src/ZF/CardinalArith.ML

 
 goalw CardinalArith.thy [eqpoll_def] "A+B eqpoll B+A";
 by (rtac exI 1);
 by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")] 
     lam_bijective 1);
-by (safe_tac (ZF_cs addSEs [sumE]));
-by (ALLGOALS (asm_simp_tac case_ss));
+by (safe_tac (!claset addSEs [sumE]));
+by (ALLGOALS (Asm_simp_tac));
 qed "sum_commute_eqpoll";
 
 goalw CardinalArith.thy [cadd_def] "i |+| j = j |+| i";
 by (rtac (sum_commute_eqpoll RS cardinal_cong) 1);
 qed "cadd_commute";

 by (rtac exI 1);
 by (rtac bij_0_sum 1);
 qed "sum_0_eqpoll";
 
 goalw CardinalArith.thy [cadd_def] "!!K. Card(K) ==> 0 |+| K = K";
-by (asm_simp_tac (ZF_ss addsimps [sum_0_eqpoll RS cardinal_cong, 
+by (asm_simp_tac (!simpset addsimps [sum_0_eqpoll RS cardinal_cong, 
                                   Card_cardinal_eq]) 1);
 qed "cadd_0";
 
 (** Addition by another cardinal **)
 
 goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A+B";
 by (res_inst_tac [("x", "lam x:A. Inl(x)")] exI 1);
-by (asm_simp_tac (sum_ss addsimps [lam_type]) 1);
+by (asm_simp_tac (!simpset addsimps [lam_type]) 1);
 qed "sum_lepoll_self";
 
 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
 goalw CardinalArith.thy [cadd_def]
     "!!K. [| Card(K);  Ord(L) |] ==> K le (K |+| L)";

 by (res_inst_tac 
       [("d", "case(%w. Inl(converse(f)`w), %y. Inr(converse(fa)`y))")] 
       lam_injective 1);
 by (typechk_tac ([inj_is_fun, case_type, InlI, InrI] @ ZF_typechecks));
 by (etac sumE 1);
-by (ALLGOALS (asm_simp_tac (sum_ss addsimps [left_inverse])));
+by (ALLGOALS (asm_simp_tac (!simpset addsimps [left_inverse])));
 qed "sum_lepoll_mono";
 
 goalw CardinalArith.thy [cadd_def]
     "!!K. [| K' le K;  L' le L |] ==> (K' |+| L') le (K |+| L)";
-by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1]));
+by (safe_tac (!claset addSDs [le_subset_iff RS iffD1]));
 by (rtac well_ord_lepoll_imp_Card_le 1);
 by (REPEAT (ares_tac [sum_lepoll_mono, subset_imp_lepoll] 2));
 by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1));
 qed "cadd_le_mono";
 

 by (rtac exI 1);
 by (res_inst_tac [("c", "%z.if(z=Inl(A),A+B,z)"), 
                   ("d", "%z.if(z=A+B,Inl(A),z)")] 
     lam_bijective 1);
 by (ALLGOALS
-    (asm_simp_tac (case_ss addsimps [succI2, mem_imp_not_eq]
+    (asm_simp_tac (!simpset addsimps [succI2, mem_imp_not_eq]
                            setloop eresolve_tac [sumE,succE])));
 qed "sum_succ_eqpoll";
 
 (*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)
 (*Unconditional version requires AC*)

 
 val [mnat,nnat] = goal CardinalArith.thy
     "[| m: nat;  n: nat |] ==> m |+| n = m#+n";
 by (cut_facts_tac [nnat] 1);
 by (nat_ind_tac "m" [mnat] 1);
-by (asm_simp_tac (arith_ss addsimps [nat_into_Card RS cadd_0]) 1);
-by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cadd_succ_lemma,
+by (asm_simp_tac (!simpset addsimps [nat_into_Card RS cadd_0]) 1);
+by (asm_simp_tac (!simpset addsimps [nat_into_Ord, cadd_succ_lemma,
                                      nat_into_Card RS Card_cardinal_eq]) 1);
 qed "nat_cadd_eq_add";
 
 
 (*** Cardinal multiplication ***)

 (*Easier to prove the two directions separately*)
 goalw CardinalArith.thy [eqpoll_def] "A*B eqpoll B*A";
 by (rtac exI 1);
 by (res_inst_tac [("c", "%<x,y>.<y,x>"), ("d", "%<x,y>.<y,x>")] 
     lam_bijective 1);
-by (safe_tac ZF_cs);
-by (ALLGOALS (asm_simp_tac ZF_ss));
+by (safe_tac (!claset));
+by (ALLGOALS (Asm_simp_tac));
 qed "prod_commute_eqpoll";
 
 goalw CardinalArith.thy [cmult_def] "i |*| j = j |*| i";
 by (rtac (prod_commute_eqpoll RS cardinal_cong) 1);
 qed "cmult_commute";

