asm_simp_tac arith_ss 1);
 by (rtac rel_cf 1);
-brr (chain_in::chain_rel::prems)1;
+brr (chain_in::chain_rel::prems) 1;
 by (rtac lam_type 1);
 by (rtac lam_type 1);
 by (rtac apply_type 1);
@@ -843,25 +843,25 @@
 \    (lam x:set(D). lub(E, lam n:nat. X ` n ` x)) : cont(D, E)";
 by (rtac contI 1);
 by (rtac lam_type 1);
-by(REPEAT(ares_tac((chain_cf RS cpo_lub RS islub_in)::prems)1));
-by(asm_simp_tac ZF_ss 1);
+by (REPEAT(ares_tac((chain_cf RS cpo_lub RS islub_in)::prems) 1));
+by (asm_simp_tac ZF_ss 1);
 by (rtac dominate_islub 1);
-by(REPEAT(ares_tac((chain_cf RS cpo_lub)::prems)2));
+by (REPEAT(ares_tac((chain_cf RS cpo_lub)::prems) 2));
 by (rtac dominateI 1);
 by (assume_tac 1);
-by(asm_simp_tac ZF_ss 1);
-by(REPEAT(ares_tac ((chain_in RS cf_cont RS cont_mono)::prems) 1));
-by(REPEAT(ares_tac ((chain_cf RS chain_fun)::prems) 1));
+by (asm_simp_tac ZF_ss 1);
+by (REPEAT(ares_tac ((chain_in RS cf_cont RS cont_mono)::prems) 1));
+by (REPEAT(ares_tac ((chain_cf RS chain_fun)::prems) 1));
 by (rtac (beta RS ssubst) 1);
-by(REPEAT(ares_tac((cpo_lub RS islub_in)::prems)1));
-by(asm_simp_tac(ZF_ss addsimps[hd prems RS chain_in RS cf_cont RS cont_lub])1);
-by(forward_tac[hd prems RS matrix_lemma RS lub_matrix_diag]1);
+by (REPEAT(ares_tac((cpo_lub RS islub_in)::prems) 1));
+by (asm_simp_tac(ZF_ss addsimps[hd prems RS chain_in RS cf_cont RS cont_lub]) 1);
+by (forward_tac[hd prems RS matrix_lemma RS lub_matrix_diag]1);
 brr prems 1;
-by(asm_full_simp_tac ZF_ss 1);
-by(asm_simp_tac(ZF_ss addsimps[chain_in RS beta])1);
-bd(hd prems RS matrix_lemma RS lub_matrix_diag_sym)1;
+by (asm_full_simp_tac ZF_ss 1);
+by (asm_simp_tac(ZF_ss addsimps[chain_in RS beta]) 1);
+by (dtac (hd prems RS matrix_lemma RS lub_matrix_diag_sym) 1);
 brr prems 1;
-by(asm_full_simp_tac ZF_ss 1);
+by (asm_full_simp_tac ZF_ss 1);
 val chain_cf_lub_cont = result();
 
 val prems = goal Limit.thy  (* islub_cf *)
@@ -872,22 +872,22 @@
 by (rtac (chain_cf_lub_cont RS cont_cf) 1);
 brr prems 1;
 by (rtac rel_cfI 1);
-by(asm_simp_tac ZF_ss 1);
-bd(hd(tl(tl prems)) RSN(2,hd prems RS chain_cf RS cpo_lub RS islub_ub))1;
+by (asm_simp_tac ZF_ss 1);
+by (dtac (hd(tl(tl prems)) RSN(2,hd prems RS chain_cf RS cpo_lub RS islub_ub)) 1);
 by (assume_tac 1);
-by(asm_full_simp_tac ZF_ss 1);
-brr(cf_cont::chain_in::prems)1;
-brr(cont_cf::chain_cf_lub_cont::prems)1;
+by (asm_full_simp_tac ZF_ss 1);
+brr(cf_cont::chain_in::prems) 1;
+brr(cont_cf::chain_cf_lub_cont::prems) 1;
 by (rtac rel_cfI 1);
-by(asm_simp_tac ZF_ss 1);
-by(forward_tac[hd(tl(tl prems)) RSN(2,hd prems RS chain_cf RS cpo_lub RS 
+by (asm_simp_tac ZF_ss 1);
+by (forward_tac[hd(tl(tl prems)) RSN(2,hd prems RS chain_cf RS cpo_lub RS 
    islub_least)]1);
 by (assume_tac 2);
 brr (chain_cf_lub_cont::isubD1::cf_cont::prems) 2;
 by (rtac isubI 1);
-brr((cf_cont RS cont_fun RS apply_type)::[isubD1])1;
-by(asm_simp_tac ZF_ss 1);
-be(isubD2 RS rel_cf)1;
+brr((cf_cont RS cont_fun RS apply_type)::[isubD1]) 1;
+by (asm_simp_tac ZF_ss 1);
+by (etac (isubD2 RS rel_cf) 1);
 brr [] 1;
 val islub_cf = result();
 
@@ -895,7 +895,7 @@
     "[| cpo(D); cpo(E)|] ==> cpo(cf(D,E))";
 by (rtac (poI RS cpoI) 1);
 by (rtac rel_cfI 1);
-brr(cpo_refl::(cf_cont RS cont_fun RS apply_type)::cf_cont::prems)1;
+brr(cpo_refl::(cf_cont RS cont_fun RS apply_type)::cf_cont::prems) 1;
 by (rtac rel_cfI 1);
 by (rtac cpo_trans 1);
 by (resolve_tac prems 1);
@@ -907,7 +907,7 @@
 by (rtac fun_extension 1);
 brr[cf_cont RS cont_fun]1;
 by (rtac cpo_antisym 1);
-br(hd(tl prems))1;
+by (rtac (hd(tl prems)) 1);
 by (etac rel_cf 1);
 by (assume_tac 1);
 by (rtac rel_cf 1);
@@ -931,32 +931,32 @@
 val prems = goal Limit.thy  (* const_cont *)
     "[|y:set(E); cpo(D); cpo(E)|] ==> (lam x:set(D).y) : cont(D,E)";
 by (rtac contI 1);
-by(asm_simp_tac ZF_ss 2);
-brr(const_fun::cpo_refl::prems)1;
-by(asm_simp_tac(ZF_ss addsimps(chain_in::(cpo_lub RS islub_in)::
-    lub_const::prems))1);
+by (asm_simp_tac ZF_ss 2);
+brr(const_fun::cpo_refl::prems) 1;
+by (asm_simp_tac(ZF_ss addsimps(chain_in::(cpo_lub RS islub_in)::
+    lub_const::prems)) 1);
 val const_cont = result();
 
 val prems = goal Limit.thy  (* cf_least *)
     "[|cpo(D); pcpo(E); y:cont(D,E)|]==>rel(cf(D,E),(lam x:set(D).bot(E)),y)";
 by (rtac rel_cfI 1);
-by(asm_simp_tac ZF_ss 1);
+by (asm_simp_tac ZF_ss 1);
 brr(bot_least::bot_in::apply_type::cont_fun::const_cont::
-    cpo_cf::(prems@[pcpo_cpo]))1;
+    cpo_cf::(prems@[pcpo_cpo])) 1;
 val cf_least = result();
 
 val prems = goal Limit.thy  (* pcpo_cf *)
     "[|cpo(D); pcpo(E)|] ==> pcpo(cf(D,E))";
 by (rtac pcpoI 1);
 brr(cf_least::bot_in::(const_cont RS cont_cf)::cf_cont::
-    cpo_cf::(prems@[pcpo_cpo]))1;
+    cpo_cf::(prems@[pcpo_cpo])) 1;
 val pcpo_cf = result();
 
 val prems = goal Limit.thy  (* bot_cf *)
     "[|cpo(D); pcpo(E)|] ==> bot(cf(D,E)) = (lam x:set(D).bot(E))";
 by (rtac (bot_unique RS sym) 1);
 brr(pcpo_cf::cf_least::(bot_in RS const_cont RS cont_cf)::
-    cf_cont::(prems@[pcpo_cpo]))1;
+    cf_cont::(prems@[pcpo_cpo])) 1;
 val bot_cf = result();
 
 (*----------------------------------------------------------------------*)
@@ -964,15 +964,15 @@
 (*----------------------------------------------------------------------*)
 
 val id_thm = prove_goalw Perm.thy [id_def] "x:X ==> (id(X)`x) = x"
-  (fn prems => [simp_tac(ZF_ss addsimps prems)1]);
+  (fn prems => [simp_tac(ZF_ss addsimps prems) 1]);
 
 val prems = goal Limit.thy  (* id_cont *)
     "cpo(D) ==> id(set(D)):cont(D,D)";
 by (rtac contI 1);
 by (rtac id_type 1);
-by(asm_simp_tac (ZF_ss addsimps[id_thm])1);
-by(asm_simp_tac(ZF_ss addsimps(id_thm::(cpo_lub RS islub_in)::
-    chain_in::(chain_fun RS eta)::prems))1);
+by (asm_simp_tac (ZF_ss addsimps[id_thm]) 1);
+by (asm_simp_tac(ZF_ss addsimps(id_thm::(cpo_lub RS islub_in)::
+    chain_in::(chain_fun RS eta)::prems)) 1);
 val id_cont = result();
 
 val comp_cont_apply = cont_fun RSN(2,cont_fun RS comp_fun_apply);
@@ -980,17 +980,17 @@
 val prems = goal Limit.thy  (* comp_pres_cont *)
     "[| f:cont(D',E); g:cont(D,D'); cpo(D)|] ==> f O g : cont(D,E)";
 by (rtac contI 1);
-br(comp_cont_apply RS ssubst)2;
-br(comp_cont_apply RS ssubst)5;
+by (rtac (comp_cont_apply RS ssubst) 2);
+by (rtac (comp_cont_apply RS ssubst) 5);
 by (rtac cont_mono 8);
 by (rtac cont_mono 9); (* 15 subgoals *)
-brr(comp_fun::cont_fun::cont_map::prems)1; (* proves all but the lub case *)
-br(comp_cont_apply RS ssubst)1;
-br(cont_lub RS ssubst)4;
-br(cont_lub RS ssubst)6;
-by(asm_full_simp_tac(ZF_ss addsimps (* RS: new subgoals contain unknowns *)
-    [hd prems RS (hd(tl prems) RS comp_cont_apply),chain_in])8);
-brr((cpo_lub RS islub_in)::cont_chain::prems)1;
+brr(comp_fun::cont_fun::cont_map::prems) 1; (* proves all but the lub case *)
+by (rtac (comp_cont_apply RS ssubst) 1);
+by (rtac (cont_lub RS ssubst) 4);
+by (rtac (cont_lub RS ssubst) 6);
+by (asm_full_simp_tac(ZF_ss addsimps (* RS: new subgoals contain unknowns *)
+    [hd prems RS (hd(tl prems) RS comp_cont_apply),chain_in]) 8);
+brr((cpo_lub RS islub_in)::cont_chain::prems) 1;
 val comp_pres_cont = result();
 
 val prems = goal Limit.thy  (* comp_mono *)
@@ -998,54 +998,54 @@
 \       rel(cf(D',E),f,f'); rel(cf(D,D'),g,g'); cpo(D); cpo(E) |] ==>   \
 \    rel(cf(D,E),f O g,f' O g')";
 by (rtac rel_cfI 1); (* extra proof obl: f O g and f' O g' cont. Extra asm cpo(D). *)
-br(comp_cont_apply RS ssubst)1;
-br(comp_cont_apply RS ssubst)4;
+by (rtac (comp_cont_apply RS ssubst) 1);
+by (rtac (comp_cont_apply RS ssubst) 4);
 by (rtac cpo_trans 7);
-brr(rel_cf::cont_mono::cont_map::comp_pres_cont::prems)1;
+brr(rel_cf::cont_mono::cont_map::comp_pres_cont::prems) 1;
 val comp_mono = result();
 
 val prems = goal Limit.thy  (* chain_cf_comp *)
     "[| chain(cf(D',E),X); chain(cf(D,D'),Y); cpo(D); cpo(E)|] ==>  \
 \    chain(cf(D,E),lam n:nat. X`n O Y`n)";
 by (rtac chainI 1);
-by(asm_simp_tac arith_ss 2);
+by (asm_simp_tac arith_ss 2);
 by (rtac rel_cfI 2);
-br(comp_cont_apply RS ssubst)2;
-br(comp_cont_apply RS ssubst)5; 
+by (rtac (comp_cont_apply RS ssubst) 2);
+by (rtac (comp_cont_apply RS ssubst) 5); 
 by (rtac cpo_trans 8); 
 by (rtac rel_cf 9);
 by (rtac cont_mono 11);
 brr(lam_type::comp_pres_cont::cont_cf::(chain_in RS cf_cont)::cont_map::
-    chain_rel::rel_cf::nat_succI::prems)1;
+    chain_rel::rel_cf::nat_succI::prems) 1;
 val chain_cf_comp = result();
 
 val prems = goal Limit.thy  (* comp_lubs *)
     "[| chain(cf(D',E),X); chain(cf(D,D'),Y); cpo(D); cpo(D'); cpo(E)|] ==>  \
 \    lub(cf(D',E),X) O lub(cf(D,D'),Y) = lub(cf(D,E),lam n:nat. X`n O Y`n)";
 by (rtac fun_extension 1);
-br(lub_cf RS ssubst)3;
+by (rtac (lub_cf RS ssubst) 3);
 brr(comp_fun::(cf_cont RS cont_fun)::(cpo_lub RS islub_in)::cpo_cf::
-    chain_cf_comp::prems)1;
-by(cut_facts_tac[hd prems,hd(tl prems)]1);
-by(asm_simp_tac(ZF_ss addsimps((chain_in RS cf_cont RSN(3,chain_in RS 
-    cf_cont RS comp_cont_apply))::(tl(tl prems))))1);
-br(comp_cont_apply RS ssubst)1;
-brr((cpo_lub RS islub_in RS cf_cont)::cpo_cf::prems)1;
-by(asm_simp_tac(ZF_ss addsimps(lub_cf::
+    chain_cf_comp::prems) 1;
+by (cut_facts_tac[hd prems,hd(tl prems)]1);
+by (asm_simp_tac(ZF_ss addsimps((chain_in RS cf_cont RSN(3,chain_in RS 
+    cf_cont RS comp_cont_apply))::(tl(tl prems)))) 1);
+by (rtac (comp_cont_apply RS ssubst) 1);
+brr((cpo_lub RS islub_in RS cf_cont)::cpo_cf::prems) 1;
+by (asm_simp_tac(ZF_ss addsimps(lub_cf::
  (hd(tl prems)RS chain_cf RSN(2,hd prems RS chain_in RS cf_cont RS cont_lub))::
- (hd(tl prems) RS chain_cf RS cpo_lub RS islub_in)::prems))1);
-by(cut_inst_tac[("M","lam xa:nat. lam xb:nat. X`xa`(Y`xb`x)")]
+ (hd(tl prems) RS chain_cf RS cpo_lub RS islub_in)::prems)) 1);
+by (cut_inst_tac[("M","lam xa:nat. lam xb:nat. X`xa`(Y`xb`x)")]
    lub_matrix_diag 1);
-by(asm_full_simp_tac ZF_ss 3);
+by (asm_full_simp_tac ZF_ss 3);
 by (rtac matrix_chainI 1);
-by(asm_simp_tac ZF_ss 1);
-by(asm_simp_tac ZF_ss 2); 
-by(forward_tac[hd(tl prems) RSN(2,(hd prems RS chain_in RS cf_cont) RS 
+by (asm_simp_tac ZF_ss 1);
+by (asm_simp_tac ZF_ss 2); 
+by (forward_tac[hd(tl prems) RSN(2,(hd prems RS chain_in RS cf_cont) RS 
     (chain_cf RSN(2,cont_chain)))]1); (* Here, Isabelle was a bitch! *)
-by(asm_full_simp_tac ZF_ss 2);
+by (asm_full_simp_tac ZF_ss 2);
 by (assume_tac 1);
 by (rtac chain_cf 1);
-brr((cont_fun RS apply_type)::(chain_in RS cf_cont)::lam_type::prems)1;
+brr((cont_fun RS apply_type)::(chain_in RS cf_cont)::lam_type::prems) 1;
 val comp_lubs = result();
 
 (*----------------------------------------------------------------------*)
@@ -1054,14 +1054,14 @@
 
 val prems = goal Arith.thy  (* le_cases *)
    "[|m:nat; n:nat|] ==> m le n | n le m";
-by(safe_tac lemmas_cs);
-brr((not_le_iff_lt RS iffD1 RS leI)::nat_into_Ord::prems)1;
+by (safe_tac lemmas_cs);
+brr((not_le_iff_lt RS iffD1 RS leI)::nat_into_Ord::prems) 1;
 val le_cases = result();
 
 val prems = goal Arith.thy  (* lt_cases *)
    "[|m:nat; n:nat|] ==> m < n | n le m";
-by(safe_tac lemmas_cs);
-brr((not_le_iff_lt RS iffD1)::nat_into_Ord::prems)1;
+by (safe_tac lemmas_cs);
+brr((not_le_iff_lt RS iffD1)::nat_into_Ord::prems) 1;
 val lt_cases = result();
 
