(*  Title:      ZF/Ordinal.thy

     ID:         $Id$

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1994  University of Cambridge

 *)

 header{*Transitive Sets and Ordinals*}

 theory Ordinal = WF + Bool + equalities:

 constdefs

   Memrel        :: "i=>i"

     "Memrel(A)   == {z: A*A . EX x y. z=<x,y> & x:y }"

   Transset  :: "i=>o"

     "Transset(i) == ALL x:i. x<=i"

   Ord  :: "i=>o"

     "Ord(i)      == Transset(i) & (ALL x:i. Transset(x))"

   lt        :: "[i,i] => o"  (infixl "<" 50)   (*less-than on ordinals*)

     "i<j         == i:j & Ord(j)"

   Limit         :: "i=>o"

     "Limit(i)    == Ord(i) & 0<i & (ALL y. y<i --> succ(y)<i)"

 syntax

   "le"          :: "[i,i] => o"  (infixl 50)   (*less-than or equals*)

 translations

   "x le y"      == "x < succ(y)"

 syntax (xsymbols)

   "op le"       :: "[i,i] => o"  (infixl "\<le>" 50)  (*less-than or equals*)

 subsection{*Rules for Transset*}

 subsubsection{*Three Neat Characterisations of Transset*}

 lemma Transset_iff_Pow: "Transset(A) <-> A<=Pow(A)"

 by (unfold Transset_def, blast)

 lemma Transset_iff_Union_succ: "Transset(A) <-> Union(succ(A)) = A"

 apply (unfold Transset_def)

 apply (blast elim!: equalityE)

 done

 lemma Transset_iff_Union_subset: "Transset(A) <-> Union(A) <= A"

 by (unfold Transset_def, blast)

 subsubsection{*Consequences of Downwards Closure*}

 lemma Transset_doubleton_D: 

     "[| Transset(C); {a,b}: C |] ==> a:C & b: C"

 by (unfold Transset_def, blast)

 lemma Transset_Pair_D:

     "[| Transset(C); <a,b>: C |] ==> a:C & b: C"

 apply (simp add: Pair_def)

 apply (blast dest: Transset_doubleton_D)

 done

 lemma Transset_includes_domain:

     "[| Transset(C); A*B <= C; b: B |] ==> A <= C"

 by (blast dest: Transset_Pair_D)

 lemma Transset_includes_range:

     "[| Transset(C); A*B <= C; a: A |] ==> B <= C"

 by (blast dest: Transset_Pair_D)

 subsubsection{*Closure Properties*}

 lemma Transset_0: "Transset(0)"

 by (unfold Transset_def, blast)

 lemma Transset_Un: 

     "[| Transset(i);  Transset(j) |] ==> Transset(i Un j)"

 by (unfold Transset_def, blast)

 lemma Transset_Int: 

     "[| Transset(i);  Transset(j) |] ==> Transset(i Int j)"

 by (unfold Transset_def, blast)

 lemma Transset_succ: "Transset(i) ==> Transset(succ(i))"

 by (unfold Transset_def, blast)

 lemma Transset_Pow: "Transset(i) ==> Transset(Pow(i))"

 by (unfold Transset_def, blast)

 lemma Transset_Union: "Transset(A) ==> Transset(Union(A))"

 by (unfold Transset_def, blast)

 lemma Transset_Union_family: 

     "[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))"

 by (unfold Transset_def, blast)

 lemma Transset_Inter_family: 

     "[| !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))"

 by (unfold Inter_def Transset_def, blast)

 lemma Transset_UN:

      "(!!x. x \<in> A ==> Transset(B(x))) ==> Transset (\<Union>x\<in>A. B(x))"

 by (rule Transset_Union_family, auto) 

 lemma Transset_INT:

      "(!!x. x \<in> A ==> Transset(B(x))) ==> Transset (\<Inter>x\<in>A. B(x))"

 by (rule Transset_Inter_family, auto) 

 subsection{*Lemmas for Ordinals*}

 lemma OrdI:

     "[| Transset(i);  !!x. x:i ==> Transset(x) |]  ==>  Ord(i)"

 by (simp add: Ord_def) 

 lemma Ord_is_Transset: "Ord(i) ==> Transset(i)"

 by (simp add: Ord_def) 

 lemma Ord_contains_Transset: 

     "[| Ord(i);  j:i |] ==> Transset(j) "

 by (unfold Ord_def, blast)

 lemma Ord_in_Ord: "[| Ord(i);  j:i |] ==> Ord(j)"

 by (unfold Ord_def Transset_def, blast)

 (*suitable for rewriting PROVIDED i has been fixed*)

 lemma Ord_in_Ord': "[| j:i; Ord(i) |] ==> Ord(j)"

 by (blast intro: Ord_in_Ord)

 (* Ord(succ(j)) ==> Ord(j) *)

 lemmas Ord_succD = Ord_in_Ord [OF _ succI1]

 lemma Ord_subset_Ord: "[| Ord(i);  Transset(j);  j<=i |] ==> Ord(j)"

 by (simp add: Ord_def Transset_def, blast)

 lemma OrdmemD: "[| j:i;  Ord(i) |] ==> j<=i"

 by (unfold Ord_def Transset_def, blast)

 lemma Ord_trans: "[| i:j;  j:k;  Ord(k) |] ==> i:k"

 by (blast dest: OrdmemD)

 lemma Ord_succ_subsetI: "[| i:j;  Ord(j) |] ==> succ(i) <= j"

 by (blast dest: OrdmemD)

 subsection{*The Construction of Ordinals: 0, succ, Union*}

 lemma Ord_0 [iff,TC]: "Ord(0)"

 by (blast intro: OrdI Transset_0)

 lemma Ord_succ [TC]: "Ord(i) ==> Ord(succ(i))"

 by (blast intro: OrdI Transset_succ Ord_is_Transset Ord_contains_Transset)

 lemmas Ord_1 = Ord_0 [THEN Ord_succ]

 lemma Ord_succ_iff [iff]: "Ord(succ(i)) <-> Ord(i)"

 by (blast intro: Ord_succ dest!: Ord_succD)

 lemma Ord_Un [intro,simp,TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Un j)"

 apply (unfold Ord_def)

 apply (blast intro!: Transset_Un)

 done

 lemma Ord_Int [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Int j)"

 apply (unfold Ord_def)

 apply (blast intro!: Transset_Int)

 done

 (*There is no set of all ordinals, for then it would contain itself*)

 lemma ON_class: "~ (ALL i. i:X <-> Ord(i))"

 apply (rule notI)

 apply (frule_tac x = X in spec)

 apply (safe elim!: mem_irrefl)

 apply (erule swap, rule OrdI [OF _ Ord_is_Transset])

 apply (simp add: Transset_def)

 apply (blast intro: Ord_in_Ord)+

 done

 subsection{*< is 'less Than' for Ordinals*}

 lemma ltI: "[| i:j;  Ord(j) |] ==> i<j"

 by (unfold lt_def, blast)

 lemma ltE:

     "[| i<j;  [| i:j;  Ord(i);  Ord(j) |] ==> P |] ==> P"

 apply (unfold lt_def)

 apply (blast intro: Ord_in_Ord)

 done

 lemma ltD: "i<j ==> i:j"

 by (erule ltE, assumption)

 lemma not_lt0 [simp]: "~ i<0"

 by (unfold lt_def, blast)

 lemma lt_Ord: "j<i ==> Ord(j)"

 by (erule ltE, assumption)

 lemma lt_Ord2: "j<i ==> Ord(i)"

 by (erule ltE, assumption)

 (* "ja le j ==> Ord(j)" *)

 lemmas le_Ord2 = lt_Ord2 [THEN Ord_succD]

 (* i<0 ==> R *)

 lemmas lt0E = not_lt0 [THEN notE, elim!]

