Theory Normal

(*  Title:      ZF/Constructible/Normal.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

section ‹Closed Unbounded Classes and Normal Functions›

theory Normal imports ZF begin

One source is the book

Frank R. Drake.
\emph{Set Theory: An Introduction to Large Cardinals}.
North-Holland, 1974.

subsection ‹Closed and Unbounded (c.u.) Classes of Ordinals›

  Closed :: "(io)  o" where
    "Closed(P)  I. I  0  (iI. Ord(i)  P(i))  P((I))"

  Unbounded :: "(io)  o" where
    "Unbounded(P)  i. Ord(i)  (j. i<j  P(j))"

  Closed_Unbounded :: "(io)  o" where
    "Closed_Unbounded(P)  Closed(P)  Unbounded(P)"

subsubsection‹Simple facts about c.u. classes›

lemma ClosedI:
  "I. I  0; iI. Ord(i)  P(i)  P((I)) 
  by (simp add: Closed_def)

lemma ClosedD:
  "Closed(P); I  0; i. iI  Ord(i); i. iI  P(i) 
  by (simp add: Closed_def)

lemma UnboundedD:
  "Unbounded(P);  Ord(i)  j. i<j  P(j)"
  by (simp add: Unbounded_def)

lemma Closed_Unbounded_imp_Unbounded: "Closed_Unbounded(C)  Unbounded(C)"
  by (simp add: Closed_Unbounded_def) 

text‹The universal class, V, is closed and unbounded.
      A bit odd, since C. U. concerns only ordinals, but it's used below!›
theorem Closed_Unbounded_V [simp]: "Closed_Unbounded(λx. True)"
  by (unfold Closed_Unbounded_def Closed_def Unbounded_def, blast)

text‹The class of ordinals, termOrd, is closed and unbounded.›
theorem Closed_Unbounded_Ord   [simp]: "Closed_Unbounded(Ord)"
  by (unfold Closed_Unbounded_def Closed_def Unbounded_def, blast)

text‹The class of limit ordinals, termLimit, is closed and unbounded.›
theorem Closed_Unbounded_Limit [simp]: "Closed_Unbounded(Limit)"
proof -
  have "j. i < j  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) 
  then show ?thesis
    by (auto simp: Closed_Unbounded_def Closed_def Unbounded_def Limit_Union)

text‹The class of cardinals, termCard, is closed and unbounded.›
theorem Closed_Unbounded_Card  [simp]: "Closed_Unbounded(Card)"
proof -
  have "i. Ord(i)  (j. i < j  Card(j))"
    by (blast intro: lt_csucc Card_csucc)
  then show ?thesis
    by (simp add: Closed_Unbounded_def Closed_def Unbounded_def)

subsubsection‹The intersection of any set-indexed family of c.u. classes is

text‹The constructions below come from Kunen, \emph{Set Theory}, page 78.›
locale cub_family =
  fixes P and A
  fixes next_greater ― ‹the next ordinal satisfying class termA
  fixes sup_greater  ― ‹sup of those ordinals over all termA
  assumes closed:    "aA  Closed(P(a))"
      and unbounded: "aA  Unbounded(P(a))"
      and A_non0: "A0"
  defines "next_greater(a,x)  μ y. x<y  P(a,y)"
      and "sup_greater(x)  aA. next_greater(a,x)"


text‹Trivial that the intersection is closed.›
lemma Closed_INT: "Closed(λx. iA. P(i,x))"
  by (blast intro: ClosedI ClosedD [OF closed])

text‹All remaining effort goes to show that the intersection is unbounded.›

lemma Ord_sup_greater:
  by (simp add: sup_greater_def next_greater_def)

lemma Ord_next_greater:
  by (simp add: next_greater_def)

texttermnext_greater works as expected: it returns a larger value
and one that belongs to class termP(a).›
  assumes "Ord(x)" "aA" 
  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))  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

lemma sup_greater_gt:
  "Ord(x)  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:
  "aA  next_greater(a,x)  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)" "aA" 
  shows "sup_greater (x) = 
          (nnat. next_greater(a, sup_greater^n (x)))"
proof (rule le_anti_sym)
  show "sup_greater (x)  (nnat. 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)  
  have "Ord(nnat. sup_greater^n (x))"
    by (blast intro: Ord_iterates Ord_sup_greater assms)  
  moreover have "next_greater(a, sup_greater^n (x)) 
             (nnat. sup_greater^n (x))" if "n  nat" for n
  proof (rule UN_upper_le)
    show "next_greater(a, sup_greater^n (x))  sup_greater^succ(n) (x)"
      using assms by (simp add: next_greater_le_sup_greater) 
    show "Ord(xanat. sup_greater^xa (x))"
      using assms by (blast intro: Ord_iterates Ord_sup_greater)  
  qed (use that in auto)
  show "(nnat. next_greater(a, sup_greater^n (x)))  sup_greater (x)"
    using assms unfolding iterates_omega_def by (blast intro: UN_least_le) 