 (*----------------------------------------------------------------------*)
@@ -1071,39 +1071,39 @@
 val prems = goalw Limit.thy [projpair_def]  (* projpairI *)
     "!!x. [| e:cont(D,E); p:cont(E,D); p O e = id(set(D));   \
 \       rel(cf(E,E))(e O p)(id(set(E)))|] ==> projpair(D,E,e,p)";
-by(fast_tac FOL_cs 1);
+by (fast_tac FOL_cs 1);
 val projpairI = result();
 
 val prems = goalw Limit.thy [projpair_def]  (* projpairE *)
     "[| projpair(D,E,e,p);   \
 \       [| e:cont(D,E); p:cont(E,D); p O e = id(set(D));   \
 \          rel(cf(E,E))(e O p)(id(set(E)))|] ==> Q |] ==> Q";
-br(hd(tl prems))1;
-by(REPEAT(asm_simp_tac(FOL_ss addsimps[hd prems])1));
+by (rtac (hd(tl prems)) 1);
+by (REPEAT(asm_simp_tac(FOL_ss addsimps[hd prems]) 1));
 val projpairE = result();
 
 val prems = goal Limit.thy  (* projpair_e_cont *)
     "projpair(D,E,e,p) ==> e:cont(D,E)";
 by (rtac projpairE 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 val projpair_e_cont = result();
 
 val prems = goal Limit.thy  (* projpair_p_cont *)
     "projpair(D,E,e,p) ==> p:cont(E,D)";
 by (rtac projpairE 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 val projpair_p_cont = result();
 
 val prems = goal Limit.thy  (* projpair_eq *)
     "projpair(D,E,e,p) ==> p O e = id(set(D))";
 by (rtac projpairE 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 val projpair_eq = result();
 
 val prems = goal Limit.thy  (* projpair_rel *)
     "projpair(D,E,e,p) ==> rel(cf(E,E))(e O p)(id(set(E)))";
 by (rtac projpairE 1);
-by(REPEAT(ares_tac prems 1));
+by (REPEAT(ares_tac prems 1));
 val projpair_rel = result();
 
 val projpairDs = [projpair_e_cont,projpair_p_cont,projpair_eq,projpair_rel];
@@ -1128,19 +1128,19 @@
    are used manually, one at a time. Therefore the following contl.     *)
 val contl = [p1 RS projpair_e_cont,p1 RS projpair_p_cont,
             p2 RS projpair_e_cont,p2 RS projpair_p_cont];
-br(p2 RS projpair_p_cont RS cont_fun RS id_comp RS subst)1;
-br(p1 RS projpair_eq RS subst)1;
+by (rtac (p2 RS projpair_p_cont RS cont_fun RS id_comp RS subst) 1);
+by (rtac (p1 RS projpair_eq RS subst) 1);
 by (rtac cpo_trans 1);
-brr(cpo_cf::prems)1; 
+brr(cpo_cf::prems) 1; 
 (* The following corresponds to EXISTS_TAC, non-trivial instantiation. *)
-by(res_inst_tac[("f","p O (e' O p')")]cont_cf 4);
-br(comp_assoc RS ssubst)1;
-brr(cpo_refl::cpo_cf::cont_cf::comp_mono::comp_pres_cont::(contl@prems))1;
-by(res_inst_tac[("P","%x. rel(cf(E,D),p O e' O p',x)")]
-    (p1 RS projpair_p_cont RS cont_fun RS comp_id RS subst)1);
+by (res_inst_tac[("f","p O (e' O p')")]cont_cf 4);
+by (rtac (comp_assoc RS ssubst) 1);
+brr(cpo_refl::cpo_cf::cont_cf::comp_mono::comp_pres_cont::(contl@prems)) 1;
+by (res_inst_tac[("P","%x. rel(cf(E,D),p O e' O p',x)")]
+    (p1 RS projpair_p_cont RS cont_fun RS comp_id RS subst) 1);
 by (rtac comp_mono 1);
 brr(cpo_refl::cpo_cf::cont_cf::comp_mono::comp_pres_cont::id_cont::
-    projpair_rel::(contl@prems))1;
+    projpair_rel::(contl@prems)) 1;
 val lemma1 = result();
 
 val prems = goal Limit.thy (* lemma2 *)
@@ -1149,17 +1149,17 @@
 val [_,_,p1,p2,_] = prems;
 val contl = [p1 RS projpair_e_cont,p1 RS projpair_p_cont,
             p2 RS projpair_e_cont,p2 RS projpair_p_cont];
-br(p1 RS projpair_e_cont RS cont_fun RS comp_id RS subst)1;
-br(p2 RS projpair_eq RS subst)1;
+by (rtac (p1 RS projpair_e_cont RS cont_fun RS comp_id RS subst) 1);
+by (rtac (p2 RS projpair_eq RS subst) 1);
 by (rtac cpo_trans 1);
-brr(cpo_cf::prems)1; 
-by(res_inst_tac[("f","(e O p) O e'")]cont_cf 4);
-br(comp_assoc RS ssubst)1;
-brr((cpo_cf RS cpo_refl)::cont_cf::comp_mono::comp_pres_cont::(contl@prems))1;
-by(res_inst_tac[("P","%x. rel(cf(D,E),(e O p) O e',x)")]
-    (p2 RS projpair_e_cont RS cont_fun RS id_comp RS subst)1);
+brr(cpo_cf::prems) 1; 
+by (res_inst_tac[("f","(e O p) O e'")]cont_cf 4);
+by (rtac (comp_assoc RS ssubst) 1);
+brr((cpo_cf RS cpo_refl)::cont_cf::comp_mono::comp_pres_cont::(contl@prems)) 1;
+by (res_inst_tac[("P","%x. rel(cf(D,E),(e O p) O e',x)")]
+    (p2 RS projpair_e_cont RS cont_fun RS id_comp RS subst) 1);
 brr((cpo_cf RS cpo_refl)::cont_cf::comp_mono::id_cont::comp_pres_cont::projpair_rel::
-    (contl@prems))1;
+    (contl@prems)) 1;
 val lemma2 = result();
 
 val prems = goal Limit.thy (* projpair_unique *)
@@ -1174,23 +1174,23 @@
 (* First some existentials are instantiated. *)
 by (resolve_tac prems 4);
 by (resolve_tac prems 4);
-by(asm_simp_tac FOL_ss 4);
-brr(cpo_cf::cpo_refl::cont_cf::projpair_e_cont::prems)1;
+by (asm_simp_tac FOL_ss 4);
+brr(cpo_cf::cpo_refl::cont_cf::projpair_e_cont::prems) 1;
 by (rtac lemma1 1);
 brr prems 1;
-by(asm_simp_tac FOL_ss 1); 
-brr(cpo_cf::cpo_refl::cont_cf::(contl @ prems))1;
+by (asm_simp_tac FOL_ss 1); 
+brr(cpo_cf::cpo_refl::cont_cf::(contl @ prems)) 1;
 by (rtac cpo_antisym 1);
 by (rtac lemma2 2);
 (* First some existentials are instantiated. *)
 by (resolve_tac prems 4);
 by (resolve_tac prems 4);
-by(asm_simp_tac FOL_ss 4);
-brr(cpo_cf::cpo_refl::cont_cf::projpair_p_cont::prems)1;
+by (asm_simp_tac FOL_ss 4);
+brr(cpo_cf::cpo_refl::cont_cf::projpair_p_cont::prems) 1;
 by (rtac lemma2 1);
 brr prems 1;
-by(asm_simp_tac FOL_ss 1); 
-brr(cpo_cf::cpo_refl::cont_cf::(contl @ prems))1;
+by (asm_simp_tac FOL_ss 1); 
+brr(cpo_cf::cpo_refl::cont_cf::(contl @ prems)) 1;
 val projpair_unique = result();
 
 (* Slightly different, more asms, since THE chooses the unique element. *)
@@ -1202,7 +1202,7 @@
 by (rtac ((hd prems) RS exE) 1);
 by (rtac ex1I 1);
 by (assume_tac 1);
-br(projpair_unique RS iffD1)1;
+by (rtac (projpair_unique RS iffD1) 1);
 by (assume_tac 3); (* To instantiate variables. *)
 brr (refl::prems) 1;
 val embRp = result();
@@ -1213,14 +1213,14 @@
 
 val prems = goal Limit.thy  (* Rp_unique *)
     "[|projpair(D,E,e,p); cpo(D); cpo(E)|] ==> Rp(D,E,e) = p";
-br(projpair_unique RS iffD1)1;
+by (rtac (projpair_unique RS iffD1) 1);
 by (rtac embRp 3); (* To instantiate variables. *)
 brr (embI::refl::prems) 1;
 val Rp_unique = result();
 
 val emb_cont = prove_goalw Limit.thy [emb_def]
     "emb(D,E,e) ==> e:cont(D,E)"
-  (fn prems => [rtac(hd prems RS exE)1,rtac projpair_e_cont 1,atac 1]);
+  (fn prems => [rtac(hd prems RS exE) 1,rtac projpair_e_cont 1,atac 1]);
 
 (* The following three theorems have cpo asms due to THE (uniqueness). *)
 
@@ -1234,10 +1234,10 @@
 
 val prems = goal Limit.thy  (* embRp_eq_thm *)
     "[|emb(D,E,e); x:set(D); cpo(D); cpo(E)|] ==> Rp(D,E,e)`(e`x) = x";
-br(comp_fun_apply RS subst)1;
-brr(Rp_cont::emb_cont::cont_fun::prems)1;
-br(embRp_eq RS ssubst)1;
-brr(id_apply::prems)1;
+by (rtac (comp_fun_apply RS subst) 1);
+brr(Rp_cont::emb_cont::cont_fun::prems) 1;
+by (rtac (embRp_eq RS ssubst) 1);
+brr(id_apply::prems) 1;
 val embRp_eq_thm = result();
 
 
@@ -1247,20 +1247,20 @@
 
 val prems = goalw Limit.thy [projpair_def]  (* projpair_id *)
     "cpo(D) ==> projpair(D,D,id(set(D)),id(set(D)))";
-by(safe_tac lemmas_cs);
-brr(id_cont::id_comp::id_type::prems)1;
+by (safe_tac lemmas_cs);
+brr(id_cont::id_comp::id_type::prems) 1;
 by (rtac (id_comp RS ssubst) 1); (* Matches almost anything *)
-brr(id_cont::id_type::cpo_refl::cpo_cf::cont_cf::prems)1;
+brr(id_cont::id_type::cpo_refl::cpo_cf::cont_cf::prems) 1;
 val projpair_id = result();
 
 val prems = goal Limit.thy  (* emb_id *)
     "cpo(D) ==> emb(D,D,id(set(D)))";
-brr(embI::projpair_id::prems)1;
+brr(embI::projpair_id::prems) 1;
 val emb_id = result();
 
 val prems = goal Limit.thy  (* Rp_id *)
     "cpo(D) ==> Rp(D,D,id(set(D))) = id(set(D))";
-brr(Rp_unique::projpair_id::prems)1;
+brr(Rp_unique::projpair_id::prems) 1;
 val Rp_id = result();
 
 (*----------------------------------------------------------------------*)
@@ -1274,29 +1274,29 @@
 val prems = goalw Limit.thy [projpair_def]  (* lemma *)
     "[|emb(D,D',e); emb(D',E,e'); cpo(D); cpo(D'); cpo(E)|] ==>  \
 \    projpair(D,E,e' O e,(Rp(D,D',e)) O (Rp(D',E,e')))";
-by(safe_tac lemmas_cs);
-brr(comp_pres_cont::Rp_cont::emb_cont::prems)1;
-br(comp_assoc RS subst)1;
-by(res_inst_tac[("t1","e'")](comp_assoc RS ssubst)1);
-br(embRp_eq RS ssubst)1; (* Matches everything due to subst/ssubst. *)
+by (safe_tac lemmas_cs);
+brr(comp_pres_cont::Rp_cont::emb_cont::prems) 1;
+by (rtac (comp_assoc RS subst) 1);
+by (res_inst_tac[("t1","e'")](comp_assoc RS ssubst) 1);
+by (rtac (embRp_eq RS ssubst) 1); (* Matches everything due to subst/ssubst. *)
 brr prems 1;
-br(comp_id RS ssubst)1;
-brr(cont_fun::Rp_cont::embRp_eq::prems)1;
-br(comp_assoc RS subst)1;
-by(res_inst_tac[("t1","Rp(D,D',e)")](comp_assoc RS ssubst)1);
+by (rtac (comp_id RS ssubst) 1);
+brr(cont_fun::Rp_cont::embRp_eq::prems) 1;
+by (rtac (comp_assoc RS subst) 1);
+by (res_inst_tac[("t1","Rp(D,D',e)")](comp_assoc RS ssubst) 1);
 by (rtac cpo_trans 1);
-brr(cpo_cf::prems)1;
+brr(cpo_cf::prems) 1;
 by (rtac comp_mono 1);
 by (rtac cpo_refl 6);
-brr(cont_cf::Rp_cont::prems)7; 
-brr(cpo_cf::prems)6;
+brr(cont_cf::Rp_cont::prems) 7; 
+brr(cpo_cf::prems) 6;
 by (rtac comp_mono 5);
-brr(embRp_rel::prems)10;
-brr((cpo_cf RS cpo_refl)::cont_cf::Rp_cont::prems)9;
-br(comp_id RS ssubst)10;
+brr(embRp_rel::prems) 10;
+brr((cpo_cf RS cpo_refl)::cont_cf::Rp_cont::prems) 9;
+by (rtac (comp_id RS ssubst) 10);
 by (rtac embRp_rel 11); 
 (* There are 16 subgoals at this point. All are proved immediately by: *)
-brr(comp_pres_cont::Rp_cont::id_cont::emb_cont::cont_fun::cont_cf::prems)1;
+brr(comp_pres_cont::Rp_cont::id_cont::emb_cont::cont_fun::cont_cf::prems) 1;
 val lemma = result();
 
 (* The use of RS is great in places like the following, both ugly in HOL. *)
@@ -1310,33 +1310,33 @@
 
 val prems = goalw Limit.thy [set_def,iprod_def]  (* iprodI *)
     "!!z. x:(PROD n:nat. set(DD`n)) ==> x:set(iprod(DD))";
-by(asm_full_simp_tac ZF_ss 1);
+by (asm_full_simp_tac ZF_ss 1);
 val iprodI = result();
 
 (* Proof with non-reflexive relation approach:
 by (rtac CollectI 1);
 by (rtac domainI 1);
 by (rtac CollectI 1);
-by(simp_tac(ZF_ss addsimps prems)1);
+by (simp_tac(ZF_ss addsimps prems) 1);
 by (rtac (hd prems) 1);
-by(simp_tac ZF_ss 1);
+by (simp_tac ZF_ss 1);
 by (rtac ballI 1);
-bd((hd prems) RS apply_type)1;
+by (dtac ((hd prems) RS apply_type) 1);
 by (etac CollectE 1);
 by (assume_tac 1);
 by (rtac rel_I 1);
 by (rtac CollectI 1);
-by(fast_tac(ZF_cs addSIs prems)1);
+by (fast_tac(ZF_cs addSIs prems) 1);
 by (rtac ballI 1);
-by(simp_tac ZF_ss 1);
-bd((hd prems) RS apply_type)1;
+by (simp_tac ZF_ss 1);
+by (dtac ((hd prems) RS apply_type) 1);
 by (etac CollectE 1);
 by (assume_tac 1);
 *)
 
 val prems = goalw Limit.thy [set_def,iprod_def]  (* iprodE *)
     "!!z. x:set(iprod(DD)) ==> x:(PROD n:nat. set(DD`n))";
-by(asm_full_simp_tac ZF_ss 1);
+by (asm_full_simp_tac ZF_ss 1);
 val iprodE = result();
 
 (* Contains typing conditions in contrast to HOL-ST *)
@@ -1345,16 +1345,16 @@
     "[|!!n. n:nat ==> rel(DD`n,f`n,g`n); f:(PROD n:nat. set(DD`n));  \
 \      g:(PROD n:nat. set(DD`n))|] ==> rel(iprod(DD),f,g)";
 by (rtac rel_I 1);
-by(simp_tac ZF_ss 1);
-by(safe_tac lemmas_cs);
+by (simp_tac ZF_ss 1);
+by (safe_tac lemmas_cs);
 brr prems 1;
 val rel_iprodI = result();
 
 val prems = goalw Limit.thy [iprod_def] (* rel_iprodE *)
     "[|rel(iprod(DD),f,g); n:nat|] ==> rel(DD`n,f`n,g`n)";
-by(cut_facts_tac[hd prems RS rel_E]1);
-by(asm_full_simp_tac ZF_ss 1);
-by(safe_tac lemmas_cs);
+by (cut_facts_tac[hd prems RS rel_E]1);
+by (asm_full_simp_tac ZF_ss 1);
+by (safe_tac lemmas_cs);
 by (etac bspec 1);
 by (resolve_tac prems 1);
 val rel_iprodE = result();
@@ -1365,63 +1365,63 @@
 val prems = goalw Limit.thy [chain_def]  (* chain_iprod *)
     "[|chain(iprod(DD),X);!!n. n:nat ==> cpo(DD`n); n:nat|] ==>  \
 \    chain(DD`n,lam m:nat.X`m`n)";
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 by (rtac lam_type 1);
 by (rtac apply_type 1);
 by (rtac iprodE 1);
-be(hd prems RS conjunct1 RS apply_type)1;
+by (etac (hd prems RS conjunct1 RS apply_type) 1);
 by (resolve_tac prems 1);
-by(asm_simp_tac(arith_ss addsimps prems)1);
+by (asm_simp_tac(arith_ss addsimps prems) 1);
 by (rtac rel_iprodE 1);
-by(asm_simp_tac (arith_ss addsimps prems) 1);
+by (asm_simp_tac (arith_ss addsimps prems) 1);
 by (resolve_tac prems 1);
 val chain_iprod = result();
 
 val prems = goalw Limit.thy [islub_def,isub_def]  (* islub_iprod *)
     "[|chain(iprod(DD),X);!!n. n:nat ==> cpo(DD`n)|] ==>   \
 \    islub(iprod(DD),X,lam n:nat. lub(DD`n,lam m:nat.X`m`n))";
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 by (rtac iprodI 1);
 by (rtac lam_type 1); 
-brr((chain_iprod RS cpo_lub RS islub_in)::prems)1;
+brr((chain_iprod RS cpo_lub RS islub_in)::prems) 1;
 by (rtac rel_iprodI 1);
-by(asm_simp_tac ZF_ss 1);
+by (asm_simp_tac ZF_ss 1);
 (* Here, HOL resolution is handy, Isabelle resolution bad. *)
-by(res_inst_tac[("P","%t. rel(DD`na,t,lub(DD`na,lam x:nat. X`x`na))"),
-    ("b1","%n. X`n`na")](beta RS subst)1);
-brr((chain_iprod RS cpo_lub RS islub_ub)::iprodE::chain_in::prems)1;
-brr(iprodI::lam_type::(chain_iprod RS cpo_lub RS islub_in)::prems)1;
+by (res_inst_tac[("P","%t. rel(DD`na,t,lub(DD`na,lam x:nat. X`x`na))"),
+    ("b1","%n. X`n`na")](beta RS subst) 1);
+brr((chain_iprod RS cpo_lub RS islub_ub)::iprodE::chain_in::prems) 1;
+brr(iprodI::lam_type::(chain_iprod RS cpo_lub RS islub_in)::prems) 1;
 by (rtac rel_iprodI 1);
-by(asm_simp_tac ZF_ss 1);
-brr(islub_least::(chain_iprod RS cpo_lub)::prems)1;
-by(rewtac isub_def);
-by(safe_tac lemmas_cs);
-be(iprodE RS apply_type)1;
+by (asm_simp_tac ZF_ss 1);
+brr(islub_least::(chain_iprod RS cpo_lub)::prems) 1;
+by (rewtac isub_def);
+by (safe_tac lemmas_cs);
+by (etac (iprodE RS apply_type) 1);
 by (assume_tac 1);
-by(asm_simp_tac ZF_ss 1);
+by (asm_simp_tac ZF_ss 1);
 by (dtac bspec 1);
 by (etac rel_iprodE 2);
-brr(lam_type::(chain_iprod RS cpo_lub RS islub_in)::iprodE::prems)1;
+brr(lam_type::(chain_iprod RS cpo_lub RS islub_in)::iprodE::prems) 1;
 val islub_iprod = result();
 
 val prems = goal Limit.thy (* cpo_iprod *)
     "(!!n. n:nat ==> cpo(DD`n)) ==> cpo(iprod(DD))";
-brr(cpoI::poI::[])1;
+brr(cpoI::poI::[]) 1;
 by (rtac rel_iprodI 1); (* not repeated: want to solve 1 and leave 2 unchanged *)
-brr(cpo_refl::(iprodE RS apply_type)::iprodE::prems)1;
+brr(cpo_refl::(iprodE RS apply_type)::iprodE::prems) 1;
 by (rtac rel_iprodI 1);
 by (dtac rel_iprodE 1);
 by (dtac rel_iprodE 2);
-brr(cpo_trans::(iprodE RS apply_type)::iprodE::prems)1;
+brr(cpo_trans::(iprodE RS apply_type)::iprodE::prems) 1;
 by (rtac fun_extension 1);
-brr(cpo_antisym::rel_iprodE::(iprodE RS apply_type)::iprodE::prems)1;
-brr(islub_iprod::prems)1;
+brr(cpo_antisym::rel_iprodE::(iprodE RS apply_type)::iprodE::prems) 1;
+brr(islub_iprod::prems) 1;
 val cpo_iprod = result();
 
 val prems = goalw Limit.thy [islub_def,isub_def]  (* lub_iprod *)
     "[|chain(iprod(DD),X);!!n. n:nat ==> cpo(DD`n)|] ==>   \
 \    lub(iprod(DD),X) = (lam n:nat. lub(DD`n,lam m:nat.X`m`n))";
-brr((cpo_lub RS islub_unique)::islub_iprod::cpo_iprod::prems)1;
+brr((cpo_lub RS islub_unique)::islub_iprod::cpo_iprod::prems) 1;
 val lub_iprod = result();
 