 lemma lt_trans: "[| i<j;  j<k |] ==> i<k"

 by (blast intro!: ltI elim!: ltE intro: Ord_trans)

 lemma lt_not_sym: "i<j ==> ~ (j<i)"

 apply (unfold lt_def)

 apply (blast elim: mem_asym)

 done

 (* [| i<j;  ~P ==> j<i |] ==> P *)

 lemmas lt_asym = lt_not_sym [THEN swap]

 lemma lt_irrefl [elim!]: "i<i ==> P"

 by (blast intro: lt_asym)

 lemma lt_not_refl: "~ i<i"

 apply (rule notI)

 apply (erule lt_irrefl)

 done

 (** le is less than or equals;  recall  i le j  abbrevs  i<succ(j) !! **)

 lemma le_iff: "i le j <-> i<j | (i=j & Ord(j))"

 by (unfold lt_def, blast)

 (*Equivalently, i<j ==> i < succ(j)*)

 lemma leI: "i<j ==> i le j"

 by (simp (no_asm_simp) add: le_iff)

 lemma le_eqI: "[| i=j;  Ord(j) |] ==> i le j"

 by (simp (no_asm_simp) add: le_iff)

 lemmas le_refl = refl [THEN le_eqI]

 lemma le_refl_iff [iff]: "i le i <-> Ord(i)"

 by (simp (no_asm_simp) add: lt_not_refl le_iff)

 lemma leCI: "(~ (i=j & Ord(j)) ==> i<j) ==> i le j"

 by (simp add: le_iff, blast)

 lemma leE:

     "[| i le j;  i<j ==> P;  [| i=j;  Ord(j) |] ==> P |] ==> P"

 by (simp add: le_iff, blast)

 lemma le_anti_sym: "[| i le j;  j le i |] ==> i=j"

 apply (simp add: le_iff)

 apply (blast elim: lt_asym)

 done

 lemma le0_iff [simp]: "i le 0 <-> i=0"

 by (blast elim!: leE)

 lemmas le0D = le0_iff [THEN iffD1, dest!]

 subsection{*Natural Deduction Rules for Memrel*}

 (*The lemmas MemrelI/E give better speed than [iff] here*)

 lemma Memrel_iff [simp]: "<a,b> : Memrel(A) <-> a:b & a:A & b:A"

 by (unfold Memrel_def, blast)

 lemma MemrelI [intro!]: "[| a: b;  a: A;  b: A |] ==> <a,b> : Memrel(A)"

 by auto

 lemma MemrelE [elim!]:

     "[| <a,b> : Memrel(A);   

         [| a: A;  b: A;  a:b |]  ==> P |]  

      ==> P"

 by auto

 lemma Memrel_type: "Memrel(A) <= A*A"

 by (unfold Memrel_def, blast)

 lemma Memrel_mono: "A<=B ==> Memrel(A) <= Memrel(B)"

 by (unfold Memrel_def, blast)

 lemma Memrel_0 [simp]: "Memrel(0) = 0"

 by (unfold Memrel_def, blast)

 lemma Memrel_1 [simp]: "Memrel(1) = 0"

 by (unfold Memrel_def, blast)

 lemma relation_Memrel: "relation(Memrel(A))"

 by (simp add: relation_def Memrel_def, blast)

 (*The membership relation (as a set) is well-founded.

   Proof idea: show A<=B by applying the foundation axiom to A-B *)

 lemma wf_Memrel: "wf(Memrel(A))"

 apply (unfold wf_def)

 apply (rule foundation [THEN disjE, THEN allI], erule disjI1, blast) 

 done

 text{*The premise @{term "Ord(i)"} does not suffice.*}

 lemma trans_Memrel: 

     "Ord(i) ==> trans(Memrel(i))"

 by (unfold Ord_def Transset_def trans_def, blast)

 text{*However, the following premise is strong enough.*}

 lemma Transset_trans_Memrel: 

     "\<forall>j\<in>i. Transset(j) ==> trans(Memrel(i))"

 by (unfold Transset_def trans_def, blast)

 (*If Transset(A) then Memrel(A) internalizes the membership relation below A*)

 lemma Transset_Memrel_iff: 

     "Transset(A) ==> <a,b> : Memrel(A) <-> a:b & b:A"

 by (unfold Transset_def, blast)

 subsection{*Transfinite Induction*}

 (*Epsilon induction over a transitive set*)

 lemma Transset_induct: 

