--- a/src/ZF/OrderType.ML Fri Jan 03 10:48:28 1997 +0100
+++ b/src/ZF/OrderType.ML Fri Jan 03 15:01:55 1997 +0100
@@ -18,7 +18,7 @@
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,
+by (asm_simp_tac (!simpset addsimps [linear_def, Memrel_iff,
[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));
@@ -31,7 +31,7 @@
The smaller ordinal is an initial segment of the larger *)
goalw OrderType.thy [pred_def, lt_def]
"!!i j. j<i ==> pred(i, j, Memrel(i)) = j";
-by (asm_simp_tac (ZF_ss addsimps [Memrel_iff]) 1);
+by (asm_simp_tac (!simpset addsimps [Memrel_iff]) 1);
by (fast_tac (eq_cs addEs [Ord_trans]) 1);
qed "lt_pred_Memrel";
@@ -46,10 +46,10 @@
by (etac ltE 1);
by (rtac (well_ord_Memrel RS well_ord_iso_predE) 1 THEN
assume_tac 3 THEN assume_tac 1);
-by (asm_full_simp_tac (ZF_ss addsimps [ord_iso_def]) 1);
+by (asm_full_simp_tac (!simpset addsimps [ord_iso_def]) 1);
(*Combining the two simplifications causes looping*)
-by (asm_simp_tac (ZF_ss addsimps [Memrel_iff]) 1);
-by (fast_tac (ZF_cs addSEs [bij_is_fun RS apply_type] addEs [Ord_trans]) 1);
+by (asm_simp_tac (!simpset addsimps [Memrel_iff]) 1);
+by (fast_tac (!claset 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*)
@@ -76,10 +76,10 @@
goalw OrderType.thy [ordermap_def, pred_def]
"!!r. [| wf[A](r); x:A |] ==> \
\ ordermap(A,r) ` x = ordermap(A,r) `` pred(A,x,r)";
-by (asm_simp_tac ZF_ss 1);
+by (Asm_simp_tac 1);
by (etac (wfrec_on RS trans) 1);
by (assume_tac 1);
-by (asm_simp_tac (ZF_ss addsimps [subset_iff, image_lam,
+by (asm_simp_tac (!simpset addsimps [subset_iff, image_lam,
vimage_singleton_iff]) 1);
qed "ordermap_eq_image";
@@ -87,7 +87,7 @@
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,
+by (asm_simp_tac (!simpset addsimps [ordermap_eq_image, pred_subset,
ordermap_type RS image_fun]) 1);
qed "ordermap_pred_unfold";
@@ -103,26 +103,26 @@
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 (safe_tac (!claset));
by (wf_on_ind_tac "x" [] 1);
-by (asm_simp_tac (ZF_ss addsimps [ordermap_pred_unfold]) 1);
+by (asm_simp_tac (!simpset addsimps [ordermap_pred_unfold]) 1);
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
by (rewrite_goals_tac [pred_def,Transset_def]);
-by (fast_tac ZF_cs 2);
-by (safe_tac ZF_cs);
+by (Fast_tac 2);
+by (safe_tac (!claset));
by (ordermap_elim_tac 1);
-by (fast_tac (ZF_cs addSEs [trans_onD]) 1);
+by (fast_tac (!claset addSEs [trans_onD]) 1);
qed "Ord_ordermap";
goalw OrderType.thy [ordertype_def]
"!!r. well_ord(A,r) ==> Ord(ordertype(A,r))";
by (stac ([ordermap_type, subset_refl] MRS image_fun) 1);
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1);
-by (fast_tac (ZF_cs addIs [Ord_ordermap]) 2);
+by (fast_tac (!claset addIs [Ord_ordermap]) 2);
by (rewrite_goals_tac [Transset_def,well_ord_def]);
-by (safe_tac ZF_cs);
+by (safe_tac (!claset));
by (ordermap_elim_tac 1);
-by (fast_tac ZF_cs 1);
+by (Fast_tac 1);
qed "Ord_ordertype";
