src/ZF/OrderType.ML

-(*  Title: 	ZF/OrderType.ML
+(*  Title:      ZF/OrderType.ML
     ID:         $Id$
-    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1994  University of Cambridge
 
 Order types and ordinal arithmetic in Zermelo-Fraenkel Set Theory 
 
 Ordinal arithmetic is traditionally defined in terms of order types, as here.

 val [prem] = goal OrderType.thy "j le i ==> well_ord(j, Memrel(i))";
 by (rtac well_ordI 1);
 by (rtac (wf_Memrel RS wf_imp_wf_on) 1);
 by (resolve_tac [prem RS ltE] 1);
 by (asm_simp_tac (ZF_ss addsimps [linear_def, Memrel_iff,
-				  [ltI, prem] MRS lt_trans2 RS ltD]) 1);
+                                  [ltI, prem] MRS lt_trans2 RS ltD]) 1);
 by (REPEAT (resolve_tac [ballI, Ord_linear] 1));
 by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));
 qed "le_well_ord_Memrel";
 
 (*"Ord(i) ==> well_ord(i, Memrel(i))"*)

 by (fast_tac (ZF_cs addSEs [bij_is_fun RS apply_type] addEs [Ord_trans]) 1);
 qed "Ord_iso_implies_eq_lemma";
 
 (*Kunen's Theorem 7.3 (ii), page 16.  Isomorphic ordinals are equal*)
 goal OrderType.thy
-    "!!i. [| Ord(i);  Ord(j);  f:  ord_iso(i,Memrel(i),j,Memrel(j))	\
+    "!!i. [| Ord(i);  Ord(j);  f:  ord_iso(i,Memrel(i),j,Memrel(j))     \
 \         |] ==> i=j";
 by (res_inst_tac [("i","i"),("j","j")] Ord_linear_lt 1);
 by (REPEAT (eresolve_tac [asm_rl, ord_iso_sym, Ord_iso_implies_eq_lemma] 1));
 qed "Ord_iso_implies_eq";
 

 (*Useful for rewriting PROVIDED pred is not unfolded until later!*)
 goal OrderType.thy 
     "!!r. [| wf[A](r);  x:A |] ==> \
 \         ordermap(A,r) ` x = {ordermap(A,r)`y . y : pred(A,x,r)}";
 by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, pred_subset, 
-				  ordermap_type RS image_fun]) 1);
+                                  ordermap_type RS image_fun]) 1);
 qed "ordermap_pred_unfold";
 
 (*pred-unfolded version.  NOT suitable for rewriting -- loops!*)
 val ordermap_unfold = rewrite_rule [pred_def] ordermap_pred_unfold;
 
 (*** Showing that ordermap, ordertype yield ordinals ***)
 
 fun ordermap_elim_tac i =
     EVERY [etac (ordermap_unfold RS equalityD1 RS subsetD RS RepFunE) i,
-	   assume_tac (i+1),
-	   assume_tac i];
+           assume_tac (i+1),
+           assume_tac i];
 
 goalw OrderType.thy [well_ord_def, tot_ord_def, part_ord_def]
     "!!r. [| well_ord(A,r);  x:A |] ==> Ord(ordermap(A,r) ` x)";
 by (safe_tac ZF_cs);
 by (wf_on_ind_tac "x" [] 1);

 qed "Ord_ordertype";
 
 (*** ordermap preserves the orderings in both directions ***)
 
 goal OrderType.thy
-    "!!r. [| <w,x>: r;  wf[A](r);  w: A; x: A |] ==>	\
+    "!!r. [| <w,x>: r;  wf[A](r);  w: A; x: A |] ==>    \
 \         ordermap(A,r)`w : ordermap(A,r)`x";
 by (eres_inst_tac [("x1", "x")] (ordermap_unfold RS ssubst) 1);
 by (assume_tac 1);
 by (fast_tac ZF_cs 1);
 qed "ordermap_mono";

 by (etac mem_asym 1);
 by (assume_tac 1);
 qed "converse_ordermap_mono";
 
 bind_thm ("ordermap_surj", 
-	  rewrite_rule [symmetric ordertype_def] 
-	      (ordermap_type RS surj_image));
+          rewrite_rule [symmetric ordertype_def] 
+              (ordermap_type RS surj_image));
 
 goalw OrderType.thy [well_ord_def, tot_ord_def, bij_def, inj_def]
     "!!r. well_ord(A,r) ==> ordermap(A,r) : bij(A, ordertype(A,r))";
 by (fast_tac (ZF_cs addSIs [ordermap_type, ordermap_surj]
-		    addEs [linearE]
-		    addDs [ordermap_mono]
+                    addEs [linearE]
+                    addDs [ordermap_mono]
                     addss (ZF_ss addsimps [mem_not_refl])) 1);
 qed "ordermap_bij";
 
 (*** Isomorphisms involving ordertype ***)
 

 by (rtac ordermap_bij 1);
 by (assume_tac 1);
 by (fast_tac (ZF_cs addSEs [MemrelE, converse_ordermap_mono]) 2);
 by (rewtac well_ord_def);
 by (fast_tac (ZF_cs addSIs [MemrelI, ordermap_mono,
-			    ordermap_type RS apply_type]) 1);
+                            ordermap_type RS apply_type]) 1);
 qed "ordertype_ord_iso";
 
 goal OrderType.thy
-    "!!f. [| f: ord_iso(A,r,B,s);  well_ord(B,s) |] ==>	\
+    "!!f. [| f: ord_iso(A,r,B,s);  well_ord(B,s) |] ==> \
 \    ordertype(A,r) = ordertype(B,s)";
 by (forward_tac [well_ord_ord_iso] 1 THEN assume_tac 1);
-by (resolve_tac [Ord_iso_implies_eq] 1
-    THEN REPEAT (eresolve_tac [Ord_ordertype] 1));
+by (rtac Ord_iso_implies_eq 1
+    THEN REPEAT (etac Ord_ordertype 1));
 by (deepen_tac (ZF_cs addIs  [ord_iso_trans, ord_iso_sym]
                       addSEs [ordertype_ord_iso]) 0 1);
 qed "ordertype_eq";
 