     "[| i: k;  Transset(k);                           

         !!x.[| x: k;  ALL y:x. P(y) |] ==> P(x) |]

      ==>  P(i)"

 apply (simp add: Transset_def) 

 apply (erule wf_Memrel [THEN wf_induct2], blast+)

 done

 (*Induction over an ordinal*)

 lemmas Ord_induct [consumes 2] = Transset_induct [OF _ Ord_is_Transset]

 lemmas Ord_induct_rule = Ord_induct [rule_format, consumes 2]

 (*Induction over the class of ordinals -- a useful corollary of Ord_induct*)

 lemma trans_induct [consumes 1]:

     "[| Ord(i);  

         !!x.[| Ord(x);  ALL y:x. P(y) |] ==> P(x) |]

      ==>  P(i)"

 apply (rule Ord_succ [THEN succI1 [THEN Ord_induct]], assumption)

 apply (blast intro: Ord_succ [THEN Ord_in_Ord]) 

 done

 lemmas trans_induct_rule = trans_induct [rule_format, consumes 1]

 (*** Fundamental properties of the epsilon ordering (< on ordinals) ***)

 subsubsection{*Proving That < is a Linear Ordering on the Ordinals*}

 lemma Ord_linear [rule_format]:

      "Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)"

 apply (erule trans_induct)

 apply (rule impI [THEN allI])

 apply (erule_tac i=j in trans_induct) 

 apply (blast dest: Ord_trans) 

 done

 (*The trichotomy law for ordinals!*)

 lemma Ord_linear_lt:

     "[| Ord(i);  Ord(j);  i<j ==> P;  i=j ==> P;  j<i ==> P |] ==> P"

 apply (simp add: lt_def) 

 apply (rule_tac i1=i and j1=j in Ord_linear [THEN disjE], blast+)

 done

 lemma Ord_linear2:

     "[| Ord(i);  Ord(j);  i<j ==> P;  j le i ==> P |]  ==> P"

 apply (rule_tac i = i and j = j in Ord_linear_lt)

 apply (blast intro: leI le_eqI sym ) +

 done

 lemma Ord_linear_le:

     "[| Ord(i);  Ord(j);  i le j ==> P;  j le i ==> P |]  ==> P"

 apply (rule_tac i = i and j = j in Ord_linear_lt)

 apply (blast intro: leI le_eqI ) +

 done

 lemma le_imp_not_lt: "j le i ==> ~ i<j"

 by (blast elim!: leE elim: lt_asym)

 lemma not_lt_imp_le: "[| ~ i<j;  Ord(i);  Ord(j) |] ==> j le i"

 by (rule_tac i = i and j = j in Ord_linear2, auto)

 subsubsection{*Some Rewrite Rules for <, le*}

 lemma Ord_mem_iff_lt: "Ord(j) ==> i:j <-> i<j"

 by (unfold lt_def, blast)

 lemma not_lt_iff_le: "[| Ord(i);  Ord(j) |] ==> ~ i<j <-> j le i"

 by (blast dest: le_imp_not_lt not_lt_imp_le)

 lemma not_le_iff_lt: "[| Ord(i);  Ord(j) |] ==> ~ i le j <-> j<i"

 by (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym])

 (*This is identical to 0<succ(i) *)

 lemma Ord_0_le: "Ord(i) ==> 0 le i"

 by (erule not_lt_iff_le [THEN iffD1], auto)

 lemma Ord_0_lt: "[| Ord(i);  i~=0 |] ==> 0<i"

 apply (erule not_le_iff_lt [THEN iffD1])

 apply (rule Ord_0, blast)

 done

 lemma Ord_0_lt_iff: "Ord(i) ==> i~=0 <-> 0<i"

 by (blast intro: Ord_0_lt)

 subsection{*Results about Less-Than or Equals*}

 (** For ordinals, j<=i (subset) implies j le i (less-than or equals) **)

 lemma zero_le_succ_iff [iff]: "0 le succ(x) <-> Ord(x)"

 by (blast intro: Ord_0_le elim: ltE)