 (*----------------------------------------------------------------------*)
@@ -1432,10 +1432,10 @@
     "[|set(D)<=set(E);  \
 \      !!x y. [|x:set(D); y:set(D)|] ==> rel(D,x,y)<->rel(E,x,y);  \
 \      !!X. chain(D,X) ==> lub(E,X) : set(D)|] ==> subcpo(D,E)";
-by(safe_tac lemmas_cs);
-by(asm_full_simp_tac(ZF_ss addsimps prems)2);
-by(asm_simp_tac(ZF_ss addsimps prems)2);
-brr(prems@[subsetD])1;
+by (safe_tac lemmas_cs);
+by (asm_full_simp_tac(ZF_ss addsimps prems) 2);
+by (asm_simp_tac(ZF_ss addsimps prems) 2);
+brr(prems@[subsetD]) 1;
 val subcpoI = result();
 
 val subcpo_subset = prove_goalw Limit.thy [subcpo_def]  
@@ -1445,7 +1445,7 @@
 val subcpo_rel_eq = prove_goalw Limit.thy [subcpo_def]  
     " [|subcpo(D,E); x:set(D); y:set(D)|] ==> rel(D,x,y)<->rel(E,x,y)"
   (fn prems => 
-   [trr((hd prems RS conjunct2 RS conjunct1 RS bspec RS bspec)::prems)1]);
+   [trr((hd prems RS conjunct2 RS conjunct1 RS bspec RS bspec)::prems) 1]);
 
 val subcpo_relD1 = subcpo_rel_eq RS iffD1;
 val subcpo_relD2 = subcpo_rel_eq RS iffD2;
@@ -1459,18 +1459,18 @@
     "[|subcpo(D,E); chain(D,X)|] ==> chain(E,X)";
 by (rtac chainI 1);
 by (rtac Pi_type 1);
-brr(chain_fun::prems)1;
+brr(chain_fun::prems) 1;
 by (rtac subsetD 1);
 brr(subcpo_subset::chain_in::(hd prems RS subcpo_relD1)::nat_succI::
-    chain_rel::prems)1;
+    chain_rel::prems) 1;
 val chain_subcpo = result();
 
 val prems = goal Limit.thy  (* ub_subcpo *)
     "[|subcpo(D,E); chain(D,X); isub(D,X,x)|] ==> isub(E,X,x)";
 brr(isubI::(hd prems RS subcpo_subset RS subsetD)::
-    (hd prems RS subcpo_relD1)::prems)1;
-brr(isubD1::prems)1;
-brr((hd prems RS subcpo_relD1)::chain_in::isubD1::isubD2::prems)1;
+    (hd prems RS subcpo_relD1)::prems) 1;
+brr(isubD1::prems) 1;
+brr((hd prems RS subcpo_relD1)::chain_in::isubD1::isubD2::prems) 1;
 val ub_subcpo = result();
         
 (* STRIP_TAC and HOL resolution is efficient sometimes. The following
@@ -1481,30 +1481,30 @@
 brr[islubI,isubI]1;
 brr(subcpo_lub::(hd prems RS subcpo_relD2)::chain_in::islub_ub::islub_least::
     cpo_lub::(hd prems RS chain_subcpo)::isubD1::(hd prems RS ub_subcpo)::
-    prems)1;
+    prems) 1;
 val islub_subcpo = result();
 
 val prems = goal Limit.thy  (* subcpo_cpo *)
     "[|subcpo(D,E); cpo(E)|] ==> cpo(D)";
 brr[cpoI,poI]1;
 (* Changing the order of the assumptions, otherwise full_simp doesn't work. *)
-by(asm_full_simp_tac(ZF_ss addsimps[hd prems RS subcpo_rel_eq])1);
-brr(cpo_refl::(hd prems RS subcpo_subset RS subsetD)::prems)1;
-bd(imp_refl RS mp)1;
-bd(imp_refl RS mp)1;
-by(asm_full_simp_tac(ZF_ss addsimps[hd prems RS subcpo_rel_eq])1);
-brr(cpo_trans::(hd prems RS subcpo_subset RS subsetD)::prems)1;
+by (asm_full_simp_tac(ZF_ss addsimps[hd prems RS subcpo_rel_eq]) 1);
+brr(cpo_refl::(hd prems RS subcpo_subset RS subsetD)::prems) 1;
+by (dtac (imp_refl RS mp) 1);
+by (dtac (imp_refl RS mp) 1);
+by (asm_full_simp_tac(ZF_ss addsimps[hd prems RS subcpo_rel_eq]) 1);
+brr(cpo_trans::(hd prems RS subcpo_subset RS subsetD)::prems) 1;
 (* Changing the order of the assumptions, otherwise full_simp doesn't work. *)
-bd(imp_refl RS mp)1;
-bd(imp_refl RS mp)1;
-by(asm_full_simp_tac(ZF_ss addsimps[hd prems RS subcpo_rel_eq])1);
-brr(cpo_antisym::(hd prems RS subcpo_subset RS subsetD)::prems)1;
-brr(islub_subcpo::prems)1;
+by (dtac (imp_refl RS mp) 1);
+by (dtac (imp_refl RS mp) 1);
+by (asm_full_simp_tac(ZF_ss addsimps[hd prems RS subcpo_rel_eq]) 1);
+brr(cpo_antisym::(hd prems RS subcpo_subset RS subsetD)::prems) 1;
+brr(islub_subcpo::prems) 1;
 val subcpo_cpo = result();
 
 val prems = goal Limit.thy  (* lub_subcpo *)
     "[|subcpo(D,E); cpo(E); chain(D,X)|] ==> lub(D,X) = lub(E,X)";
-brr((cpo_lub RS islub_unique)::islub_subcpo::(hd prems RS subcpo_cpo)::prems)1;
+brr((cpo_lub RS islub_unique)::islub_subcpo::(hd prems RS subcpo_cpo)::prems) 1;
 val lub_subcpo = result();
 
 (*----------------------------------------------------------------------*)
@@ -1513,44 +1513,44 @@
 
 val prems = goalw Limit.thy [set_def,mkcpo_def]  (* mkcpoI *)
     "!!z. [|x:set(D); P(x)|] ==> x:set(mkcpo(D,P))";
-by(simp_tac ZF_ss 1);
-brr(conjI::prems)1;
+by (simp_tac ZF_ss 1);
+brr(conjI::prems) 1;
 val mkcpoI = result();
 
 (* Old proof where cpos are non-reflexive relations.
-by(rewtac set_def); (* Annoying, cannot just rewrite once. *)
+by (rewtac set_def); (* Annoying, cannot just rewrite once. *)
 by (rtac CollectI 1);
 by (rtac domainI 1);
 by (rtac CollectI 1);
 (* Now, work on subgoal 2 (and 3) to instantiate unknown. *)
-by(simp_tac ZF_ss 2);
+by (simp_tac ZF_ss 2);
 by (rtac conjI 2);
 by (rtac conjI 3);
 by (resolve_tac prems 3);
-by(simp_tac(ZF_ss addsimps [rewrite_rule[set_def](hd prems)])1);
+by (simp_tac(ZF_ss addsimps [rewrite_rule[set_def](hd prems)]) 1);
 by (resolve_tac prems 1);
 by (rtac cpo_refl 1);
 by (resolve_tac prems 1);
 by (rtac rel_I 1);
 by (rtac CollectI 1);
-by(fast_tac(ZF_cs addSIs [rewrite_rule[set_def](hd prems)])1);
-by(simp_tac ZF_ss 1);
-brr(conjI::cpo_refl::prems)1;
+by (fast_tac(ZF_cs addSIs [rewrite_rule[set_def](hd prems)]) 1);
+by (simp_tac ZF_ss 1);
+brr(conjI::cpo_refl::prems) 1;
 *)
 
 val prems = goalw Limit.thy [set_def,mkcpo_def]  (* mkcpoD1 *)
     "!!z. x:set(mkcpo(D,P))==> x:set(D)";
-by(asm_full_simp_tac ZF_ss 1);
+by (asm_full_simp_tac ZF_ss 1);
 val mkcpoD1 = result();
 
 val prems = goalw Limit.thy [set_def,mkcpo_def]  (* mkcpoD2 *)
     "!!z. x:set(mkcpo(D,P))==> P(x)";
-by(asm_full_simp_tac ZF_ss 1);
+by (asm_full_simp_tac ZF_ss 1);
 val mkcpoD2 = result();
 
 val prems = goalw Limit.thy [rel_def,mkcpo_def]  (* rel_mkcpoE *)
     "!!a. rel(mkcpo(D,P),x,y) ==> rel(D,x,y)";
-by(asm_full_simp_tac ZF_ss 1);
+by (asm_full_simp_tac ZF_ss 1);
 val rel_mkcpoE = result();
 
 val rel_mkcpo = prove_goalw Limit.thy [mkcpo_def,rel_def,set_def]
@@ -1565,25 +1565,25 @@
 by (rtac chainI 1);
 (*---begin additional---*)
 by (rtac Pi_type 1);
-brr(chain_fun::prems)1;
-brr((chain_in RS mkcpoD1)::prems)1;
+brr(chain_fun::prems) 1;
+brr((chain_in RS mkcpoD1)::prems) 1;
 (*---end additional---*)
-br(rel_mkcpo RS iffD1)1;
+by (rtac (rel_mkcpo RS iffD1) 1);
 (*---begin additional---*)
 by (rtac mkcpoD1 1); 
 by (rtac mkcpoD1 2); 
-brr(chain_in::nat_succI::prems)1; 
+brr(chain_in::nat_succI::prems) 1; 
 (*---end additional---*)
-brr(chain_rel::prems)1;
+brr(chain_rel::prems) 1;
 val chain_mkcpo = result();
 
 val prems = goal Limit.thy  (* subcpo_mkcpo *)
     "[|!!X. chain(mkcpo(D,P),X) ==> P(lub(D,X)); cpo(D)|] ==>   \
 \    subcpo(mkcpo(D,P),D)";
-brr(subcpoI::subsetI::prems)1;
+brr(subcpoI::subsetI::prems) 1;
 by (rtac rel_mkcpo 2);
-by(REPEAT(etac mkcpoD1 1)); 
-brr(mkcpoI::(cpo_lub RS islub_in)::chain_mkcpo::prems)1;
+by (REPEAT(etac mkcpoD1 1)); 
+brr(mkcpoI::(cpo_lub RS islub_in)::chain_mkcpo::prems) 1;
 val subcpo_mkcpo = result();
 
 (*----------------------------------------------------------------------*)
@@ -1593,7 +1593,7 @@
 val prems = goalw Limit.thy [emb_chain_def]  (* emb_chainI *)
     "[|!!n. n:nat ==> cpo(DD`n);   \
 \      !!n. n:nat ==> emb(DD`n,DD`succ(n),ee`n)|] ==> emb_chain(DD,ee)";
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 brr prems 1;
 val emb_chainI = result();
 
@@ -1613,7 +1613,7 @@
     "[|x:(PROD n:nat. set(DD`n));  \
 \      !!n. n:nat ==> Rp(DD`n,DD`succ(n),ee`n)`(x`succ(n)) = x`n|] ==>   \
 \    x:set(Dinf(DD,ee))";
-brr(mkcpoI::iprodI::ballI::prems)1;
+brr(mkcpoI::iprodI::ballI::prems) 1;
 val DinfI = result();
 
 val prems = goalw Limit.thy [Dinf_def]  (* DinfD1 *)
@@ -1627,7 +1627,7 @@
 val prems = goalw Limit.thy [Dinf_def]  (* DinfD2 *)
     "[|x:set(Dinf(DD,ee)); n:nat|] ==>   \
 \    Rp(DD`n,DD`succ(n),ee`n)`(x`succ(n)) = x`n";
-by(asm_simp_tac(ZF_ss addsimps[(hd prems RS mkcpoD2),hd(tl prems)])1);
+by (asm_simp_tac(ZF_ss addsimps[(hd prems RS mkcpoD2),hd(tl prems)]) 1);
 val DinfD2 = result();
 val Dinf_eq = DinfD2;
 
@@ -1635,9 +1635,9 @@
 val prems = goalw Limit.thy [Dinf_def]  (* rel_DinfI *)
     "[|!!n. n:nat ==> rel(DD`n,x`n,y`n);  \
 \      x:set(Dinf(DD,ee)); y:set(Dinf(DD,ee))|] ==> rel(Dinf(DD,ee),x,y)";
-br(rel_mkcpo RS iffD2)1;
+by (rtac (rel_mkcpo RS iffD2) 1);
 brr prems 1;
-brr(rel_iprodI::rewrite_rule[Dinf_def]DinfD1::prems)1;
+brr(rel_iprodI::rewrite_rule[Dinf_def]DinfD1::prems) 1;
 val rel_DinfI = result();
 *)
 
@@ -1645,33 +1645,33 @@
     "[|!!n. n:nat ==> rel(DD`n,x`n,y`n);  \
 \      x:(PROD n:nat. set(DD`n)); y:(PROD n:nat. set(DD`n))|] ==>   \
 \    rel(Dinf(DD,ee),x,y)";
-br(rel_mkcpo RS iffD2)1;
-brr(rel_iprodI::iprodI::prems)1;
+by (rtac (rel_mkcpo RS iffD2) 1);
+brr(rel_iprodI::iprodI::prems) 1;
 val rel_DinfI = result();
 
 val prems = goalw Limit.thy [Dinf_def]  (* rel_Dinf *)
     "[|rel(Dinf(DD,ee),x,y); n:nat|] ==> rel(DD`n,x`n,y`n)";
-br(hd prems RS rel_mkcpoE RS rel_iprodE)1;
+by (rtac (hd prems RS rel_mkcpoE RS rel_iprodE) 1);
 by (resolve_tac prems 1);
 val rel_Dinf = result();
 
 val chain_Dinf = prove_goalw Limit.thy [Dinf_def] 
     "chain(Dinf(DD,ee),X) ==> chain(iprod(DD),X)"
-  (fn prems => [rtac(hd prems RS chain_mkcpo)1]);
+  (fn prems => [rtac(hd prems RS chain_mkcpo) 1]);
 
 val prems = goalw Limit.thy [Dinf_def]  (* subcpo_Dinf *)
     "emb_chain(DD,ee) ==> subcpo(Dinf(DD,ee),iprod(DD))";
 by (rtac subcpo_mkcpo 1);
-by(fold_tac [Dinf_def]);
+by (fold_tac [Dinf_def]);
 by (rtac ballI 1);
-br(lub_iprod RS ssubst)1;
-brr(chain_Dinf::(hd prems RS emb_chain_cpo)::[])1;
-by(asm_simp_tac arith_ss 1);
-br(Rp_cont RS cont_lub RS ssubst)1;
-brr(emb_chain_cpo::emb_chain_emb::nat_succI::chain_iprod::chain_Dinf::prems)1;
+by (rtac (lub_iprod RS ssubst) 1);
+brr(chain_Dinf::(hd prems RS emb_chain_cpo)::[]) 1;
+by (asm_simp_tac arith_ss 1);
+by (rtac (Rp_cont RS cont_lub RS ssubst) 1);
+brr(emb_chain_cpo::emb_chain_emb::nat_succI::chain_iprod::chain_Dinf::prems) 1;
 (* Useful simplification, ugly in HOL. *)
-by(asm_simp_tac(arith_ss addsimps(DinfD2::chain_in::[]))1);
-brr(cpo_iprod::emb_chain_cpo::prems)1;
+by (asm_simp_tac(arith_ss addsimps(DinfD2::chain_in::[])) 1);
+brr(cpo_iprod::emb_chain_cpo::prems) 1;
 val subcpo_Dinf = result();
 
 (* Simple example of existential reasoning in Isabelle versus HOL. *)
@@ -1679,7 +1679,7 @@
 val prems = goal Limit.thy  (* cpo_Dinf *)
   "emb_chain(DD,ee) ==> cpo(Dinf(DD,ee))";
 by (rtac subcpo_cpo 1);
-brr(subcpo_Dinf::cpo_iprod::emb_chain_cpo::prems)1;;
+brr(subcpo_Dinf::cpo_iprod::emb_chain_cpo::prems) 1;;
 val cpo_Dinf = result();
 
 (* Again and again the proofs are much easier to WRITE in Isabelle, but 
@@ -1688,8 +1688,8 @@
 val prems = goal Limit.thy  (* lub_Dinf *)
     "[|chain(Dinf(DD,ee),X); emb_chain(DD,ee)|] ==>  \
 \    lub(Dinf(DD,ee),X) = (lam n:nat. lub(DD`n,lam m:nat. X`m`n))";
-br(subcpo_Dinf RS lub_subcpo RS ssubst)1;
-brr(cpo_iprod::emb_chain_cpo::lub_iprod::chain_Dinf::prems)1;
+by (rtac (subcpo_Dinf RS lub_subcpo RS ssubst) 1);
+brr(cpo_iprod::emb_chain_cpo::lub_iprod::chain_Dinf::prems) 1;
 val lub_Dinf = result();
 
