(* Title: ZF/Constructible/Normal.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
*)
section \<open>Closed Unbounded Classes and Normal Functions\<close>
theory Normal imports ZF begin
text\<open>
One source is the book
Frank R. Drake.
\emph{Set Theory: An Introduction to Large Cardinals}.
North-Holland, 1974.
\<close>
subsection \<open>Closed and Unbounded (c.u.) Classes of Ordinals\<close>
definition
Closed :: "(i\<Rightarrow>o) \<Rightarrow> o" where
"Closed(P) \<equiv> \<forall>I. I \<noteq> 0 \<longrightarrow> (\<forall>i\<in>I. Ord(i) \<and> P(i)) \<longrightarrow> P(\<Union>(I))"
definition
Unbounded :: "(i\<Rightarrow>o) \<Rightarrow> o" where
"Unbounded(P) \<equiv> \<forall>i. Ord(i) \<longrightarrow> (\<exists>j. i<j \<and> P(j))"
definition
Closed_Unbounded :: "(i\<Rightarrow>o) \<Rightarrow> o" where
"Closed_Unbounded(P) \<equiv> Closed(P) \<and> Unbounded(P)"
subsubsection\<open>Simple facts about c.u. classes\<close>
lemma ClosedI:
"\<lbrakk>\<And>I. \<lbrakk>I \<noteq> 0; \<forall>i\<in>I. Ord(i) \<and> P(i)\<rbrakk> \<Longrightarrow> P(\<Union>(I))\<rbrakk>
\<Longrightarrow> Closed(P)"
by (simp add: Closed_def)
lemma ClosedD:
"\<lbrakk>Closed(P); I \<noteq> 0; \<And>i. i\<in>I \<Longrightarrow> Ord(i); \<And>i. i\<in>I \<Longrightarrow> P(i)\<rbrakk>
\<Longrightarrow> P(\<Union>(I))"
by (simp add: Closed_def)
lemma UnboundedD:
"\<lbrakk>Unbounded(P); Ord(i)\<rbrakk> \<Longrightarrow> \<exists>j. i<j \<and> P(j)"
by (simp add: Unbounded_def)
lemma Closed_Unbounded_imp_Unbounded: "Closed_Unbounded(C) \<Longrightarrow> Unbounded(C)"
by (simp add: Closed_Unbounded_def)
text\<open>The universal class, V, is closed and unbounded.
A bit odd, since C. U. concerns only ordinals, but it's used below!\<close>
theorem Closed_Unbounded_V [simp]: "Closed_Unbounded(\<lambda>x. True)"
by (unfold Closed_Unbounded_def Closed_def Unbounded_def, blast)
text\<open>The class of ordinals, \<^term>\<open>Ord\<close>, is closed and unbounded.\<close>
theorem Closed_Unbounded_Ord [simp]: "Closed_Unbounded(Ord)"
by (unfold Closed_Unbounded_def Closed_def Unbounded_def, blast)
text\<open>The class of limit ordinals, \<^term>\<open>Limit\<close>, is closed and unbounded.\<close>
theorem Closed_Unbounded_Limit [simp]: "Closed_Unbounded(Limit)"
proof -
have "\<exists>j. i < j \<and> Limit(j)" if "Ord(i)" for i
apply (rule_tac x="i++nat" in exI)
apply (blast intro: oadd_lt_self oadd_LimitI Limit_has_0 that)
done
then show ?thesis
by (auto simp: Closed_Unbounded_def Closed_def Unbounded_def Limit_Union)
qed
text\<open>The class of cardinals, \<^term>\<open>Card\<close>, is closed and unbounded.\<close>
theorem Closed_Unbounded_Card [simp]: "Closed_Unbounded(Card)"
proof -
have "\<forall>i. Ord(i) \<longrightarrow> (\<exists>j. i < j \<and> Card(j))"
by (blast intro: lt_csucc Card_csucc)
then show ?thesis
by (simp add: Closed_Unbounded_def Closed_def Unbounded_def)
qed
subsubsection\<open>The intersection of any set-indexed family of c.u. classes is
c.u.\<close>
text\<open>The constructions below come from Kunen, \emph{Set Theory}, page 78.\<close>
locale cub_family =
fixes P and A
fixes next_greater \<comment> \<open>the next ordinal satisfying class \<^term>\<open>A\<close>\<close>