 goal OrderType.thy
-    "!!A B. [| ordertype(A,r) = ordertype(B,s);	\
+    "!!A B. [| ordertype(A,r) = ordertype(B,s); \
 \              well_ord(A,r);  well_ord(B,s) \
 \           |] ==> EX f. f: ord_iso(A,r,B,s)";
-by (resolve_tac [exI] 1);
+by (rtac exI 1);
 by (resolve_tac [ordertype_ord_iso RS ord_iso_trans] 1);
 by (assume_tac 1);
-by (eresolve_tac [ssubst] 1);
+by (etac ssubst 1);
 by (eresolve_tac [ordertype_ord_iso RS ord_iso_sym] 1);
 qed "ordertype_eq_imp_ord_iso";
 
 (*** Basic equalities for ordertype ***)
 
 (*Ordertype of Memrel*)
 goal OrderType.thy "!!i. j le i ==> ordertype(j,Memrel(i)) = j";
 by (resolve_tac [Ord_iso_implies_eq RS sym] 1);
-by (eresolve_tac [ltE] 1);
+by (etac ltE 1);
 by (REPEAT (ares_tac [le_well_ord_Memrel, Ord_ordertype] 1));
-by (resolve_tac [ord_iso_trans] 1);
+by (rtac ord_iso_trans 1);
 by (eresolve_tac [le_well_ord_Memrel RS ordertype_ord_iso] 2);
 by (resolve_tac [id_bij RS ord_isoI] 1);
 by (asm_simp_tac (ZF_ss addsimps [id_conv, Memrel_iff]) 1);
 by (fast_tac (ZF_cs addEs [ltE, Ord_in_Ord, Ord_trans]) 1);
 qed "le_ordertype_Memrel";

 bind_thm ("ordertype_Memrel", le_refl RS le_ordertype_Memrel);
 
 goal OrderType.thy "ordertype(0,r) = 0";
 by (resolve_tac [id_bij RS ord_isoI RS ordertype_eq RS trans] 1);
 by (etac emptyE 1);
-by (resolve_tac [well_ord_0] 1);
+by (rtac well_ord_0 1);
 by (resolve_tac [Ord_0 RS ordertype_Memrel] 1);
 qed "ordertype_0";
 
 (*Ordertype of rvimage:  [| f: bij(A,B);  well_ord(B,s) |] ==>
                          ordertype(A, rvimage(A,f,s)) = ordertype(B,s) *)

 
 (*** A fundamental unfolding law for ordertype. ***)
 
 (*Ordermap returns the same result if applied to an initial segment*)
 goal OrderType.thy
-    "!!r. [| well_ord(A,r);  y:A;  z: pred(A,y,r) |] ==>	\
-\	  ordermap(pred(A,y,r), r) ` z = ordermap(A, r) ` z";
+    "!!r. [| well_ord(A,r);  y:A;  z: pred(A,y,r) |] ==>        \
+\         ordermap(pred(A,y,r), r) ` z = ordermap(A, r) ` z";
 by (forward_tac [[well_ord_is_wf, pred_subset] MRS wf_on_subset_A] 1);
 by (wf_on_ind_tac "z" [] 1);
 by (safe_tac (ZF_cs addSEs [predE]));
 by (asm_simp_tac
     (ZF_ss addsimps [ordermap_pred_unfold, well_ord_is_wf, pred_iff]) 1);

     "!!r. [| well_ord(A,r);  x:A |] ==>  \
 \         ordertype(pred(A,x,r),r) <= ordertype(A,r)";
 by (asm_simp_tac (ZF_ss addsimps [ordertype_unfold, 
                   pred_subset RSN (2, well_ord_subset)]) 1);
 by (fast_tac (ZF_cs addIs [ordermap_pred_eq_ordermap, RepFun_eqI]
-	            addEs [predE]) 1);
+                    addEs [predE]) 1);
 qed "ordertype_pred_subset";
 
 goal OrderType.thy
     "!!r. [| well_ord(A,r);  x:A |] ==>  \
 \         ordertype(pred(A,x,r),r) < ordertype(A,r)";
 by (resolve_tac [ordertype_pred_subset RS subset_imp_le RS leE] 1);
 by (REPEAT (ares_tac [Ord_ordertype, well_ord_subset, pred_subset] 1));
 by (eresolve_tac [sym RS ordertype_eq_imp_ord_iso RS exE] 1);
-by (eresolve_tac [well_ord_iso_predE] 3);
+by (etac well_ord_iso_predE 3);
 by (REPEAT (ares_tac [pred_subset, well_ord_subset] 1));
 qed "ordertype_pred_lt";
 
 (*May rewrite with this -- provided no rules are supplied for proving that
- 	well_ord(pred(A,x,r), r) *)
+        well_ord(pred(A,x,r), r) *)
 goal OrderType.thy
     "!!A r. well_ord(A,r) ==>  \
 \           ordertype(A,r) = {ordertype(pred(A,x,r),r). x:A}";
 by (safe_tac (eq_cs addSIs [ordertype_pred_lt RS ltD]));
 by (fast_tac
     (ZF_cs addss
      (ZF_ss addsimps [ordertype_def, 
-		      well_ord_is_wf RS ordermap_eq_image, 
-		      ordermap_type RS image_fun, 
-		      ordermap_pred_eq_ordermap, 
-		      pred_subset]))
+                      well_ord_is_wf RS ordermap_eq_image, 
+                      ordermap_type RS image_fun, 
+                      ordermap_pred_eq_ordermap, 
+                      pred_subset]))
     1);
 qed "ordertype_pred_unfold";
 
