--- a/src/ZF/CardinalArith.thy Fri May 17 16:47:24 2002 +0200
+++ b/src/ZF/CardinalArith.thy Fri May 17 16:48:11 2002 +0200
@@ -52,16 +52,16 @@
(*** The following really belong in Order ***)
lemma subset_ord_iso_Memrel:
- "\<lbrakk>f: ord_iso(A,Memrel(B),C,r); A<=B\<rbrakk> \<Longrightarrow> f: ord_iso(A,Memrel(A),C,r)"
+ "[| f: ord_iso(A,Memrel(B),C,r); A<=B |] ==> f: ord_iso(A,Memrel(A),C,r)"
apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN fun_is_rel])
apply (frule ord_iso_trans [OF id_ord_iso_Memrel], assumption)
apply (simp add: right_comp_id)
done
lemma restrict_ord_iso:
- "\<lbrakk>f \<in> ord_iso(i, Memrel(i), Order.pred(A,a,r), r); a \<in> A; j < i;
- trans[A](r)\<rbrakk>
- \<Longrightarrow> restrict(f,j) \<in> ord_iso(j, Memrel(j), Order.pred(A,f`j,r), r)"
+ "[| f \<in> ord_iso(i, Memrel(i), Order.pred(A,a,r), r); a \<in> A; j < i;
+ trans[A](r) |]
+ ==> restrict(f,j) \<in> ord_iso(j, Memrel(j), Order.pred(A,f`j,r), r)"
apply (frule ltD)
apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption)
apply (frule ord_iso_restrict_pred, assumption)
@@ -70,21 +70,21 @@
done
lemma restrict_ord_iso2:
- "\<lbrakk>f \<in> ord_iso(Order.pred(A,a,r), r, i, Memrel(i)); a \<in> A;
- j < i; trans[A](r)\<rbrakk>
- \<Longrightarrow> converse(restrict(converse(f), j))
+ "[| f \<in> ord_iso(Order.pred(A,a,r), r, i, Memrel(i)); a \<in> A;
+ j < i; trans[A](r) |]
+ ==> converse(restrict(converse(f), j))
\<in> ord_iso(Order.pred(A, converse(f)`j, r), r, j, Memrel(j))"
by (blast intro: restrict_ord_iso ord_iso_sym ltI)
(*** The following really belong in OrderType ***)
-lemma oadd_eq_0_iff: "\<lbrakk>Ord(i); Ord(j)\<rbrakk> \<Longrightarrow> (i ++ j) = 0 <-> i=0 & j=0"
+lemma oadd_eq_0_iff: "[| Ord(i); Ord(j) |] ==> (i ++ j) = 0 <-> i=0 & j=0"
apply (erule trans_induct3 [of j])
apply (simp_all add: oadd_Limit)
apply (simp add: Union_empty_iff Limit_def lt_def, blast)
done
-lemma oadd_eq_lt_iff: "\<lbrakk>Ord(i); Ord(j)\<rbrakk> \<Longrightarrow> 0 < (i ++ j) <-> 0<i | 0<j"
+lemma oadd_eq_lt_iff: "[| Ord(i); Ord(j) |] ==> 0 < (i ++ j) <-> 0<i | 0<j"
by (simp add: Ord_0_lt_iff [symmetric] oadd_eq_0_iff)
lemma oadd_lt_self: "[| Ord(i); 0<j |] ==> i < i++j"
@@ -94,7 +94,7 @@
apply (blast intro: succ_leI oadd_le_mono)
done
-lemma oadd_LimitI: "\<lbrakk>Ord(i); Limit(j)\<rbrakk> \<Longrightarrow> Limit(i ++ j)"
+lemma oadd_LimitI: "[| Ord(i); Limit(j) |] ==> Limit(i ++ j)"
apply (simp add: oadd_Limit)
apply (frule Limit_has_1 [THEN ltD])
apply (rule increasing_LimitI)