fixes sup_greater \<comment> \<open>sup of those ordinals over all \<^term>\<open>A\<close>\<close>
assumes closed: "a\<in>A \<Longrightarrow> Closed(P(a))"
and unbounded: "a\<in>A \<Longrightarrow> Unbounded(P(a))"
and A_non0: "A\<noteq>0"
defines "next_greater(a,x) \<equiv> \<mu> y. x<y \<and> P(a,y)"
and "sup_greater(x) \<equiv> \<Union>a\<in>A. next_greater(a,x)"
begin
text\<open>Trivial that the intersection is closed.\<close>
lemma Closed_INT: "Closed(\<lambda>x. \<forall>i\<in>A. P(i,x))"
by (blast intro: ClosedI ClosedD [OF closed])
text\<open>All remaining effort goes to show that the intersection is unbounded.\<close>
lemma Ord_sup_greater:
"Ord(sup_greater(x))"
by (simp add: sup_greater_def next_greater_def)
lemma Ord_next_greater:
"Ord(next_greater(a,x))"
by (simp add: next_greater_def)
text\<open>\<^term>\<open>next_greater\<close> works as expected: it returns a larger value
and one that belongs to class \<^term>\<open>P(a)\<close>.\<close>
lemma
assumes "Ord(x)" "a\<in>A"
shows next_greater_in_P: "P(a, next_greater(a,x))"
and next_greater_gt: "x < next_greater(a,x)"
proof -
obtain y where "x < y" "P(a,y)"
using assms UnboundedD [OF unbounded] by blast
then have "P(a, next_greater(a,x)) \<and> x < next_greater(a,x)"
unfolding next_greater_def
by (blast intro: LeastI2 lt_Ord2)
then show "P(a, next_greater(a,x))" "x < next_greater(a,x)"
by auto
qed
lemma sup_greater_gt:
"Ord(x) \<Longrightarrow> x < sup_greater(x)"
using A_non0 unfolding sup_greater_def
by (blast intro: UN_upper_lt next_greater_gt Ord_next_greater)
lemma next_greater_le_sup_greater:
"a\<in>A \<Longrightarrow> next_greater(a,x) \<le> sup_greater(x)"
unfolding sup_greater_def
by (force intro: UN_upper_le Ord_next_greater)
lemma omega_sup_greater_eq_UN:
assumes "Ord(x)" "a\<in>A"
shows "sup_greater^\<omega> (x) =
(\<Union>n\<in>nat. next_greater(a, sup_greater^n (x)))"
proof (rule le_anti_sym)
show "sup_greater^\<omega> (x) \<le> (\<Union>n\<in>nat. next_greater(a, sup_greater^n (x)))"
using assms
unfolding iterates_omega_def
by (blast intro: leI le_implies_UN_le_UN next_greater_gt Ord_iterates Ord_sup_greater)
next
have "Ord(\<Union>n\<in>nat. sup_greater^n (x))"
by (blast intro: Ord_iterates Ord_sup_greater assms)
moreover have "next_greater(a, sup_greater^n (x)) \<le>
(\<Union>n\<in>nat. sup_greater^n (x))" if "n \<in> nat" for n
proof (rule UN_upper_le)
show "next_greater(a, sup_greater^n (x)) \<le> sup_greater^succ(n) (x)"
using assms by (simp add: next_greater_le_sup_greater)
show "Ord(\<Union>xa\<in>nat. sup_greater^xa (x))"
using assms by (blast intro: Ord_iterates Ord_sup_greater)
qed (use that in auto)
ultimately
show "(\<Union>n\<in>nat. next_greater(a, sup_greater^n (x))) \<le> sup_greater^\<omega> (x)"
using assms unfolding iterates_omega_def by (blast intro: UN_least_le)
qed
lemma P_omega_sup_greater:
"\<lbrakk>Ord(x); a\<in>A\<rbrakk> \<Longrightarrow> P(a, sup_greater^\<omega> (x))"
apply (simp add: omega_sup_greater_eq_UN)
apply (rule ClosedD [OF closed])
apply (blast intro: ltD, auto)
apply (blast intro: Ord_iterates Ord_next_greater Ord_sup_greater)
apply (blast intro: next_greater_in_P Ord_iterates Ord_sup_greater)
done
lemma omega_sup_greater_gt:
"Ord(x) \<Longrightarrow> x < sup_greater^\<omega> (x)"
apply (simp add: iterates_omega_def)
