(* Title: ZF/AC/OrdQuant.thy
ID: $Id$
Authors: Krzysztof Grabczewski and L C Paulson
Quantifiers and union operator for ordinals.
*)
theory OrdQuant = Ordinal:
constdefs
(* Ordinal Quantifiers *)
oall :: "[i, i => o] => o"
"oall(A, P) == ALL x. x<A --> P(x)"
oex :: "[i, i => o] => o"
"oex(A, P) == EX x. x<A & P(x)"
(* Ordinal Union *)
OUnion :: "[i, i => i] => i"
"OUnion(i,B) == {z: UN x:i. B(x). Ord(i)}"
syntax
"@oall" :: "[idt, i, o] => o" ("(3ALL _<_./ _)" 10)
"@oex" :: "[idt, i, o] => o" ("(3EX _<_./ _)" 10)
"@OUNION" :: "[idt, i, i] => i" ("(3UN _<_./ _)" 10)
translations
"ALL x<a. P" == "oall(a, %x. P)"
"EX x<a. P" == "oex(a, %x. P)"
"UN x<a. B" == "OUnion(a, %x. B)"
syntax (xsymbols)
"@oall" :: "[idt, i, o] => o" ("(3\<forall>_<_./ _)" 10)
"@oex" :: "[idt, i, o] => o" ("(3\<exists>_<_./ _)" 10)
"@OUNION" :: "[idt, i, i] => i" ("(3\<Union>_<_./ _)" 10)
declare Ord_Un [intro,simp,TC]
declare Ord_UN [intro,simp,TC]
declare Ord_Union [intro,simp,TC]
(** These mostly belong to theory Ordinal **)
lemma Union_upper_le:
"\<lbrakk>j: J; i\<le>j; Ord(\<Union>(J))\<rbrakk> \<Longrightarrow> i \<le> \<Union>J"
apply (subst Union_eq_UN)
apply (rule UN_upper_le, auto)
done
lemma zero_not_Limit [iff]: "~ Limit(0)"
by (simp add: Limit_def)
lemma Limit_has_1: "Limit(i) \<Longrightarrow> 1 < i"
by (blast intro: Limit_has_0 Limit_has_succ)
lemma Limit_Union [rule_format]: "\<lbrakk>I \<noteq> 0; \<forall>i\<in>I. Limit(i)\<rbrakk> \<Longrightarrow> Limit(\<Union>I)"
apply (simp add: Limit_def lt_def)
apply (blast intro!: equalityI)
done
lemma increasing_LimitI: "\<lbrakk>0<l; \<forall>x\<in>l. \<exists>y\<in>l. x<y\<rbrakk> \<Longrightarrow> Limit(l)"
apply (simp add: Limit_def lt_Ord2, clarify)
apply (drule_tac i=y in ltD)
apply (blast intro: lt_trans1 succ_leI ltI lt_Ord2)
done
lemma UN_upper_lt:
"\<lbrakk>a\<in> A; i < b(a); Ord(\<Union>x\<in>A. b(x))\<rbrakk> \<Longrightarrow> i < (\<Union>x\<in>A. b(x))"
by (unfold lt_def, blast)
lemma lt_imp_0_lt: "j<i \<Longrightarrow> 0<i"
by (blast intro: lt_trans1 Ord_0_le [OF lt_Ord])
lemma Ord_set_cases:
"\<forall>i\<in>I. Ord(i) \<Longrightarrow> I=0 \<or> \<Union>(I) \<in> I \<or> (\<Union>(I) \<notin> I \<and> Limit(\<Union>(I)))"
apply (clarify elim!: not_emptyE)
apply (cases "\<Union>(I)" rule: Ord_cases)
apply (blast intro: Ord_Union)
apply (blast intro: subst_elem)
apply auto
apply (clarify elim!: equalityE succ_subsetE)
apply (simp add: Union_subset_iff)
apply (subgoal_tac "B = succ(j)", blast )
apply (rule le_anti_sym)
apply (simp add: le_subset_iff)
apply (simp add: ltI)
done
lemma Ord_Union_eq_succD: "[|\<forall>x\<in>X. Ord(x); \<Union>X = succ(j)|] ==> succ(j) \<in> X"
by (drule Ord_set_cases, auto)
(*See also Transset_iff_Union_succ*)
lemma Ord_Union_succ_eq: "Ord(i) \<Longrightarrow> \<Union>(succ(i)) = i"
by (blast intro: Ord_trans)
lemma lt_Union_iff: "\<forall>i\<in>A. Ord(i) \<Longrightarrow> (j < \<Union>(A)) <-> (\<exists>i\<in>A. j<i)"
by (auto simp: lt_def Ord_Union)
lemma Un_upper1_lt: "[|k < i; Ord(j)|] ==> k < i Un j"
by (simp add: lt_def)
lemma Un_upper2_lt: "[|k < j; Ord(i)|] ==> k < i Un j"
by (simp add: lt_def)
lemma Ord_OUN [intro,simp]:
"\<lbrakk>!!x. x<A \<Longrightarrow> Ord(B(x))\<rbrakk> \<Longrightarrow> Ord(\<Union>x<A. B(x))"
by (simp add: OUnion_def ltI Ord_UN)
lemma OUN_upper_lt:
"\<lbrakk>a<A; i < b(a); Ord(\<Union>x<A. b(x))\<rbrakk> \<Longrightarrow> i < (\<Union>x<A. b(x))"
by (unfold OUnion_def lt_def, blast )
lemma OUN_upper_le:
"\<lbrakk>a<A; i\<le>b(a); Ord(\<Union>x<A. b(x))\<rbrakk> \<Longrightarrow> i \<le> (\<Union>x<A. b(x))"
apply (unfold OUnion_def, auto)
apply (rule UN_upper_le )
apply (auto simp add: lt_def)
done
lemma Limit_OUN_eq: "Limit(i) ==> (UN x<i. x) = i"
by (simp add: OUnion_def Limit_Union_eq Limit_is_Ord)
(* No < version; consider (UN i:nat.i)=nat *)
lemma OUN_least:
"(!!x. x<A ==> B(x) \<subseteq> C) ==> (UN x<A. B(x)) \<subseteq> C"
by (simp add: OUnion_def UN_least ltI)
(* No < version; consider (UN i:nat.i)=nat *)
lemma OUN_least_le:
"[| Ord(i); !!x. x<A ==> b(x) \<le> i |] ==> (UN x<A. b(x)) \<le> i"
by (simp add: OUnion_def UN_least_le ltI Ord_0_le)
lemma le_implies_OUN_le_OUN:
"[| !!x. x<A ==> c(x) \<le> d(x) |] ==> (UN x<A. c(x)) \<le> (UN x<A. d(x))"
by (blast intro: OUN_least_le OUN_upper_le le_Ord2 Ord_OUN)
lemma OUN_UN_eq:
"(!!x. x:A ==> Ord(B(x)))
==> (UN z < (UN x:A. B(x)). C(z)) = (UN x:A. UN z < B(x). C(z))"
by (simp add: OUnion_def)
lemma OUN_Union_eq:
"(!!x. x:X ==> Ord(x))
==> (UN z < Union(X). C(z)) = (UN x:X. UN z < x. C(z))"
by (simp add: OUnion_def)
(*So that rule_format will get rid of ALL x<A...*)
lemma atomize_oall [symmetric, rulify]:
"(!!x. x<A ==> P(x)) == Trueprop (ALL x<A. P(x))"
by (simp add: oall_def atomize_all atomize_imp)
end