 (** Multiplication by 0 yields 0 **)
 
 goalw CardinalArith.thy [eqpoll_def] "0*A eqpoll 0";
 by (rtac exI 1);
 by (rtac lam_bijective 1);
-by (safe_tac ZF_cs);
+by (safe_tac (!claset));
 qed "prod_0_eqpoll";
 
 goalw CardinalArith.thy [cmult_def] "0 |*| i = 0";
-by (asm_simp_tac (ZF_ss addsimps [prod_0_eqpoll RS cardinal_cong, 
+by (asm_simp_tac (!simpset addsimps [prod_0_eqpoll RS cardinal_cong, 
                                   Card_0 RS Card_cardinal_eq]) 1);
 qed "cmult_0";
 
 (** 1 is the identity for multiplication **)
 

 by (rtac exI 1);
 by (resolve_tac [singleton_prod_bij RS bij_converse_bij] 1);
 qed "prod_singleton_eqpoll";
 
 goalw CardinalArith.thy [cmult_def, succ_def] "!!K. Card(K) ==> 1 |*| K = K";
-by (asm_simp_tac (ZF_ss addsimps [prod_singleton_eqpoll RS cardinal_cong, 
+by (asm_simp_tac (!simpset addsimps [prod_singleton_eqpoll RS cardinal_cong, 
                                   Card_cardinal_eq]) 1);
 qed "cmult_1";
 
 (*** Some inequalities for multiplication ***)
 
 goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A*A";
 by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1);
-by (simp_tac (ZF_ss addsimps [lam_type]) 1);
+by (simp_tac (!simpset addsimps [lam_type]) 1);
 qed "prod_square_lepoll";
 
 (*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
 goalw CardinalArith.thy [cmult_def] "!!K. Card(K) ==> K le K |*| K";
 by (rtac le_trans 1);
 by (rtac well_ord_lepoll_imp_Card_le 2);
 by (rtac prod_square_lepoll 3);
 by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2));
-by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
+by (asm_simp_tac (!simpset addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1);
 qed "cmult_square_le";
 
 (** Multiplication by a non-zero cardinal **)
 
 goalw CardinalArith.thy [lepoll_def, inj_def] "!!b. b: B ==> A lepoll A*B";
 by (res_inst_tac [("x", "lam x:A. <x,b>")] exI 1);
-by (asm_simp_tac (ZF_ss addsimps [lam_type]) 1);
+by (asm_simp_tac (!simpset addsimps [lam_type]) 1);
 qed "prod_lepoll_self";
 
 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
 goalw CardinalArith.thy [cmult_def]
     "!!K. [| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)";

 by (res_inst_tac [("x", "lam <w,y>:A*B. <f`w, fa`y>")] exI 1);
 by (res_inst_tac [("d", "%<w,y>.<converse(f)`w, converse(fa)`y>")] 
                   lam_injective 1);
 by (typechk_tac (inj_is_fun::ZF_typechecks));
 by (etac SigmaE 1);
-by (asm_simp_tac (ZF_ss addsimps [left_inverse]) 1);
+by (asm_simp_tac (!simpset addsimps [left_inverse]) 1);
 qed "prod_lepoll_mono";
 
 goalw CardinalArith.thy [cmult_def]
     "!!K. [| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)";
-by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1]));
+by (safe_tac (!claset addSDs [le_subset_iff RS iffD1]));
 by (rtac well_ord_lepoll_imp_Card_le 1);
 by (REPEAT (ares_tac [prod_lepoll_mono, subset_imp_lepoll] 2));
 by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1));
 qed "cmult_le_mono";
 

 goalw CardinalArith.thy [eqpoll_def] "succ(A)*B eqpoll B + A*B";
 by (rtac exI 1);
 by (res_inst_tac [("c", "%<x,y>. if(x=A, Inl(y), Inr(<x,y>))"), 
                   ("d", "case(%y. <A,y>, %z.z)")] 
     lam_bijective 1);
-by (safe_tac (ZF_cs addSEs [sumE]));
+by (safe_tac (!claset addSEs [sumE]));
 by (ALLGOALS
-    (asm_simp_tac (case_ss addsimps [succI2, if_type, mem_imp_not_eq])));
+    (asm_simp_tac (!simpset addsimps [succI2, if_type, mem_imp_not_eq])));
 qed "prod_succ_eqpoll";
 