 (*----------------------------------------------------------------------*)
@@ -1699,7 +1699,7 @@
 
 val prems = goalw Limit.thy [e_less_def]  (* e_less_eq *)
     "!!x. m:nat ==> e_less(DD,ee,m,m) = id(set(DD`m))";
-by(asm_simp_tac (arith_ss addsimps[diff_self_eq_0]) 1);
+by (asm_simp_tac (arith_ss addsimps[diff_self_eq_0]) 1);
 val e_less_eq = result();
  
 (* ARITH_CONV proves the following in HOL. Would like something similar 
@@ -1708,13 +1708,13 @@
 goalw Arith.thy []  (* lemma_succ_sub *)
     "!!z. [|n:nat; m:nat|] ==> succ(m#+n)#-m = succ(n)";
 (*Uses add_succ_right the wrong way round!*)
-by(asm_simp_tac(nat_ss addsimps [add_succ_right RS sym, diff_add_inverse])1);
+by (asm_simp_tac(nat_ss addsimps [add_succ_right RS sym, diff_add_inverse]) 1);
 val lemma_succ_sub = result();
 
 val prems = goalw Limit.thy [e_less_def] (* e_less_add *)
     "!!x. [|m:nat; k:nat|] ==>    \
 \         e_less(DD,ee,m,succ(m#+k)) = (ee`(m#+k))O(e_less(DD,ee,m,m#+k))";
-by(asm_simp_tac (arith_ss addsimps [lemma_succ_sub,diff_add_inverse]) 1);
+by (asm_simp_tac (arith_ss addsimps [lemma_succ_sub,diff_add_inverse]) 1);
 val e_less_add = result();
 
 (* Again, would like more theorems about arithmetic. *)
@@ -1727,51 +1727,51 @@
 
 val prems = goal Limit.thy  (* succ_sub1 *)
     "x:nat ==> 0 < x --> succ(x#-1)=x";
-by(res_inst_tac[("n","x")]nat_induct 1);
+by (res_inst_tac[("n","x")]nat_induct 1);
 by (resolve_tac prems 1);
-by(fast_tac lt_cs 1);
-by(safe_tac lemmas_cs);
-by(asm_simp_tac arith_ss 1);
-by(asm_simp_tac arith_ss 1);
+by (fast_tac lt_cs 1);
+by (safe_tac lemmas_cs);
+by (asm_simp_tac arith_ss 1);
+by (asm_simp_tac arith_ss 1);
 val succ_sub1 = result();
 
 val prems = goal Limit.thy (* succ_le_pos *)
     "[|m:nat; k:nat|] ==> succ(m) le m #+ k --> 0 < k";
-by(res_inst_tac[("n","m")]nat_induct 1);
+by (res_inst_tac[("n","m")]nat_induct 1);
 by (resolve_tac prems 1);
 by (rtac impI 1);
-by(asm_full_simp_tac(arith_ss addsimps prems)1);
-by(safe_tac lemmas_cs);
-by(asm_full_simp_tac(arith_ss addsimps prems)1); (* Surprise, surprise. *)
+by (asm_full_simp_tac(arith_ss addsimps prems) 1);
+by (safe_tac lemmas_cs);
+by (asm_full_simp_tac(arith_ss addsimps prems) 1); (* Surprise, surprise. *)
 val succ_le_pos = result();
 
 val prems = goal Limit.thy  (* lemma_le_exists *)
     "!!z. [|n:nat; m:nat|] ==> m le n --> (EX k:nat. n = m #+ k)";
-by(res_inst_tac[("n","m")]nat_induct 1);
+by (res_inst_tac[("n","m")]nat_induct 1);
 by (assume_tac 1);
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 by (rtac bexI 1);
-br(add_0 RS sym)1;
+by (rtac (add_0 RS sym) 1);
 by (assume_tac 1);
-by(asm_full_simp_tac arith_ss 1);
+by (asm_full_simp_tac arith_ss 1);
 (* Great, by luck I found lt_cs. Such cs's and ss's should be documented. *)
-by(fast_tac lt_cs 1); 
-by(asm_simp_tac (nat_ss addsimps[add_succ, add_succ_right RS sym]) 1);
+by (fast_tac lt_cs 1); 
+by (asm_simp_tac (nat_ss addsimps[add_succ, add_succ_right RS sym]) 1);
 by (rtac bexI 1);
-br(succ_sub1 RS mp RS ssubst)1;
+by (rtac (succ_sub1 RS mp RS ssubst) 1);
 (* Instantiation. *)
 by (rtac refl 3);
 by (assume_tac 1);
-br(succ_le_pos RS mp)1;
+by (rtac (succ_le_pos RS mp) 1);
 by (assume_tac 3); (* Instantiation *)
 brr[]1;
-by(asm_simp_tac arith_ss 1);
+by (asm_simp_tac arith_ss 1);
 val lemma_le_exists = result();
 
 val prems = goal Limit.thy  (* le_exists *)
     "[|m le n;!!x. [|n=m#+x; x:nat|] ==> Q; m:nat; n:nat|] ==> Q";
-br(lemma_le_exists RS mp RS bexE)1;
-br(hd(tl prems))4;
+by (rtac (lemma_le_exists RS mp RS bexE) 1);
+by (rtac (hd(tl prems)) 4);
 by (assume_tac 4);
 brr prems 1;
 val le_exists = result();
@@ -1781,7 +1781,7 @@
 \    e_less(DD,ee,m,succ(n)) = ee`n O e_less(DD,ee,m,n)";
 by (rtac le_exists 1);
 by (resolve_tac prems 1);
-by(asm_simp_tac(ZF_ss addsimps(e_less_add::prems))1);
+by (asm_simp_tac(ZF_ss addsimps(e_less_add::prems)) 1);
 brr prems 1;
 val e_less_le = result();
 
@@ -1789,15 +1789,15 @@
 
 val prems = goal Limit.thy  (* e_less_succ *)
     "m:nat ==> e_less(DD,ee,m,succ(m)) = ee`m O id(set(DD`m))";
-by(asm_simp_tac(arith_ss addsimps(e_less_le::e_less_eq::prems))1);
+by (asm_simp_tac(arith_ss addsimps(e_less_le::e_less_eq::prems)) 1);
 val e_less_succ = result();
 
 val prems = goal Limit.thy  (* e_less_succ_emb *)
     "[|!!n. n:nat ==> emb(DD`n,DD`succ(n),ee`n); m:nat|] ==>   \
 \    e_less(DD,ee,m,succ(m)) = ee`m";
-by(asm_simp_tac(arith_ss addsimps(e_less_succ::prems))1);
-br(comp_id RS ssubst)1;
-brr(emb_cont::cont_fun::refl::prems)1;
+by (asm_simp_tac(arith_ss addsimps(e_less_succ::prems)) 1);
+by (rtac (comp_id RS ssubst) 1);
+brr(emb_cont::cont_fun::refl::prems) 1;
 val e_less_succ_emb = result();
 
 (* Compare this proof with the HOL one, here we do type checking. *)
@@ -1806,12 +1806,12 @@
 val prems = goal Limit.thy  (* emb_e_less_add *)
     "[|emb_chain(DD,ee); m:nat; k:nat|] ==>   \
 \    emb(DD`m,DD`(m#+k),e_less(DD,ee,m,m#+k))";
-by(res_inst_tac[("n","k")]nat_induct 1);
+by (res_inst_tac[("n","k")]nat_induct 1);
 by (resolve_tac prems 1);
-by(asm_simp_tac(ZF_ss addsimps(add_0_right::e_less_eq::prems))1);
-brr(emb_id::emb_chain_cpo::prems)1;
-by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_less_add::prems))1);
-brr(emb_comp::emb_chain_emb::emb_chain_cpo::add_type::nat_succI::prems)1;
+by (asm_simp_tac(ZF_ss addsimps(add_0_right::e_less_eq::prems)) 1);
+brr(emb_id::emb_chain_cpo::prems) 1;
+by (asm_simp_tac(ZF_ss addsimps(add_succ_right::e_less_add::prems)) 1);
+brr(emb_comp::emb_chain_emb::emb_chain_cpo::add_type::nat_succI::prems) 1;
 val emb_e_less_add = result();
 
 val prems = goal Limit.thy  (* emb_e_less *)
@@ -1820,7 +1820,7 @@
 (* same proof as e_less_le *)
 by (rtac le_exists 1);
 by (resolve_tac prems 1);
-by(asm_simp_tac(ZF_ss addsimps(emb_e_less_add::prems))1);
+by (asm_simp_tac(ZF_ss addsimps(emb_e_less_add::prems)) 1);
 brr prems 1;
 val emb_e_less = result();
 
@@ -1838,44 +1838,44 @@
     "[| emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==>  \
 \    n le k --> \
 \    e_less(DD,ee,m,m#+k) = e_less(DD,ee,m#+n,m#+k) O e_less(DD,ee,m,m#+n)";
-by(res_inst_tac[("n","k")]nat_induct 1);
+by (res_inst_tac[("n","k")]nat_induct 1);
 by (resolve_tac prems 1);
 by (rtac impI 1);
-by(asm_full_simp_tac(ZF_ss addsimps
-    (le0_iff::add_0_right::e_less_eq::(id_type RS id_comp)::prems))1);
-by(asm_simp_tac(ZF_ss addsimps[le_succ_iff])1);
+by (asm_full_simp_tac(ZF_ss addsimps
+    (le0_iff::add_0_right::e_less_eq::(id_type RS id_comp)::prems)) 1);
+by (asm_simp_tac(ZF_ss addsimps[le_succ_iff]) 1);
 by (rtac impI 1);
 by (etac disjE 1);
 by (etac impE 1);
 by (assume_tac 1);
-by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_less_add::
-    add_type::nat_succI::prems))1);
+by (asm_simp_tac(ZF_ss addsimps(add_succ_right::e_less_add::
+    add_type::nat_succI::prems)) 1);
 (* Again and again, simplification is a pain. When does it work, when not? *)
-br(e_less_le RS ssubst)1;
-brr(add_le_mono::nat_le_refl::add_type::nat_succI::prems)1;
-br(comp_assoc RS ssubst)1;
-brr(comp_mono_eq::refl::[])1;
+by (rtac (e_less_le RS ssubst) 1);
+brr(add_le_mono::nat_le_refl::add_type::nat_succI::prems) 1;
+by (rtac (comp_assoc RS ssubst) 1);
+brr(comp_mono_eq::refl::[]) 1;
 (* by(asm_simp_tac ZF_ss 1); *)
-by(asm_simp_tac(ZF_ss addsimps(e_less_eq::add_type::nat_succI::prems))1);
-br(id_comp RS ssubst)1; (* simp cannot unify/inst right, use brr below(?). *)
-brr((emb_e_less_add RS emb_cont RS cont_fun)::refl::nat_succI::prems)1;
+by (asm_simp_tac(ZF_ss addsimps(e_less_eq::add_type::nat_succI::prems)) 1);
+by (rtac (id_comp RS ssubst) 1); (* simp cannot unify/inst right, use brr below(?). *)
+brr((emb_e_less_add RS emb_cont RS cont_fun)::refl::nat_succI::prems) 1;
 val e_less_split_add_lemma = result();
 
 val e_less_split_add = prove_goal Limit.thy  
     "[| n le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==>  \
 \    e_less(DD,ee,m,m#+k) = e_less(DD,ee,m#+n,m#+k) O e_less(DD,ee,m,m#+n)"
-  (fn prems => [trr((e_less_split_add_lemma RS mp)::prems)1]);
+  (fn prems => [trr((e_less_split_add_lemma RS mp)::prems) 1]);
 
 val prems = goalw Limit.thy [e_gr_def]  (* e_gr_eq *)
     "!!x. m:nat ==> e_gr(DD,ee,m,m) = id(set(DD`m))";
-by(asm_simp_tac (arith_ss addsimps[diff_self_eq_0]) 1);
+by (asm_simp_tac (arith_ss addsimps[diff_self_eq_0]) 1);
 val e_gr_eq = result();
 
 val prems = goalw Limit.thy [e_gr_def] (* e_gr_add *)
     "!!x. [|n:nat; k:nat|] ==>    \
 \         e_gr(DD,ee,succ(n#+k),n) =   \
 \         e_gr(DD,ee,n#+k,n) O Rp(DD`(n#+k),DD`succ(n#+k),ee`(n#+k))";
-by(asm_simp_tac (arith_ss addsimps [lemma_succ_sub,diff_add_inverse]) 1);
+by (asm_simp_tac (arith_ss addsimps [lemma_succ_sub,diff_add_inverse]) 1);
 val e_gr_add = result();
 
 val prems = goal Limit.thy  (* e_gr_le *)
@@ -1883,14 +1883,14 @@
 \    e_gr(DD,ee,succ(m),n) = e_gr(DD,ee,m,n) O Rp(DD`m,DD`succ(m),ee`m)";
 by (rtac le_exists 1);
 by (resolve_tac prems 1);
-by(asm_simp_tac(ZF_ss addsimps(e_gr_add::prems))1);
+by (asm_simp_tac(ZF_ss addsimps(e_gr_add::prems)) 1);
 brr prems 1;
 val e_gr_le = result();
 
 val prems = goal Limit.thy  (* e_gr_succ *)
     "m:nat ==>   \
 \    e_gr(DD,ee,succ(m),m) = id(set(DD`m)) O Rp(DD`m,DD`succ(m),ee`m)";
-by(asm_simp_tac(arith_ss addsimps(e_gr_le::e_gr_eq::prems))1);
+by (asm_simp_tac(arith_ss addsimps(e_gr_le::e_gr_eq::prems)) 1);
 val e_gr_succ = result();
 
 (* Cpo asm's due to THE uniqueness. *)
@@ -1898,20 +1898,20 @@
 val prems = goal Limit.thy  (* e_gr_succ_emb *)
     "[|emb_chain(DD,ee); m:nat|] ==>   \
 \    e_gr(DD,ee,succ(m),m) = Rp(DD`m,DD`succ(m),ee`m)";
-by(asm_simp_tac(arith_ss addsimps(e_gr_succ::prems))1);
-br(id_comp RS ssubst)1;
-brr(Rp_cont::cont_fun::refl::emb_chain_cpo::emb_chain_emb::nat_succI::prems)1;
+by (asm_simp_tac(arith_ss addsimps(e_gr_succ::prems)) 1);
+by (rtac (id_comp RS ssubst) 1);
+brr(Rp_cont::cont_fun::refl::emb_chain_cpo::emb_chain_emb::nat_succI::prems) 1;
 val e_gr_succ_emb = result();
 
 val prems = goal Limit.thy  (* e_gr_fun_add *)
     "[|emb_chain(DD,ee); n:nat; k:nat|] ==>   \
 \    e_gr(DD,ee,n#+k,n): set(DD`(n#+k))->set(DD`n)";
-by(res_inst_tac[("n","k")]nat_induct 1);
+by (res_inst_tac[("n","k")]nat_induct 1);
 by (resolve_tac prems 1);
-by(asm_simp_tac(ZF_ss addsimps(add_0_right::e_gr_eq::id_type::prems))1);
-by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_add::prems))1);
+by (asm_simp_tac(ZF_ss addsimps(add_0_right::e_gr_eq::id_type::prems)) 1);
+by (asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_add::prems)) 1);
 brr(comp_fun::Rp_cont::cont_fun::emb_chain_emb::emb_chain_cpo::add_type::
-    nat_succI::prems)1;
+    nat_succI::prems) 1;
 val e_gr_fun_add = result();
 
 val prems = goal Limit.thy  (* e_gr_fun *)
@@ -1919,7 +1919,7 @@
 \    e_gr(DD,ee,m,n): set(DD`m)->set(DD`n)";
 by (rtac le_exists 1);
 by (resolve_tac prems 1);
-by(asm_simp_tac(ZF_ss addsimps(e_gr_fun_add::prems))1);
+by (asm_simp_tac(ZF_ss addsimps(e_gr_fun_add::prems)) 1);
 brr prems 1;
 val e_gr_fun = result();
 