(*** ordermap preserves the orderings in both directions ***)
@@ -132,16 +132,16 @@
\ 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);
+by (Fast_tac 1);
qed "ordermap_mono";
(*linearity of r is crucial here*)
goalw OrderType.thy [well_ord_def, tot_ord_def]
"!!r. [| ordermap(A,r)`w : ordermap(A,r)`x; well_ord(A,r); \
\ w: A; x: A |] ==> <w,x>: r";
-by (safe_tac ZF_cs);
+by (safe_tac (!claset));
by (linear_case_tac 1);
-by (fast_tac (ZF_cs addSEs [mem_not_refl RS notE]) 1);
+by (fast_tac (!claset addSEs [mem_not_refl RS notE]) 1);
by (dtac ordermap_mono 1);
by (REPEAT_SOME assume_tac);
by (etac mem_asym 1);
@@ -154,10 +154,10 @@
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]
+by (fast_tac (!claset addSIs [ordermap_type, ordermap_surj]
addEs [linearE]
addDs [ordermap_mono]
- addss (ZF_ss addsimps [mem_not_refl])) 1);
+ addss (!simpset addsimps [mem_not_refl])) 1);
qed "ordermap_bij";
(*** Isomorphisms involving ordertype ***)
@@ -165,12 +165,12 @@
goalw OrderType.thy [ord_iso_def]
"!!r. well_ord(A,r) ==> \
\ ordermap(A,r) : ord_iso(A,r, ordertype(A,r), Memrel(ordertype(A,r)))";
-by (safe_tac ZF_cs);
+by (safe_tac (!claset));
by (rtac ordermap_bij 1);
by (assume_tac 1);
-by (fast_tac (ZF_cs addSEs [MemrelE, converse_ordermap_mono]) 2);
+by (fast_tac (!claset addSEs [MemrelE, converse_ordermap_mono]) 2);
by (rewtac well_ord_def);
-by (fast_tac (ZF_cs addSIs [MemrelI, ordermap_mono,
+by (fast_tac (!claset addSIs [MemrelI, ordermap_mono,
ordermap_type RS apply_type]) 1);
qed "ordertype_ord_iso";
@@ -180,7 +180,7 @@
by (forward_tac [well_ord_ord_iso] 1 THEN assume_tac 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]
+by (deepen_tac (!claset addIs [ord_iso_trans, ord_iso_sym]
addSEs [ordertype_ord_iso]) 0 1);
qed "ordertype_eq";
@@ -205,8 +205,8 @@
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);
+by (asm_simp_tac (!simpset addsimps [id_conv, Memrel_iff]) 1);
+by (fast_tac (!claset addEs [ltE, Ord_in_Ord, Ord_trans]) 1);
qed "le_ordertype_Memrel";
(*"Ord(i) ==> ordertype(i, Memrel(i)) = i"*)
@@ -219,6 +219,8 @@
by (resolve_tac [Ord_0 RS ordertype_Memrel] 1);
qed "ordertype_0";
+Addsimps [ordertype_0];
+
(*Ordertype of rvimage: [| f: bij(A,B); well_ord(B,s) |] ==>
ordertype(A, rvimage(A,f,s)) = ordertype(B,s) *)
bind_thm ("bij_ordertype_vimage", ord_iso_rvimage RS ordertype_eq);
@@ -231,12 +233,12 @@
\ 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 (safe_tac (!claset addSEs [predE]));
by (asm_simp_tac
- (ZF_ss addsimps [ordermap_pred_unfold, well_ord_is_wf, pred_iff]) 1);
+ (!simpset addsimps [ordermap_pred_unfold, well_ord_is_wf, pred_iff]) 1);
(*combining these two simplifications LOOPS! *)
-by (asm_simp_tac (ZF_ss addsimps [pred_pred_eq]) 1);
-by (asm_full_simp_tac (ZF_ss addsimps [pred_def]) 1);
+by (asm_simp_tac (!simpset addsimps [pred_pred_eq]) 1);
+by (asm_full_simp_tac (!simpset addsimps [pred_def]) 1);