 (*Unconditional version requires AC*)
 goalw CardinalArith.thy [cmult_def, cadd_def]
     "!!m n. [| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)";

 
 val [mnat,nnat] = goal CardinalArith.thy
     "[| m: nat;  n: nat |] ==> m |*| n = m#*n";
 by (cut_facts_tac [nnat] 1);
 by (nat_ind_tac "m" [mnat] 1);
-by (asm_simp_tac (arith_ss addsimps [cmult_0]) 1);
-by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cmult_succ_lemma,
+by (asm_simp_tac (!simpset addsimps [cmult_0]) 1);
+by (asm_simp_tac (!simpset addsimps [nat_into_Ord, cmult_succ_lemma,
                                      nat_cadd_eq_add]) 1);
 qed "nat_cmult_eq_mult";
 
 goal CardinalArith.thy "!!m n. Card(n) ==> 2 |*| n = n |+| n";
 by (asm_simp_tac 
-    (ZF_ss addsimps [Ord_0, Ord_succ, cmult_0, cmult_succ_lemma, Card_is_Ord,
+    (!simpset addsimps [Ord_0, Ord_succ, cmult_0, cmult_succ_lemma, Card_is_Ord,
                      read_instantiate [("j","0")] cadd_commute, cadd_0]) 1);
 qed "cmult_2";
 
 
 (*** Infinite Cardinals are Limit Ordinals ***)

     "lam z:cons(u,A). if(z=u, f`0,      \
 \                        if(z: range(f), f`succ(converse(f)`z), z))")] exI 1);
 by (res_inst_tac [("d", "%y. if(y: range(f),    \
 \                               nat_case(u, %z.f`z, converse(f)`y), y)")] 
     lam_injective 1);
-by (fast_tac (ZF_cs addSIs [if_type, nat_0I, nat_succI, apply_type]
+by (fast_tac (!claset addSIs [if_type, nat_0I, nat_succI, apply_type]
                     addIs  [inj_is_fun, inj_converse_fun]) 1);
 by (asm_simp_tac 
-    (ZF_ss addsimps [inj_is_fun RS apply_rangeI,
+    (!simpset addsimps [inj_is_fun RS apply_rangeI,
                      inj_converse_fun RS apply_rangeI,
                      inj_converse_fun RS apply_funtype,
                      left_inverse, right_inverse, nat_0I, nat_succI, 
                      nat_case_0, nat_case_succ]
            setloop split_tac [expand_if]) 1);

 goalw CardinalArith.thy [succ_def] "!!i. nat <= A ==> succ(A) eqpoll A";
 by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1);
 qed "nat_succ_eqpoll";
 
 goalw CardinalArith.thy [InfCard_def] "InfCard(nat)";
-by (fast_tac (ZF_cs addIs [Card_nat, le_refl, Card_is_Ord]) 1);
+by (fast_tac (!claset addIs [Card_nat, le_refl, Card_is_Ord]) 1);
 qed "InfCard_nat";
 
 goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Card(K)";
 by (etac conjunct1 1);
 qed "InfCard_is_Card";
 
 goalw CardinalArith.thy [InfCard_def]
     "!!K L. [| InfCard(K);  Card(L) |] ==> InfCard(K Un L)";
-by (asm_simp_tac (ZF_ss addsimps [Card_Un, Un_upper1_le RSN (2,le_trans), 
+by (asm_simp_tac (!simpset addsimps [Card_Un, Un_upper1_le RSN (2,le_trans), 
                                   Card_is_Ord]) 1);
 qed "InfCard_Un";
 
 (*Kunen's Lemma 10.11*)
 goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Limit(K)";
 by (etac conjE 1);
 by (forward_tac [Card_is_Ord] 1);
 by (rtac (ltI RS non_succ_LimitI) 1);
 by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1);
-by (safe_tac (ZF_cs addSDs [Limit_nat RS Limit_le_succD]));
+by (safe_tac (!claset addSDs [Limit_nat RS Limit_le_succD]));
 by (rewtac Card_def);
 by (dtac trans 1);
 by (etac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1);
 by (etac (Ord_succD RS Ord_cardinal_le RS lt_trans2 RS lt_irrefl) 1);
 by (REPEAT (ares_tac [le_eqI, Ord_cardinal] 1));

 
 (*A general fact about ordermap*)
 goalw Cardinal.thy [eqpoll_def]
     "!!A. [| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x eqpoll pred(A,x,r)";
 by (rtac exI 1);
-by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, well_ord_is_wf]) 1);
+by (asm_simp_tac (!simpset addsimps [ordermap_eq_image, well_ord_is_wf]) 1);
 by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1);
 by (rtac pred_subset 1);
 qed "ordermap_eqpoll_pred";
 