 
 (**** Alternative definition of ordinal ****)
 
 (*proof by Krzysztof Grabczewski*)
 goalw OrderType.thy [Ord_alt_def] "!!i. Ord(i) ==> Ord_alt(i)";
-by (resolve_tac [conjI] 1);
-by (eresolve_tac [well_ord_Memrel] 1);
+by (rtac conjI 1);
+by (etac well_ord_Memrel 1);
 by (rewrite_goals_tac [Ord_def, Transset_def, pred_def, Memrel_def]);
 by (fast_tac eq_cs 1);
 qed "Ord_is_Ord_alt";
 
 (*proof by lcp*)
 goalw OrderType.thy [Ord_alt_def, Ord_def, Transset_def, well_ord_def, 
-		     tot_ord_def, part_ord_def, trans_on_def] 
+                     tot_ord_def, part_ord_def, trans_on_def] 
     "!!i. Ord_alt(i) ==> Ord(i)";
 by (asm_full_simp_tac (ZF_ss addsimps [Memrel_iff, pred_Memrel]) 1);
 by (safe_tac ZF_cs);
 by (fast_tac (ZF_cs addSDs [equalityD1]) 1);
 by (subgoal_tac "xa: i" 1);

 (** Initial segments of radd.  Statements by Grabczewski **)
 
 (*In fact, pred(A+B, Inl(a), radd(A,r,B,s)) = pred(A,a,r)+0 *)
 goalw OrderType.thy [pred_def]
  "!!A B. a:A ==>  \
-\        (lam x:pred(A,a,r). Inl(x))	\
+\        (lam x:pred(A,a,r). Inl(x))    \
 \        : bij(pred(A,a,r), pred(A+B, Inl(a), radd(A,r,B,s)))";
 by (res_inst_tac [("d", "case(%x.x, %y.y)")] lam_bijective 1);
 by (safe_tac sum_cs);
 by (ALLGOALS
     (asm_full_simp_tac 

 by (asm_full_simp_tac (ZF_ss addsimps [radd_Inl_iff, pred_def]) 1);
 qed "ordertype_pred_Inl_eq";
 
 goalw OrderType.thy [pred_def, id_def]
  "!!A B. b:B ==>  \
-\        id(A+pred(B,b,s))	\
+\        id(A+pred(B,b,s))      \
 \        : bij(A+pred(B,b,s), pred(A+B, Inr(b), radd(A,r,B,s)))";
 by (res_inst_tac [("d", "%z.z")] lam_bijective 1);
 by (safe_tac sum_cs);
 by (ALLGOALS (asm_full_simp_tac radd_ss));
 qed "pred_Inr_bij";

 
 (** Ordinal addition with zero **)
 
 goalw OrderType.thy [oadd_def] "!!i. Ord(i) ==> i++0 = i";
 by (asm_simp_tac (ZF_ss addsimps [Memrel_0, ordertype_sum_0_eq, 
-				  ordertype_Memrel, well_ord_Memrel]) 1);
+                                  ordertype_Memrel, well_ord_Memrel]) 1);
 qed "oadd_0";
 
 goalw OrderType.thy [oadd_def] "!!i. Ord(i) ==> 0++i = i";
 by (asm_simp_tac (ZF_ss addsimps [Memrel_0, ordertype_0_sum_eq, 
-				  ordertype_Memrel, well_ord_Memrel]) 1);
+                                  ordertype_Memrel, well_ord_Memrel]) 1);
 qed "oadd_0_left";
 
 
 (*** Further properties of ordinal addition.  Statements by Grabczewski,
     proofs by lcp. ***)
 
 goalw OrderType.thy [oadd_def] "!!i j k. [| k<i;  Ord(j) |] ==> k < i++j";
-by (resolve_tac [ltE] 1 THEN assume_tac 1);
-by (resolve_tac [ltI] 1);
+by (rtac ltE 1 THEN assume_tac 1);
+by (rtac ltI 1);
 by (REPEAT (ares_tac [Ord_ordertype, well_ord_radd, well_ord_Memrel] 2));
 by (asm_simp_tac 
     (ZF_ss addsimps [ordertype_pred_unfold, 
-		     well_ord_radd, well_ord_Memrel,
-		     ordertype_pred_Inl_eq, 
-		     lt_pred_Memrel, leI RS le_ordertype_Memrel]
-	   setloop rtac (InlI RSN (2,RepFun_eqI))) 1);
+                     well_ord_radd, well_ord_Memrel,
+                     ordertype_pred_Inl_eq, 
+                     lt_pred_Memrel, leI RS le_ordertype_Memrel]
+           setloop rtac (InlI RSN (2,RepFun_eqI))) 1);
 qed "lt_oadd1";
 
 (*Thus also we obtain the rule  i++j = k ==> i le k *)
 goal OrderType.thy "!!i j. [| Ord(i);  Ord(j) |] ==> i le i++j";
-by (resolve_tac [all_lt_imp_le] 1);
+by (rtac all_lt_imp_le 1);
 by (REPEAT (ares_tac [Ord_oadd, lt_oadd1] 1));
 qed "oadd_le_self";
 