@@ -1927,61 +1927,61 @@
     "[| emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==>  \
 \    m le k --> \
 \    e_gr(DD,ee,n#+k,n) = e_gr(DD,ee,n#+m,n) O e_gr(DD,ee,n#+k,n#+m)";
-by(res_inst_tac[("n","k")]nat_induct 1);
+by (res_inst_tac[("n","k")]nat_induct 1);
 by (resolve_tac prems 1);
 by (rtac impI 1);
-by(asm_full_simp_tac(ZF_ss addsimps
-    (le0_iff::add_0_right::e_gr_eq::(id_type RS comp_id)::prems))1);
-by(asm_simp_tac(ZF_ss addsimps[le_succ_iff])1);
+by (asm_full_simp_tac(ZF_ss addsimps
+    (le0_iff::add_0_right::e_gr_eq::(id_type RS comp_id)::prems)) 1);
+by (asm_simp_tac(ZF_ss addsimps[le_succ_iff]) 1);
 by (rtac impI 1);
 by (etac disjE 1);
 by (etac impE 1);
 by (assume_tac 1);
-by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_add::
-    add_type::nat_succI::prems))1);
+by (asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_add::
+    add_type::nat_succI::prems)) 1);
 (* Again and again, simplification is a pain. When does it work, when not? *)
-br(e_gr_le RS ssubst)1;
-brr(add_le_mono::nat_le_refl::add_type::nat_succI::prems)1;
-br(comp_assoc RS ssubst)1;
-brr(comp_mono_eq::refl::[])1;
+by (rtac (e_gr_le RS ssubst) 1);
+brr(add_le_mono::nat_le_refl::add_type::nat_succI::prems) 1;
+by (rtac (comp_assoc RS ssubst) 1);
+brr(comp_mono_eq::refl::[]) 1;
 (* New direct subgoal *)
-by(asm_simp_tac(ZF_ss addsimps(e_gr_eq::add_type::nat_succI::prems))1);
-br(comp_id RS ssubst)1; (* simp cannot unify/inst right, use brr below(?). *)
-brr(e_gr_fun::add_type::refl::add_le_self::nat_succI::prems)1;
+by (asm_simp_tac(ZF_ss addsimps(e_gr_eq::add_type::nat_succI::prems)) 1);
+by (rtac (comp_id RS ssubst) 1); (* simp cannot unify/inst right, use brr below(?). *)
+brr(e_gr_fun::add_type::refl::add_le_self::nat_succI::prems) 1;
 val e_gr_split_add_lemma = result();
 
 val e_gr_split_add = prove_goal Limit.thy  
     "[| m le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==>  \
 \    e_gr(DD,ee,n#+k,n) = e_gr(DD,ee,n#+m,n) O e_gr(DD,ee,n#+k,n#+m)"
-  (fn prems => [trr((e_gr_split_add_lemma RS mp)::prems)1]);
+  (fn prems => [trr((e_gr_split_add_lemma RS mp)::prems) 1]);
 
 val e_less_cont = prove_goal Limit.thy  
     "[|m le n; emb_chain(DD,ee); m:nat; n:nat|] ==>   \
 \    e_less(DD,ee,m,n):cont(DD`m,DD`n)"
-  (fn prems => [trr(emb_cont::emb_e_less::prems)1]);
+  (fn prems => [trr(emb_cont::emb_e_less::prems) 1]);
 
 val prems = goal Limit.thy  (* e_gr_cont_lemma *)
     "[|emb_chain(DD,ee); m:nat; n:nat|] ==>   \
 \    n le m --> e_gr(DD,ee,m,n):cont(DD`m,DD`n)";
-by(res_inst_tac[("n","m")]nat_induct 1);
+by (res_inst_tac[("n","m")]nat_induct 1);
 by (resolve_tac prems 1);
-by(asm_full_simp_tac(ZF_ss addsimps
-    (le0_iff::e_gr_eq::nat_0I::prems))1);
-brr(impI::id_cont::emb_chain_cpo::nat_0I::prems)1;
-by(asm_full_simp_tac(ZF_ss addsimps[le_succ_iff])1);
+by (asm_full_simp_tac(ZF_ss addsimps
+    (le0_iff::e_gr_eq::nat_0I::prems)) 1);
+brr(impI::id_cont::emb_chain_cpo::nat_0I::prems) 1;
+by (asm_full_simp_tac(ZF_ss addsimps[le_succ_iff]) 1);
 by (etac disjE 1);
 by (etac impE 1);
 by (assume_tac 1);
-by(asm_simp_tac(ZF_ss addsimps(e_gr_le::prems))1);
-brr(comp_pres_cont::Rp_cont::emb_chain_cpo::emb_chain_emb::nat_succI::prems)1;
-by(asm_simp_tac(ZF_ss addsimps(e_gr_eq::nat_succI::prems))1);
-brr(id_cont::emb_chain_cpo::nat_succI::prems)1;
+by (asm_simp_tac(ZF_ss addsimps(e_gr_le::prems)) 1);
+brr(comp_pres_cont::Rp_cont::emb_chain_cpo::emb_chain_emb::nat_succI::prems) 1;
+by (asm_simp_tac(ZF_ss addsimps(e_gr_eq::nat_succI::prems)) 1);
+brr(id_cont::emb_chain_cpo::nat_succI::prems) 1;
 val e_gr_cont_lemma = result();
 
 val prems = goal Limit.thy  (* e_gr_cont *)
     "[|n le m; emb_chain(DD,ee); m:nat; n:nat|] ==>   \
 \    e_gr(DD,ee,m,n):cont(DD`m,DD`n)";
-brr((e_gr_cont_lemma RS mp)::prems)1;
+brr((e_gr_cont_lemma RS mp)::prems) 1;
 val e_gr_cont = result();
 
 (* Considerably shorter.... 57 against 26 *)
@@ -1992,26 +1992,26 @@
 (* Use mp to prepare for induction. *)
 by (rtac mp 1);
 by (resolve_tac prems 2);
-by(res_inst_tac[("n","k")]nat_induct 1);
+by (res_inst_tac[("n","k")]nat_induct 1);
 by (resolve_tac prems 1);
-by(asm_full_simp_tac(ZF_ss addsimps
-    (le0_iff::add_0_right::e_gr_eq::e_less_eq::(id_type RS id_comp)::prems))1);
-by(simp_tac(ZF_ss addsimps[le_succ_iff])1);
+by (asm_full_simp_tac(ZF_ss addsimps
+    (le0_iff::add_0_right::e_gr_eq::e_less_eq::(id_type RS id_comp)::prems)) 1);
+by (simp_tac(ZF_ss addsimps[le_succ_iff]) 1);
 by (rtac impI 1);
 by (etac disjE 1);
 by (etac impE 1);
 by (assume_tac 1);
-by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_le::e_less_le::
-   add_le_self::nat_le_refl::add_le_mono::add_type::prems))1);
-br(comp_assoc RS ssubst)1;
-by(res_inst_tac[("s1","ee`(m#+x)")](comp_assoc RS subst)1);
-br(embRp_eq RS ssubst)1;
-brr(emb_chain_emb::add_type::emb_chain_cpo::nat_succI::prems)1;
-br(id_comp RS ssubst)1;
-brr((e_less_cont RS cont_fun)::add_type::add_le_self::refl::prems)1;
-by(asm_full_simp_tac(ZF_ss addsimps(e_gr_eq::nat_succI::add_type::prems))1);
-br(id_comp RS ssubst)1;
-brr((e_less_cont RS cont_fun)::add_type::nat_succI::add_le_self::refl::prems)1;
+by (asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_le::e_less_le::
+   add_le_self::nat_le_refl::add_le_mono::add_type::prems)) 1);
+by (rtac (comp_assoc RS ssubst) 1);
+by (res_inst_tac[("s1","ee`(m#+x)")](comp_assoc RS subst) 1);
+by (rtac (embRp_eq RS ssubst) 1);
+brr(emb_chain_emb::add_type::emb_chain_cpo::nat_succI::prems) 1;
+by (rtac (id_comp RS ssubst) 1);
+brr((e_less_cont RS cont_fun)::add_type::add_le_self::refl::prems) 1;
+by (asm_full_simp_tac(ZF_ss addsimps(e_gr_eq::nat_succI::add_type::prems)) 1);
+by (rtac (id_comp RS ssubst) 1);
+brr((e_less_cont RS cont_fun)::add_type::nat_succI::add_le_self::refl::prems) 1;
 val e_less_e_gr_split_add = result();
 
 (* Again considerably shorter, and easy to obtain from the previous thm. *)
@@ -2022,100 +2022,100 @@
 (* Use mp to prepare for induction. *)
 by (rtac mp 1);
 by (resolve_tac prems 2);
-by(res_inst_tac[("n","k")]nat_induct 1);
+by (res_inst_tac[("n","k")]nat_induct 1);
 by (resolve_tac prems 1);
-by(asm_full_simp_tac(arith_ss addsimps
-    (add_0_right::e_gr_eq::e_less_eq::(id_type RS id_comp)::prems))1);
-by(simp_tac(ZF_ss addsimps[le_succ_iff])1);
+by (asm_full_simp_tac(arith_ss addsimps
+    (add_0_right::e_gr_eq::e_less_eq::(id_type RS id_comp)::prems)) 1);
+by (simp_tac(ZF_ss addsimps[le_succ_iff]) 1);
 by (rtac impI 1);
 by (etac disjE 1);
 by (etac impE 1);
 by (assume_tac 1);
-by(asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_le::e_less_le::
-   add_le_self::nat_le_refl::add_le_mono::add_type::prems))1);
-br(comp_assoc RS ssubst)1;
-by(res_inst_tac[("s1","ee`(n#+x)")](comp_assoc RS subst)1);
-br(embRp_eq RS ssubst)1;
-brr(emb_chain_emb::add_type::emb_chain_cpo::nat_succI::prems)1;
-br(id_comp RS ssubst)1;
+by (asm_simp_tac(ZF_ss addsimps(add_succ_right::e_gr_le::e_less_le::
+   add_le_self::nat_le_refl::add_le_mono::add_type::prems)) 1);
+by (rtac (comp_assoc RS ssubst) 1);
+by (res_inst_tac[("s1","ee`(n#+x)")](comp_assoc RS subst) 1);
+by (rtac (embRp_eq RS ssubst) 1);
+brr(emb_chain_emb::add_type::emb_chain_cpo::nat_succI::prems) 1;
+by (rtac (id_comp RS ssubst) 1);
 brr((e_less_cont RS cont_fun)::add_type::add_le_mono::nat_le_refl::refl::
-    prems)1;
-by(asm_full_simp_tac(ZF_ss addsimps(e_less_eq::nat_succI::add_type::prems))1);
-br(comp_id RS ssubst)1;
-brr((e_gr_cont RS cont_fun)::add_type::nat_succI::add_le_self::refl::prems)1;
+    prems) 1;
+by (asm_full_simp_tac(ZF_ss addsimps(e_less_eq::nat_succI::add_type::prems)) 1);
+by (rtac (comp_id RS ssubst) 1);
+brr((e_gr_cont RS cont_fun)::add_type::nat_succI::add_le_self::refl::prems) 1;
 val e_gr_e_less_split_add = result();
 
 val prems = goalw Limit.thy [eps_def]  (* emb_eps *)
     "[|m le n; emb_chain(DD,ee); m:nat; n:nat|] ==>   \
 \    emb(DD`m,DD`n,eps(DD,ee,m,n))";
-by(asm_simp_tac(ZF_ss addsimps prems)1);
-brr(emb_e_less::prems)1;
+by (asm_simp_tac(ZF_ss addsimps prems) 1);
+brr(emb_e_less::prems) 1;
 val emb_eps = result();
 
 val prems = goalw Limit.thy [eps_def]  (* eps_fun *)
     "[|emb_chain(DD,ee); m:nat; n:nat|] ==>   \
 \    eps(DD,ee,m,n): set(DD`m)->set(DD`n)";
-br(expand_if RS iffD2)1;
-by(safe_tac lemmas_cs);
-brr((e_less_cont RS cont_fun)::prems)1;
-brr((not_le_iff_lt RS iffD1 RS leI)::e_gr_fun::nat_into_Ord::prems)1;
+by (rtac (expand_if RS iffD2) 1);
+by (safe_tac lemmas_cs);
+brr((e_less_cont RS cont_fun)::prems) 1;
+brr((not_le_iff_lt RS iffD1 RS leI)::e_gr_fun::nat_into_Ord::prems) 1;
 val eps_fun = result();
 
 val eps_id = prove_goalw Limit.thy [eps_def]  
     "n:nat ==> eps(DD,ee,n,n) = id(set(DD`n))"
-  (fn prems => [simp_tac(ZF_ss addsimps(e_less_eq::nat_le_refl::prems))1]);
+  (fn prems => [simp_tac(ZF_ss addsimps(e_less_eq::nat_le_refl::prems)) 1]);
 
 val eps_e_less_add = prove_goalw Limit.thy [eps_def]
     "[|m:nat; n:nat|] ==> eps(DD,ee,m,m#+n) = e_less(DD,ee,m,m#+n)"
-  (fn prems => [simp_tac(ZF_ss addsimps(add_le_self::prems))1]);
+  (fn prems => [simp_tac(ZF_ss addsimps(add_le_self::prems)) 1]);
 
 val eps_e_less = prove_goalw Limit.thy [eps_def]
     "[|m le n; m:nat; n:nat|] ==> eps(DD,ee,m,n) = e_less(DD,ee,m,n)"
-  (fn prems => [simp_tac(ZF_ss addsimps prems)1]);
+  (fn prems => [simp_tac(ZF_ss addsimps prems) 1]);
 
 val shift_asm = imp_refl RS mp;
 
 val prems = goalw Limit.thy [eps_def]  (* eps_e_gr_add *)
     "[|n:nat; k:nat|] ==> eps(DD,ee,n#+k,n) = e_gr(DD,ee,n#+k,n)";
-br(expand_if RS iffD2)1;
-by(safe_tac lemmas_cs);
+by (rtac (expand_if RS iffD2) 1);
+by (safe_tac lemmas_cs);
 by (etac leE 1);
 (* Must control rewriting by instantiating a variable. *)
-by(asm_full_simp_tac(arith_ss addsimps
+by (asm_full_simp_tac(arith_ss addsimps
      ((hd prems RS nat_into_Ord RS not_le_iff_lt RS iff_sym)::nat_into_Ord::
-      add_le_self::prems))1);
-by(asm_simp_tac(ZF_ss addsimps(e_less_eq::e_gr_eq::prems))1);
+      add_le_self::prems)) 1);
+by (asm_simp_tac(ZF_ss addsimps(e_less_eq::e_gr_eq::prems)) 1);
 val eps_e_gr_add = result();
 
 val prems = goalw Limit.thy []  (* eps_e_gr *)
     "[|n le m; m:nat; n:nat|] ==> eps(DD,ee,m,n) = e_gr(DD,ee,m,n)";
 by (rtac le_exists 1);
 by (resolve_tac prems 1);
-by(asm_simp_tac(ZF_ss addsimps(eps_e_gr_add::prems))1);
+by (asm_simp_tac(ZF_ss addsimps(eps_e_gr_add::prems)) 1);
 brr prems 1;
 val eps_e_gr = result();
 
 val prems = goal Limit.thy  (* eps_succ_ee *)
     "[|!!n. n:nat ==> emb(DD`n,DD`succ(n),ee`n); m:nat|] ==>  \
 \    eps(DD,ee,m,succ(m)) = ee`m";
-by(asm_simp_tac(arith_ss addsimps(eps_e_less::le_succ_iff::e_less_succ_emb::
-   prems))1);
+by (asm_simp_tac(arith_ss addsimps(eps_e_less::le_succ_iff::e_less_succ_emb::
+   prems)) 1);
 val eps_succ_ee = result();
 
 val prems = goal Limit.thy  (* eps_succ_Rp *)
     "[|emb_chain(DD,ee); m:nat|] ==>  \
 \    eps(DD,ee,succ(m),m) = Rp(DD`m,DD`succ(m),ee`m)";
-by(asm_simp_tac(arith_ss addsimps(eps_e_gr::le_succ_iff::e_gr_succ_emb::
-   prems))1);
+by (asm_simp_tac(arith_ss addsimps(eps_e_gr::le_succ_iff::e_gr_succ_emb::
+   prems)) 1);
 val eps_succ_Rp = result();
 
 val prems = goal Limit.thy  (* eps_cont *)
     "[|emb_chain(DD,ee); m:nat; n:nat|] ==> eps(DD,ee,m,n): cont(DD`m,DD`n)";
-br(le_cases RS disjE)1;
+by (rtac (le_cases RS disjE) 1);
 by (resolve_tac prems 1);
-br(hd(rev prems))1;
-by(asm_simp_tac(ZF_ss addsimps(eps_e_less::e_less_cont::prems))1);
-by(asm_simp_tac(ZF_ss addsimps(eps_e_gr::e_gr_cont::prems))1);
+by (rtac (hd(rev prems)) 1);
+by (asm_simp_tac(ZF_ss addsimps(eps_e_less::e_less_cont::prems)) 1);
+by (asm_simp_tac(ZF_ss addsimps(eps_e_gr::e_gr_cont::prems)) 1);
 val eps_cont = result();
 
 (* Theorems about splitting. *)
@@ -2123,33 +2123,33 @@
 val prems = goal Limit.thy  (* eps_split_add_left *)
     "[|n le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==>  \
 \    eps(DD,ee,m,m#+k) = eps(DD,ee,m#+n,m#+k) O eps(DD,ee,m,m#+n)";
-by(asm_simp_tac(arith_ss addsimps 
-    (eps_e_less::add_le_self::add_le_mono::prems))1);
-brr(e_less_split_add::prems)1;
+by (asm_simp_tac(arith_ss addsimps 
+    (eps_e_less::add_le_self::add_le_mono::prems)) 1);
+brr(e_less_split_add::prems) 1;
 val eps_split_add_left = result();
 
 val prems = goal Limit.thy  (* eps_split_add_left_rev *)
     "[|n le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==>  \
 \    eps(DD,ee,m,m#+n) = eps(DD,ee,m#+k,m#+n) O eps(DD,ee,m,m#+k)";
-by(asm_simp_tac(arith_ss addsimps 
-    (eps_e_less_add::eps_e_gr::add_le_self::add_le_mono::prems))1);
-brr(e_less_e_gr_split_add::prems)1;
+by (asm_simp_tac(arith_ss addsimps 
+    (eps_e_less_add::eps_e_gr::add_le_self::add_le_mono::prems)) 1);
+brr(e_less_e_gr_split_add::prems) 1;
 val eps_split_add_left_rev = result();
 
 val prems = goal Limit.thy  (* eps_split_add_right *)
     "[|m le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==>  \
 \    eps(DD,ee,n#+k,n) = eps(DD,ee,n#+m,n) O eps(DD,ee,n#+k,n#+m)";
-by(asm_simp_tac(arith_ss addsimps 
-    (eps_e_gr::add_le_self::add_le_mono::prems))1);
-brr(e_gr_split_add::prems)1;
+by (asm_simp_tac(arith_ss addsimps 
+    (eps_e_gr::add_le_self::add_le_mono::prems)) 1);
+brr(e_gr_split_add::prems) 1;
 val eps_split_add_right = result();
 
 val prems = goal Limit.thy  (* eps_split_add_right_rev *)
     "[|m le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==>  \
 \    eps(DD,ee,n#+m,n) = eps(DD,ee,n#+k,n) O eps(DD,ee,n#+m,n#+k)";
-by(asm_simp_tac(arith_ss addsimps 
-    (eps_e_gr_add::eps_e_less::add_le_self::add_le_mono::prems))1);
-brr(e_gr_e_less_split_add::prems)1;
+by (asm_simp_tac(arith_ss addsimps 
+    (eps_e_gr_add::eps_e_less::add_le_self::add_le_mono::prems)) 1);
+brr(e_gr_e_less_split_add::prems) 1;
 val eps_split_add_right_rev = result();
 
 (* Arithmetic, little support in Isabelle/ZF. *)
@@ -2158,19 +2158,19 @@
     "[|m#+n le m#+k; m:nat; n:nat; k:nat|] ==> n le k";
 by (rtac mp 1);
 by (resolve_tac prems 2);
-by(res_inst_tac[("n","n")]nat_induct 1);
+by (res_inst_tac[("n","n")]nat_induct 1);
 by (resolve_tac prems 1);
-by(simp_tac (arith_ss addsimps prems) 1);
-by(safe_tac lemmas_cs);
-by(asm_full_simp_tac (ZF_ss addsimps 
+by (simp_tac (arith_ss addsimps prems) 1);
+by (safe_tac lemmas_cs);
+by (asm_full_simp_tac (ZF_ss addsimps 
     (not_le_iff_lt::succ_le_iff::add_succ::add_succ_right::
      add_type::nat_into_Ord::prems)) 1);
 by (etac lt_asym 1);
 by (assume_tac 1);
-by(asm_full_simp_tac (ZF_ss addsimps 
+by (asm_full_simp_tac (ZF_ss addsimps 
     (succ_le_iff::add_succ::add_succ_right::le_iff::
      add_type::nat_into_Ord::prems)) 1);
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 by (etac lt_irrefl 1);
 val add_le_elim1 = result();
 