 (** Establishing the well-ordering **)
 
 goalw CardinalArith.thy [inj_def]
  "!!K. Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)";
-by (fast_tac (ZF_cs addss ZF_ss
+by (fast_tac (!claset addss (!simpset)
                     addIs [lam_type, Un_least_lt RS ltD, ltI]) 1);
 qed "csquare_lam_inj";
 
 goalw CardinalArith.thy [csquare_rel_def]
  "!!K. Ord(K) ==> well_ord(K*K, csquare_rel(K))";

 
 goalw CardinalArith.thy [csquare_rel_def]
  "!!K. [| x<K;  y<K;  z<K |] ==> \
 \      <<x,y>, <z,z>> : csquare_rel(K) --> x le z & y le z";
 by (REPEAT (etac ltE 1));
-by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
+by (asm_simp_tac (!simpset addsimps [rvimage_iff, rmult_iff, Memrel_iff,
                                   Un_absorb, Un_least_mem_iff, ltD]) 1);
-by (safe_tac (ZF_cs addSEs [mem_irrefl] 
+by (safe_tac (!claset addSEs [mem_irrefl] 
                     addSIs [Un_upper1_le, Un_upper2_le]));
-by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [lt_def, succI2, Ord_succ])));
+by (ALLGOALS (asm_simp_tac (!simpset addsimps [lt_def, succI2, Ord_succ])));
 val csquareD_lemma = result();
 
 bind_thm ("csquareD", csquareD_lemma RS mp);
 
 goalw CardinalArith.thy [pred_def]
  "!!K. z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)";
-by (safe_tac (lemmas_cs addSEs [SigmaE]));      (*avoids using succCI*)
+by (safe_tac (claset_of"ZF" addSEs [SigmaE]));  (*avoids using succCI,...*)
 by (rtac (csquareD RS conjE) 1);
 by (rewtac lt_def);
 by (assume_tac 4);
-by (ALLGOALS (fast_tac ZF_cs));
+by (ALLGOALS Fast_tac);
 qed "pred_csquare_subset";
 
 goalw CardinalArith.thy [csquare_rel_def]
  "!!K. [| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)";
 by (subgoals_tac ["x<K", "y<K"] 1);
 by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2));
 by (REPEAT (etac ltE 1));
-by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
-                                  Un_absorb, Un_least_mem_iff, ltD]) 1);
+by (asm_simp_tac (!simpset addsimps [rvimage_iff, rmult_iff, Memrel_iff,
+				     Un_absorb, Un_least_mem_iff, ltD]) 1);
 qed "csquare_ltI";
 
 (*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
 goalw CardinalArith.thy [csquare_rel_def]
  "!!K. [| x le z;  y le z;  z<K |] ==> \
 \      <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z";
 by (subgoals_tac ["x<K", "y<K"] 1);
 by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2));
 by (REPEAT (etac ltE 1));
-by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff,
+by (asm_simp_tac (!simpset addsimps [rvimage_iff, rmult_iff, Memrel_iff,
                                   Un_absorb, Un_least_mem_iff, ltD]) 1);
 by (REPEAT_FIRST (etac succE));
 by (ALLGOALS
-    (asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iff_sym, 
+    (asm_simp_tac (!simpset addsimps [subset_Un_iff RS iff_sym, 
                                    subset_Un_iff2 RS iff_sym, OrdmemD])));
 qed "csquare_or_eqI";
 
 (** The cardinality of initial segments **)
 

     "!!K. [| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==> \
 \         ordermap(K*K, csquare_rel(K)) ` <x,y> <               \
 \         ordermap(K*K, csquare_rel(K)) ` <z,z>";
 by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1);
 by (etac (Limit_is_Ord RS well_ord_csquare) 2);
-by (fast_tac (ZF_cs addSIs [Un_least_lt, Limit_has_succ]) 2);
+by (fast_tac (!claset addSIs [Un_least_lt, Limit_has_succ]) 2);
 by (rtac (csquare_ltI RS ordermap_mono RS ltI) 1);
 by (etac well_ord_is_wf 4);
 by (ALLGOALS 
-    (fast_tac (ZF_cs addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap] 
+    (fast_tac (!claset addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap] 
                      addSEs [ltE])));
 qed "ordermap_z_lt";
 