 (** A couple of strange but necessary results! **)
 

 by (asm_simp_tac (ZF_ss addsimps [id_conv, Memrel_iff]) 1);
 by (fast_tac ZF_cs 1);
 qed "id_ord_iso_Memrel";
 
 goal OrderType.thy
-    "!!k. [| well_ord(A,r);  k<j |] ==>			\
-\            ordertype(A+k, radd(A, r, k, Memrel(j))) =	\
+    "!!k. [| well_ord(A,r);  k<j |] ==>                 \
+\            ordertype(A+k, radd(A, r, k, Memrel(j))) = \
 \            ordertype(A+k, radd(A, r, k, Memrel(k)))";
-by (eresolve_tac [ltE] 1);
+by (etac ltE 1);
 by (resolve_tac [ord_iso_refl RS sum_ord_iso_cong RS ordertype_eq] 1);
 by (eresolve_tac [OrdmemD RS id_ord_iso_Memrel RS ord_iso_sym] 1);
 by (REPEAT_FIRST (ares_tac [well_ord_radd, well_ord_Memrel]));
 qed "ordertype_sum_Memrel";
 
 goalw OrderType.thy [oadd_def] "!!i j k. [| k<j;  Ord(i) |] ==> i++k < i++j";
-by (resolve_tac [ltE] 1 THEN assume_tac 1);
+by (rtac ltE 1 THEN assume_tac 1);
 by (resolve_tac [ordertype_pred_unfold RS equalityD2 RS subsetD RS ltI] 1);
 by (REPEAT_FIRST (ares_tac [Ord_ordertype, well_ord_radd, well_ord_Memrel]));
-by (resolve_tac [RepFun_eqI] 1);
-by (eresolve_tac [InrI] 2);
+by (rtac RepFun_eqI 1);
+by (etac InrI 2);
 by (asm_simp_tac 
     (ZF_ss addsimps [ordertype_pred_Inr_eq, well_ord_Memrel, 
-		     lt_pred_Memrel, leI RS le_ordertype_Memrel,
-		     ordertype_sum_Memrel]) 1);
+                     lt_pred_Memrel, leI RS le_ordertype_Memrel,
+                     ordertype_sum_Memrel]) 1);
 qed "oadd_lt_mono2";
 
 goal OrderType.thy
     "!!i j k. [| i++j < i++k;  Ord(i);  Ord(j); Ord(k) |] ==> j<k";
 by (rtac Ord_linear_lt 1);

     "!!i j k. [| k < i++j;  Ord(i);  Ord(j) |] ==> k<i | (EX l:j. k = i++l )";
 (*Rotate the hypotheses so that simplification will work*)
 by (etac revcut_rl 1);
 by (asm_full_simp_tac 
     (ZF_ss addsimps [ordertype_pred_unfold, well_ord_radd,
-		     well_ord_Memrel]) 1);
+                     well_ord_Memrel]) 1);
 by (eresolve_tac [ltD RS RepFunE] 1);
 by (fast_tac (sum_cs addss 
-	      (ZF_ss addsimps [ordertype_pred_Inl_eq, well_ord_Memrel, 
-			       ltI, lt_pred_Memrel, le_ordertype_Memrel, leI,
-			       ordertype_pred_Inr_eq, 
-			       ordertype_sum_Memrel])) 1);
+              (ZF_ss addsimps [ordertype_pred_Inl_eq, well_ord_Memrel, 
+                               ltI, lt_pred_Memrel, le_ordertype_Memrel, leI,
+                               ordertype_pred_Inr_eq, 
+                               ordertype_sum_Memrel])) 1);
 qed "lt_oadd_disj";
 
 
 (*** Ordinal addition with successor -- via associativity! ***)
 
 goalw OrderType.thy [oadd_def]
     "!!i j k. [| Ord(i);  Ord(j);  Ord(k) |] ==> (i++j)++k = i++(j++k)";
 by (resolve_tac [ordertype_eq RS trans] 1);
 by (rtac ([ordertype_ord_iso RS ord_iso_sym, ord_iso_refl] MRS 
-	  sum_ord_iso_cong) 1);
+          sum_ord_iso_cong) 1);
 by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Ord_ordertype] 1));
 by (resolve_tac [sum_assoc_ord_iso RS ordertype_eq RS trans] 1);
 by (rtac ([ord_iso_refl, ordertype_ord_iso] MRS sum_ord_iso_cong RS 
-	  ordertype_eq) 2);
+          ordertype_eq) 2);
 by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Ord_ordertype] 1));
 qed "oadd_assoc";
 
 goal OrderType.thy
     "!!i j. [| Ord(i);  Ord(j) |] ==> i++j = i Un (UN k:j. {i++k})";
 by (rtac (subsetI RS equalityI) 1);
 by (eresolve_tac [ltI RS lt_oadd_disj RS disjE] 1);
 by (REPEAT (ares_tac [Ord_oadd] 1));
 by (fast_tac (ZF_cs addIs [lt_oadd1, oadd_lt_mono2]
-	            addss (ZF_ss addsimps [Ord_mem_iff_lt, Ord_oadd])) 3);
+                    addss (ZF_ss addsimps [Ord_mem_iff_lt, Ord_oadd])) 3);
 by (fast_tac ZF_cs 2);
 by (fast_tac (ZF_cs addSEs [ltE]) 1);
 qed "oadd_unfold";
 
 goal OrderType.thy "!!i. Ord(i) ==> i++1 = succ(i)";

 
 val prems = goal OrderType.thy
     "[| Ord(i);  !!x. x:A ==> Ord(j(x));  a:A |] ==> \
 \    i ++ (UN x:A. j(x)) = (UN x:A. i++j(x))";
 by (fast_tac (eq_cs addIs (prems @ [ltI, Ord_UN, Ord_oadd, 
-				    lt_oadd1 RS ltD, oadd_lt_mono2 RS ltD])
+                                    lt_oadd1 RS ltD, oadd_lt_mono2 RS ltD])
                      addSEs [ltE, ltI RS lt_oadd_disj RS disjE]) 1);
 qed "oadd_UN";
 
 goal OrderType.thy 
     "!!i j. [| Ord(i);  Limit(j) |] ==> i++j = (UN k:j. i++k)";
 by (forward_tac [Limit_has_0 RS ltD] 1);
 by (asm_simp_tac (ZF_ss addsimps [Limit_is_Ord RS Ord_in_Ord,
-				  oadd_UN RS sym, Union_eq_UN RS sym, 
-				  Limit_Union_eq]) 1);
+                                  oadd_UN RS sym, Union_eq_UN RS sym, 
+                                  Limit_Union_eq]) 1);
 qed "oadd_Limit";
 