by (rtac (refl RSN (2,RepFun_cong)) 1);
by (dtac well_ord_is_trans_on 1);
by (fast_tac (eq_cs addSEs [trans_onD]) 1);
@@ -252,9 +254,9 @@
goal OrderType.thy
"!!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,
+by (asm_simp_tac (!simpset addsimps [ordertype_unfold,
pred_subset RSN (2, well_ord_subset)]) 1);
-by (fast_tac (ZF_cs addIs [ordermap_pred_eq_ordermap, RepFun_eqI]
+by (fast_tac (!claset addIs [ordermap_pred_eq_ordermap, RepFun_eqI]
addEs [predE]) 1);
qed "ordertype_pred_subset";
@@ -275,8 +277,8 @@
\ 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,
+ (!claset addss
+ (!simpset addsimps [ordertype_def,
well_ord_is_wf RS ordermap_eq_image,
ordermap_type RS image_fun,
ordermap_pred_eq_ordermap,
@@ -299,12 +301,12 @@
goalw OrderType.thy [Ord_alt_def, Ord_def, Transset_def, well_ord_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 (asm_full_simp_tac (!simpset addsimps [Memrel_iff, pred_Memrel]) 1);
+by (safe_tac (!claset));
+by (fast_tac (!claset addSDs [equalityD1]) 1);
by (subgoal_tac "xa: i" 1);
-by (fast_tac (ZF_cs addSDs [equalityD1]) 2);
-by (fast_tac (ZF_cs addSDs [equalityD1]
+by (fast_tac (!claset addSDs [equalityD1]) 2);
+by (fast_tac (!claset addSDs [equalityD1]
addSEs [bspec RS bspec RS bspec RS mp RS mp]) 1);
qed "Ord_alt_is_Ord";
@@ -318,27 +320,27 @@
goal OrderType.thy "(lam z:A+0. case(%x.x, %y.y, z)) : bij(A+0, A)";
by (res_inst_tac [("d", "Inl")] lam_bijective 1);
by (safe_tac sum_cs);
-by (ALLGOALS (asm_simp_tac sum_ss));
+by (ALLGOALS (Asm_simp_tac));
qed "bij_sum_0";
goal OrderType.thy
"!!A r. well_ord(A,r) ==> ordertype(A+0, radd(A,r,0,s)) = ordertype(A,r)";
by (resolve_tac [bij_sum_0 RS ord_isoI RS ordertype_eq] 1);
by (assume_tac 2);
-by (fast_tac (sum_cs addss (sum_ss addsimps [radd_Inl_iff, Memrel_iff])) 1);
+by (fast_tac (sum_cs addss (!simpset addsimps [radd_Inl_iff, Memrel_iff])) 1);
qed "ordertype_sum_0_eq";
goal OrderType.thy "(lam z:0+A. case(%x.x, %y.y, z)) : bij(0+A, A)";
by (res_inst_tac [("d", "Inr")] lam_bijective 1);
by (safe_tac sum_cs);
-by (ALLGOALS (asm_simp_tac sum_ss));
+by (ALLGOALS (Asm_simp_tac));
qed "bij_0_sum";
goal OrderType.thy
"!!A r. well_ord(A,r) ==> ordertype(0+A, radd(0,s,A,r)) = ordertype(A,r)";
by (resolve_tac [bij_0_sum RS ord_isoI RS ordertype_eq] 1);
by (assume_tac 2);
-by (fast_tac (sum_cs addss (sum_ss addsimps [radd_Inr_iff, Memrel_iff])) 1);
+by (fast_tac (sum_cs addss (!simpset addsimps [radd_Inr_iff, Memrel_iff])) 1);
qed "ordertype_0_sum_eq";
(** Initial segments of radd. Statements by Grabczewski **)
@@ -352,7 +354,7 @@
by (safe_tac sum_cs);
by (ALLGOALS
(asm_full_simp_tac
- (sum_ss addsimps [radd_Inl_iff, radd_Inr_Inl_iff])));
+ (!simpset addsimps [radd_Inl_iff, radd_Inr_Inl_iff])));
qed "pred_Inl_bij";
goal OrderType.thy
@@ -361,7 +363,7 @@
\ ordertype(pred(A,a,r), r)";
by (resolve_tac [pred_Inl_bij RS ord_isoI RS ord_iso_sym RS ordertype_eq] 1);