apply (rule UN_upper_lt [of 1], simp_all)
apply (blast intro: sup_greater_gt)
apply (blast intro: Ord_iterates Ord_sup_greater)
done
lemma Unbounded_INT: "Unbounded(\<lambda>x. \<forall>a\<in>A. P(a,x))"
unfolding Unbounded_def
by (blast intro!: omega_sup_greater_gt P_omega_sup_greater)
lemma Closed_Unbounded_INT:
"Closed_Unbounded(\<lambda>x. \<forall>a\<in>A. P(a,x))"
by (simp add: Closed_Unbounded_def Closed_INT Unbounded_INT)
end
theorem Closed_Unbounded_INT:
assumes "\<And>a. a\<in>A \<Longrightarrow> Closed_Unbounded(P(a))"
shows "Closed_Unbounded(\<lambda>x. \<forall>a\<in>A. P(a, x))"
proof (cases "A=0")
case False
with assms [unfolded Closed_Unbounded_def] show ?thesis
by (intro cub_family.Closed_Unbounded_INT [OF cub_family.intro]) auto
qed auto
lemma Int_iff_INT2:
"P(x) \<and> Q(x) \<longleftrightarrow> (\<forall>i\<in>2. (i=0 \<longrightarrow> P(x)) \<and> (i=1 \<longrightarrow> Q(x)))"
by auto
theorem Closed_Unbounded_Int:
"\<lbrakk>Closed_Unbounded(P); Closed_Unbounded(Q)\<rbrakk>
\<Longrightarrow> Closed_Unbounded(\<lambda>x. P(x) \<and> Q(x))"
unfolding Int_iff_INT2
by (rule Closed_Unbounded_INT, auto)
subsection \<open>Normal Functions\<close>
definition
mono_le_subset :: "(i\<Rightarrow>i) \<Rightarrow> o" where
"mono_le_subset(M) \<equiv> \<forall>i j. i\<le>j \<longrightarrow> M(i) \<subseteq> M(j)"
definition
mono_Ord :: "(i\<Rightarrow>i) \<Rightarrow> o" where
"mono_Ord(F) \<equiv> \<forall>i j. i<j \<longrightarrow> F(i) < F(j)"
definition
cont_Ord :: "(i\<Rightarrow>i) \<Rightarrow> o" where
"cont_Ord(F) \<equiv> \<forall>l. Limit(l) \<longrightarrow> F(l) = (\<Union>i<l. F(i))"
definition
Normal :: "(i\<Rightarrow>i) \<Rightarrow> o" where
"Normal(F) \<equiv> mono_Ord(F) \<and> cont_Ord(F)"
subsubsection\<open>Immediate properties of the definitions\<close>
lemma NormalI:
"\<lbrakk>\<And>i j. i<j \<Longrightarrow> F(i) < F(j); \<And>l. Limit(l) \<Longrightarrow> F(l) = (\<Union>i<l. F(i))\<rbrakk>
\<Longrightarrow> Normal(F)"
by (simp add: Normal_def mono_Ord_def cont_Ord_def)
lemma mono_Ord_imp_Ord: "\<lbrakk>Ord(i); mono_Ord(F)\<rbrakk> \<Longrightarrow> Ord(F(i))"
apply (auto simp add: mono_Ord_def)
apply (blast intro: lt_Ord)
done
lemma mono_Ord_imp_mono: "\<lbrakk>i<j; mono_Ord(F)\<rbrakk> \<Longrightarrow> F(i) < F(j)"
by (simp add: mono_Ord_def)
lemma Normal_imp_Ord [simp]: "\<lbrakk>Normal(F); Ord(i)\<rbrakk> \<Longrightarrow> Ord(F(i))"
by (simp add: Normal_def mono_Ord_imp_Ord)
lemma Normal_imp_cont: "\<lbrakk>Normal(F); Limit(l)\<rbrakk> \<Longrightarrow> F(l) = (\<Union>i<l. F(i))"
by (simp add: Normal_def cont_Ord_def)
lemma Normal_imp_mono: "\<lbrakk>i<j; Normal(F)\<rbrakk> \<Longrightarrow> F(i) < F(j)"
by (simp add: Normal_def mono_Ord_def)
lemma Normal_increasing:
assumes i: "Ord(i)" and F: "Normal(F)" shows"i \<le> F(i)"
using i
proof (induct i rule: trans_induct3)
case 0 thus ?case by (simp add: subset_imp_le F)
next
case (succ i)
hence "F(i) < F(succ(i))" using F
by (simp add: Normal_def mono_Ord_def)
thus ?case using succ.hyps
by (blast intro: lt_trans1)
next
case (limit l)
hence "l = (\<Union>y<l. y)"
by (simp add: Limit_OUN_eq)
also have "... \<le> (\<Union>y<l. F(y))" using limit
by (blast intro: ltD le_implies_OUN_le_OUN)
finally have "l \<le> (\<Union>y<l. F(y))" .