 (*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
 goalw CardinalArith.thy [cmult_def]
   "!!K. [| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==> \
 \       | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|";
 by (rtac (well_ord_rmult RS well_ord_lepoll_imp_Card_le) 1);
 by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1));
 by (subgoals_tac ["z<K"] 1);
-by (fast_tac (ZF_cs addSIs [Un_least_lt, Limit_has_succ]) 2);
+by (fast_tac (!claset addSIs [Un_least_lt, Limit_has_succ]) 2);
 by (rtac (ordermap_z_lt RS leI RS le_imp_lepoll RS lepoll_trans) 1);
 by (REPEAT_SOME assume_tac);
 by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1);
 by (etac (Limit_is_Ord RS well_ord_csquare) 1);
-by (fast_tac (ZF_cs addIs [ltD]) 1);
+by (fast_tac (!claset addIs [ltD]) 1);
 by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN
     assume_tac 1);
 by (REPEAT_FIRST (etac ltE));
 by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1);
 by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll)));

 by (assume_tac 1);
 by (etac (well_ord_csquare RS Ord_ordertype) 1);
 by (rtac Card_lt_imp_lt 1);
 by (etac InfCard_is_Card 3);
 by (etac ltE 2 THEN assume_tac 2);
-by (asm_full_simp_tac (ZF_ss addsimps [ordertype_unfold]) 1);
-by (safe_tac (ZF_cs addSEs [ltE]));
+by (asm_full_simp_tac (!simpset addsimps [ordertype_unfold]) 1);
+by (safe_tac (!claset addSEs [ltE]));
 by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1);
 by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2));
 by (rtac (InfCard_is_Limit RS ordermap_csquare_le RS lt_trans1) 1  THEN
     REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1));
 by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1  THEN
     REPEAT (ares_tac [Ord_Un, Ord_nat] 1));
 (*the finite case: xb Un y < nat *)
 by (res_inst_tac [("j", "nat")] lt_trans2 1);
-by (asm_full_simp_tac (FOL_ss addsimps [InfCard_def]) 2);
+by (asm_full_simp_tac (!simpset addsimps [InfCard_def]) 2);
 by (asm_full_simp_tac
-    (ZF_ss addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type,
+    (!simpset addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type,
                      nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1);
 (*case nat le (xb Un y) *)
 by (asm_full_simp_tac
-    (ZF_ss addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong,
+    (!simpset addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong,
                      le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt, 
                      Ord_Un, ltI, nat_le_cardinal,
                      Ord_cardinal_le RS lt_trans1 RS ltD]) 1);
 qed "ordertype_csquare_le";
 

 by (etac (InfCard_is_Card RS cmult_square_le) 2);
 by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1);
 by (assume_tac 2);
 by (assume_tac 2);
 by (asm_simp_tac 
-    (ZF_ss addsimps [cmult_def, Ord_cardinal_le,
+    (!simpset addsimps [cmult_def, Ord_cardinal_le,
                      well_ord_csquare RS ordermap_bij RS 
                           bij_imp_eqpoll RS cardinal_cong,
                      well_ord_csquare RS Ord_ordertype]) 1);
 qed "InfCard_csquare_eq";
 

     "!!A. [| well_ord(A,r);  InfCard(|A|) |] ==> A*A eqpoll A";
 by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1);
 by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1));
 by (rtac well_ord_cardinal_eqE 1);
 by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1));
-by (asm_simp_tac (ZF_ss addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1);
+by (asm_simp_tac (!simpset addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1);
 qed "well_ord_InfCard_square_eq";
 
 (** Toward's Kunen's Corollary 10.13 (1) **)
 
 goal CardinalArith.thy "!!K. [| InfCard(K);  L le K;  0<L |] ==> K |*| L = K";
 by (rtac le_anti_sym 1);
 by (etac ltE 2 THEN
     REPEAT (ares_tac [cmult_le_self, InfCard_is_Card] 2));
 by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
 by (resolve_tac [cmult_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
-by (asm_simp_tac (ZF_ss addsimps [InfCard_csquare_eq]) 1);
+by (asm_simp_tac (!simpset addsimps [InfCard_csquare_eq]) 1);
 qed "InfCard_le_cmult_eq";
 
 (*Corollary 10.13 (1), for cardinal multiplication*)
 goal CardinalArith.thy
     "!!K. [| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L";

 by (typechk_tac [InfCard_is_Card, Card_is_Ord]);
 by (resolve_tac [cmult_commute RS ssubst] 1);
 by (resolve_tac [Un_commute RS ssubst] 1);
 by (ALLGOALS
     (asm_simp_tac 
-     (ZF_ss addsimps [InfCard_is_Limit RS Limit_has_0, InfCard_le_cmult_eq,
+     (!simpset addsimps [InfCard_is_Limit RS Limit_has_0, InfCard_le_cmult_eq,
                       subset_Un_iff2 RS iffD1, le_imp_subset])));
 qed "InfCard_cmult_eq";
 