 (** Order/monotonicity properties of ordinal addition **)
 
 goal OrderType.thy "!!i j. [| Ord(i);  Ord(j) |] ==> i le j++i";
 by (eres_inst_tac [("i","i")] trans_induct3 1);
 by (asm_simp_tac (ZF_ss addsimps [oadd_0, Ord_0_le]) 1);
 by (asm_simp_tac (ZF_ss addsimps [oadd_succ, succ_leI]) 1);
 by (asm_simp_tac (ZF_ss addsimps [oadd_Limit]) 1);
-by (resolve_tac [le_trans] 1);
-by (resolve_tac [le_implies_UN_le_UN] 2);
+by (rtac le_trans 1);
+by (rtac le_implies_UN_le_UN 2);
 by (fast_tac ZF_cs 2);
 by (asm_simp_tac (ZF_ss addsimps [Union_eq_UN RS sym, Limit_Union_eq, 
-				  le_refl, Limit_is_Ord]) 1);
+                                  le_refl, Limit_is_Ord]) 1);
 qed "oadd_le_self2";
 
 goal OrderType.thy "!!i j k. [| k le j;  Ord(i) |] ==> k++i le j++i";
 by (forward_tac [lt_Ord] 1);
 by (forward_tac [le_Ord2] 1);
-by (eresolve_tac [trans_induct3] 1);
+by (etac trans_induct3 1);
 by (asm_simp_tac (ZF_ss addsimps [oadd_0]) 1);
 by (asm_simp_tac (ZF_ss addsimps [oadd_succ, succ_le_iff]) 1);
 by (asm_simp_tac (ZF_ss addsimps [oadd_Limit]) 1);
-by (resolve_tac [le_implies_UN_le_UN] 1);
+by (rtac le_implies_UN_le_UN 1);
 by (fast_tac ZF_cs 1);
 qed "oadd_le_mono1";
 
 goal OrderType.thy "!!i j. [| i' le i;  j'<j |] ==> i'++j' < i++j";
-by (resolve_tac [lt_trans1] 1);
+by (rtac lt_trans1 1);
 by (REPEAT (eresolve_tac [asm_rl, oadd_le_mono1, oadd_lt_mono2, ltE,
-			  Ord_succD] 1));
+                          Ord_succD] 1));
 qed "oadd_lt_mono";
 
 goal OrderType.thy "!!i j. [| i' le i;  j' le j |] ==> i'++j' le i++j";
 by (asm_simp_tac (ZF_ss addsimps [oadd_succ RS sym, le_Ord2, oadd_lt_mono]) 1);
 qed "oadd_le_mono";
 
 goal OrderType.thy
     "!!i j k. [| Ord(i); Ord(j); Ord(k) |] ==> i++j le i++k <-> j le k";
 by (asm_simp_tac (ZF_ss addsimps [oadd_lt_iff2, oadd_succ RS sym, 
-				  Ord_succ]) 1);
+                                  Ord_succ]) 1);
 qed "oadd_le_iff2";
 
 
 (** Ordinal subtraction; the difference is ordertype(j-i, Memrel(j)). 
     Probably simpler to define the difference recursively!

 goal OrderType.thy
     "!!A B. A<=B ==> (lam y:B. if(y:A, Inl(y), Inr(y))) : bij(B, A+(B-A))";
 by (res_inst_tac [("d", "case(%x.x, %y.y)")] lam_bijective 1);
 by (fast_tac (sum_cs addSIs [if_type]) 1);
 by (fast_tac (ZF_cs addSIs [case_type]) 1);
-by (eresolve_tac [sumE] 2);
+by (etac sumE 2);
 by (ALLGOALS (asm_simp_tac (sum_ss setloop split_tac [expand_if])));
 qed "bij_sum_Diff";
 
 goal OrderType.thy
-    "!!i j. i le j ==>	\
-\           ordertype(i+(j-i), radd(i,Memrel(j),j-i,Memrel(j))) = 	\
+    "!!i j. i le j ==>  \
+\           ordertype(i+(j-i), radd(i,Memrel(j),j-i,Memrel(j))) =       \
 \           ordertype(j, Memrel(j))";
 by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1]));
 by (resolve_tac [bij_sum_Diff RS ord_isoI RS ord_iso_sym RS ordertype_eq] 1);
-by (eresolve_tac [well_ord_Memrel] 3);
+by (etac well_ord_Memrel 3);
 by (assume_tac 1);
 by (asm_simp_tac 
      (radd_ss setloop split_tac [expand_if] addsimps [Memrel_iff]) 1);
 by (forw_inst_tac [("j", "y")] Ord_in_Ord 1 THEN assume_tac 1);
 by (forw_inst_tac [("j", "x")] Ord_in_Ord 1 THEN assume_tac 1);
 by (asm_simp_tac (ZF_ss addsimps [Ord_mem_iff_lt, lt_Ord, not_lt_iff_le]) 1);
 by (fast_tac (ZF_cs addEs [lt_trans2, lt_trans]) 1);
 qed "ordertype_sum_Diff";
 
 goalw OrderType.thy [oadd_def, odiff_def]
-    "!!i j. i le j ==> 	\
+    "!!i j. i le j ==>  \
 \           i ++ (j--i) = ordertype(i+(j-i), radd(i,Memrel(j),j-i,Memrel(j)))";
 by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1]));
 by (resolve_tac [sum_ord_iso_cong RS ordertype_eq] 1);
-by (eresolve_tac [id_ord_iso_Memrel] 1);
+by (etac id_ord_iso_Memrel 1);
 by (resolve_tac [ordertype_ord_iso RS ord_iso_sym] 1);
 by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel RS well_ord_subset,
-		      Diff_subset] 1));
+                      Diff_subset] 1));
 qed "oadd_ordertype_Diff";
 
 goal OrderType.thy "!!i j. i le j ==> i ++ (j--i) = j";
 by (asm_simp_tac (ZF_ss addsimps [oadd_ordertype_Diff, ordertype_sum_Diff, 
-				  ordertype_Memrel, lt_Ord2 RS Ord_succD]) 1);
+                                  ordertype_Memrel, lt_Ord2 RS Ord_succD]) 1);
 qed "oadd_odiff_inverse";
 
 goalw OrderType.thy [odiff_def] 
     "!!i j. [| Ord(i);  Ord(j) |] ==> Ord(i--j)";
 by (REPEAT (ares_tac [Ord_ordertype, well_ord_Memrel RS well_ord_subset, 
-		      Diff_subset] 1));
+                      Diff_subset] 1));
 qed "Ord_odiff";
 