moreover have "(\<Union>y<l. F(y)) \<le> F(l)" using limit F
by (simp add: Normal_imp_cont lt_Ord)
ultimately show ?case
by (blast intro: le_trans)
qed
subsubsection\<open>The class of fixedpoints is closed and unbounded\<close>
text\<open>The proof is from Drake, pages 113--114.\<close>
lemma mono_Ord_imp_le_subset: "mono_Ord(F) \<Longrightarrow> mono_le_subset(F)"
apply (simp add: mono_le_subset_def, clarify)
apply (subgoal_tac "F(i)\<le>F(j)", blast dest: le_imp_subset)
apply (simp add: le_iff)
apply (blast intro: lt_Ord2 mono_Ord_imp_Ord mono_Ord_imp_mono)
done
text\<open>The following equation is taken for granted in any set theory text.\<close>
lemma cont_Ord_Union:
"\<lbrakk>cont_Ord(F); mono_le_subset(F); X=0 \<longrightarrow> F(0)=0; \<forall>x\<in>X. Ord(x)\<rbrakk>
\<Longrightarrow> F(\<Union>(X)) = (\<Union>y\<in>X. F(y))"
apply (frule Ord_set_cases)
apply (erule disjE, force)
apply (thin_tac "X=0 \<longrightarrow> Q" for Q, auto)
txt\<open>The trival case of \<^term>\<open>\<Union>X \<in> X\<close>\<close>
apply (rule equalityI, blast intro: Ord_Union_eq_succD)
apply (simp add: mono_le_subset_def UN_subset_iff le_subset_iff)
apply (blast elim: equalityE)
txt\<open>The limit case, \<^term>\<open>Limit(\<Union>X)\<close>:
@{subgoals[display,indent=0,margin=65]}
\<close>
apply (simp add: OUN_Union_eq cont_Ord_def)
apply (rule equalityI)
txt\<open>First inclusion:\<close>
apply (rule UN_least [OF OUN_least])
apply (simp add: mono_le_subset_def, blast intro: leI)
txt\<open>Second inclusion:\<close>
apply (rule UN_least)
apply (frule Union_upper_le, blast, blast)
apply (erule leE, drule ltD, elim UnionE)
apply (simp add: OUnion_def)
apply blast+
done
lemma Normal_Union:
"\<lbrakk>X\<noteq>0; \<forall>x\<in>X. Ord(x); Normal(F)\<rbrakk> \<Longrightarrow> F(\<Union>(X)) = (\<Union>y\<in>X. F(y))"
unfolding Normal_def
by (blast intro: mono_Ord_imp_le_subset cont_Ord_Union)
lemma Normal_imp_fp_Closed: "Normal(F) \<Longrightarrow> Closed(\<lambda>i. F(i) = i)"
apply (simp add: Closed_def ball_conj_distrib, clarify)
apply (frule Ord_set_cases)
apply (auto simp add: Normal_Union)
done
lemma iterates_Normal_increasing:
"\<lbrakk>n\<in>nat; x < F(x); Normal(F)\<rbrakk>
\<Longrightarrow> F^n (x) < F^(succ(n)) (x)"
by (induct n rule: nat_induct) (simp_all add: Normal_imp_mono)
lemma Ord_iterates_Normal:
"\<lbrakk>n\<in>nat; Normal(F); Ord(x)\<rbrakk> \<Longrightarrow> Ord(F^n (x))"
by (simp)
text\<open>THIS RESULT IS UNUSED\<close>
lemma iterates_omega_Limit:
"\<lbrakk>Normal(F); x < F(x)\<rbrakk> \<Longrightarrow> Limit(F^\<omega> (x))"
apply (frule lt_Ord)
apply (simp add: iterates_omega_def)
apply (rule increasing_LimitI)
\<comment> \<open>this lemma is @{thm increasing_LimitI [no_vars]}\<close>
apply (blast intro: UN_upper_lt [of "1"] Normal_imp_Ord
Ord_iterates lt_imp_0_lt
iterates_Normal_increasing, clarify)
apply (rule bexI)
apply (blast intro: Ord_in_Ord [OF Ord_iterates_Normal])
apply (rule UN_I, erule nat_succI)
apply (blast intro: iterates_Normal_increasing Ord_iterates_Normal
ltD [OF lt_trans1, OF succ_leI, OF ltI])
done
lemma iterates_omega_fixedpoint:
"\<lbrakk>Normal(F); Ord(a)\<rbrakk> \<Longrightarrow> F(F^\<omega> (a)) = F^\<omega> (a)"
apply (frule Normal_increasing, assumption)
apply (erule leE)
apply (simp_all add: iterates_omega_triv [OF sym]) (*for subgoal 2*)
apply (simp add: iterates_omega_def Normal_Union)
apply (rule equalityI, force simp add: nat_succI)
txt\<open>Opposite inclusion:
@{subgoals[display,indent=0,margin=65]}
\<close>
apply clarify
apply (rule UN_I, assumption)
apply (frule iterates_Normal_increasing, assumption, assumption, simp)
apply (blast intro: Ord_trans ltD Ord_iterates_Normal Normal_imp_Ord [of F])
done
lemma iterates_omega_increasing:
"\<lbrakk>Normal(F); Ord(a)\<rbrakk> \<Longrightarrow> a \<le> F^\<omega> (a)"
by (simp add: iterates_omega_def UN_upper_le [of 0])
lemma Normal_imp_fp_Unbounded: "Normal(F) \<Longrightarrow> Unbounded(\<lambda>i. F(i) = i)"
apply (unfold Unbounded_def, clarify)
apply (rule_tac x="F^\<omega> (succ(i))" in exI)
apply (simp add: iterates_omega_fixedpoint)
apply (blast intro: lt_trans2 [OF _ iterates_omega_increasing])
done
theorem Normal_imp_fp_Closed_Unbounded:
"Normal(F) \<Longrightarrow> Closed_Unbounded(\<lambda>i. F(i) = i)"
by (simp add: Closed_Unbounded_def Normal_imp_fp_Closed Normal_imp_fp_Unbounded)
subsubsection\<open>Function \<open>normalize\<close>\<close>
text\<open>Function \<open>normalize\<close> maps a function \<open>F\<close> to a
normal function that bounds it above. The result is normal if and
only if \<open>F\<close> is continuous: succ is not bounded above by any
normal function, by @{thm [source] Normal_imp_fp_Unbounded}.
