src/ZF/CardinalArith.thy
changeset 13118 336b0bcbd27c
parent 12820 02e2ff3e4d37
child 13161 a40db0418145
     1.1 --- a/src/ZF/CardinalArith.thy	Wed May 08 09:14:56 2002 +0200
     1.2 +++ b/src/ZF/CardinalArith.thy	Wed May 08 10:12:57 2002 +0200
     1.3 @@ -40,6 +40,42 @@
     1.4    "op |*|"     :: "[i,i] => i"          (infixl "\<otimes>" 70)
     1.5  
     1.6  
     1.7 +(*** The following really belong early in the development ***)
     1.8 +
     1.9 +lemma relation_converse_converse [simp]:
    1.10 +     "relation(r) ==> converse(converse(r)) = r"
    1.11 +by (simp add: relation_def, blast) 
    1.12 +
    1.13 +lemma relation_restrict [simp]:  "relation(restrict(r,A))"
    1.14 +by (simp add: restrict_def relation_def, blast) 
    1.15 +
    1.16 +(*** The following really belong in Order ***)
    1.17 +
    1.18 +lemma subset_ord_iso_Memrel:
    1.19 +     "\<lbrakk>f: ord_iso(A,Memrel(B),C,r); A<=B\<rbrakk> \<Longrightarrow> f: ord_iso(A,Memrel(A),C,r)"
    1.20 +apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN fun_is_rel]) 
    1.21 +apply (frule ord_iso_trans [OF id_ord_iso_Memrel], assumption) 
    1.22 +apply (simp add: right_comp_id) 
    1.23 +done
    1.24 +
    1.25 +lemma restrict_ord_iso:
    1.26 +     "\<lbrakk>f \<in> ord_iso(i, Memrel(i), Order.pred(A,a,r), r);  a \<in> A; j < i; 
    1.27 +       trans[A](r)\<rbrakk>
    1.28 +      \<Longrightarrow> restrict(f,j) \<in> ord_iso(j, Memrel(j), Order.pred(A,f`j,r), r)"
    1.29 +apply (frule ltD) 
    1.30 +apply (frule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption) 
    1.31 +apply (frule ord_iso_restrict_pred, assumption) 
    1.32 +apply (simp add: pred_iff trans_pred_pred_eq lt_pred_Memrel)
    1.33 +apply (blast intro!: subset_ord_iso_Memrel le_imp_subset [OF leI]) 
    1.34 +done
    1.35 +
    1.36 +lemma restrict_ord_iso2:
    1.37 +     "\<lbrakk>f \<in> ord_iso(Order.pred(A,a,r), r, i, Memrel(i));  a \<in> A; 
    1.38 +       j < i; trans[A](r)\<rbrakk>
    1.39 +      \<Longrightarrow> converse(restrict(converse(f), j)) 
    1.40 +          \<in> ord_iso(Order.pred(A, converse(f)`j, r), r, j, Memrel(j))"
    1.41 +by (blast intro: restrict_ord_iso ord_iso_sym ltI)
    1.42 +
    1.43  (*** The following really belong in OrderType ***)
    1.44  
    1.45  lemma oadd_eq_0_iff: "\<lbrakk>Ord(i); Ord(j)\<rbrakk> \<Longrightarrow> (i ++ j) = 0 <-> i=0 & j=0"