by (REPEAT_FIRST (ares_tac [pred_subset, well_ord_subset]));
-by (asm_full_simp_tac (ZF_ss addsimps [radd_Inl_iff, pred_def]) 1);
+by (asm_full_simp_tac (!simpset addsimps [radd_Inl_iff, pred_def]) 1);
qed "ordertype_pred_Inl_eq";
goalw OrderType.thy [pred_def, id_def]
@@ -370,7 +372,7 @@
\ : 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));
+by (ALLGOALS (Asm_full_simp_tac));
qed "pred_Inr_bij";
goal OrderType.thy
@@ -378,7 +380,7 @@
\ ordertype(pred(A+B, Inr(b), radd(A,r,B,s)), radd(A,r,B,s)) = \
\ ordertype(A+pred(B,b,s), radd(A,r,pred(B,b,s),s))";
by (resolve_tac [pred_Inr_bij RS ord_isoI RS ord_iso_sym RS ordertype_eq] 1);
-by (fast_tac (sum_cs addss (radd_ss addsimps [pred_def, id_def])) 2);
+by (fast_tac (sum_cs addss (!simpset addsimps [pred_def, id_def])) 2);
by (REPEAT_FIRST (ares_tac [well_ord_radd, pred_subset, well_ord_subset]));
qed "ordertype_pred_Inr_eq";
@@ -392,15 +394,16 @@
(** 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,
+by (asm_simp_tac (!simpset addsimps [Memrel_0, ordertype_sum_0_eq,
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,
+by (asm_simp_tac (!simpset addsimps [Memrel_0, ordertype_0_sum_eq,
ordertype_Memrel, well_ord_Memrel]) 1);
qed "oadd_0_left";
+Addsimps [oadd_0, oadd_0_left];
(*** Further properties of ordinal addition. Statements by Grabczewski,
proofs by lcp. ***)
@@ -410,11 +413,11 @@
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);
+ (!simpset 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,bexI))) 1);
qed "lt_oadd1";
(*Thus also we obtain the rule i++j = k ==> i le k *)
@@ -428,8 +431,8 @@
goal OrderType.thy
"!!A B. A<=B ==> id(A) : ord_iso(A, Memrel(A), A, Memrel(B))";
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 1);
+by (asm_simp_tac (!simpset addsimps [id_conv, Memrel_iff]) 1);
+by (Fast_tac 1);
qed "id_ord_iso_Memrel";
goal OrderType.thy
@@ -449,7 +452,7 @@
by (rtac RepFun_eqI 1);
by (etac InrI 2);
by (asm_simp_tac
- (ZF_ss addsimps [ordertype_pred_Inr_eq, well_ord_Memrel,
+ (!simpset addsimps [ordertype_pred_Inr_eq, well_ord_Memrel,
lt_pred_Memrel, leI RS le_ordertype_Memrel,
ordertype_sum_Memrel]) 1);
qed "oadd_lt_mono2";
@@ -459,12 +462,12 @@
by (rtac Ord_linear_lt 1);
by (REPEAT_SOME assume_tac);
by (ALLGOALS
- (fast_tac (ZF_cs addDs [oadd_lt_mono2] addEs [lt_irrefl, lt_asym])));
+ (fast_tac (!claset addDs [oadd_lt_mono2] addEs [lt_irrefl, lt_asym])));
qed "oadd_lt_cancel2";
goal OrderType.thy
"!!i j k. [| Ord(i); Ord(j); Ord(k) |] ==> i++j < i++k <-> j<k";
-by (fast_tac (ZF_cs addSIs [oadd_lt_mono2] addSEs [oadd_lt_cancel2]) 1);
+by (fast_tac (!claset addSIs [oadd_lt_mono2] addSEs [oadd_lt_cancel2]) 1);
qed "oadd_lt_iff2";
goal OrderType.thy
@@ -472,8 +475,8 @@
by (rtac Ord_linear_lt 1);
by (REPEAT_SOME assume_tac);
by (ALLGOALS
- (fast_tac (ZF_cs addDs [oadd_lt_mono2]
- addss (ZF_ss addsimps [lt_not_refl]))));
+ (fast_tac (!claset addDs [oadd_lt_mono2]