@@ -2178,8 +2178,8 @@
     "[|n le k; k le m;  \
 \      !!p q. [|p le q; k=n#+p; m=n#+q; p:nat; q:nat|] ==> R; \
 \      m:nat; n:nat; k:nat|]==>R";
-br(hd prems RS le_exists)1;
-br(le_exists)1;
+by (rtac (hd prems RS le_exists) 1);
+by (rtac (le_exists) 1);
 by (rtac le_trans 1);
 (* Careful *)
 by (resolve_tac prems 1);
@@ -2187,8 +2187,8 @@
 by (resolve_tac prems 1);
 by (assume_tac 2);
 by (assume_tac 2);
-by(cut_facts_tac[hd prems,hd(tl prems)]1);
-by(asm_full_simp_tac arith_ss 1);
+by (cut_facts_tac[hd prems,hd(tl prems)]1);
+by (asm_full_simp_tac arith_ss 1);
 by (etac add_le_elim1 1);
 brr prems 1;
 val le_exists_lemma = result();
@@ -2198,8 +2198,8 @@
 \    eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
 by (rtac le_exists_lemma 1);
 brr prems 1;
-by(asm_simp_tac ZF_ss 1);
-brr(eps_split_add_left::prems)1;
+by (asm_simp_tac ZF_ss 1);
+brr(eps_split_add_left::prems) 1;
 val eps_split_left_le = result();
 
 val prems = goal Limit.thy  (* eps_split_left_le_rev *)
@@ -2207,8 +2207,8 @@
 \    eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
 by (rtac le_exists_lemma 1);
 brr prems 1;
-by(asm_simp_tac ZF_ss 1);
-brr(eps_split_add_left_rev::prems)1;
+by (asm_simp_tac ZF_ss 1);
+brr(eps_split_add_left_rev::prems) 1;
 val eps_split_left_le_rev = result();
 
 val prems = goal Limit.thy  (* eps_split_right_le *)
@@ -2216,8 +2216,8 @@
 \    eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
 by (rtac le_exists_lemma 1);
 brr prems 1;
-by(asm_simp_tac ZF_ss 1);
-brr(eps_split_add_right::prems)1;
+by (asm_simp_tac ZF_ss 1);
+brr(eps_split_add_right::prems) 1;
 val eps_split_right_le = result();
 
 val prems = goal Limit.thy  (* eps_split_right_le_rev *)
@@ -2225,8 +2225,8 @@
 \    eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
 by (rtac le_exists_lemma 1);
 brr prems 1;
-by(asm_simp_tac ZF_ss 1);
-brr(eps_split_add_right_rev::prems)1;
+by (asm_simp_tac ZF_ss 1);
+brr(eps_split_add_right_rev::prems) 1;
 val eps_split_right_le_rev = result();
 
 (* The desired two theorems about `splitting'. *)
@@ -2234,10 +2234,10 @@
 val prems = goal Limit.thy  (* eps_split_left *)
     "[|m le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==>  \
 \    eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
-br(le_cases RS disjE)1;
+by (rtac (le_cases RS disjE) 1);
 by (rtac eps_split_right_le_rev 4);
 by (assume_tac 4);
-br(le_cases RS disjE)3;
+by (rtac (le_cases RS disjE) 3);
 by (rtac eps_split_left_le 5);
 by (assume_tac 6);
 by (rtac eps_split_left_le_rev 10);
@@ -2247,10 +2247,10 @@
 val prems = goal Limit.thy  (* eps_split_right *)
     "[|n le k; emb_chain(DD,ee); m:nat; n:nat; k:nat|] ==>  \
 \    eps(DD,ee,m,n) = eps(DD,ee,k,n) O eps(DD,ee,m,k)";
-br(le_cases RS disjE)1;
+by (rtac (le_cases RS disjE) 1);
 by (rtac eps_split_left_le_rev 3);
 by (assume_tac 3);
-br(le_cases RS disjE)8;
+by (rtac (le_cases RS disjE) 8);
 by (rtac eps_split_right_le 10);
 by (assume_tac 11);
 by (rtac eps_split_right_le_rev 15);
@@ -2266,35 +2266,35 @@
 val prems = goalw Limit.thy [rho_emb_def] (* rho_emb_fun *)
     "[|emb_chain(DD,ee); n:nat|] ==>   \
 \    rho_emb(DD,ee,n): set(DD`n) -> set(Dinf(DD,ee))";
-brr(lam_type::DinfI::(eps_cont RS cont_fun RS apply_type)::prems)1;
-by(asm_simp_tac arith_ss 1);
-br(le_cases RS disjE)1;
+brr(lam_type::DinfI::(eps_cont RS cont_fun RS apply_type)::prems) 1;
+by (asm_simp_tac arith_ss 1);
+by (rtac (le_cases RS disjE) 1);
 by (rtac nat_succI 1);
 by (assume_tac 1);
 by (resolve_tac prems 1);
 (* The easiest would be to apply add1 everywhere also in the assumptions, 
    but since x le y is x<succ(y) simplification does too much with this thm. *)
-br(eps_split_right_le RS ssubst)1;
+by (rtac (eps_split_right_le RS ssubst) 1);
 by (assume_tac 2);
-by(asm_simp_tac(ZF_ss addsimps (add1::[])) 1);
-brr(add_le_self::nat_0I::nat_succI::prems)1;
-by(asm_simp_tac(ZF_ss addsimps(eps_succ_Rp::prems))1);
-br(comp_fun_apply RS ssubst)1;
+by (asm_simp_tac(ZF_ss addsimps (add1::[])) 1);
+brr(add_le_self::nat_0I::nat_succI::prems) 1;
+by (asm_simp_tac(ZF_ss addsimps(eps_succ_Rp::prems)) 1);
+by (rtac (comp_fun_apply RS ssubst) 1);
 brr(eps_fun::nat_succI::(Rp_cont RS cont_fun)::emb_chain_emb::
-    emb_chain_cpo::refl::prems)1;
+    emb_chain_cpo::refl::prems) 1;
 (* Now the second part of the proof. Slightly different than HOL. *)
-by(asm_simp_tac(ZF_ss addsimps(eps_e_less::nat_succI::prems))1);
-be(le_iff RS iffD1 RS disjE)1;
-by(asm_simp_tac(ZF_ss addsimps(e_less_le::prems))1);
-br(comp_fun_apply RS ssubst)1;
-brr(e_less_cont::cont_fun::emb_chain_emb::emb_cont::prems)1;
-br(embRp_eq_thm RS ssubst)1;
+by (asm_simp_tac(ZF_ss addsimps(eps_e_less::nat_succI::prems)) 1);
+by (etac (le_iff RS iffD1 RS disjE) 1);
+by (asm_simp_tac(ZF_ss addsimps(e_less_le::prems)) 1);
+by (rtac (comp_fun_apply RS ssubst) 1);
+brr(e_less_cont::cont_fun::emb_chain_emb::emb_cont::prems) 1;
+by (rtac (embRp_eq_thm RS ssubst) 1);
 brr(emb_chain_emb::(e_less_cont RS cont_fun RS apply_type)::emb_chain_cpo::
-    nat_succI::prems)1;
-by(asm_simp_tac(ZF_ss addsimps(eps_e_less::prems))1);
+    nat_succI::prems) 1;
+by (asm_simp_tac(ZF_ss addsimps(eps_e_less::prems)) 1);
 by (dtac shift_asm 1);
-by(asm_full_simp_tac(ZF_ss addsimps(eps_succ_Rp::e_less_eq::id_apply::
-   nat_succI::prems))1);
+by (asm_full_simp_tac(ZF_ss addsimps(eps_succ_Rp::e_less_eq::id_apply::
+   nat_succI::prems)) 1);
 val rho_emb_fun = result();
 
 val rho_emb_apply1 = prove_goalw Limit.thy [rho_emb_def]
@@ -2307,7 +2307,7 @@
 
 val rho_emb_id = prove_goal Limit.thy 
   "!!z. [| x:set(DD`n); n:nat|] ==> rho_emb(DD,ee,n)`x`n = x"
-  (fn prems => [asm_simp_tac(ZF_ss addsimps[rho_emb_apply2,eps_id,id_thm])1]);
+  (fn prems => [asm_simp_tac(ZF_ss addsimps[rho_emb_apply2,eps_id,id_thm]) 1]);
 
 (* Shorter proof, 23 against 62. *)
 
@@ -2315,30 +2315,30 @@
     "[|emb_chain(DD,ee); n:nat|] ==>   \
 \    rho_emb(DD,ee,n): cont(DD`n,Dinf(DD,ee))";
 by (rtac contI 1);
-brr(rho_emb_fun::prems)1;
+brr(rho_emb_fun::prems) 1;
 by (rtac rel_DinfI 1);
-by(SELECT_GOAL(rewtac rho_emb_def)1);
-by(asm_simp_tac ZF_ss 1);
-brr((eps_cont RS cont_mono)::Dinf_prod::apply_type::rho_emb_fun::prems)1;
+by (SELECT_GOAL(rewtac rho_emb_def) 1);
+by (asm_simp_tac ZF_ss 1);
+brr((eps_cont RS cont_mono)::Dinf_prod::apply_type::rho_emb_fun::prems) 1;
 (* Continuity, different order, slightly different proofs. *)
-br(lub_Dinf RS ssubst)1;
+by (rtac (lub_Dinf RS ssubst) 1);
 by (rtac chainI 1);
-brr(lam_type::(rho_emb_fun RS apply_type)::chain_in::prems)1;
-by(asm_simp_tac arith_ss 1);
+brr(lam_type::(rho_emb_fun RS apply_type)::chain_in::prems) 1;
+by (asm_simp_tac arith_ss 1);
 by (rtac rel_DinfI 1);
-by(asm_simp_tac(arith_ss addsimps (rho_emb_apply2::chain_in::[]))1);
+by (asm_simp_tac(arith_ss addsimps (rho_emb_apply2::chain_in::[])) 1);
 brr((eps_cont RS cont_mono)::chain_rel::Dinf_prod::
-    (rho_emb_fun RS apply_type)::chain_in::nat_succI::prems)1;
+    (rho_emb_fun RS apply_type)::chain_in::nat_succI::prems) 1;
 (* Now, back to the result of applying lub_Dinf *)
-by(asm_simp_tac(arith_ss addsimps (rho_emb_apply2::chain_in::[]))1);
+by (asm_simp_tac(arith_ss addsimps (rho_emb_apply2::chain_in::[])) 1);
 by (rtac (rho_emb_apply1 RS ssubst) 1);
-brr((cpo_lub RS islub_in)::emb_chain_cpo::prems)1;
+brr((cpo_lub RS islub_in)::emb_chain_cpo::prems) 1;
 by (rtac fun_extension 1);
 brr(lam_type::(eps_cont RS cont_fun RS apply_type)::(cpo_lub RS islub_in)::
-    emb_chain_cpo::prems)1;
-brr(cont_chain::eps_cont::emb_chain_cpo::prems)1;
-by(asm_simp_tac arith_ss 1);
-by(asm_simp_tac(ZF_ss addsimps((eps_cont RS cont_lub)::prems))1);
+    emb_chain_cpo::prems) 1;
+brr(cont_chain::eps_cont::emb_chain_cpo::prems) 1;
+by (asm_simp_tac arith_ss 1);
+by (asm_simp_tac(ZF_ss addsimps((eps_cont RS cont_lub)::prems)) 1);
 val rho_emb_cont = result();
 
 (* 32 vs 61, using safe_tac with imp in asm would be unfortunate (5steps) *)
@@ -2346,40 +2346,40 @@
 val prems = goalw Limit.thy [] (* lemma1 *)
     "[|m le n; emb_chain(DD,ee); x:set(Dinf(DD,ee)); m:nat; n:nat|] ==>   \
 \    rel(DD`n,e_less(DD,ee,m,n)`(x`m),x`n)";
-by(rtac impE 1 THEN atac 3 THEN rtac(hd prems)2);  (* For induction proof *)
-by(res_inst_tac[("n","n")]nat_induct 1);
+by (rtac impE 1 THEN atac 3 THEN rtac(hd prems) 2);  (* For induction proof *)
+by (res_inst_tac[("n","n")]nat_induct 1);
 by (rtac impI 2);
-by(asm_full_simp_tac (arith_ss addsimps (e_less_eq::prems)) 2);
-br(id_thm RS ssubst)2;
-brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_0I::prems)1;
-by(asm_full_simp_tac (arith_ss addsimps [le_succ_iff]) 1);
+by (asm_full_simp_tac (arith_ss addsimps (e_less_eq::prems)) 2);
+by (rtac (id_thm RS ssubst) 2);
+brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_0I::prems) 1;
+by (asm_full_simp_tac (arith_ss addsimps [le_succ_iff]) 1);
 by (rtac impI 1);
 by (etac disjE 1);
-by(dtac mp 1 THEN atac 1);
+by (dtac mp 1 THEN atac 1);
 by (rtac cpo_trans 1);
-br(e_less_le RS ssubst)2;
-brr(emb_chain_cpo::nat_succI::prems)1;
-br(comp_fun_apply RS ssubst)1;
+by (rtac (e_less_le RS ssubst) 2);
+brr(emb_chain_cpo::nat_succI::prems) 1;
+by (rtac (comp_fun_apply RS ssubst) 1);
 brr((emb_chain_emb RS emb_cont)::e_less_cont::cont_fun::apply_type::
-    Dinf_prod::prems)1;
-by(res_inst_tac[("y","x`xa")](emb_chain_emb RS emb_cont RS cont_mono)1);
-brr((e_less_cont RS cont_fun)::apply_type::Dinf_prod::prems)1;
-by(res_inst_tac[("x1","x"),("n1","xa")](Dinf_eq RS subst)1);
-br(comp_fun_apply RS subst)3;
-by(res_inst_tac
+    Dinf_prod::prems) 1;
+by (res_inst_tac[("y","x`xa")](emb_chain_emb RS emb_cont RS cont_mono) 1);
+brr((e_less_cont RS cont_fun)::apply_type::Dinf_prod::prems) 1;
+by (res_inst_tac[("x1","x"),("n1","xa")](Dinf_eq RS subst) 1);
+by (rtac (comp_fun_apply RS subst) 3);
+by (res_inst_tac
   [("P",
     "%z. rel(DD ` succ(xa), \
 \    (ee ` xa O Rp(?DD46(xa) ` xa,?DD46(xa) ` succ(xa),?ee46(xa) ` xa)) ` \
-\            (x ` succ(xa)),z)")](id_thm RS subst)6);
+\            (x ` succ(xa)),z)")](id_thm RS subst) 6);
 by (rtac rel_cf 7); 
 (* Dinf and cont_fun doesn't go well together, both Pi(_,%x._). *)
 (* brr solves 11 of 12 subgoals *)
 brr((hd(tl(tl prems)) RS Dinf_prod RS apply_type)::cont_fun::Rp_cont::
     e_less_cont::emb_cont::emb_chain_emb::emb_chain_cpo::apply_type::
-    embRp_rel::(disjI1 RS (le_succ_iff RS iffD2))::nat_succI::prems)1;
-by(asm_full_simp_tac (arith_ss addsimps (e_less_eq::prems)) 1);
-br(id_thm RS ssubst)1;
-brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_succI::prems)1;
+    embRp_rel::(disjI1 RS (le_succ_iff RS iffD2))::nat_succI::prems) 1;
+by (asm_full_simp_tac (arith_ss addsimps (e_less_eq::prems)) 1);
+by (rtac (id_thm RS ssubst) 1);
+brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_succI::prems) 1;
 val lemma1 = result();
 
 (* 18 vs 40 *)
@@ -2387,39 +2387,39 @@
 val prems = goalw Limit.thy [] (* lemma2 *)
     "[|n le m; emb_chain(DD,ee); x:set(Dinf(DD,ee)); m:nat; n:nat|] ==>   \
 \    rel(DD`n,e_gr(DD,ee,m,n)`(x`m),x`n)";
-by(rtac impE 1 THEN atac 3 THEN rtac(hd prems)2);  (* For induction proof *)
-by(res_inst_tac[("n","m")]nat_induct 1);
+by (rtac impE 1 THEN atac 3 THEN rtac(hd prems) 2);  (* For induction proof *)
+by (res_inst_tac[("n","m")]nat_induct 1);
 by (rtac impI 2);
-by(asm_full_simp_tac (arith_ss addsimps (e_gr_eq::prems)) 2);
-br(id_thm RS ssubst)2;
-brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_0I::prems)1;
-by(asm_full_simp_tac (arith_ss addsimps [le_succ_iff]) 1);
+by (asm_full_simp_tac (arith_ss addsimps (e_gr_eq::prems)) 2);
+by (rtac (id_thm RS ssubst) 2);
+brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_0I::prems) 1;
+by (asm_full_simp_tac (arith_ss addsimps [le_succ_iff]) 1);
 by (rtac impI 1);
 by (etac disjE 1);
-by(dtac mp 1 THEN atac 1);
-br(e_gr_le RS ssubst)1;
-br(comp_fun_apply RS ssubst)4;
-br(Dinf_eq RS ssubst)7;
+by (dtac mp 1 THEN atac 1);
+by (rtac (e_gr_le RS ssubst) 1);
+by (rtac (comp_fun_apply RS ssubst) 4);
+by (rtac (Dinf_eq RS ssubst) 7);
 brr(emb_chain_emb::emb_chain_cpo::Rp_cont::e_gr_cont::cont_fun::emb_cont::
-    apply_type::Dinf_prod::nat_succI::prems)1;
-by(asm_full_simp_tac (arith_ss addsimps (e_gr_eq::prems)) 1);
-br(id_thm RS ssubst)1;
-brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_succI::prems)1;
+    apply_type::Dinf_prod::nat_succI::prems) 1;
+by (asm_full_simp_tac (arith_ss addsimps (e_gr_eq::prems)) 1);
+by (rtac (id_thm RS ssubst) 1);
+brr(apply_type::Dinf_prod::cpo_refl::emb_chain_cpo::nat_succI::prems) 1;
 val lemma2 = result();
 
 val prems = goalw ZF.thy [if_def]
   "[| P==>R(a); ~P==>R(b) |] ==> R(if(P,a,b))";
-by(excluded_middle_tac"P"1);
-by(ALLGOALS(asm_simp_tac ZF_ss THEN' rtac theI2));
-by(ALLGOALS(asm_full_simp_tac FOL_ss));
-brr(ex1I::refl::prems)1;
+by (excluded_middle_tac"P"1);
+by (ALLGOALS(asm_simp_tac ZF_ss THEN' rtac theI2));
+by (ALLGOALS(asm_full_simp_tac FOL_ss));
+brr(ex1I::refl::prems) 1;
 val if_case = result();
 
 val prems = goalw Limit.thy [eps_def] (* eps1 *)
     "[|emb_chain(DD,ee); x:set(Dinf(DD,ee)); m:nat; n:nat|] ==>   \
 \    rel(DD`n,eps(DD,ee,m,n)`(x`m),x`n)";
 by (rtac if_case 1);
-brr(lemma1::(not_le_iff_lt RS iffD1 RS leI RS lemma2)::nat_into_Ord::prems)1;
+brr(lemma1::(not_le_iff_lt RS iffD1 RS leI RS lemma2)::nat_into_Ord::prems) 1;
 val eps1 = result();
 
