--- a/src/ZF/OrderType.ML Mon Jan 29 14:16:13 1996 +0100
+++ b/src/ZF/OrderType.ML Tue Jan 30 13:42:57 1996 +0100
@@ -1,6 +1,6 @@
-(* 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
@@ -19,7 +19,7 @@
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";
@@ -54,7 +54,7 @@
(*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));
@@ -88,7 +88,7 @@
"!!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!*)
@@ -98,8 +98,8 @@
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)";
@@ -128,7 +128,7 @@
(*** 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);
@@ -149,14 +149,14 @@
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";
@@ -171,27 +171,27 @@
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";
@@ -200,9 +200,9 @@
(*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);
@@ -215,7 +215,7 @@
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";
@@ -227,8 +227,8 @@
(*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]));
@@ -255,7 +255,7 @@
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
@@ -264,12 +264,12 @@
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}";
@@ -277,10 +277,10 @@
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";
@@ -289,15 +289,15 @@
(*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);
@@ -346,7 +346,7 @@
(*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);
@@ -366,7 +366,7 @@
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);
@@ -393,12 +393,12 @@
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";
@@ -406,20 +406,20 @@
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";
@@ -433,25 +433,25 @@
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
@@ -482,13 +482,13 @@
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";
@@ -498,11 +498,11 @@
"!!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";
@@ -512,7 +512,7 @@
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";
@@ -535,7 +535,7 @@
"[| 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";
@@ -543,8 +543,8 @@
"!!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 **)
@@ -554,28 +554,28 @@
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";
@@ -585,7 +585,7 @@
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";
@@ -598,17 +598,17 @@
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);
@@ -619,32 +619,32 @@
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";
@@ -654,9 +654,9 @@
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";
@@ -671,7 +671,7 @@
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])));
@@ -681,11 +681,11 @@
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";
@@ -696,14 +696,14 @@
\ 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);
@@ -714,23 +714,23 @@
"!!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);
@@ -740,7 +740,7 @@
"!!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);
@@ -764,7 +764,7 @@
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";
@@ -772,7 +772,7 @@
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";
@@ -782,14 +782,14 @@
"!!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";
@@ -803,12 +803,12 @@
"!!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";
@@ -826,7 +826,7 @@
"!!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";
@@ -836,52 +836,52 @@
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";
@@ -889,16 +889,16 @@
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";