+ addss (!simpset addsimps [lt_not_refl]))));
qed "oadd_inject";
goalw OrderType.thy [oadd_def]
@@ -481,11 +484,11 @@
(*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,
+ (!simpset addsimps [ordertype_pred_unfold, well_ord_radd,
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,
+ (!simpset addsimps [ordertype_pred_Inl_eq, well_ord_Memrel,
ltI, lt_pred_Memrel, le_ordertype_Memrel, leI,
ordertype_pred_Inr_eq,
ordertype_sum_Memrel])) 1);
@@ -511,21 +514,22 @@
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);
-by (fast_tac ZF_cs 2);
-by (fast_tac (ZF_cs addSEs [ltE]) 1);
+by (fast_tac (!claset addIs [lt_oadd1, oadd_lt_mono2]
+ addss (!simpset addsimps [Ord_mem_iff_lt, Ord_oadd])) 3);
+by (Fast_tac 2);
+by (fast_tac (!claset addSEs [ltE]) 1);
qed "oadd_unfold";
goal OrderType.thy "!!i. Ord(i) ==> i++1 = succ(i)";
-by (asm_simp_tac (ZF_ss addsimps [oadd_unfold, Ord_1, oadd_0]) 1);
+by (asm_simp_tac (!simpset addsimps [oadd_unfold, Ord_1, oadd_0]) 1);
by (fast_tac eq_cs 1);
qed "oadd_1";
goal OrderType.thy
"!!i. [| Ord(i); Ord(j) |] ==> i++succ(j) = succ(i++j)";
-by (asm_simp_tac
- (ZF_ss addsimps [oadd_1 RS sym, Ord_oadd, oadd_assoc, Ord_1]) 1);
+ (*ZF_ss prevents looping*)
+by (asm_simp_tac (ZF_ss addsimps [Ord_oadd, oadd_1 RS sym]) 1);
+by (asm_simp_tac (!simpset addsimps [oadd_1, oadd_assoc, Ord_1]) 1);
qed "oadd_succ";
@@ -542,7 +546,7 @@
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,
+by (asm_simp_tac (!simpset addsimps [Limit_is_Ord RS Ord_in_Ord,
oadd_UN RS sym, Union_eq_UN RS sym,
Limit_Union_eq]) 1);
qed "oadd_Limit";
@@ -551,25 +555,25 @@
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 (asm_simp_tac (!simpset addsimps [Ord_0_le]) 1);
+by (asm_simp_tac (!simpset addsimps [oadd_succ, succ_leI]) 1);
+by (asm_simp_tac (!simpset addsimps [oadd_Limit]) 1);
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);
+by (Fast_tac 2);
+by (asm_simp_tac (!simpset addsimps [Union_eq_UN RS sym, Limit_Union_eq,
+ 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 (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 (Asm_simp_tac 1);
+by (asm_simp_tac (!simpset addsimps [oadd_succ, succ_le_iff]) 1);
+by (asm_simp_tac (!simpset addsimps [oadd_Limit]) 1);
by (rtac le_implies_UN_le_UN 1);
-by (fast_tac ZF_cs 1);
+by (Fast_tac 1);
qed "oadd_le_mono1";
goal OrderType.thy "!!i j. [| i' le i; j'<j |] ==> i'++j' < i++j";
@@ -579,12 +583,12 @@
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);
+by (asm_simp_tac (!simpset 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,
+by (asm_simp_tac (!simpset addsimps [oadd_lt_iff2, oadd_succ RS sym,
Ord_succ]) 1);
qed "oadd_le_iff2";
@@ -597,31 +601,31 @@
"!!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 (fast_tac (!claset addSIs [case_type]) 1);
by (etac sumE 2);
-by (ALLGOALS (asm_simp_tac (sum_ss setloop split_tac [expand_if])));