 (* The following theorem is needed/useful due to type check for rel_cfI, 
@@ -2430,53 +2430,53 @@
   "[| emb_chain(DD,ee); n:nat |] ==> \
 \  (lam x:set(Dinf(DD,ee)). x`n) : cont(Dinf(DD,ee),DD`n)";
 by (rtac contI 1);
-brr(lam_type::apply_type::Dinf_prod::prems)1;
-by(asm_simp_tac ZF_ss 1);
-brr(rel_Dinf::prems)1;
-br(beta RS ssubst)1;
-brr(cpo_Dinf::islub_in::cpo_lub::prems)1;
-by(asm_simp_tac(ZF_ss addsimps(chain_in::lub_Dinf::prems))1);
+brr(lam_type::apply_type::Dinf_prod::prems) 1;
+by (asm_simp_tac ZF_ss 1);
+brr(rel_Dinf::prems) 1;
+by (rtac (beta RS ssubst) 1);
+brr(cpo_Dinf::islub_in::cpo_lub::prems) 1;
+by (asm_simp_tac(ZF_ss addsimps(chain_in::lub_Dinf::prems)) 1);
 val lam_Dinf_cont = result();
 
 val prems = goalw Limit.thy  [rho_proj_def] (* rho_projpair *)
     "[| emb_chain(DD,ee); n:nat |] ==> \
 \    projpair(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n),rho_proj(DD,ee,n))";
 by (rtac projpairI 1);
-brr(rho_emb_cont::prems)1;
+brr(rho_emb_cont::prems) 1;
 (* lemma used, introduced because same fact needed below due to rel_cfI. *)
-brr(lam_Dinf_cont::prems)1;
+brr(lam_Dinf_cont::prems) 1;
 (*-----------------------------------------------*)
 (* This part is 7 lines, but 30 in HOL (75% reduction!) *)
 by (rtac fun_extension 1);
-br(id_thm RS ssubst)3;
-br(comp_fun_apply RS ssubst)4;
-br(beta RS ssubst)7;
-br(rho_emb_id RS ssubst)8;
+by (rtac (id_thm RS ssubst) 3);
+by (rtac (comp_fun_apply RS ssubst) 4);
+by (rtac (beta RS ssubst) 7);
+by (rtac (rho_emb_id RS ssubst) 8);
 brr(comp_fun::id_type::lam_type::rho_emb_fun::(Dinf_prod RS apply_type)::
-    apply_type::refl::prems)1;
+    apply_type::refl::prems) 1;
 (*^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
 by (rtac rel_cfI 1); (* ------------------>>>Yields type cond, not in HOL *)
-br(id_thm RS ssubst)1;
-br(comp_fun_apply RS ssubst)2;
-br(beta RS ssubst)5;
-br(rho_emb_apply1 RS ssubst)6;
+by (rtac (id_thm RS ssubst) 1);
+by (rtac (comp_fun_apply RS ssubst) 2);
+by (rtac (beta RS ssubst) 5);
+by (rtac (rho_emb_apply1 RS ssubst) 6);
 by (rtac rel_DinfI 7); (* ------------------>>>Yields type cond, not in HOL *)
-br(beta RS ssubst)7;
+by (rtac (beta RS ssubst) 7);
 brr(eps1::lam_type::rho_emb_fun::eps_fun:: (* Dinf_prod bad with lam_type *)
-    (Dinf_prod RS apply_type)::refl::prems)1;
+    (Dinf_prod RS apply_type)::refl::prems) 1;
 brr(apply_type::eps_fun::Dinf_prod::comp_pres_cont::rho_emb_cont::
-    lam_Dinf_cont::id_cont::cpo_Dinf::emb_chain_cpo::prems)1;
+    lam_Dinf_cont::id_cont::cpo_Dinf::emb_chain_cpo::prems) 1;
 val rho_projpair = result();
 
 val prems = goalw Limit.thy [emb_def]
   "[| emb_chain(DD,ee); n:nat |] ==> emb(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n))";
-brr(exI::rho_projpair::prems)1;
+brr(exI::rho_projpair::prems) 1;
 val emb_rho_emb = result();
 
 val prems = goal Limit.thy 
   "[| emb_chain(DD,ee); n:nat |] ==>   \
 \  rho_proj(DD,ee,n) : cont(Dinf(DD,ee),DD`n)";
-brr(rho_projpair::projpair_p_cont::prems)1;
+brr(rho_projpair::projpair_p_cont::prems) 1;
 val rho_proj_cont = result();
 
 (*----------------------------------------------------------------------*)
@@ -2487,19 +2487,19 @@
   "[| !!n. n:nat ==> emb(DD`n,E,r(n));   \
 \     !!m n. [|m le n; m:nat; n:nat|] ==> r(n) O eps(DD,ee,m,n) = r(m) |] ==>  \
 \  commute(DD,ee,E,r)";
-by(safe_tac lemmas_cs);
+by (safe_tac lemmas_cs);
 brr prems 1;
 val commuteI = result();
 
 val prems = goalw Limit.thy [commute_def]  (* commute_emb *)
   "!!z. [| commute(DD,ee,E,r); n:nat |] ==> emb(DD`n,E,r(n))";
-by(fast_tac ZF_cs 1);
+by (fast_tac ZF_cs 1);
 val commute_emb = result();
 
 val prems = goalw Limit.thy [commute_def]  (* commute_eq *)
   "!!z. [| commute(DD,ee,E,r); m le n; m:nat; n:nat |] ==>   \
 \       r(n) O eps(DD,ee,m,n) = r(m) ";
-by(fast_tac ZF_cs 1);
+by (fast_tac ZF_cs 1);
 val commute_eq = result();
 
 (* Shorter proof: 11 vs 46 lines. *)
@@ -2507,22 +2507,22 @@
 val prems = goal Limit.thy (* rho_emb_commute *)
   "emb_chain(DD,ee) ==> commute(DD,ee,Dinf(DD,ee),rho_emb(DD,ee))";
 by (rtac commuteI 1);
-brr(emb_rho_emb::prems)1;
+brr(emb_rho_emb::prems) 1;
 by (rtac fun_extension 1);       (* Manual instantiation in HOL. *)
-br(comp_fun_apply RS ssubst)3;
+by (rtac (comp_fun_apply RS ssubst) 3);
 by (rtac fun_extension 6); (* Next, clean up and instantiate unknowns *)
-brr(comp_fun::rho_emb_fun::eps_fun::Dinf_prod::apply_type::prems)1; 
-by(asm_simp_tac
-    (ZF_ss addsimps(rho_emb_apply2::(eps_fun RS apply_type)::prems))1);
-br(comp_fun_apply RS subst)1;
-br(eps_split_left RS subst)4;
-brr(eps_fun::refl::prems)1;
+brr(comp_fun::rho_emb_fun::eps_fun::Dinf_prod::apply_type::prems) 1; 
+by (asm_simp_tac
+    (ZF_ss addsimps(rho_emb_apply2::(eps_fun RS apply_type)::prems)) 1);
+by (rtac (comp_fun_apply RS subst) 1);
+by (rtac (eps_split_left RS subst) 4);
+brr(eps_fun::refl::prems) 1;
 val rho_emb_commute = result();
 
 val le_succ = prove_goal Arith.thy "n:nat ==> n le succ(n)"
  (fn prems =>
    [REPEAT (ares_tac
-    ((disjI1 RS(le_succ_iff RS iffD2))::le_refl::nat_into_Ord::prems)1)]);
+    ((disjI1 RS(le_succ_iff RS iffD2))::le_refl::nat_into_Ord::prems) 1)]);
 
 (* Shorter proof: 21 vs 83 (106 - 23, due to OAssoc complication) *)
 
@@ -2532,31 +2532,31 @@
 val emb_r = hd prems RS commute_emb; (* To avoid BACKTRACKING !! *)
 by (rtac chainI 1);
 brr(lam_type::cont_cf::comp_pres_cont::emb_r::Rp_cont::emb_cont::
-    emb_chain_cpo::prems)1;
-by(asm_simp_tac arith_ss 1);
-by(res_inst_tac[("r1","r"),("m1","n")](commute_eq RS subst)1);
-brr(le_succ::nat_succI::prems)1;
-br(Rp_comp RS ssubst)1;
-brr(emb_eps::emb_r::emb_chain_cpo::le_succ::nat_succI::prems)1;
-br(comp_assoc RS subst)1;   (* Remember that comp_assoc is simpler in Isa *)
-by(res_inst_tac[("r1","r(succ(n))")](comp_assoc RS ssubst)1);
+    emb_chain_cpo::prems) 1;
+by (asm_simp_tac arith_ss 1);
+by (res_inst_tac[("r1","r"),("m1","n")](commute_eq RS subst) 1);
+brr(le_succ::nat_succI::prems) 1;
+by (rtac (Rp_comp RS ssubst) 1);
+brr(emb_eps::emb_r::emb_chain_cpo::le_succ::nat_succI::prems) 1;
+by (rtac (comp_assoc RS subst) 1);   (* Remember that comp_assoc is simpler in Isa *)
+by (res_inst_tac[("r1","r(succ(n))")](comp_assoc RS ssubst) 1);
 by (rtac comp_mono 1);
 brr(comp_pres_cont::eps_cont::emb_eps::emb_r::Rp_cont::emb_cont::
-    emb_chain_cpo::le_succ::nat_succI::prems)1;
-by(res_inst_tac[("b","r(succ(n))")](comp_id RS subst)1); (* 1 subst too much *)
+    emb_chain_cpo::le_succ::nat_succI::prems) 1;
+by (res_inst_tac[("b","r(succ(n))")](comp_id RS subst) 1); (* 1 subst too much *)
 by (rtac comp_mono 2);
 brr(comp_pres_cont::eps_cont::emb_eps::emb_id::emb_r::Rp_cont::emb_cont::
-    cont_fun::emb_chain_cpo::le_succ::nat_succI::prems)1;
-br(comp_id RS ssubst)1; (* Undo's "1 subst too much", typing next anyway *)
+    cont_fun::emb_chain_cpo::le_succ::nat_succI::prems) 1;
+by (rtac (comp_id RS ssubst) 1); (* Undo's "1 subst too much", typing next anyway *)
 brr(cont_fun::Rp_cont::emb_cont::emb_r::cpo_refl::cont_cf::cpo_cf::
-    emb_chain_cpo::embRp_rel::emb_eps::le_succ::nat_succI::prems)1;
+    emb_chain_cpo::embRp_rel::emb_eps::le_succ::nat_succI::prems) 1;
 val commute_chain = result();
 
 val prems = goal Limit.thy (* rho_emb_chain *)
   "emb_chain(DD,ee) ==>  \
 \  chain(cf(Dinf(DD,ee),Dinf(DD,ee)),   \
 \        lam n:nat. rho_emb(DD,ee,n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n)))";
-brr(commute_chain::rho_emb_commute::cpo_Dinf::prems)1;
+brr(commute_chain::rho_emb_commute::cpo_Dinf::prems) 1;
 val rho_emb_chain = result();
 
 val prems = goal Limit.thy (* rho_emb_chain_apply1 *)
@@ -2564,14 +2564,14 @@
 \  chain(Dinf(DD,ee),   \
 \        lam n:nat.   \
 \         (rho_emb(DD,ee,n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n)))`x)";
-by(cut_facts_tac[hd(tl prems) RS (hd prems RS (rho_emb_chain RS chain_cf))]1);
-by(asm_full_simp_tac ZF_ss 1);
+by (cut_facts_tac[hd(tl prems) RS (hd prems RS (rho_emb_chain RS chain_cf))]1);
+by (asm_full_simp_tac ZF_ss 1);
 val rho_emb_chain_apply1 = result();
 
 val prems = goal Limit.thy
   "[| chain(iprod(DD),X); emb_chain(DD,ee); n:nat |] ==>  \
 \  chain(DD`n,lam m:nat. X `m `n)";
-brr(chain_iprod::emb_chain_cpo::prems)1;
+brr(chain_iprod::emb_chain_cpo::prems) 1;
 val chain_iprod_emb_chain = result();
 
 val prems = goal Limit.thy (* rho_emb_chain_apply2 *)
@@ -2581,10 +2581,10 @@
 \    lam xa:nat.  \
 \     (rho_emb(DD, ee, xa) O Rp(DD ` xa, Dinf(DD, ee),rho_emb(DD, ee, xa))) ` \
 \      x ` n)";
-by(cut_facts_tac
+by (cut_facts_tac
    [hd(tl(tl prems)) RS (hd prems RS (hd(tl prems) RS (hd prems RS 
     (rho_emb_chain_apply1 RS chain_Dinf RS chain_iprod_emb_chain))))]1);
-by(asm_full_simp_tac ZF_ss 1);
+by (asm_full_simp_tac ZF_ss 1);
 val rho_emb_chain_apply2 = result();
 
 (* Shorter proof: 32 vs 72 (roughly), Isabelle proof has lemmas. *)
@@ -2596,36 +2596,36 @@
 \  id(set(Dinf(DD,ee)))";
 by (rtac cpo_antisym 1);
 by (rtac cpo_cf 1); (* Instantiate variable, continued below (would loop otherwise) *)
-brr(cpo_Dinf::prems)1; 
+brr(cpo_Dinf::prems) 1; 
 by (rtac islub_least 1);
-brr(cpo_lub::rho_emb_chain::cpo_cf::cpo_Dinf::isubI::cont_cf::id_cont::prems)1;
-by(asm_simp_tac ZF_ss 1);
-brr(embRp_rel::emb_rho_emb::emb_chain_cpo::cpo_Dinf::prems)1;
+brr(cpo_lub::rho_emb_chain::cpo_cf::cpo_Dinf::isubI::cont_cf::id_cont::prems) 1;
+by (asm_simp_tac ZF_ss 1);
+brr(embRp_rel::emb_rho_emb::emb_chain_cpo::cpo_Dinf::prems) 1;
 by (rtac rel_cfI 1);
-by(asm_simp_tac
-    (ZF_ss addsimps(id_thm::lub_cf::rho_emb_chain::cpo_Dinf::prems))1);
+by (asm_simp_tac
+    (ZF_ss addsimps(id_thm::lub_cf::rho_emb_chain::cpo_Dinf::prems)) 1);
 by (rtac rel_DinfI 1); (* Addtional assumptions *)
-br(lub_Dinf RS ssubst)1;
-brr(rho_emb_chain_apply1::prems)1;  
+by (rtac (lub_Dinf RS ssubst) 1);
+brr(rho_emb_chain_apply1::prems) 1;  
 brr(Dinf_prod::(cpo_lub RS islub_in)::id_cont::cpo_Dinf::cpo_cf::cf_cont::
-    rho_emb_chain::rho_emb_chain_apply1::(id_cont RS cont_cf)::prems)2;
-by(asm_simp_tac ZF_ss 1);
+    rho_emb_chain::rho_emb_chain_apply1::(id_cont RS cont_cf)::prems) 2;
+by (asm_simp_tac ZF_ss 1);
 by (rtac dominate_islub 1);
 by (rtac cpo_lub 3);
-brr(rho_emb_chain_apply2::emb_chain_cpo::prems)3;
-by(res_inst_tac[("x1","x`n")](chain_const RS chain_fun)3);
+brr(rho_emb_chain_apply2::emb_chain_cpo::prems) 3;
+by (res_inst_tac[("x1","x`n")](chain_const RS chain_fun) 3);
 brr(islub_const::apply_type::Dinf_prod::emb_chain_cpo::chain_fun::
-    rho_emb_chain_apply2::prems)2;
+    rho_emb_chain_apply2::prems) 2;
 by (rtac dominateI 1);
 by (assume_tac 1); 
-by(asm_simp_tac ZF_ss 1);
-br(comp_fun_apply RS ssubst)1;
-brr(cont_fun::Rp_cont::emb_cont::emb_rho_emb::cpo_Dinf::emb_chain_cpo::prems)1;
-br((rho_projpair RS Rp_unique) RS ssubst)1;
-by(SELECT_GOAL(rewtac rho_proj_def)5);
-by(asm_simp_tac ZF_ss 5);
-br(rho_emb_id RS ssubst)5;
-brr(cpo_refl::cpo_Dinf::apply_type::Dinf_prod::emb_chain_cpo::prems)1;
+by (asm_simp_tac ZF_ss 1);
+by (rtac (comp_fun_apply RS ssubst) 1);
+brr(cont_fun::Rp_cont::emb_cont::emb_rho_emb::cpo_Dinf::emb_chain_cpo::prems) 1;
+by (rtac ((rho_projpair RS Rp_unique) RS ssubst) 1);
+by (SELECT_GOAL(rewtac rho_proj_def) 5);
+by (asm_simp_tac ZF_ss 5);
+by (rtac (rho_emb_id RS ssubst) 5);
+brr(cpo_refl::cpo_Dinf::apply_type::Dinf_prod::emb_chain_cpo::prems) 1;
 val rho_emb_lub = result();
 
