author lcp Thu, 12 Jan 1995 03:00:38 +0100 changeset 849 013a16d3addb parent 848 b1dc15d86081 child 850 a744f9749885
Proved equivalence of Ord and Ord_alt. Proved ordertype_eq_imp_ord_iso, le_well_ord_Memrel, le_ordertype_Memrel, lt_oadd1, oadd_le_self, bij_0_sum, oadd_0, oadd_assoc, id_ord_iso_Memrel, ordertype_0. Now well_ord_Memrel follows from le_well_ord_Memrel and ordertype_Memrel follows from le_ordertype_Memrel. Proved simpler versions of Krzysztof's theorems Ord_oadd, ordertype_pred_subset, ordertype_pred_lt, ordertype_pred_unfold, bij_sum_0, bij_sum_succ, ordertype_sum_Memrel, lt_oadd_disj, oadd_inject. Deleted ordertype_subset: subsumed by ordertype_pred_unfold. Proved ordinal multiplication theorems Ord_omult, lt_omult, omult_oadd_lt, omult_unfold, omult_0, omult_0_left, omult_1, omult_1_left, oadd_omult_distrib, omult_succ, omult_assoc, omult_UN, omult_Limit, lt_omult1, omult_le_self, omult_le_mono1, omult_lt_mono2, omult_le_mono2, omult_le_mono, omult_lt_mono, omult_le_self2, omult_inject.
 src/ZF/OrderType.ML file | annotate | diff | comparison | revisions
```--- a/src/ZF/OrderType.ML	Wed Jan 11 18:47:03 1995 +0100
+++ b/src/ZF/OrderType.ML	Thu Jan 12 03:00:38 1995 +0100
@@ -3,42 +3,49 @@
Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright   1994  University of Cambridge

-Order types in Zermelo-Fraenkel Set Theory
+Order types and ordinal arithmetic in Zermelo-Fraenkel Set Theory
+
+Ordinal arithmetic is traditionally defined in terms of order types, as here.
+But a definition by transfinite recursion would be much simpler!
*)

open OrderType;

-(*** Proofs needing the combination of Ordinal.thy and Order.thy ***)
+(**** Proofs needing the combination of Ordinal.thy and Order.thy ****)

-goal OrderType.thy "!!i. Ord(i) ==> well_ord(i, Memrel(i))";
+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 (asm_simp_tac (ZF_ss addsimps [linear_def, Memrel_iff]) 1);
-by (REPEAT (resolve_tac [ballI, Ord_linear] 1));;
-by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));;
-qed "well_ord_Memrel";
+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);
+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))"*)
+bind_thm ("well_ord_Memrel", le_refl RS le_well_ord_Memrel);

(*Kunen's Theorem 7.3 (i), page 16;  see also Ordinal/Ord_in_Ord
The smaller ordinal is an initial segment of the larger *)
goalw OrderType.thy [pred_def, lt_def]
-    "!!i j. j<i ==> j = pred(i, j, Memrel(i))";
+    "!!i j. j<i ==> pred(i, j, Memrel(i)) = j";
by (asm_simp_tac (ZF_ss addsimps [Memrel_iff]) 1);
by (fast_tac (eq_cs addEs [Ord_trans]) 1);
-qed "lt_eq_pred";
+qed "lt_pred_Memrel";

goalw OrderType.thy [pred_def,Memrel_def]
-      "!!A x. x:A ==> pred(A, x, Memrel(A)) = {b:A. b:x}";
+      "!!A x. x:A ==> pred(A, x, Memrel(A)) = A Int x";
by (fast_tac eq_cs 1);
qed "pred_Memrel";

goal OrderType.thy
"!!i. [| j<i;  f: ord_iso(i,Memrel(i),j,Memrel(j)) |] ==> R";
-by (forward_tac [lt_eq_pred] 1);
+by (forward_tac [lt_pred_Memrel] 1);
by (etac ltE 1);
by (rtac (well_ord_Memrel RS well_ord_iso_predE) 1 THEN
assume_tac 3 THEN assume_tac 1);
-by (etac subst 1);
by (asm_full_simp_tac (ZF_ss addsimps [ord_iso_def]) 1);
(*Combining the two simplifications causes looping*)
by (asm_simp_tac (ZF_ss addsimps [Memrel_iff]) 1);
@@ -53,7 +60,8 @@
by (REPEAT (eresolve_tac [asm_rl, ord_iso_sym, Ord_iso_implies_eq_lemma] 1));
qed "Ord_iso_implies_eq";

-(*** Ordermap and ordertype ***)
+
+(**** Ordermap and ordertype ****)

goalw OrderType.thy [ordermap_def,ordertype_def]