 (*By oadd_inject, the difference between i and j is unique.  Note that we get
   i++j = k  ==>  j = k--i.  *)
 goal OrderType.thy
     "!!i j. [| Ord(i); Ord(j) |] ==> (i++j) -- i = j";
-br oadd_inject 1;
+by (rtac oadd_inject 1);
 by (REPEAT (ares_tac [Ord_ordertype, Ord_oadd, Ord_odiff] 2));
 by (asm_simp_tac (ZF_ss addsimps [oadd_odiff_inverse, oadd_le_self]) 1);
 qed "odiff_oadd_inverse";
 
 val [i_lt_j, k_le_i] = goal OrderType.thy
     "[| i<j;  k le i |] ==> i--k < j--k";
 by (rtac (k_le_i RS lt_Ord RSN (2,oadd_lt_cancel2)) 1);
 by (simp_tac
     (ZF_ss addsimps [i_lt_j, k_le_i, [k_le_i, leI] MRS le_trans,
-		     oadd_odiff_inverse]) 1);
+                     oadd_odiff_inverse]) 1);
 by (REPEAT (resolve_tac (Ord_odiff :: 
-			 ([i_lt_j, k_le_i] RL [lt_Ord, lt_Ord2])) 1));
+                         ([i_lt_j, k_le_i] RL [lt_Ord, lt_Ord2])) 1));
 qed "odiff_lt_mono2";
 
 
 (**** Ordinal Multiplication ****)
 

 
 (*** A useful unfolding law ***)
 
 goalw OrderType.thy [pred_def]
  "!!A B. [| a:A;  b:B |] ==>  \
-\        pred(A*B, <a,b>, rmult(A,r,B,s)) =	\
+\        pred(A*B, <a,b>, rmult(A,r,B,s)) =     \
 \        pred(A,a,r)*B Un ({a} * pred(B,b,s))";
 by (safe_tac eq_cs);
 by (ALLGOALS (asm_full_simp_tac (ZF_ss addsimps [rmult_iff])));
 by (ALLGOALS (fast_tac ZF_cs));
 qed "pred_Pair_eq";
 
 goal OrderType.thy
  "!!A B. [| a:A;  b:B;  well_ord(A,r);  well_ord(B,s) |] ==>  \
 \        ordertype(pred(A*B, <a,b>, rmult(A,r,B,s)), rmult(A,r,B,s)) = \
-\        ordertype(pred(A,a,r)*B + pred(B,b,s), 		\
+\        ordertype(pred(A,a,r)*B + pred(B,b,s),                 \
 \                 radd(A*B, rmult(A,r,B,s), B, s))";
 by (asm_simp_tac (ZF_ss addsimps [pred_Pair_eq]) 1);
 by (resolve_tac [ordertype_eq RS sym] 1);
-by (resolve_tac [prod_sum_singleton_ord_iso] 1);
+by (rtac prod_sum_singleton_ord_iso 1);
 by (REPEAT_FIRST (ares_tac [pred_subset, well_ord_rmult RS well_ord_subset]));
 by (fast_tac (ZF_cs addSEs [predE]) 1);
 qed "ordertype_pred_Pair_eq";
 
 goalw OrderType.thy [oadd_def, omult_def]
  "!!i j. [| i'<i;  j'<j |] ==>  \
 \        ordertype(pred(i*j, <i',j'>, rmult(i,Memrel(i),j,Memrel(j))), \
 \                  rmult(i,Memrel(i),j,Memrel(j))) = \
 \        j**i' ++ j'";
 by (asm_simp_tac (ZF_ss addsimps [ordertype_pred_Pair_eq, lt_pred_Memrel, 
-				  ltD, lt_Ord2, well_ord_Memrel]) 1);
-by (resolve_tac [trans] 1);
+                                  ltD, lt_Ord2, well_ord_Memrel]) 1);
+by (rtac trans 1);
 by (resolve_tac [ordertype_ord_iso RS sum_ord_iso_cong RS ordertype_eq] 2);
-by (resolve_tac [ord_iso_refl] 3);
+by (rtac ord_iso_refl 3);
 by (resolve_tac [id_bij RS ord_isoI RS ordertype_eq] 1);
 by (REPEAT_FIRST (eresolve_tac [SigmaE, sumE, ltE, ssubst]));
 by (REPEAT_FIRST (ares_tac [well_ord_rmult, well_ord_radd, well_ord_Memrel, 
-			    Ord_ordertype]));
+                            Ord_ordertype]));
 by (ALLGOALS 
     (asm_simp_tac (radd_ss addsimps [rmult_iff, id_conv, Memrel_iff])));
 by (safe_tac ZF_cs);
 by (ALLGOALS (fast_tac (ZF_cs addEs [Ord_trans])));
 qed "ordertype_pred_Pair_lemma";
 