 (*This proof appear to be the simplest!*)
 goal CardinalArith.thy "!!K. InfCard(K) ==> K |+| K = K";
 by (asm_simp_tac
-    (ZF_ss addsimps [cmult_2 RS sym, InfCard_is_Card, cmult_commute]) 1);
+    (!simpset addsimps [cmult_2 RS sym, InfCard_is_Card, cmult_commute]) 1);
 by (rtac InfCard_le_cmult_eq 1);
 by (typechk_tac [Ord_0, le_refl, leI]);
 by (typechk_tac [InfCard_is_Limit, Limit_has_0, Limit_has_succ]);
 qed "InfCard_cdouble_eq";
 

 by (rtac le_anti_sym 1);
 by (etac ltE 2 THEN
     REPEAT (ares_tac [cadd_le_self, InfCard_is_Card] 2));
 by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1);
 by (resolve_tac [cadd_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1));
-by (asm_simp_tac (ZF_ss addsimps [InfCard_cdouble_eq]) 1);
+by (asm_simp_tac (!simpset addsimps [InfCard_cdouble_eq]) 1);
 qed "InfCard_le_cadd_eq";
 
 goal CardinalArith.thy
     "!!K. [| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L";
 by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1);
 by (typechk_tac [InfCard_is_Card, Card_is_Ord]);
 by (resolve_tac [cadd_commute RS ssubst] 1);
 by (resolve_tac [Un_commute RS ssubst] 1);
 by (ALLGOALS
     (asm_simp_tac 
-     (ZF_ss addsimps [InfCard_le_cadd_eq,
+     (!simpset addsimps [InfCard_le_cadd_eq,
                       subset_Un_iff2 RS iffD1, le_imp_subset])));
 qed "InfCard_cadd_eq";
 
 (*The other part, Corollary 10.13 (2), refers to the cardinality of the set
   of all n-tuples of elements of K.  A better version for the Isabelle theory

 (*** For every cardinal number there exists a greater one
      [Kunen's Theorem 10.16, which would be trivial using AC] ***)
 
 goalw CardinalArith.thy [jump_cardinal_def] "Ord(jump_cardinal(K))";
 by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
-by (fast_tac (ZF_cs addSIs [Ord_ordertype]) 2);
+by (fast_tac (!claset addSIs [Ord_ordertype]) 2);
 by (rewtac Transset_def);
 by (safe_tac subset_cs);
-by (asm_full_simp_tac (ZF_ss addsimps [ordertype_pred_unfold]) 1);
-by (safe_tac ZF_cs);
+by (asm_full_simp_tac (!simpset addsimps [ordertype_pred_unfold]) 1);
+by (safe_tac (!claset));
 by (rtac UN_I 1);
 by (rtac ReplaceI 2);
-by (ALLGOALS (fast_tac (ZF_cs addSEs [well_ord_subset, predE])));
+by (ALLGOALS (fast_tac (!claset addSEs [well_ord_subset, predE])));
 qed "Ord_jump_cardinal";
 
 (*Allows selective unfolding.  Less work than deriving intro/elim rules*)
 goalw CardinalArith.thy [jump_cardinal_def]
      "i : jump_cardinal(K) <->   \

 goal CardinalArith.thy "!!K. Ord(K) ==> K < jump_cardinal(K)";
 by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1);
 by (resolve_tac [jump_cardinal_iff RS iffD2] 1);
 by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel]));
 by (rtac subset_refl 2);
-by (asm_simp_tac (ZF_ss addsimps [Memrel_def, subset_iff]) 1);
-by (asm_simp_tac (ZF_ss addsimps [ordertype_Memrel]) 1);
+by (asm_simp_tac (!simpset addsimps [Memrel_def, subset_iff]) 1);
+by (asm_simp_tac (!simpset addsimps [ordertype_Memrel]) 1);
 qed "K_lt_jump_cardinal";
 
 (*The proof by contradiction: the bijection f yields a wellordering of X
   whose ordertype is jump_cardinal(K).  *)
 goal CardinalArith.thy

 by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1);
 by (REPEAT (assume_tac 1));
 by (etac (bij_is_inj RS well_ord_rvimage) 1);
 by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1);
 by (asm_simp_tac
-    (ZF_ss addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage), 
+    (!simpset addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage), 
                      ordertype_Memrel, Ord_jump_cardinal]) 1);
 qed "Card_jump_cardinal_lemma";
 