+by (ALLGOALS (asm_simp_tac (!simpset 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))) = \
\ ordertype(j, Memrel(j))";
-by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1]));
+by (safe_tac (!claset addSDs [le_subset_iff RS iffD1]));
by (resolve_tac [bij_sum_Diff RS ord_isoI RS ord_iso_sym RS ordertype_eq] 1);
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);
+ (!simpset 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);
+by (asm_simp_tac (!simpset addsimps [Ord_mem_iff_lt, lt_Ord, not_lt_iff_le]) 1);
+by (fast_tac (!claset 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) = ordertype(i+(j-i), radd(i,Memrel(j),j-i,Memrel(j)))";
-by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1]));
+by (safe_tac (!claset addSDs [le_subset_iff RS iffD1]));
by (resolve_tac [sum_ord_iso_cong RS ordertype_eq] 1);
by (etac id_ord_iso_Memrel 1);
by (resolve_tac [ordertype_ord_iso RS ord_iso_sym] 1);
@@ -630,7 +634,7 @@
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,
+by (asm_simp_tac (!simpset addsimps [oadd_ordertype_Diff, ordertype_sum_Diff,
ordertype_Memrel, lt_Ord2 RS Ord_succD]) 1);
qed "oadd_odiff_inverse";
@@ -646,14 +650,14 @@
"!!i j. [| Ord(i); Ord(j) |] ==> (i++j) -- i = j";
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);
+by (asm_simp_tac (!simpset 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,
+ (!simpset addsimps [i_lt_j, k_le_i, [k_le_i, leI] MRS le_trans,
oadd_odiff_inverse]) 1);
by (REPEAT (resolve_tac (Ord_odiff ::
([i_lt_j, k_le_i] RL [lt_Ord, lt_Ord2])) 1));
@@ -674,8 +678,8 @@
\ 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));
+by (ALLGOALS (asm_full_simp_tac (!simpset addsimps [rmult_iff])));
+by (ALLGOALS (Fast_tac));
qed "pred_Pair_eq";
goal OrderType.thy
@@ -683,11 +687,11 @@
\ 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), \
\ radd(A*B, rmult(A,r,B,s), B, s))";
-by (asm_simp_tac (ZF_ss addsimps [pred_Pair_eq]) 1);
+by (asm_simp_tac (!simpset addsimps [pred_Pair_eq]) 1);
by (resolve_tac [ordertype_eq RS sym] 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);
+by (fast_tac (!claset addSEs [predE]) 1);
qed "ordertype_pred_Pair_eq";
goalw OrderType.thy [oadd_def, omult_def]
@@ -695,7 +699,7 @@
\ 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,
+by (asm_simp_tac (!simpset addsimps [ordertype_pred_Pair_eq, lt_pred_Memrel,
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);
@@ -705,35 +709,36 @@
by (REPEAT_FIRST (ares_tac [well_ord_rmult, well_ord_radd, well_ord_Memrel,
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])));
+ (asm_simp_tac (!simpset addsimps [rmult_iff, id_conv, Memrel_iff])));
+by (safe_tac (!claset));
+by (ALLGOALS (fast_tac (!claset 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,
+by (asm_full_simp_tac (!simpset addsimps [ordertype_pred_unfold,
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,
+by (step_tac (!claset addSEs [ltE]) 1);