 goalw OrderType.thy [omult_def]
  "!!i j. [| Ord(i);  Ord(j);  k<j**i |] ==>  \
 \        EX j' i'. k = j**i' ++ j' & j'<j & i'<i";
 by (asm_full_simp_tac (ZF_ss addsimps [ordertype_pred_unfold, 
-				       well_ord_rmult, well_ord_Memrel]) 1);
+                                       well_ord_rmult, well_ord_Memrel]) 1);
 by (step_tac (ZF_cs addSEs [ltE]) 1);
 by (asm_simp_tac (ZF_ss addsimps [ordertype_pred_Pair_lemma, ltI,
-				  symmetric omult_def]) 1);
+                                  symmetric omult_def]) 1);
 by (fast_tac (ZF_cs addIs [ltI]) 1);
 qed "lt_omult";
 
 goalw OrderType.thy [omult_def]
  "!!i j. [| j'<j;  i'<i |] ==> j**i' ++ j'  <  j**i";
-by (resolve_tac [ltI] 1);
+by (rtac ltI 1);
 by (asm_simp_tac
     (ZF_ss addsimps [Ord_ordertype, well_ord_rmult, well_ord_Memrel, 
-		     lt_Ord2]) 2);
+                     lt_Ord2]) 2);
 by (asm_simp_tac 
     (ZF_ss addsimps [ordertype_pred_unfold, 
-		     well_ord_rmult, well_ord_Memrel, lt_Ord2]) 1);
-by (resolve_tac [RepFun_eqI] 1);
+                     well_ord_rmult, well_ord_Memrel, lt_Ord2]) 1);
+by (rtac RepFun_eqI 1);
 by (fast_tac (ZF_cs addSEs [ltE]) 2);
 by (asm_simp_tac 
     (ZF_ss addsimps [ordertype_pred_Pair_lemma, ltI, symmetric omult_def]) 1);
 qed "omult_oadd_lt";
 
 goal OrderType.thy
  "!!i j. [| Ord(i);  Ord(j) |] ==> j**i = (UN j':j. UN i':i. {j**i' ++ j'})";
 by (rtac (subsetI RS equalityI) 1);
 by (resolve_tac [lt_omult RS exE] 1);
-by (eresolve_tac [ltI] 3);
+by (etac ltI 3);
 by (REPEAT (ares_tac [Ord_omult] 1));
 by (fast_tac (ZF_cs addSEs [ltE]) 1);
 by (fast_tac (ZF_cs addIs [omult_oadd_lt RS ltD, ltI]) 1);
 qed "omult_unfold";
 

 
 goalw OrderType.thy [omult_def] "!!i. Ord(i) ==> i**1 = i";
 by (resolve_tac [ord_isoI RS ordertype_eq RS trans] 1);
 by (res_inst_tac [("c", "snd"), ("d", "%z.<0,z>")] lam_bijective 1);
 by (REPEAT_FIRST (eresolve_tac [snd_type, SigmaE, succE, emptyE, 
-				well_ord_Memrel, ordertype_Memrel]));
+                                well_ord_Memrel, ordertype_Memrel]));
 by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [rmult_iff, Memrel_iff])));
 qed "omult_1";
 
 goalw OrderType.thy [omult_def] "!!i. Ord(i) ==> 1**i = i";
 by (resolve_tac [ord_isoI RS ordertype_eq RS trans] 1);
 by (res_inst_tac [("c", "fst"), ("d", "%z.<z,0>")] lam_bijective 1);
 by (REPEAT_FIRST (eresolve_tac [fst_type, SigmaE, succE, emptyE, 
-				well_ord_Memrel, ordertype_Memrel]));
+                                well_ord_Memrel, ordertype_Memrel]));
 by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [rmult_iff, Memrel_iff])));
 qed "omult_1_left";
 
 (** Distributive law for ordinal multiplication and addition **)
 
 goalw OrderType.thy [omult_def, oadd_def]
     "!!i. [| Ord(i);  Ord(j);  Ord(k) |] ==> i**(j++k) = (i**j)++(i**k)";
 by (resolve_tac [ordertype_eq RS trans] 1);
 by (rtac ([ordertype_ord_iso RS ord_iso_sym, ord_iso_refl] MRS 
-	  prod_ord_iso_cong) 1);
+          prod_ord_iso_cong) 1);
 by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd, well_ord_Memrel, 
-		      Ord_ordertype] 1));
+                      Ord_ordertype] 1));
 by (rtac (sum_prod_distrib_ord_iso RS ordertype_eq RS trans) 1);
 by (rtac ordertype_eq 2);
 by (rtac ([ordertype_ord_iso, ordertype_ord_iso] MRS sum_ord_iso_cong) 2);
 by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd, well_ord_Memrel, 
-		      Ord_ordertype] 1));
+                      Ord_ordertype] 1));
 qed "oadd_omult_distrib";
 
 goal OrderType.thy "!!i. [| Ord(i);  Ord(j) |] ==> i**succ(j) = (i**j)++i";
 by (asm_simp_tac 
     (ZF_ss addsimps [oadd_1 RS sym, omult_1, oadd_omult_distrib, Ord_1]) 1);

 
 goalw OrderType.thy [omult_def]
     "!!i j k. [| Ord(i);  Ord(j);  Ord(k) |] ==> (i**j)**k = i**(j**k)";
 by (resolve_tac [ordertype_eq RS trans] 1);
 by (rtac ([ord_iso_refl, ordertype_ord_iso RS ord_iso_sym] MRS 
-	  prod_ord_iso_cong) 1);
+          prod_ord_iso_cong) 1);
 by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
 by (resolve_tac [prod_assoc_ord_iso RS ord_iso_sym RS 
-		 ordertype_eq RS trans] 1);
+                 ordertype_eq RS trans] 1);
 by (rtac ([ordertype_ord_iso, ord_iso_refl] MRS prod_ord_iso_cong RS
-	  ordertype_eq) 2);
+          ordertype_eq) 2);
 by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Ord_ordertype] 1));
 qed "omult_assoc";
 