 (*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
 goal CardinalArith.thy "Card(jump_cardinal(K))";
 by (rtac (Ord_jump_cardinal RS CardI) 1);
 by (rewtac eqpoll_def);
-by (safe_tac (ZF_cs addSDs [ltD, jump_cardinal_iff RS iffD1]));
+by (safe_tac (!claset addSDs [ltD, jump_cardinal_iff RS iffD1]));
 by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1));
 qed "Card_jump_cardinal";
 
 (*** Basic properties of successor cardinals ***)
 

 qed "lt_csucc_iff";
 
 goal CardinalArith.thy
     "!!K' K. [| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K";
 by (asm_simp_tac 
-    (ZF_ss addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1);
+    (!simpset addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1);
 qed "Card_lt_csucc_iff";
 
 goalw CardinalArith.thy [InfCard_def]
     "!!K. InfCard(K) ==> InfCard(csucc(K))";
-by (asm_simp_tac (ZF_ss addsimps [Card_csucc, Card_is_Ord, 
+by (asm_simp_tac (!simpset addsimps [Card_csucc, Card_is_Ord, 
                                   lt_csucc RS leI RSN (2,le_trans)]) 1);
 qed "InfCard_csucc";
 
 
 (*** Finite sets ***)
 
 goal CardinalArith.thy
     "!!n. n: nat ==> ALL A. A eqpoll n --> A : Fin(A)";
 by (etac nat_induct 1);
-by (simp_tac (ZF_ss addsimps (eqpoll_0_iff::Fin.intrs)) 1);
-by (step_tac ZF_cs 1);
+by (simp_tac (!simpset addsimps (eqpoll_0_iff::Fin.intrs)) 1);
+by (step_tac (!claset) 1);
 by (subgoal_tac "EX u. u:A" 1);
 by (etac exE 1);
 by (resolve_tac [Diff_sing_eqpoll RS revcut_rl] 1);
 by (assume_tac 2);
 by (assume_tac 1);
 by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
 by (assume_tac 1);
 by (resolve_tac [Fin.consI] 1);
-by (fast_tac ZF_cs 1);
-by (deepen_tac (ZF_cs addIs [Fin_mono RS subsetD]) 0 1);  (*SLOW*)
+by (Fast_tac 1);
+by (fast_tac (!claset addIs [subset_consI  RS Fin_mono RS subsetD]) 1); 
 (*Now for the lemma assumed above*)
 by (rewtac eqpoll_def);
-by (fast_tac (ZF_cs addSEs [bij_converse_bij RS bij_is_fun RS apply_type]) 1);
+by (best_tac (!claset addSEs [bij_converse_bij RS bij_is_fun RS apply_type]) 1);
 val lemma = result();
 
 goalw CardinalArith.thy [Finite_def] "!!A. Finite(A) ==> A : Fin(A)";
-by (fast_tac (ZF_cs addIs [lemma RS spec RS mp]) 1);
+by (fast_tac (!claset addIs [lemma RS spec RS mp]) 1);
 qed "Finite_into_Fin";
 
 goal CardinalArith.thy "!!A. A : Fin(U) ==> Finite(A)";
-by (fast_tac (ZF_cs addSIs [Finite_0, Finite_cons] addEs [Fin.induct]) 1);
+by (fast_tac (!claset addSIs [Finite_0, Finite_cons] addEs [Fin.induct]) 1);
 qed "Fin_into_Finite";
 
 goal CardinalArith.thy "Finite(A) <-> A : Fin(A)";
-by (fast_tac (ZF_cs addIs [Finite_into_Fin] addEs [Fin_into_Finite]) 1);
+by (fast_tac (!claset addIs [Finite_into_Fin] addEs [Fin_into_Finite]) 1);
 qed "Finite_Fin_iff";
 
 goal CardinalArith.thy
     "!!A. [| Finite(A); Finite(B) |] ==> Finite(A Un B)";
-by (fast_tac (ZF_cs addSIs [Fin_into_Finite, Fin_UnI] 
+by (fast_tac (!claset addSIs [Fin_into_Finite, Fin_UnI] 
                     addSDs [Finite_into_Fin]
                     addSEs [Un_upper1 RS Fin_mono RS subsetD,
                             Un_upper2 RS Fin_mono RS subsetD]) 1);
 qed "Finite_Un";
 

