(* 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 Main 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=>o) => o" where
"Closed(P) == \<forall>I. I \<noteq> 0 \<longrightarrow> (\<forall>i\<in>I. Ord(i) \<and> P(i)) \<longrightarrow> P(\<Union>(I))"
definition
Unbounded :: "(i=>o) => o" where
"Unbounded(P) == \<forall>i. Ord(i) \<longrightarrow> (\<exists>j. i<j \<and> P(j))"
definition
Closed_Unbounded :: "(i=>o) => o" where
"Closed_Unbounded(P) == Closed(P) \<and> Unbounded(P)"
subsubsection\<open>Simple facts about c.u. classes\<close>
lemma ClosedI:
"[| !!I. [| I \<noteq> 0; \<forall>i\<in>I. Ord(i) \<and> P(i) |] ==> P(\<Union>(I)) |]
==> Closed(P)"
by (simp add: Closed_def)
lemma ClosedD:
"[| Closed(P); I \<noteq> 0; !!i. i\<in>I ==> Ord(i); !!i. i\<in>I ==> P(i) |]
==> P(\<Union>(I))"
by (simp add: Closed_def)
lemma UnboundedD:
"[| Unbounded(P); Ord(i) |] ==> \<exists>j. i<j \<and> P(j)"
by (simp add: Unbounded_def)
lemma Closed_Unbounded_imp_Unbounded: "Closed_Unbounded(C) ==> 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 Ord}, 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 Limit}, is closed and unbounded.\<close>
theorem Closed_Unbounded_Limit [simp]: "Closed_Unbounded(Limit)"
apply (simp add: Closed_Unbounded_def Closed_def Unbounded_def Limit_Union,
clarify)
apply (rule_tac x="i++nat" in exI)
apply (blast intro: oadd_lt_self oadd_LimitI Limit_nat Limit_has_0)
done
text\<open>The class of cardinals, @{term Card}, is closed and unbounded.\<close>
theorem Closed_Unbounded_Card [simp]: "Closed_Unbounded(Card)"
apply (simp add: Closed_Unbounded_def Closed_def Unbounded_def Card_Union)
apply (blast intro: lt_csucc Card_csucc)
done
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> "the next ordinal satisfying class @{term A}"
fixes sup_greater \<comment> "sup of those ordinals over all @{term A}"
assumes closed: "a\<in>A ==> Closed(P(a))"
and unbounded: "a\<in>A ==> Unbounded(P(a))"
and A_non0: "A\<noteq>0"
defines "next_greater(a,x) == \<mu> y. x<y \<and> P(a,y)"
and "sup_greater(x) == \<Union>a\<in>A. next_greater(a,x)"
text\<open>Trivial that the intersection is closed.\<close>
lemma (in cub_family) 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 (in cub_family) Ord_sup_greater:
"Ord(sup_greater(x))"
by (simp add: sup_greater_def next_greater_def)
lemma (in cub_family) Ord_next_greater:
"Ord(next_greater(a,x))"
by (simp add: next_greater_def Ord_Least)
text\<open>@{term next_greater} works as expected: it returns a larger value
and one that belongs to class @{term "P(a)"}.\<close>
lemma (in cub_family) next_greater_lemma:
"[| Ord(x); a\<in>A |] ==> P(a, next_greater(a,x)) \<and> x < next_greater(a,x)"
apply (simp add: next_greater_def)
apply (rule exE [OF UnboundedD [OF unbounded]])
apply assumption+
apply (blast intro: LeastI2 lt_Ord2)
done
lemma (in cub_family) next_greater_in_P:
"[| Ord(x); a\<in>A |] ==> P(a, next_greater(a,x))"
by (blast dest: next_greater_lemma)
lemma (in cub_family) next_greater_gt:
"[| Ord(x); a\<in>A |] ==> x < next_greater(a,x)"
by (blast dest: next_greater_lemma)
lemma (in cub_family) sup_greater_gt:
"Ord(x) ==> x < sup_greater(x)"
apply (simp add: sup_greater_def)
apply (insert A_non0)
apply (blast intro: UN_upper_lt next_greater_gt Ord_next_greater)
done
lemma (in cub_family) next_greater_le_sup_greater:
"a\<in>A ==> next_greater(a,x) \<le> sup_greater(x)"
apply (simp add: sup_greater_def)
apply (blast intro: UN_upper_le Ord_next_greater)
done
lemma (in cub_family) omega_sup_greater_eq_UN:
"[| Ord(x); a\<in>A |]
==> sup_greater^\<omega> (x) =
(\<Union>n\<in>nat. next_greater(a, sup_greater^n (x)))"
apply (simp add: iterates_omega_def)
apply (rule le_anti_sym)
apply (rule le_implies_UN_le_UN)
apply (blast intro: leI next_greater_gt Ord_iterates Ord_sup_greater)
txt\<open>Opposite bound:
@{subgoals[display,indent=0,margin=65]}
\<close>
apply (rule UN_least_le)
apply (blast intro: Ord_UN Ord_iterates Ord_sup_greater)
apply (rule_tac a="succ(n)" in UN_upper_le)
apply (simp_all add: next_greater_le_sup_greater)
apply (blast intro: Ord_UN Ord_iterates Ord_sup_greater)
done
lemma (in cub_family) P_omega_sup_greater:
"[| Ord(x); a\<in>A |] ==> 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 (in cub_family) omega_sup_greater_gt:
"Ord(x) ==> 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_UN Ord_iterates Ord_sup_greater)
done
lemma (in cub_family) Unbounded_INT: "Unbounded(\<lambda>x. \<forall>a\<in>A. P(a,x))"
apply (unfold Unbounded_def)
apply (blast intro!: omega_sup_greater_gt P_omega_sup_greater)
done
lemma (in cub_family) Closed_Unbounded_INT:
"Closed_Unbounded(\<lambda>x. \<forall>a\<in>A. P(a,x))"
by (simp add: Closed_Unbounded_def Closed_INT Unbounded_INT)
theorem Closed_Unbounded_INT:
"(!!a. a\<in>A ==> Closed_Unbounded(P(a)))
==> Closed_Unbounded(\<lambda>x. \<forall>a\<in>A. P(a, x))"
apply (case_tac "A=0", simp)
apply (rule cub_family.Closed_Unbounded_INT [OF cub_family.intro])
apply (simp_all add: Closed_Unbounded_def)
done
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:
"[| Closed_Unbounded(P); Closed_Unbounded(Q) |]
==> Closed_Unbounded(\<lambda>x. P(x) \<and> Q(x))"
apply (simp only: Int_iff_INT2)
apply (rule Closed_Unbounded_INT, auto)
done
subsection \<open>Normal Functions\<close>
definition
mono_le_subset :: "(i=>i) => o" where
"mono_le_subset(M) == \<forall>i j. i\<le>j \<longrightarrow> M(i) \<subseteq> M(j)"
definition
mono_Ord :: "(i=>i) => o" where
"mono_Ord(F) == \<forall>i j. i<j \<longrightarrow> F(i) < F(j)"
definition
cont_Ord :: "(i=>i) => o" where
"cont_Ord(F) == \<forall>l. Limit(l) \<longrightarrow> F(l) = (\<Union>i<l. F(i))"
definition
Normal :: "(i=>i) => o" where
"Normal(F) == mono_Ord(F) \<and> cont_Ord(F)"
subsubsection\<open>Immediate properties of the definitions\<close>
lemma NormalI:
"[|!!i j. i<j ==> F(i) < F(j); !!l. Limit(l) ==> F(l) = (\<Union>i<l. F(i))|]
==> Normal(F)"
by (simp add: Normal_def mono_Ord_def cont_Ord_def)
lemma mono_Ord_imp_Ord: "[| Ord(i); mono_Ord(F) |] ==> Ord(F(i))"
apply (auto simp add: mono_Ord_def)
apply (blast intro: lt_Ord)
done
lemma mono_Ord_imp_mono: "[| i<j; mono_Ord(F) |] ==> F(i) < F(j)"
by (simp add: mono_Ord_def)
lemma Normal_imp_Ord [simp]: "[| Normal(F); Ord(i) |] ==> Ord(F(i))"