+by (asm_simp_tac (!simpset addsimps [ordertype_pred_Pair_lemma, ltI,
symmetric omult_def]) 1);
-by (fast_tac (ZF_cs addIs [ltI]) 1);
+by (fast_tac (!claset addIs [ltI]) 1);
qed "lt_omult";
goalw OrderType.thy [omult_def]
"!!i j. [| j'<j; i'<i |] ==> j**i' ++ j' < j**i";
by (rtac ltI 1);
by (asm_simp_tac
- (ZF_ss addsimps [Ord_ordertype, well_ord_rmult, well_ord_Memrel,
- lt_Ord2]) 2);
+ (!simpset addsimps [Ord_ordertype, well_ord_rmult, well_ord_Memrel,
+ lt_Ord2]) 2);
by (asm_simp_tac
- (ZF_ss addsimps [ordertype_pred_unfold,
+ (!simpset addsimps [ordertype_pred_unfold,
well_ord_rmult, well_ord_Memrel, lt_Ord2]) 1);
-by (rtac RepFun_eqI 1);
-by (fast_tac (ZF_cs addSEs [ltE]) 2);
+by (rtac bexI 1);
+by (fast_tac (!claset addSEs [ltE]) 2);
by (asm_simp_tac
- (ZF_ss addsimps [ordertype_pred_Pair_lemma, ltI, symmetric omult_def]) 1);
+ (!simpset addsimps [ordertype_pred_Pair_lemma, ltI,
+ symmetric omult_def]) 1);
qed "omult_oadd_lt";
goal OrderType.thy
@@ -742,8 +747,8 @@
by (resolve_tac [lt_omult RS exE] 1);
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);
+by (fast_tac (!claset addSEs [ltE]) 1);
+by (fast_tac (!claset addIs [omult_oadd_lt RS ltD, ltI]) 1);
qed "omult_unfold";
(*** Basic laws for ordinal multiplication ***)
@@ -751,13 +756,15 @@
(** Ordinal multiplication by zero **)
goalw OrderType.thy [omult_def] "i**0 = 0";
-by (asm_simp_tac (ZF_ss addsimps [ordertype_0]) 1);
+by (Asm_simp_tac 1);
qed "omult_0";
goalw OrderType.thy [omult_def] "0**i = 0";
-by (asm_simp_tac (ZF_ss addsimps [ordertype_0]) 1);
+by (Asm_simp_tac 1);
qed "omult_0_left";
+Addsimps [omult_0, omult_0_left];
+
(** Ordinal multiplication by 1 **)
goalw OrderType.thy [omult_def] "!!i. Ord(i) ==> i**1 = i";
@@ -765,7 +772,7 @@
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]));
-by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [rmult_iff, Memrel_iff])));
+by (ALLGOALS (asm_simp_tac (!simpset addsimps [rmult_iff, Memrel_iff])));
qed "omult_1";
goalw OrderType.thy [omult_def] "!!i. Ord(i) ==> 1**i = i";
@@ -773,9 +780,11 @@
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]));
-by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [rmult_iff, Memrel_iff])));
+by (ALLGOALS (asm_simp_tac (!simpset addsimps [rmult_iff, Memrel_iff])));
qed "omult_1_left";
+Addsimps [omult_1, omult_1_left];
+
(** Distributive law for ordinal multiplication and addition **)
goalw OrderType.thy [omult_def, oadd_def]
@@ -793,8 +802,10 @@
qed "oadd_omult_distrib";
goal OrderType.thy "!!i. [| Ord(i); Ord(j) |] ==> i**succ(j) = (i**j)++i";
+ (*ZF_ss prevents looping*)
+by (asm_simp_tac (ZF_ss addsimps [oadd_1 RS sym]) 1);
by (asm_simp_tac
- (ZF_ss addsimps [oadd_1 RS sym, omult_1, oadd_omult_distrib, Ord_1]) 1);
+ (!simpset addsimps [omult_1, oadd_omult_distrib, Ord_1]) 1);
qed "omult_succ";
(** Associative law **)
@@ -818,14 +829,14 @@
val prems = goal OrderType.thy
"[| Ord(i); !!x. x:A ==> Ord(j(x)) |] ==> \