 (** Removing elements from a finite set decreases its cardinality **)
 
 goal CardinalArith.thy
     "!!A. A: Fin(U) ==> x~:A --> ~ cons(x,A) lepoll A";
 by (etac Fin_induct 1);
-by (simp_tac (ZF_ss addsimps [lepoll_0_iff]) 1);
+by (simp_tac (!simpset addsimps [lepoll_0_iff]) 1);
 by (subgoal_tac "cons(x,cons(xa,y)) = cons(xa,cons(x,y))" 1);
-by (asm_simp_tac ZF_ss 1);
-by (fast_tac (ZF_cs addSDs [cons_lepoll_consD]) 1);
+by (Asm_simp_tac 1);
+by (fast_tac (!claset addSDs [cons_lepoll_consD]) 1);
 by (fast_tac eq_cs 1);
 qed "Fin_imp_not_cons_lepoll";
 
 goal CardinalArith.thy
     "!!a A. [| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)";
 by (rewtac cardinal_def);
 by (rtac Least_equality 1);
 by (fold_tac [cardinal_def]);
-by (simp_tac (ZF_ss addsimps [succ_def]) 1);
-by (fast_tac (ZF_cs addIs [cons_eqpoll_cong, well_ord_cardinal_eqpoll] 
+by (simp_tac (!simpset addsimps [succ_def]) 1);
+by (fast_tac (!claset addIs [cons_eqpoll_cong, well_ord_cardinal_eqpoll] 
                     addSEs [mem_irrefl]
                     addSDs [Finite_imp_well_ord]) 1);
-by (fast_tac (ZF_cs addIs [Ord_succ, Card_cardinal, Card_is_Ord]) 1);
+by (fast_tac (!claset addIs [Ord_succ, Card_cardinal, Card_is_Ord]) 1);
 by (rtac notI 1);
 by (resolve_tac [Finite_into_Fin RS Fin_imp_not_cons_lepoll RS mp RS notE] 1);
 by (assume_tac 1);
 by (assume_tac 1);
 by (eresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_trans] 1);
 by (eresolve_tac [le_imp_lepoll RS lepoll_trans] 1);
-by (fast_tac (ZF_cs addIs [well_ord_cardinal_eqpoll RS eqpoll_imp_lepoll] 
+by (fast_tac (!claset addIs [well_ord_cardinal_eqpoll RS eqpoll_imp_lepoll] 
                     addSDs [Finite_imp_well_ord]) 1);
 qed "Finite_imp_cardinal_cons";
 
 
 goal CardinalArith.thy "!!a A. [| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|";
 by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1);
 by (assume_tac 1);
-by (asm_simp_tac (ZF_ss addsimps [Finite_imp_cardinal_cons,
+by (asm_simp_tac (!simpset addsimps [Finite_imp_cardinal_cons,
                                   Diff_subset RS subset_Finite]) 1);
-by (asm_simp_tac (ZF_ss addsimps [cons_Diff]) 1);
+by (asm_simp_tac (!simpset addsimps [cons_Diff]) 1);
 qed "Finite_imp_succ_cardinal_Diff";
 
 goal CardinalArith.thy "!!a A. [| Finite(A);  a:A |] ==> |A-{a}| < |A|";
 by (rtac succ_leE 1);
-by (asm_simp_tac (ZF_ss addsimps [Finite_imp_succ_cardinal_Diff, 
+by (asm_simp_tac (!simpset addsimps [Finite_imp_succ_cardinal_Diff, 
                                   Ord_cardinal RS le_refl]) 1);
 qed "Finite_imp_cardinal_Diff";
 
 
 (** Thanks to Krzysztof Grabczewski **)

 by (eresolve_tac [nat_implies_well_ord RS (
                   nat_implies_well_ord RSN (2,
                   well_ord_radd RS well_ord_cardinal_eqpoll)) RS eqpoll_sym] 1 
     THEN (assume_tac 1));
 by (eresolve_tac [nat_cadd_eq_add RS subst] 1 THEN (assume_tac 1));
-by (asm_full_simp_tac (ZF_ss addsimps [cadd_def, eqpoll_refl]) 1);
+by (asm_full_simp_tac (!simpset addsimps [cadd_def, eqpoll_refl]) 1);
 qed "nat_sum_eqpoll_sum";
 
 goal Nat.thy "!!m. [| m le n; n:nat |] ==> m:nat";
-by (fast_tac (ZF_cs addSDs [nat_succI RS (Ord_nat RSN (2, OrdmemD))]
+by (fast_tac (!claset addSDs [nat_succI RS (Ord_nat RSN (2, OrdmemD))]
         addSEs [ltE]) 1);
 qed "le_in_nat";