 
 (** Ordinal multiplication with limit ordinals **)

 
 goal OrderType.thy 
     "!!i j. [| Ord(i);  Limit(j) |] ==> i**j = (UN k:j. i**k)";
 by (asm_simp_tac 
     (ZF_ss addsimps [Limit_is_Ord RS Ord_in_Ord, omult_UN RS sym, 
-		     Union_eq_UN RS sym, Limit_Union_eq]) 1);
+                     Union_eq_UN RS sym, Limit_Union_eq]) 1);
 qed "omult_Limit";
 
 
 (*** Ordering/monotonicity properties of ordinal multiplication ***)
 
 (*As a special case we have "[| 0<i;  0<j |] ==> 0 < i**j" *)
 goal OrderType.thy "!!i j. [| k<i;  0<j |] ==> k < i**j";
 by (safe_tac (ZF_cs addSEs [ltE] addSIs [ltI, Ord_omult]));
 by (asm_simp_tac (ZF_ss addsimps [omult_unfold]) 1);
-by (REPEAT (eresolve_tac [UN_I] 1));
+by (REPEAT (etac UN_I 1));
 by (asm_simp_tac (ZF_ss addsimps [omult_0, oadd_0_left]) 1);
 qed "lt_omult1";
 
 goal OrderType.thy "!!i j. [| Ord(i);  0<j |] ==> i le i**j";
-by (resolve_tac [all_lt_imp_le] 1);
+by (rtac all_lt_imp_le 1);
 by (REPEAT (ares_tac [Ord_omult, lt_omult1, lt_Ord2] 1));
 qed "omult_le_self";
 
 goal OrderType.thy "!!i j k. [| k le j;  Ord(i) |] ==> k**i le j**i";
 by (forward_tac [lt_Ord] 1);
 by (forward_tac [le_Ord2] 1);
-by (eresolve_tac [trans_induct3] 1);
+by (etac trans_induct3 1);
 by (asm_simp_tac (ZF_ss addsimps [omult_0, le_refl, Ord_0]) 1);
 by (asm_simp_tac (ZF_ss addsimps [omult_succ, oadd_le_mono]) 1);
 by (asm_simp_tac (ZF_ss addsimps [omult_Limit]) 1);
-by (resolve_tac [le_implies_UN_le_UN] 1);
+by (rtac le_implies_UN_le_UN 1);
 by (fast_tac ZF_cs 1);
 qed "omult_le_mono1";
 
 goal OrderType.thy "!!i j k. [| k<j;  0<i |] ==> i**k < i**j";
-by (resolve_tac [ltI] 1);
+by (rtac ltI 1);
 by (asm_simp_tac (ZF_ss addsimps [omult_unfold, lt_Ord2]) 1);
 by (safe_tac (ZF_cs addSEs [ltE] addSIs [Ord_omult]));
-by (REPEAT (eresolve_tac [UN_I] 1));
+by (REPEAT (etac UN_I 1));
 by (asm_simp_tac (ZF_ss addsimps [omult_0, oadd_0, Ord_omult]) 1);
 qed "omult_lt_mono2";
 
 goal OrderType.thy "!!i j k. [| k le j;  Ord(i) |] ==> i**k le i**j";
-by (resolve_tac [subset_imp_le] 1);
+by (rtac subset_imp_le 1);
 by (safe_tac (ZF_cs addSEs [ltE, make_elim Ord_succD] addSIs [Ord_omult]));
 by (asm_full_simp_tac (ZF_ss addsimps [omult_unfold]) 1);
 by (deepen_tac (ZF_cs addEs [Ord_trans, UN_I]) 0 1);
 qed "omult_le_mono2";
 
 goal OrderType.thy "!!i j. [| i' le i;  j' le j |] ==> i'**j' le i**j";
-by (resolve_tac [le_trans] 1);
+by (rtac le_trans 1);
 by (REPEAT (eresolve_tac [asm_rl, omult_le_mono1, omult_le_mono2, ltE,
-			  Ord_succD] 1));
+                          Ord_succD] 1));
 qed "omult_le_mono";
 
 goal OrderType.thy
       "!!i j. [| i' le i;  j'<j;  0<i |] ==> i'**j' < i**j";
-by (resolve_tac [lt_trans1] 1);
+by (rtac lt_trans1 1);
 by (REPEAT (eresolve_tac [asm_rl, omult_le_mono1, omult_lt_mono2, ltE,
-			  Ord_succD] 1));
+                          Ord_succD] 1));
 qed "omult_lt_mono";
 
 goal OrderType.thy "!!i j. [| Ord(i);  0<j |] ==> i le j**i";
 by (forward_tac [lt_Ord2] 1);
 by (eres_inst_tac [("i","i")] trans_induct3 1);
 by (asm_simp_tac (ZF_ss addsimps [omult_0, Ord_0 RS le_refl]) 1);
 by (asm_simp_tac (ZF_ss addsimps [omult_succ, succ_le_iff]) 1);
-by (eresolve_tac [lt_trans1] 1);
+by (etac lt_trans1 1);
 by (res_inst_tac [("b", "j**x")] (oadd_0 RS subst) 1 THEN 
     rtac oadd_lt_mono2 2);
 by (REPEAT (ares_tac [Ord_omult] 1));
 by (asm_simp_tac (ZF_ss addsimps [omult_Limit]) 1);
-by (resolve_tac [le_trans] 1);
-by (resolve_tac [le_implies_UN_le_UN] 2);
+by (rtac le_trans 1);
+by (rtac le_implies_UN_le_UN 2);
 by (fast_tac ZF_cs 2);
 by (asm_simp_tac (ZF_ss addsimps [Union_eq_UN RS sym, Limit_Union_eq, 
-				  Limit_is_Ord RS le_refl]) 1);
+                                  Limit_is_Ord RS le_refl]) 1);
 qed "omult_le_self2";
 
 
 (** Further properties of ordinal multiplication **)