\ i ** (UN x:A. j(x)) = (UN x:A. i**j(x))";
-by (asm_simp_tac (ZF_ss addsimps (prems@[Ord_UN, omult_unfold])) 1);
+by (asm_simp_tac (!simpset addsimps (prems@[Ord_UN, omult_unfold])) 1);
by (fast_tac eq_cs 1);
qed "omult_UN";
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,
+ (!simpset addsimps [Limit_is_Ord RS Ord_in_Ord, omult_UN RS sym,
Union_eq_UN RS sym, Limit_Union_eq]) 1);
qed "omult_Limit";
@@ -834,10 +845,10 @@
(*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 (etac UN_I 1));
-by (asm_simp_tac (ZF_ss addsimps [omult_0, oadd_0_left]) 1);
+by (safe_tac (!claset addSEs [ltE] addSIs [ltI, Ord_omult]));
+by (asm_simp_tac (!simpset addsimps [omult_unfold]) 1);
+by (REPEAT_FIRST (ares_tac [bexI]));
+by (Asm_simp_tac 1);
qed "lt_omult1";
goal OrderType.thy "!!i j. [| Ord(i); 0<j |] ==> i le i**j";
@@ -849,26 +860,26 @@
by (forward_tac [lt_Ord] 1);
by (forward_tac [le_Ord2] 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 (asm_simp_tac (!simpset addsimps [le_refl, Ord_0]) 1);
+by (asm_simp_tac (!simpset addsimps [omult_succ, oadd_le_mono]) 1);
+by (asm_simp_tac (!simpset addsimps [omult_Limit]) 1);
by (rtac le_implies_UN_le_UN 1);
-by (fast_tac ZF_cs 1);
+by (Fast_tac 1);
qed "omult_le_mono1";
goal OrderType.thy "!!i j k. [| k<j; 0<i |] ==> i**k < i**j";
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 (etac UN_I 1));
-by (asm_simp_tac (ZF_ss addsimps [omult_0, oadd_0, Ord_omult]) 1);
+by (asm_simp_tac (!simpset addsimps [omult_unfold, lt_Ord2]) 1);
+by (safe_tac (!claset addSEs [ltE] addSIs [Ord_omult]));
+by (REPEAT_FIRST (ares_tac [bexI]));
+by (asm_simp_tac (!simpset addsimps [Ord_omult]) 1);
qed "omult_lt_mono2";
goal OrderType.thy "!!i j k. [| k le j; Ord(i) |] ==> i**k le i**j";
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);
+by (safe_tac (!claset addSEs [ltE, make_elim Ord_succD] addSIs [Ord_omult]));
+by (asm_full_simp_tac (!simpset addsimps [omult_unfold]) 1);
+by (deepen_tac (!claset addEs [Ord_trans]) 0 1);
qed "omult_le_mono2";
goal OrderType.thy "!!i j. [| i' le i; j' le j |] ==> i'**j' le i**j";
@@ -887,18 +898,18 @@
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 (asm_simp_tac (!simpset addsimps [omult_0, Ord_0 RS le_refl]) 1);
+by (asm_simp_tac (!simpset addsimps [omult_succ, succ_le_iff]) 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 (asm_simp_tac (!simpset addsimps [omult_Limit]) 1);
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);
+by (Fast_tac 2);
+by (asm_simp_tac (!simpset addsimps [Union_eq_UN RS sym, Limit_Union_eq,
+ Limit_is_Ord RS le_refl]) 1);
qed "omult_le_self2";
@@ -908,8 +919,8 @@
by (rtac Ord_linear_lt 1);
by (REPEAT_SOME assume_tac);
by (ALLGOALS
- (fast_tac (ZF_cs addDs [omult_lt_mono2]
- addss (ZF_ss addsimps [lt_not_refl]))));
+ (best_tac (!claset addDs [omult_lt_mono2]
+ addss (!simpset addsimps [lt_not_refl]))));
qed "omult_inject";