"ordermap(A,r) : A -> ordertype(A,r)";
@@ -62,7 +70,7 @@
by (REPEAT (assume_tac 1));
qed "ordermap_type";

-(** Unfolding of ordermap **)
+(*** Unfolding of ordermap ***)

(*Useful for cardinality reasoning; see CardinalArith.ML*)
goalw OrderType.thy [ordermap_def, pred_def]
@@ -86,7 +94,7 @@
(*pred-unfolded version.  NOT suitable for rewriting -- loops!*)
val ordermap_unfold = rewrite_rule [pred_def] ordermap_pred_unfold;

-(** Showing that ordermap, ordertype yield ordinals **)
+(*** Showing that ordermap, ordertype yield ordinals ***)

fun ordermap_elim_tac i =
EVERY [etac (ordermap_unfold RS equalityD1 RS subsetD RS RepFunE) i,
@@ -117,7 +125,7 @@
by (fast_tac ZF_cs 1);
qed "Ord_ordertype";

-(** ordermap preserves the orderings in both directions **)
+(*** ordermap preserves the orderings in both directions ***)

goal OrderType.thy
"!!r. [| <w,x>: r;  wf[A](r);  w: A; x: A |] ==>	\
@@ -156,7 +164,7 @@
by (ALLGOALS (asm_full_simp_tac (ZF_ss addsimps [mem_not_refl])));
qed "ordermap_bij";

-(** Isomorphisms involving ordertype **)
+(*** Isomorphisms involving ordertype ***)

goalw OrderType.thy [ord_iso_def]
"!!r. well_ord(A,r) ==> \
@@ -180,24 +188,47 @@
addSEs [ordertype_ord_iso]) 0 1);
qed "ordertype_eq";

-
-(** Unfolding of ordertype **)
+goal OrderType.thy
+    "!!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 (resolve_tac [ordertype_ord_iso RS ord_iso_trans] 1);
+by (assume_tac 1);
+by (eresolve_tac [ssubst] 1);
+by (eresolve_tac [ordertype_ord_iso RS ord_iso_sym] 1);
+qed "ordertype_eq_imp_ord_iso";

-goalw OrderType.thy [ordertype_def]
-    "ordertype(A,r) = {ordermap(A,r)`y . y : A}";
-by (rtac ([ordermap_type, subset_refl] MRS image_fun) 1);
-qed "ordertype_unfold";
+(*** Basic equalities for ordertype ***)

(*Ordertype of Memrel*)
-goal OrderType.thy "!!i. Ord(i) ==> ordertype(i,Memrel(i)) = i";
+goal OrderType.thy "!!i. j le i ==> ordertype(j,Memrel(i)) = j";
by (resolve_tac [Ord_iso_implies_eq RS sym] 1);
-by (resolve_tac [ordertype_ord_iso] 3);
-by (REPEAT (ares_tac [well_ord_Memrel, Ord_ordertype] 1));
-qed "ordertype_Memrel";
+by (eresolve_tac [ltE] 1);
+by (REPEAT (ares_tac [le_well_ord_Memrel, Ord_ordertype] 1));
+by (resolve_tac [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";
+
+(*"Ord(i) ==> ordertype(i, Memrel(i)) = i"*)
+bind_thm ("ordertype_Memrel", le_refl RS le_ordertype_Memrel);

-(*Ordertype of rvimage*)
+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 (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) *)
bind_thm ("bij_ordertype_vimage", ord_iso_rvimage RS ordertype_eq);

+(*** 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) |] ==>	\
@@ -215,15 +246,653 @@
by (fast_tac (eq_cs addSEs [trans_onD]) 1);
qed "ordermap_pred_eq_ordermap";

-(*Lemma for proving there exist ever larger cardinals*)
+goalw OrderType.thy [ordertype_def]
+    "ordertype(A,r) = {ordermap(A,r)`y . y : A}";
+by (rtac ([ordermap_type, subset_refl] MRS image_fun) 1);
+qed "ordertype_unfold";
+
+(** Theorems by Krzysztof Grabczewski; proofs simplified by lcp **)
+
+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,
+                  pred_subset RSN (2, well_ord_subset)]) 1);
+by (fast_tac (ZF_cs addIs [ordermap_pred_eq_ordermap, RepFun_eqI]
+	            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 (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) *)
+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 (asm_full_simp_tac
+    (ZF_ss addsimps [ordertype_def,