by (simp add: Normal_def mono_Ord_imp_Ord)
lemma Normal_imp_cont: "[| Normal(F); Limit(l) |] ==> F(l) = (\<Union>i<l. F(i))"
by (simp add: Normal_def cont_Ord_def)
lemma Normal_imp_mono: "[| i<j; Normal(F) |] ==> 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) ==> 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:
"[| cont_Ord(F); mono_le_subset(F); X=0 \<longrightarrow> F(0)=0; \<forall>x\<in>X. Ord(x) |]
==> 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 "\<Union>X \<in> X"}\<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 "Limit(\<Union>X)"}:
@{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 intro: Ord_Union)
apply (erule leE, drule ltD, elim UnionE)
apply (simp add: OUnion_def)
apply blast+
done
lemma Normal_Union:
"[| X\<noteq>0; \<forall>x\<in>X. Ord(x); Normal(F) |] ==> F(\<Union>(X)) = (\<Union>y\<in>X. F(y))"
apply (simp add: Normal_def)
apply (blast intro: mono_Ord_imp_le_subset cont_Ord_Union)
done
lemma Normal_imp_fp_Closed: "Normal(F) ==> 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:
"[| n\<in>nat; x < F(x); Normal(F) |]
==> F^n (x) < F^(succ(n)) (x)"
apply (induct n rule: nat_induct)
apply (simp_all add: Normal_imp_mono)
done
lemma Ord_iterates_Normal:
"[| n\<in>nat; Normal(F); Ord(x) |] ==> Ord(F^n (x))"
by (simp add: Ord_iterates)
text\<open>THIS RESULT IS UNUSED\<close>
lemma iterates_omega_Limit:
"[| Normal(F); x < F(x) |] ==> Limit(F^\<omega> (x))"
apply (frule lt_Ord)
apply (simp add: iterates_omega_def)
apply (rule increasing_LimitI)
\<comment>"this lemma is @{thm increasing_LimitI [no_vars]}"
apply (blast intro: UN_upper_lt [of "1"] Normal_imp_Ord
Ord_UN 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:
"[| Normal(F); Ord(a) |] ==> 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:
"[| Normal(F); Ord(a) |] ==> a \<le> F^\<omega> (a)"
apply (unfold iterates_omega_def)
apply (rule UN_upper_le [of 0], simp_all)
done
lemma Normal_imp_fp_Unbounded: "Normal(F) ==> 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) ==> 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=>i, i] => i" where
"normalize(F,a) == transrec2(a, F(0), \<lambda>x r. F(succ(x)) \<union> succ(r))"
lemma Ord_normalize [simp, intro]:
"[| Ord(a); !!x. Ord(x) ==> Ord(F(x)) |] ==> Ord(normalize(F, a))"
apply (induct a rule: trans_induct3)
apply (simp_all add: ltD def_transrec2 [OF normalize_def])
done
lemma normalize_increasing:
assumes ab: "a < b" and F: "!!x. Ord(x) ==> 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:
"(!!x. Ord(x) ==> Ord(F(x))) ==> Normal(normalize(F))"
apply (rule NormalI)
apply (blast intro!: normalize_increasing)
apply (simp add: def_transrec2 [OF normalize_def])
done
theorem le_normalize:
assumes a: "Ord(a)" and coF: "cont_Ord(F)" and F: "!!x. Ord(x) ==> 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 => i" ("\<aleph>_" [90] 90) where
"Aleph(a) == transrec2(a, nat, \<lambda>x r. csucc(r))"
lemma Card_Aleph [simp, intro]:
"Ord(a) ==> 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)"
apply (rule NormalI)
apply (blast intro!: Aleph_increasing)
apply (simp add: def_transrec2 [OF Aleph_def])
done
end