 val prems = goal Limit.thy (* theta_chain, almost same prf as commute_chain *)
@@ -2636,25 +2636,25 @@
 val emb_f = hd(tl prems) RS commute_emb; (* Used in the rest of the FILE *)
 by (rtac chainI 1);
 brr(lam_type::cont_cf::comp_pres_cont::emb_r::emb_f::
-    Rp_cont::emb_cont::emb_chain_cpo::prems)1;
-by(asm_simp_tac arith_ss 1);
-by(res_inst_tac[("r1","r"),("m1","n")](commute_eq RS subst)1);
-by(res_inst_tac[("r1","f"),("m1","n")](commute_eq RS subst)5);
-brr(le_succ::nat_succI::prems)1;
-br(Rp_comp RS ssubst)1;
-brr(emb_eps::emb_r::emb_chain_cpo::le_succ::nat_succI::prems)1;
-br(comp_assoc RS subst)1;   (* Remember that comp_assoc is simpler in Isa *)
-by(res_inst_tac[("r1","f(succ(n))")](comp_assoc RS ssubst)1);
+    Rp_cont::emb_cont::emb_chain_cpo::prems) 1;
+by (asm_simp_tac arith_ss 1);
+by (res_inst_tac[("r1","r"),("m1","n")](commute_eq RS subst) 1);
+by (res_inst_tac[("r1","f"),("m1","n")](commute_eq RS subst) 5);
+brr(le_succ::nat_succI::prems) 1;
+by (rtac (Rp_comp RS ssubst) 1);
+brr(emb_eps::emb_r::emb_chain_cpo::le_succ::nat_succI::prems) 1;
+by (rtac (comp_assoc RS subst) 1);   (* Remember that comp_assoc is simpler in Isa *)
+by (res_inst_tac[("r1","f(succ(n))")](comp_assoc RS ssubst) 1);
 by (rtac comp_mono 1);
 brr(comp_pres_cont::eps_cont::emb_eps::emb_r::emb_f::Rp_cont::
-    emb_cont::emb_chain_cpo::le_succ::nat_succI::prems)1;
-by(res_inst_tac[("b","f(succ(n))")](comp_id RS subst)1); (* 1 subst too much *)
+    emb_cont::emb_chain_cpo::le_succ::nat_succI::prems) 1;
+by (res_inst_tac[("b","f(succ(n))")](comp_id RS subst) 1); (* 1 subst too much *)
 by (rtac comp_mono 2);
 brr(comp_pres_cont::eps_cont::emb_eps::emb_id::emb_r::emb_f::Rp_cont::
-    emb_cont::cont_fun::emb_chain_cpo::le_succ::nat_succI::prems)1;
-br(comp_id RS ssubst)1; (* Undo's "1 subst too much", typing next anyway *)
+    emb_cont::cont_fun::emb_chain_cpo::le_succ::nat_succI::prems) 1;
+by (rtac (comp_id RS ssubst) 1); (* Undo's "1 subst too much", typing next anyway *)
 brr(cont_fun::Rp_cont::emb_cont::emb_r::emb_f::cpo_refl::cont_cf::
-    cpo_cf::emb_chain_cpo::embRp_rel::emb_eps::le_succ::nat_succI::prems)1;
+    cpo_cf::emb_chain_cpo::embRp_rel::emb_eps::le_succ::nat_succI::prems) 1;
 val theta_chain = result();
 
 val prems = goal Limit.thy (* theta_proj_chain, same prf as theta_chain *)
@@ -2663,25 +2663,25 @@
 \  chain(cf(G,E),lam n:nat. r(n) O Rp(DD`n,G,f(n)))";
 by (rtac chainI 1);
 brr(lam_type::cont_cf::comp_pres_cont::emb_r::emb_f::
-    Rp_cont::emb_cont::emb_chain_cpo::prems)1;
-by(asm_simp_tac arith_ss 1);
-by(res_inst_tac[("r1","r"),("m1","n")](commute_eq RS subst)1);
-by(res_inst_tac[("r1","f"),("m1","n")](commute_eq RS subst)5);
-brr(le_succ::nat_succI::prems)1;
-br(Rp_comp RS ssubst)1;
-brr(emb_eps::emb_f::emb_chain_cpo::le_succ::nat_succI::prems)1;
-br(comp_assoc RS subst)1;   (* Remember that comp_assoc is simpler in Isa *)
-by(res_inst_tac[("r1","r(succ(n))")](comp_assoc RS ssubst)1);
+    Rp_cont::emb_cont::emb_chain_cpo::prems) 1;
+by (asm_simp_tac arith_ss 1);
+by (res_inst_tac[("r1","r"),("m1","n")](commute_eq RS subst) 1);
+by (res_inst_tac[("r1","f"),("m1","n")](commute_eq RS subst) 5);
+brr(le_succ::nat_succI::prems) 1;
+by (rtac (Rp_comp RS ssubst) 1);
+brr(emb_eps::emb_f::emb_chain_cpo::le_succ::nat_succI::prems) 1;
+by (rtac (comp_assoc RS subst) 1);   (* Remember that comp_assoc is simpler in Isa *)
+by (res_inst_tac[("r1","r(succ(n))")](comp_assoc RS ssubst) 1);
 by (rtac comp_mono 1);
 brr(comp_pres_cont::eps_cont::emb_eps::emb_r::emb_f::Rp_cont::
-    emb_cont::emb_chain_cpo::le_succ::nat_succI::prems)1;
-by(res_inst_tac[("b","r(succ(n))")](comp_id RS subst)1); (* 1 subst too much *)
+    emb_cont::emb_chain_cpo::le_succ::nat_succI::prems) 1;
+by (res_inst_tac[("b","r(succ(n))")](comp_id RS subst) 1); (* 1 subst too much *)
 by (rtac comp_mono 2);
 brr(comp_pres_cont::eps_cont::emb_eps::emb_id::emb_r::emb_f::Rp_cont::
-    emb_cont::cont_fun::emb_chain_cpo::le_succ::nat_succI::prems)1;
-br(comp_id RS ssubst)1; (* Undo's "1 subst too much", typing next anyway *)
+    emb_cont::cont_fun::emb_chain_cpo::le_succ::nat_succI::prems) 1;
+by (rtac (comp_id RS ssubst) 1); (* Undo's "1 subst too much", typing next anyway *)
 brr(cont_fun::Rp_cont::emb_cont::emb_r::emb_f::cpo_refl::cont_cf::
-    cpo_cf::emb_chain_cpo::embRp_rel::emb_eps::le_succ::nat_succI::prems)1;
+    cpo_cf::emb_chain_cpo::embRp_rel::emb_eps::le_succ::nat_succI::prems) 1;
 val theta_proj_chain = result();
 
 (* Simplification with comp_assoc is possible inside a lam-abstraction,
@@ -2696,10 +2696,10 @@
 \     emb_chain(DD,ee); cpo(E); cpo(G); x:nat |] ==>  \
 \  r(x) O Rp(DD ` x, G, f(x)) O f(x) O Rp(DD ` x, E, r(x)) =  \
 \  r(x) O Rp(DD ` x, E, r(x))";
-by(res_inst_tac[("s1","f(x)")](comp_assoc RS subst)1);
-br(embRp_eq RS ssubst)1;
-br(id_comp RS ssubst)4;
-brr(cont_fun::Rp_cont::emb_r::emb_f::emb_chain_cpo::refl::prems)1;
+by (res_inst_tac[("s1","f(x)")](comp_assoc RS subst) 1);
+by (rtac (embRp_eq RS ssubst) 1);
+by (rtac (id_comp RS ssubst) 4);
+brr(cont_fun::Rp_cont::emb_r::emb_f::emb_chain_cpo::refl::prems) 1;
 val lemma = result();
 
 val lemma_assoc = prove_goal Limit.thy "a O b O c O d = a O (b O c) O d"
@@ -2717,26 +2717,26 @@
 \   (E,G,   \
 \    lub(cf(E,G), lam n:nat. f(n) O Rp(DD`n,E,r(n))),  \
 \    lub(cf(G,E), lam n:nat. r(n) O Rp(DD`n,G,f(n))))";
-by(safe_tac lemmas_cs);
-br(comp_lubs RS ssubst)3;
+by (safe_tac lemmas_cs);
+by (rtac (comp_lubs RS ssubst) 3);
 (* The following one line is 15 lines in HOL, and includes existentials. *)
-brr(cf_cont::islub_in::cpo_lub::cpo_cf::theta_chain::theta_proj_chain::prems)1;
-by(simp_tac (ZF_ss addsimps[comp_assoc]) 1);
-by(simp_tac (ZF_ss addsimps[(tl prems) MRS lemma]) 1);
-br(comp_lubs RS ssubst)2;
-brr(cf_cont::islub_in::cpo_lub::cpo_cf::theta_chain::theta_proj_chain::prems)1;
-by(simp_tac (ZF_ss addsimps[comp_assoc]) 1);
-by(simp_tac (ZF_ss addsimps[
+brr(cf_cont::islub_in::cpo_lub::cpo_cf::theta_chain::theta_proj_chain::prems) 1;
+by (simp_tac (ZF_ss addsimps[comp_assoc]) 1);
+by (simp_tac (ZF_ss addsimps[(tl prems) MRS lemma]) 1);
+by (rtac (comp_lubs RS ssubst) 2);
+brr(cf_cont::islub_in::cpo_lub::cpo_cf::theta_chain::theta_proj_chain::prems) 1;
+by (simp_tac (ZF_ss addsimps[comp_assoc]) 1);
+by (simp_tac (ZF_ss addsimps[
    [elem 3 prems,elem 2 prems,elem 4 prems,elem 6 prems, elem 5 prems] 
    MRS lemma]) 1);
 by (rtac dominate_islub 1);
 by (rtac cpo_lub 2);
 brr(commute_chain::emb_f::islub_const::cont_cf::id_cont::cpo_cf::
-    chain_fun::chain_const::prems)2;
+    chain_fun::chain_const::prems) 2;
 by (rtac dominateI 1);
 by (assume_tac 1); 
-by(asm_simp_tac ZF_ss 1);
-brr(embRp_rel::emb_f::emb_chain_cpo::prems)1;
+by (asm_simp_tac ZF_ss 1);
+brr(embRp_rel::emb_f::emb_chain_cpo::prems) 1;
 val theta_projpair = result();
 
 val prems = goalw Limit.thy [emb_def] (* emb_theta *)
@@ -2745,19 +2745,19 @@
 \     emb_chain(DD,ee); cpo(E); cpo(G) |] ==>  \  
 \  emb(E,G,lub(cf(E,G), lam n:nat. f(n) O Rp(DD`n,E,r(n))))";
 by (rtac exI 1);
-br(prems MRS theta_projpair)1;
+by (rtac (prems MRS theta_projpair) 1);
 val emb_theta = result();
 
 val prems = goal Limit.thy (* mono_lemma *)
   "[| g:cont(D,D'); cpo(D); cpo(D'); cpo(E) |] ==>  \
 \  (lam f : cont(D',E). f O g) : mono(cf(D',E),cf(D,E))";
 by (rtac monoI 1);
-by(REPEAT(dtac cf_cont 2));
-by(asm_simp_tac ZF_ss 2);
+by (REPEAT(dtac cf_cont 2));
+by (asm_simp_tac ZF_ss 2);
 by (rtac comp_mono 2);
-by(SELECT_GOAL(rewrite_goals_tac[set_def,cf_def])1);
-by(asm_simp_tac ZF_ss 1);
-brr(lam_type::comp_pres_cont::cpo_cf::cpo_refl::cont_cf::prems)1;
+by (SELECT_GOAL(rewrite_goals_tac[set_def,cf_def]) 1);
+by (asm_simp_tac ZF_ss 1);
+brr(lam_type::comp_pres_cont::cpo_cf::cpo_refl::cont_cf::prems) 1;
 val mono_lemma = result();
 
 (* PAINFUL: wish condrew with difficult conds on term bound in lam-abs. *)
@@ -2770,25 +2770,25 @@
 \  ((lam n:nat. f(n) O Rp(DD ` n, E, r(n))) ` na)) =  \
 \   (lam na:nat. (f(na) O Rp(DD ` na, E, r(na))) O r(n))";
 by (rtac fun_extension 1);
-br(beta RS ssubst)3;
-br(beta RS ssubst)4;
-br(beta RS ssubst)5;
+by (rtac (beta RS ssubst) 3);
+by (rtac (beta RS ssubst) 4);
+by (rtac (beta RS ssubst) 5);
 by (rtac lam_type 1);
-br(beta RS ssubst)1;
-by(ALLGOALS(asm_simp_tac (ZF_ss addsimps prems)));
+by (rtac (beta RS ssubst) 1);
+by (ALLGOALS(asm_simp_tac (ZF_ss addsimps prems)));
 brr(lam_type::comp_pres_cont::Rp_cont::emb_cont::emb_r::emb_f::
-    emb_chain_cpo::prems)1;
+    emb_chain_cpo::prems) 1;
 val lemma = result();
 
 val prems = goal Limit.thy (* chain_lemma *)
   "[| commute(DD,ee,E,r); commute(DD,ee,G,f);   \
 \     emb_chain(DD,ee); cpo(E); cpo(G); n:nat |] ==>  \  
 \  chain(cf(DD`n,G),lam x:nat. (f(x) O Rp(DD ` x, E, r(x))) O r(n))";
-by(cut_facts_tac[(rev(tl(rev prems)) MRS theta_chain) RS 
+by (cut_facts_tac[(rev(tl(rev prems)) MRS theta_chain) RS 
     (elem 5 prems RS (elem 4 prems RS ((elem 6 prems RS 
     (elem 3 prems RS emb_chain_cpo)) RS (elem 6 prems RS 
     (emb_r RS emb_cont RS mono_lemma RS mono_chain)))))]1);
-br((prems MRS lemma) RS subst)1;
+by (rtac ((prems MRS lemma) RS subst) 1);
 by (assume_tac 1);
 val chain_lemma = result();
 
@@ -2796,19 +2796,19 @@
   "[| commute(DD,ee,E,r); commute(DD,ee,G,f);   \
 \     emb_chain(DD,ee); cpo(E); cpo(G); cpo(DD`x); x:nat |] ==>  \  
 \  suffix(lam n:nat. (f(n) O Rp(DD`n,E,r(n))) O r(x),x) = (lam n:nat. f(x))";
-by(simp_tac (arith_ss addsimps prems) 1);
+by (simp_tac (arith_ss addsimps prems) 1);
 by (rtac fun_extension 1); 
 brr(lam_type::comp_fun::cont_fun::Rp_cont::emb_cont::emb_r::emb_f::
-    add_type::emb_chain_cpo::prems)1;
-by(asm_simp_tac ZF_ss 1);
-by(res_inst_tac[("r1","r"),("m1","x")](commute_eq RS subst)1);
-brr(emb_r::add_le_self::add_type::prems)1;
-br(comp_assoc RS ssubst)1;
-br(lemma_assoc RS ssubst)1;
-br(embRp_eq RS ssubst)1;
-br(id_comp RS ssubst)4;
-br((hd(tl prems) RS commute_eq) RS ssubst)5; (* avoid eta_contraction:=true. *)
-brr(emb_r::add_type::eps_fun::add_le_self::refl::emb_chain_cpo::prems)1;
+    add_type::emb_chain_cpo::prems) 1;
+by (asm_simp_tac ZF_ss 1);
+by (res_inst_tac[("r1","r"),("m1","x")](commute_eq RS subst) 1);
+brr(emb_r::add_le_self::add_type::prems) 1;
+by (rtac (comp_assoc RS ssubst) 1);
+by (rtac (lemma_assoc RS ssubst) 1);
+by (rtac (embRp_eq RS ssubst) 1);
+by (rtac (id_comp RS ssubst) 4);
+by (rtac ((hd(tl prems) RS commute_eq) RS ssubst) 5); (* avoid eta_contraction:=true. *)
+brr(emb_r::add_type::eps_fun::add_le_self::refl::emb_chain_cpo::prems) 1;
 val suffix_lemma = result();
 
 val mediatingI = prove_goalw Limit.thy [mediating_def]
@@ -2828,17 +2828,17 @@
 \     commute(DD,ee,E,r); commute(DD,ee,G,f);   \
 \     emb_chain(DD,ee); cpo(E); cpo(G) |] ==>  \  
 \  mediating(E,G,r,f,lub(cf(E,G), lam n:nat. f(n) O Rp(DD`n,E,r(n))))";
-brr(mediatingI::emb_theta::prems)1;
-by(res_inst_tac[("b","r(n)")](lub_const RS subst)1);
-br(comp_lubs RS ssubst)3;
+brr(mediatingI::emb_theta::prems) 1;
+by (res_inst_tac[("b","r(n)")](lub_const RS subst) 1);
+by (rtac (comp_lubs RS ssubst) 3);
 brr(cont_cf::emb_cont::emb_r::cpo_cf::theta_chain::chain_const::
-    emb_chain_cpo::prems)1;
-by(simp_tac ZF_ss 1);
-br(lub_suffix RS subst)1;
-brr(chain_lemma::cpo_cf::emb_chain_cpo::prems)1;
-br((tl prems MRS suffix_lemma) RS ssubst)1;
-br(lub_const RS ssubst)3;
-brr(cont_cf::emb_cont::emb_f::cpo_cf::emb_chain_cpo::refl::prems)1;
+    emb_chain_cpo::prems) 1;
+by (simp_tac ZF_ss 1);
+by (rtac (lub_suffix RS subst) 1);
+brr(chain_lemma::cpo_cf::emb_chain_cpo::prems) 1;
+by (rtac ((tl prems MRS suffix_lemma) RS ssubst) 1);
+by (rtac (lub_const RS ssubst) 3);
+brr(cont_cf::emb_cont::emb_f::cpo_cf::emb_chain_cpo::refl::prems) 1;
 val lub_universal_mediating = result();
 
 val prems = goal Limit.thy (* lub_universal_unique *)
@@ -2847,13 +2847,13 @@
 \     commute(DD,ee,E,r); commute(DD,ee,G,f);   \
 \     emb_chain(DD,ee); cpo(E); cpo(G) |] ==>   \
 \  t = lub(cf(E,G), lam n:nat. f(n) O Rp(DD`n,E,r(n)))";
-by(res_inst_tac[("b","t")](comp_id RS subst)1);
-br(hd(tl prems) RS subst)2;
-by(res_inst_tac[("b","t")](lub_const RS subst)2);
-br(comp_lubs RS ssubst)4;
-by(simp_tac (ZF_ss addsimps(comp_assoc::(hd prems RS mediating_eq)::prems))9);
+by (res_inst_tac[("b","t")](comp_id RS subst) 1);
+by (rtac (hd(tl prems) RS subst) 2);
+by (res_inst_tac[("b","t")](lub_const RS subst) 2);
+by (rtac (comp_lubs RS ssubst) 4);
+by (simp_tac (ZF_ss addsimps(comp_assoc::(hd prems RS mediating_eq)::prems)) 9);
 brr(cont_fun::emb_cont::mediating_emb::cont_cf::cpo_cf::chain_const::
-    commute_chain::emb_chain_cpo::prems)1;
+    commute_chain::emb_chain_cpo::prems) 1;
 val lub_universal_unique = result();
 
 (*---------------------------------------------------------------------*)
@@ -2870,8 +2870,8 @@
 \  (ALL t. mediating(Dinf(DD,ee),G,rho_emb(DD,ee),f,t) -->  \
 \    t = lub(cf(Dinf(DD,ee),G),   \
 \        lam n:nat. f(n) O Rp(DD`n,Dinf(DD,ee),rho_emb(DD,ee,n))))";
-by(safe_tac lemmas_cs);
-brr(lub_universal_mediating::rho_emb_commute::rho_emb_lub::cpo_Dinf::prems)1;
-brr(lub_universal_unique::rho_emb_commute::rho_emb_lub::cpo_Dinf::prems)1;
+by (safe_tac lemmas_cs);
+brr(lub_universal_mediating::rho_emb_commute::rho_emb_lub::cpo_Dinf::prems) 1;
+brr(lub_universal_unique::rho_emb_commute::rho_emb_lub::cpo_Dinf::prems) 1;
 val Dinf_universal = result();