lemma P_omega_sup_greater:
  "Ord(x); aA  P(a, sup_greater (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)

lemma omega_sup_greater_gt:
  "Ord(x)  x < sup_greater (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)

lemma Unbounded_INT: "Unbounded(λx. aA. P(a,x))"
    unfolding Unbounded_def  
    by (blast intro!: omega_sup_greater_gt P_omega_sup_greater) 

lemma Closed_Unbounded_INT: 
  "Closed_Unbounded(λx. aA. P(a,x))"
  by (simp add: Closed_Unbounded_def Closed_INT Unbounded_INT)


theorem Closed_Unbounded_INT:
  assumes "a. aA  Closed_Unbounded(P(a))"
  shows "Closed_Unbounded(λx. aA. 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)  Q(x)    (i2. (i=0  P(x))  (i=1  Q(x)))"
  by auto

theorem Closed_Unbounded_Int:
  "Closed_Unbounded(P); Closed_Unbounded(Q) 
       Closed_Unbounded(λx. P(x)  Q(x))"
  unfolding Int_iff_INT2 
  by (rule Closed_Unbounded_INT, auto) 

subsection ‹Normal Functions› 

  mono_le_subset :: "(ii)  o" where
    "mono_le_subset(M)  i j. ij  M(i)  M(j)"

  mono_Ord :: "(ii)  o" where
    "mono_Ord(F)  i j. i<j  F(i) < F(j)"

  cont_Ord :: "(ii)  o" where
    "cont_Ord(F)  l. Limit(l)  F(l) = (i<l. F(i))"

  Normal :: "(ii)  o" where
    "Normal(F)  mono_Ord(F)  cont_Ord(F)"

subsubsection‹Immediate properties of the definitions›

lemma NormalI:
  "i j. i<j  F(i) < F(j);  l. Limit(l)  F(l) = (i<l. F(i))
  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) 

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) = (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  F(i)"
using i
proof (induct i rule: trans_induct3)
  case 0 thus ?case by (simp add: subset_imp_le F)
  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)
  case (limit l) 
  hence "l = (y<l. y)" 
    by (simp add: Limit_OUN_eq)
  also have "...  (y<l. F(y))" using limit
    by (blast intro: ltD le_implies_OUN_le_OUN)
  finally have "l  (y<l. F(y))" .
  moreover have "(y<l. F(y))  F(l)" using limit F
    by (simp add: Normal_imp_cont lt_Ord)
  ultimately show ?case
    by (blast intro: le_trans) 

subsubsection‹The class of fixedpoints is closed and unbounded›

text‹The proof is from Drake, pages 113--114.›

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)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) 

text‹The following equation is taken for granted in any set theory text.›
lemma cont_Ord_Union:
  "cont_Ord(F); mono_le_subset(F); X=0  F(0)=0; xX. Ord(x) 
       F((X)) = (yX. F(y))"
  apply (frule Ord_set_cases)
  apply (erule disjE, force) 
  apply (thin_tac "X=0  Q" for Q, auto)
  txt‹The trival case of termX  X
   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‹The limit case, termLimit(X):
  apply (simp add: OUN_Union_eq cont_Ord_def)
  apply (rule equalityI) 
  txt‹First inclusion:›
   apply (rule UN_least [OF OUN_least])
   apply (simp add: mono_le_subset_def, blast intro: leI) 
  txt‹Second inclusion:›
  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+

lemma Normal_Union:
  "X0; xX. Ord(x); Normal(F)  F((X)) = (yX. F(y))"
  unfolding Normal_def
  by (blast intro: mono_Ord_imp_le_subset cont_Ord_Union) 

lemma Normal_imp_fp_Closed: "Normal(F)  Closed(λi. F(i) = i)"
  apply (simp add: Closed_def ball_conj_distrib, clarify)
  apply (frule Ord_set_cases)
  apply (auto simp add: Normal_Union)

