src/ZF/OrdQuant.thy
 changeset 13172 03a5afa7b888 parent 13170 9e23faed6c97 child 13174 85d3c0981a16
```     1.1 --- a/src/ZF/OrdQuant.thy	Wed May 22 17:26:34 2002 +0200
1.2 +++ b/src/ZF/OrdQuant.thy	Wed May 22 18:11:57 2002 +0200
1.3 @@ -58,100 +58,16 @@
1.4  apply (blast intro: lt_Ord2)
1.5  done
1.6
1.7 -declare Ord_Un [intro,simp,TC]
1.8 -declare Ord_UN [intro,simp,TC]
1.9 -declare Ord_Union [intro,simp,TC]
1.10 -
1.11  (** Now some very basic ZF theorems **)
1.12
1.13 +(*FIXME: move to ZF.thy or even to FOL.thy??*)
1.14  lemma [simp]: "((P-->Q) <-> (P-->R)) <-> (P --> (Q<->R))"
1.15  by blast
1.16
1.17 -lemma [simp]: "cons(a,cons(a,B)) = cons(a,B)"
1.18 -by blast
1.19 -
1.20 +(*FIXME: move to Rel.thy*)
1.21  lemma trans_imp_trans_on: "trans(r) ==> trans[A](r)"
1.22  by (unfold trans_def trans_on_def, blast)
1.23
1.24 -lemma image_is_UN: "[| function(g); x <= domain(g) |] ==> g``x = (UN k:x. {g`k})"
1.25 -by (blast intro: function_apply_equality [THEN sym] function_apply_Pair)
1.26 -
1.27 -lemma functionI:
1.28 -     "[| !!x y y'. [| <x,y>:r; <x,y'>:r |] ==> y=y' |] ==> function(r)"
1.29 -by (simp add: function_def, blast)
1.30 -
1.31 -lemma function_lam: "function (lam x:A. b(x))"
1.32 -by (simp add: function_def lam_def)
1.33 -
1.34 -lemma relation_lam: "relation (lam x:A. b(x))"
1.35 -by (simp add: relation_def lam_def)
1.36 -
1.37 -lemma restrict_iff: "z \<in> restrict(r,A) \<longleftrightarrow> z \<in> r & (\<exists>x\<in>A. \<exists>y. z = \<langle>x, y\<rangle>)"
1.39 -
1.40 -(** These mostly belong to theory Ordinal **)
1.41 -
1.42 -lemma Union_upper_le:
1.43 -     "[| j: J;  i\<le>j;  Ord(\<Union>(J)) |] ==> i \<le> \<Union>J"
1.44 -apply (subst Union_eq_UN)
1.45 -apply (rule UN_upper_le, auto)
1.46 -done
1.47 -
1.48 -lemma zero_not_Limit [iff]: "~ Limit(0)"
1.50 -
1.51 -lemma Limit_has_1: "Limit(i) ==> 1 < i"
1.52 -by (blast intro: Limit_has_0 Limit_has_succ)
1.53 -
1.54 -lemma Limit_Union [rule_format]: "[| I \<noteq> 0;  \<forall>i\<in>I. Limit(i) |] ==> Limit(\<Union>I)"
1.55 -apply (simp add: Limit_def lt_def)
1.56 -apply (blast intro!: equalityI)
1.57 -done
1.58 -
1.59 -lemma increasing_LimitI: "[| 0<l; \<forall>x\<in>l. \<exists>y\<in>l. x<y |] ==> Limit(l)"
1.60 -apply (simp add: Limit_def lt_Ord2, clarify)
1.61 -apply (drule_tac i=y in ltD)
1.62 -apply (blast intro: lt_trans1 [OF _ ltI] lt_Ord2)
1.63 -done
1.64 -
1.65 -lemma UN_upper_lt:
1.66 -     "[| a\<in>A;  i < b(a);  Ord(\<Union>x\<in>A. b(x)) |] ==> i < (\<Union>x\<in>A. b(x))"
1.67 -by (unfold lt_def, blast)
1.68 -
1.69 -lemma lt_imp_0_lt: "j<i ==> 0<i"
1.70 -by (blast intro: lt_trans1 Ord_0_le [OF lt_Ord])
1.71 -
1.72 -lemma Ord_set_cases:
1.73 -   "\<forall>i\<in>I. Ord(i) ==> I=0 \<or> \<Union>(I) \<in> I \<or> (\<Union>(I) \<notin> I \<and> Limit(\<Union>(I)))"
1.74 -apply (clarify elim!: not_emptyE)
1.75 -apply (cases "\<Union>(I)" rule: Ord_cases)
1.76 -   apply (blast intro: Ord_Union)
1.77 -  apply (blast intro: subst_elem)
1.78 - apply auto
1.79 -apply (clarify elim!: equalityE succ_subsetE)
1.81 -apply (subgoal_tac "B = succ(j)", blast )
1.82 -apply (rule le_anti_sym)
1.83 - apply (simp add: le_subset_iff)
1.85 -done
1.86 -
1.87 -lemma Ord_Union_eq_succD: "[|\<forall>x\<in>X. Ord(x);  \<Union>X = succ(j)|] ==> succ(j) \<in> X"
1.88 -by (drule Ord_set_cases, auto)
1.89 -
1.91 -lemma Ord_Union_succ_eq: "Ord(i) ==> \<Union>(succ(i)) = i"
1.92 -by (blast intro: Ord_trans)
1.93 -
1.94 -lemma lt_Union_iff: "\<forall>i\<in>A. Ord(i) ==> (j < \<Union>(A)) <-> (\<exists>i\<in>A. j<i)"
1.95 -by (auto simp: lt_def Ord_Union)
1.96 -
1.97 -lemma Un_upper1_lt: "[|k < i; Ord(j)|] ==> k < i Un j"
1.99 -
1.100 -lemma Un_upper2_lt: "[|k < j; Ord(i)|] ==> k < i Un j"