+		     ordermap_bij RS bij_is_fun RS image_fun]) 1);
+by (eresolve_tac [RepFunE] 1);
+by (asm_full_simp_tac
+    (ZF_ss addsimps [well_ord_is_wf, ordermap_eq_image,
+		     ordermap_type RS image_fun,
+		     ordermap_pred_eq_ordermap,
+		     pred_subset, subset_refl]) 1);
+by (eresolve_tac [RepFunI] 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 (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]
+    "!!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);
+by (fast_tac (ZF_cs addSDs [equalityD1]) 2);
+by (fast_tac (ZF_cs addSDs [equalityD1]
+                    addSEs [bspec RS bspec RS bspec RS mp RS mp]) 1);
+qed "Ord_alt_is_Ord";
+
+
+(**** Ordinal Addition ****)
+
+(*** Order Type calculations for radd ***)
+
+(** Addition with 0 **)
+
+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));
+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 (asm_simp_tac ZF_ss 1);
+by (REPEAT_FIRST (eresolve_tac [sumE, emptyE]));
+by (asm_simp_tac (sum_ss 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));
+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 (asm_simp_tac ZF_ss 1);
+by (REPEAT_FIRST (eresolve_tac [sumE, emptyE]));
+by (asm_simp_tac (sum_ss addsimps [radd_Inr_iff, Memrel_iff]) 1);
+qed "ordertype_0_sum_eq";
+
+(** 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))	\
+\        : 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
+qed "pred_Inl_bij";
+
+goal OrderType.thy
+ "!!A B. [| a:A;  well_ord(A,r) |] ==>  \
+\        ordertype(pred(A+B, Inl(a), radd(A,r,B,s)), radd(A,r,B,s)) = \
+\        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);
+qed "ordertype_pred_Inl_eq";
+
+goalw OrderType.thy [pred_def, id_def]
+ "!!A B. b:B ==>  \
+\        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";
+
+goal OrderType.thy
+ "!!A B. [| b:B;  well_ord(A,r);  well_ord(B,s) |] ==>  \
+\        ordertype(pred(A+B, Inr(b), radd(A,r,B,s)), radd(A,r,B,s)) = \
+by (resolve_tac [pred_Inr_bij RS ord_isoI RS ord_iso_sym RS ordertype_eq] 1);
+by (REPEAT_FIRST (ares_tac [well_ord_radd, pred_subset, well_ord_subset]));
+by (asm_full_simp_tac (ZF_ss addsimps [pred_def, id_def]) 1);
+by (REPEAT_FIRST (eresolve_tac [sumE]));
+by (ALLGOALS (asm_simp_tac radd_ss));
+qed "ordertype_pred_Inr_eq";
+
+(*** Basic laws for ordinal addition ***)
+
+    "!!i j. [| Ord(i);  Ord(j) |] ==> Ord(i++j)";
+by (REPEAT (ares_tac [Ord_ordertype, well_ord_radd, well_ord_Memrel] 1));
+
+(** 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);
+
+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);
+
+
+(*** 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 (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]) 1);
+by (resolve_tac [RepFun_eqI] 1);
+by (eresolve_tac [InlI] 2);
+by (asm_simp_tac
+    (ZF_ss addsimps [ordertype_pred_Inl_eq, well_ord_Memrel,
+		     lt_pred_Memrel, leI RS le_ordertype_Memrel]) 1);
+
+goal OrderType.thy "!!i j. [| Ord(i);  Ord(j) |] ==> i le i++j";
+by (resolve_tac [all_lt_imp_le] 1);
+
+(** A couple of strange but necessary results! **)
+
+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);
+qed "id_ord_iso_Memrel";
+
+goal OrderType.thy
+    "!!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 (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 (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 (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);
+
+goal OrderType.thy "!!i j. [| i++j = i++k; Ord(i); Ord(j); Ord(k) |] ==> j=k";
+by (rtac Ord_linear_lt 1);
+by (REPEAT_SOME assume_tac);