lemma iterates_Normal_increasing:
  "nnat;  x < F(x);  Normal(F) 
       F^n (x) < F^(succ(n)) (x)"  
  by (induct n rule: nat_induct) (simp_all add: Normal_imp_mono)

lemma Ord_iterates_Normal:
     "nnat;  Normal(F);  Ord(x)  Ord(F^n (x))"  
by (simp) 

lemma iterates_omega_Limit:
  "Normal(F);  x < F(x)  Limit(F (x))"  
  apply (frule lt_Ord) 
  apply (simp add: iterates_omega_def)
  apply (rule increasing_LimitI) 
    ― ‹this lemma is @{thm increasing_LimitI [no_vars]}
   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]) 

lemma iterates_omega_fixedpoint:
     "Normal(F); Ord(a)  F(F (a)) = F (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‹Opposite inclusion:
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]) 

lemma iterates_omega_increasing:
     "Normal(F); Ord(a)  a  F (a)"   
  by (simp add: iterates_omega_def UN_upper_le [of 0])

lemma Normal_imp_fp_Unbounded: "Normal(F)  Unbounded(λi. F(i) = i)"
apply (unfold Unbounded_def, clarify)
apply (rule_tac x="F (succ(i))" in exI)
apply (simp add: iterates_omega_fixedpoint) 
apply (blast intro: lt_trans2 [OF _ iterates_omega_increasing])

theorem Normal_imp_fp_Closed_Unbounded: 
     "Normal(F)  Closed_Unbounded(λi. F(i) = i)"
  by (simp add: Closed_Unbounded_def Normal_imp_fp_Closed Normal_imp_fp_Unbounded)

subsubsection‹Function normalize›

text‹Function normalize› maps a function F› to a 
      normal function that bounds it above.  The result is normal if and
      only if F› is continuous: succ is not bounded above by any 
      normal function, by @{thm [source] Normal_imp_fp_Unbounded}.
  normalize :: "[ii, i]  i" where
    "normalize(F,a)  transrec2(a, F(0), λx r. F(succ(x))  succ(r))"

lemma Ord_normalize [simp, intro]:
     "Ord(a); x. Ord(x)  Ord(F(x))  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: "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  normalize(F,x) < normalize(F,b)"
    proof (induct b arbitrary: x rule: trans_induct3)
      case 0 thus ?case by simp
      case (succ b)
      thus ?case
        by (auto simp add: le_iff def_transrec2 [OF normalize_def] intro: Un_upper2_lt F)
      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) < (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])
  } thus ?thesis using ab .

theorem Normal_normalize:
  assumes "x. Ord(x)  Ord(F(x))" shows "Normal(normalize(F))"
proof (rule NormalI) 
  show "i j. i < j  normalize(F,i) < normalize(F,j)"
    using assms by (blast intro!: normalize_increasing)
  show "l. Limit(l)  normalize(F, l) = (i<l. normalize(F,i))"
    by (simp add: def_transrec2 [OF normalize_def])

theorem le_normalize:
  assumes a: "Ord(a)" and coF: "cont_Ord(F)" and F: "x. Ord(x)  Ord(F(x))"
  shows "F(a)  normalize(F,a)"
using a
proof (induct a rule: trans_induct3)
  case 0 thus ?case by (simp add: F def_transrec2 [OF normalize_def])
  case (succ a)
  thus ?case
    by (simp add: def_transrec2 [OF normalize_def] Un_upper1_le F )
  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) 

subsection ‹The Alephs›
text ‹This is the well-known transfinite enumeration of the cardinal 

  Aleph :: "i  i"  (_› [90] 90) where
    "Aleph(a)  transrec2(a, nat, λx 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])

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  Aleph(x) < Aleph(b)"
    proof (induct b arbitrary: x rule: trans_induct3)
      case 0 thus ?case by simp
      case (succ b)
      thus ?case
        by (force simp add: le_iff def_transrec2 [OF Aleph_def] 
                  intro: lt_trans lt_csucc Card_is_Ord)
      case (limit l)
      hence sc: "succ(x) < l" 
        by (blast intro: Limit_has_succ) 
      hence " x < (j<l. 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])
  } thus ?thesis using ab .

theorem Normal_Aleph: "Normal(Aleph)"
proof (rule NormalI) 
  show "i j. i < j  i < j"
    by (blast intro!: Aleph_increasing)
  show "l. Limit(l)  l = (i<l. i)"
    by (simp add: def_transrec2 [OF Aleph_def])