\<close>
definition
normalize :: "[i\<Rightarrow>i, i] \<Rightarrow> i" where
"normalize(F,a) \<equiv> transrec2(a, F(0), \<lambda>x r. F(succ(x)) \<union> succ(r))"
lemma Ord_normalize [simp, intro]:
"\<lbrakk>Ord(a); \<And>x. Ord(x) \<Longrightarrow> Ord(F(x))\<rbrakk> \<Longrightarrow> Ord(normalize(F, a))"
proof (induct a rule: trans_induct3)
qed (simp_all add: ltD def_transrec2 [OF normalize_def])
lemma normalize_increasing:
assumes ab: "a < b" and F: "\<And>x. Ord(x) \<Longrightarrow> Ord(F(x))"
shows "normalize(F,a) < normalize(F,b)"
proof -
{ fix x
have "Ord(b)" using ab by (blast intro: lt_Ord2)
hence "x < b \<Longrightarrow> normalize(F,x) < normalize(F,b)"
proof (induct b arbitrary: x rule: trans_induct3)
case 0 thus ?case by simp
next
case (succ b)
thus ?case
by (auto simp add: le_iff def_transrec2 [OF normalize_def] intro: Un_upper2_lt F)
next
case (limit l)
hence sc: "succ(x) < l"
by (blast intro: Limit_has_succ)
hence "normalize(F,x) < normalize(F,succ(x))"
by (blast intro: limit elim: ltE)
hence "normalize(F,x) < (\<Union>j<l. normalize(F,j))"
by (blast intro: OUN_upper_lt lt_Ord F sc)
thus ?case using limit
by (simp add: def_transrec2 [OF normalize_def])
qed
} thus ?thesis using ab .
qed
theorem Normal_normalize:
assumes "\<And>x. Ord(x) \<Longrightarrow> Ord(F(x))" shows "Normal(normalize(F))"
proof (rule NormalI)
show "\<And>i j. i < j \<Longrightarrow> normalize(F,i) < normalize(F,j)"
using assms by (blast intro!: normalize_increasing)
show "\<And>l. Limit(l) \<Longrightarrow> normalize(F, l) = (\<Union>i<l. normalize(F,i))"
by (simp add: def_transrec2 [OF normalize_def])
qed
theorem le_normalize:
assumes a: "Ord(a)" and coF: "cont_Ord(F)" and F: "\<And>x. Ord(x) \<Longrightarrow> Ord(F(x))"
shows "F(a) \<le> normalize(F,a)"
using a
proof (induct a rule: trans_induct3)
case 0 thus ?case by (simp add: F def_transrec2 [OF normalize_def])
next
case (succ a)
thus ?case
by (simp add: def_transrec2 [OF normalize_def] Un_upper1_le F )
next
case (limit l)
thus ?case using F coF [unfolded cont_Ord_def]
by (simp add: def_transrec2 [OF normalize_def] le_implies_OUN_le_OUN ltD)
qed
subsection \<open>The Alephs\<close>
text \<open>This is the well-known transfinite enumeration of the cardinal
numbers.\<close>
definition
Aleph :: "i \<Rightarrow> i" (\<open>\<aleph>_\<close> [90] 90) where
"Aleph(a) \<equiv> transrec2(a, nat, \<lambda>x r. csucc(r))"
lemma Card_Aleph [simp, intro]:
"Ord(a) \<Longrightarrow> Card(Aleph(a))"
apply (erule trans_induct3)
apply (simp_all add: Card_csucc Card_nat Card_is_Ord
def_transrec2 [OF Aleph_def])
done
lemma Aleph_increasing:
assumes ab: "a < b" shows "Aleph(a) < Aleph(b)"
proof -
{ fix x
have "Ord(b)" using ab by (blast intro: lt_Ord2)
hence "x < b \<Longrightarrow> Aleph(x) < Aleph(b)"
proof (induct b arbitrary: x rule: trans_induct3)
case 0 thus ?case by simp
next
case (succ b)
thus ?case
by (force simp add: le_iff def_transrec2 [OF Aleph_def]
intro: lt_trans lt_csucc Card_is_Ord)
next
case (limit l)
hence sc: "succ(x) < l"
by (blast intro: Limit_has_succ)
hence "\<aleph> x < (\<Union>j<l. \<aleph>j)" using limit
by (blast intro: OUN_upper_lt Card_is_Ord ltD lt_Ord)
thus ?case using limit
by (simp add: def_transrec2 [OF Aleph_def])
qed
} thus ?thesis using ab .
qed
theorem Normal_Aleph: "Normal(Aleph)"
proof (rule NormalI)
show "\<And>i j. i < j \<Longrightarrow> \<aleph>i < \<aleph>j"
by (blast intro!: Aleph_increasing)
show "\<And>l. Limit(l) \<Longrightarrow> \<aleph>l = (\<Union>i<l. \<aleph>i)"
by (simp add: def_transrec2 [OF Aleph_def])
qed
end