+by (ALLGOALS
+    (dresolve_tac [oadd_lt_mono2] THEN' assume_tac THEN'
+     asm_full_simp_tac (ZF_ss addsimps [lt_not_refl])));
+
+    "!!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
+		     well_ord_Memrel]) 1);
+by (eresolve_tac [ltD RS RepFunE] 1);
+by (eresolve_tac [sumE] 1);
+by (asm_simp_tac
+    (ZF_ss addsimps [ordertype_pred_Inl_eq, well_ord_Memrel,
+		     ltI, lt_pred_Memrel, le_ordertype_Memrel, leI]) 1);
+by (asm_simp_tac
+    (ZF_ss addsimps [ordertype_pred_Inr_eq, well_ord_Memrel,
+		     ltI, lt_pred_Memrel, ordertype_sum_Memrel]) 1);
+by (fast_tac ZF_cs 1);
+
+
+(*** Ordinal addition with successor -- via associativity! ***)
+
+    "!!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);
+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);
+by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Ord_ordertype] 1));
+
+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 addSEs [ltE]) 1);
+by (fast_tac ZF_cs 1);
+by (safe_tac ZF_cs);
+by (ALLGOALS
+
+goal OrderType.thy "!!i. Ord(i) ==> i++1 = succ(i)";
+by (fast_tac eq_cs 1);
+
goal OrderType.thy
-    "!!r. [| well_ord(A,r);  i: ordertype(A,r) |] ==>	\
-\	  EX B. B<=A & i = ordertype(B,r)";
-by (dresolve_tac [ordertype_unfold RS equalityD1 RS subsetD] 1);
-by (etac RepFunE 1);
-by (res_inst_tac [("x", "pred(A,y,r)")] exI 1);
+    "!!i. [| Ord(i);  Ord(j) |] ==> i++succ(j) = succ(i++j)";
+by (asm_simp_tac
+
+
+(** Ordinal addition with limit ordinals **)
+
+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])
+                     addSEs [ltE, ltI RS lt_oadd_disj RS disjE]) 1);
+
+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);
+
+(** 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 (resolve_tac [le_trans] 1);
+by (resolve_tac [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);
+
+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 (asm_simp_tac (ZF_ss addsimps [oadd_succ, succ_le_iff]) 1);
+by (resolve_tac [le_implies_UN_le_UN] 1);
+by (fast_tac ZF_cs 1);
+
+goal OrderType.thy "!!i j. [| i' le i;  j'<j |] ==> i'++j' < i++j";
+by (resolve_tac [lt_trans1] 1);
+by (REPEAT (eresolve_tac [asm_rl, oadd_le_mono1, oadd_lt_mono2, ltE,
+			  Ord_succD] 1));
+
+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);
+
+
+(** 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 (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))) = 	\
+\           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 (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";
+
+    "!!i j. i le j ==> 	\
+\           i ++ ordertype(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 (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));
+
+goal OrderType.thy
+    "!!i j. i le j ==> i ++ ordertype(j-i, Memrel(j)) = j";
+				  ordertype_Memrel, lt_Ord2 RS Ord_succD]) 1);
+
+(*By oadd_inject, the difference between i and j is unique.*)
+
+
+(**** Ordinal Multiplication ****)
+
+goalw OrderType.thy [omult_def]
+    "!!i j. [| Ord(i);  Ord(j) |] ==> Ord(i**j)";
+by (REPEAT (ares_tac [Ord_ordertype, well_ord_rmult, well_ord_Memrel] 1));
+qed "Ord_omult";
+
+(*** 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,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), 		\
+\                 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 (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);
+by (resolve_tac [ordertype_ord_iso RS sum_ord_iso_cong RS ordertype_eq] 2);
+by (resolve_tac [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]));
+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);
+by (step_tac (ZF_cs addSEs [ltE]) 1);
+by (asm_simp_tac (ZF_ss addsimps [ordertype_pred_Pair_lemma, ltI,
+				  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 (asm_full_simp_tac
+    (ZF_ss addsimps [ordertype_pred_unfold,
+		     well_ord_rmult, well_ord_Memrel, lt_Ord2]) 1);
+by (resolve_tac [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);
by (asm_simp_tac
-    (ZF_ss addsimps [pred_subset, well_ord_is_wf RS ordermap_pred_unfold,
-		     ordertype_unfold, ordermap_pred_eq_ordermap]) 1);
-qed "ordertype_subset";
+    (ZF_ss addsimps [Ord_ordertype, well_ord_rmult, well_ord_Memrel,
+		     lt_Ord2]) 1);
+
+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 (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";
+
+(*** Basic laws for ordinal multiplication ***)
+
+(** Ordinal multiplication by zero **)
+
+goalw OrderType.thy [omult_def] "i**0 = 0";
+by (asm_simp_tac (ZF_ss addsimps [ordertype_0]) 1);
+qed "omult_0";
+
+goalw OrderType.thy [omult_def] "0**i = 0";
+by (asm_simp_tac (ZF_ss addsimps [ordertype_0]) 1);
+qed "omult_0_left";
+
+(** Ordinal multiplication by 1 **)
+
+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]));
+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]));
+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);
+by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd, well_ord_Memrel,
+		      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));
+
+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);
+qed "omult_succ";
+
+(** Associative law **)
+
+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);
+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);
+by (rtac ([ordertype_ord_iso, ord_iso_refl] MRS prod_ord_iso_cong RS
+	  ordertype_eq) 2);
+by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Ord_ordertype] 1));
+qed "omult_assoc";
+
+
+(** Ordinal multiplication with limit ordinals **)
+
+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 (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,
+		     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 (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 (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 (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 (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 (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 (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 (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 (safe_tac ZF_cs);
+by (eresolve_tac [UN_I] 1);
+by (deepen_tac (ZF_cs 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";
+by (resolve_tac [le_trans] 1);
+by (REPEAT (eresolve_tac [asm_rl, omult_le_mono1, omult_le_mono2, ltE,
+			  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 (REPEAT (eresolve_tac [asm_rl, omult_le_mono1, omult_lt_mono2, ltE,
+			  Ord_succD] 1));
+qed "omult_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);
+
+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 (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 (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);
+qed "omult_le_self2";
+
+
+(** Further properties of ordinal multiplication **)
+
+goal OrderType.thy "!!i j. [| i**j = i**k;  0<i;  Ord(j);  Ord(k) |] ==> j=k";
+by (rtac Ord_linear_lt 1);
+by (REPEAT_SOME assume_tac);
+by (ALLGOALS
+    (dresolve_tac [omult_lt_mono2] THEN' assume_tac THEN'
+     asm_full_simp_tac (ZF_ss addsimps [lt_not_refl])));
+qed "omult_inject";
+
+```