 lemma subset_imp_le: "[| j<=i;  Ord(i);  Ord(j) |] ==> j le i"

 apply (rule not_lt_iff_le [THEN iffD1], assumption+)

 apply (blast elim: ltE mem_irrefl)

 done

 lemma le_imp_subset: "i le j ==> i<=j"

 by (blast dest: OrdmemD elim: ltE leE)

 lemma le_subset_iff: "j le i <-> j<=i & Ord(i) & Ord(j)"

 by (blast dest: subset_imp_le le_imp_subset elim: ltE)

 lemma le_succ_iff: "i le succ(j) <-> i le j | i=succ(j) & Ord(i)"

 apply (simp (no_asm) add: le_iff)

 apply blast

 done

 (*Just a variant of subset_imp_le*)

 lemma all_lt_imp_le: "[| Ord(i);  Ord(j);  !!x. x<j ==> x<i |] ==> j le i"

 by (blast intro: not_lt_imp_le dest: lt_irrefl)

 subsubsection{*Transitivity Laws*}

 lemma lt_trans1: "[| i le j;  j<k |] ==> i<k"

 by (blast elim!: leE intro: lt_trans)

 lemma lt_trans2: "[| i<j;  j le k |] ==> i<k"

 by (blast elim!: leE intro: lt_trans)

 lemma le_trans: "[| i le j;  j le k |] ==> i le k"

 by (blast intro: lt_trans1)

 lemma succ_leI: "i<j ==> succ(i) le j"

 apply (rule not_lt_iff_le [THEN iffD1]) 

 apply (blast elim: ltE leE lt_asym)+

 done

 (*Identical to  succ(i) < succ(j) ==> i<j  *)

 lemma succ_leE: "succ(i) le j ==> i<j"

 apply (rule not_le_iff_lt [THEN iffD1])

 apply (blast elim: ltE leE lt_asym)+

 done

 lemma succ_le_iff [iff]: "succ(i) le j <-> i<j"

 by (blast intro: succ_leI succ_leE)

 lemma succ_le_imp_le: "succ(i) le succ(j) ==> i le j"

 by (blast dest!: succ_leE)

 lemma lt_subset_trans: "[| i <= j;  j<k;  Ord(i) |] ==> i<k"

 apply (rule subset_imp_le [THEN lt_trans1]) 

 apply (blast intro: elim: ltE) +

 done

 lemma lt_imp_0_lt: "j<i ==> 0<i"

 by (blast intro: lt_trans1 Ord_0_le [OF lt_Ord]) 

 lemma succ_lt_iff: "succ(i) < j <-> i<j & succ(i) \<noteq> j"

 apply auto 

 apply (blast intro: lt_trans le_refl dest: lt_Ord) 

 apply (frule lt_Ord) 

 apply (rule not_le_iff_lt [THEN iffD1]) 

   apply (blast intro: lt_Ord2)

  apply blast  

 apply (simp add: lt_Ord lt_Ord2 le_iff) 

 apply (blast dest: lt_asym) 

 done

 lemma Ord_succ_mem_iff: "Ord(j) ==> succ(i) \<in> succ(j) <-> i\<in>j"

 apply (insert succ_le_iff [of i j]) 

 apply (simp add: lt_def) 

 done

 subsubsection{*Union and Intersection*}

 lemma Un_upper1_le: "[| Ord(i); Ord(j) |] ==> i le i Un j"

 by (rule Un_upper1 [THEN subset_imp_le], auto)

 lemma Un_upper2_le: "[| Ord(i); Ord(j) |] ==> j le i Un j"

 by (rule Un_upper2 [THEN subset_imp_le], auto)

 (*Replacing k by succ(k') yields the similar rule for le!*)

 lemma Un_least_lt: "[| i<k;  j<k |] ==> i Un j < k"

 apply (rule_tac i = i and j = j in Ord_linear_le)

 apply (auto simp add: Un_commute le_subset_iff subset_Un_iff lt_Ord) 

 done

 lemma Un_least_lt_iff: "[| Ord(i); Ord(j) |] ==> i Un j < k  <->  i<k & j<k"

 apply (safe intro!: Un_least_lt)

 apply (rule_tac [2] Un_upper2_le [THEN lt_trans1])

 apply (rule Un_upper1_le [THEN lt_trans1], auto) 

 done

 lemma Un_least_mem_iff:

     "[| Ord(i); Ord(j); Ord(k) |] ==> i Un j : k  <->  i:k & j:k"

 apply (insert Un_least_lt_iff [of i j k]) 

 apply (simp add: lt_def)

 done

 (*Replacing k by succ(k') yields the similar rule for le!*)

 lemma Int_greatest_lt: "[| i<k;  j<k |] ==> i Int j < k"

 apply (rule_tac i = i and j = j in Ord_linear_le)

 apply (auto simp add: Int_commute le_subset_iff subset_Int_iff lt_Ord) 

 done

 lemma Ord_Un_if:

      "[| Ord(i); Ord(j) |] ==> i \<union> j = (if j<i then i else j)"

 by (simp add: not_lt_iff_le le_imp_subset leI

               subset_Un_iff [symmetric]  subset_Un_iff2 [symmetric]) 

 lemma succ_Un_distrib:

      "[| Ord(i); Ord(j) |] ==> succ(i \<union> j) = succ(i) \<union> succ(j)"

 by (simp add: Ord_Un_if lt_Ord le_Ord2) 

 lemma lt_Un_iff:

      "[| Ord(i); Ord(j) |] ==> k < i \<union> j <-> k < i | k < j";

 apply (simp add: Ord_Un_if not_lt_iff_le) 

 apply (blast intro: leI lt_trans2)+ 

 done

 lemma le_Un_iff:

      "[| Ord(i); Ord(j) |] ==> k \<le> i \<union> j <-> k \<le> i | k \<le> j";

 by (simp add: succ_Un_distrib lt_Un_iff [symmetric]) 

 lemma Un_upper1_lt: "[|k < i; Ord(j)|] ==> k < i Un j"

 by (simp add: lt_Un_iff lt_Ord2) 

 lemma Un_upper2_lt: "[|k < j; Ord(i)|] ==> k < i Un j"

 by (simp add: lt_Un_iff lt_Ord2) 

 (*See also Transset_iff_Union_succ*)

 lemma Ord_Union_succ_eq: "Ord(i) ==> \<Union>(succ(i)) = i"

 by (blast intro: Ord_trans)

 subsection{*Results about Limits*}

 lemma Ord_Union [intro,simp,TC]: "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))"

 apply (rule Ord_is_Transset [THEN Transset_Union_family, THEN OrdI])

 apply (blast intro: Ord_contains_Transset)+

 done

 lemma Ord_UN [intro,simp,TC]:

      "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(\<Union>x\<in>A. B(x))"

 by (rule Ord_Union, blast)

 lemma Ord_Inter [intro,simp,TC]:

     "[| !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))" 

 apply (rule Transset_Inter_family [THEN OrdI])

 apply (blast intro: Ord_is_Transset) 

 apply (simp add: Inter_def) 

 apply (blast intro: Ord_contains_Transset) 

 done

 lemma Ord_INT [intro,simp,TC]:

     "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(\<Inter>x\<in>A. B(x))"

 by (rule Ord_Inter, blast) 

 (* No < version; consider (\<Union>i\<in>nat.i)=nat *)

 lemma UN_least_le:

     "[| Ord(i);  !!x. x:A ==> b(x) le i |] ==> (\<Union>x\<in>A. b(x)) le i"

 apply (rule le_imp_subset [THEN UN_least, THEN subset_imp_le])

 apply (blast intro: Ord_UN elim: ltE)+

 done

 lemma UN_succ_least_lt:

     "[| j<i;  !!x. x:A ==> b(x)<j |] ==> (\<Union>x\<in>A. succ(b(x))) < i"

 apply (rule ltE, assumption)

 apply (rule UN_least_le [THEN lt_trans2])

 apply (blast intro: succ_leI)+

 done

 lemma UN_upper_lt:

      "[| a\<in>A;  i < b(a);  Ord(\<Union>x\<in>A. b(x)) |] ==> i < (\<Union>x\<in>A. b(x))"

 by (unfold lt_def, blast) 

 lemma UN_upper_le:

      "[| a: A;  i le b(a);  Ord(\<Union>x\<in>A. b(x)) |] ==> i le (\<Union>x\<in>A. b(x))"

 apply (frule ltD)

 apply (rule le_imp_subset [THEN subset_trans, THEN subset_imp_le])

 apply (blast intro: lt_Ord UN_upper)+

 done

 lemma lt_Union_iff: "\<forall>i\<in>A. Ord(i) ==> (j < \<Union>(A)) <-> (\<exists>i\<in>A. j<i)"

 by (auto simp: lt_def Ord_Union)

 lemma Union_upper_le:

      "[| j: J;  i\<le>j;  Ord(\<Union>(J)) |] ==> i \<le> \<Union>J"

 apply (subst Union_eq_UN)  

 apply (rule UN_upper_le, auto)

 done

 lemma le_implies_UN_le_UN:

     "[| !!x. x:A ==> c(x) le d(x) |] ==> (\<Union>x\<in>A. c(x)) le (\<Union>x\<in>A. d(x))"

 apply (rule UN_least_le)

 apply (rule_tac [2] UN_upper_le)

 apply (blast intro: Ord_UN le_Ord2)+ 

 done

 lemma Ord_equality: "Ord(i) ==> (\<Union>y\<in>i. succ(y)) = i"

 by (blast intro: Ord_trans)

 (*Holds for all transitive sets, not just ordinals*)

 lemma Ord_Union_subset: "Ord(i) ==> Union(i) <= i"

 by (blast intro: Ord_trans)

 subsection{*Limit Ordinals -- General Properties*}

 lemma Limit_Union_eq: "Limit(i) ==> Union(i) = i"

 apply (unfold Limit_def)

 apply (fast intro!: ltI elim!: ltE elim: Ord_trans)

 done

 lemma Limit_is_Ord: "Limit(i) ==> Ord(i)"

 apply (unfold Limit_def)

 apply (erule conjunct1)

 done

 lemma Limit_has_0: "Limit(i) ==> 0 < i"

 apply (unfold Limit_def)

 apply (erule conjunct2 [THEN conjunct1])

 done

 lemma Limit_nonzero: "Limit(i) ==> i ~= 0"

 by (drule Limit_has_0, blast)

 lemma Limit_has_succ: "[| Limit(i);  j<i |] ==> succ(j) < i"

 by (unfold Limit_def, blast)

 lemma Limit_succ_lt_iff [simp]: "Limit(i) ==> succ(j) < i <-> (j<i)"

 apply (safe intro!: Limit_has_succ)

 apply (frule lt_Ord)

 apply (blast intro: lt_trans)   

 done

 lemma zero_not_Limit [iff]: "~ Limit(0)"

 by (simp add: Limit_def)

 lemma Limit_has_1: "Limit(i) ==> 1 < i"

 by (blast intro: Limit_has_0 Limit_has_succ)

 lemma increasing_LimitI: "[| 0<l; \<forall>x\<in>l. \<exists>y\<in>l. x<y |] ==> Limit(l)"

 apply (unfold Limit_def, simp add: lt_Ord2, clarify)

 apply (drule_tac i=y in ltD) 

 apply (blast intro: lt_trans1 [OF _ ltI] lt_Ord2)

 done

 lemma non_succ_LimitI: 

     "[| 0<i;  ALL y. succ(y) ~= i |] ==> Limit(i)"

 apply (unfold Limit_def)

 apply (safe del: subsetI)

 apply (rule_tac [2] not_le_iff_lt [THEN iffD1])

 apply (simp_all add: lt_Ord lt_Ord2) 

 apply (blast elim: leE lt_asym)

 done

 lemma succ_LimitE [elim!]: "Limit(succ(i)) ==> P"

 apply (rule lt_irrefl)

 apply (rule Limit_has_succ, assumption)

 apply (erule Limit_is_Ord [THEN Ord_succD, THEN le_refl])

 done

 lemma not_succ_Limit [simp]: "~ Limit(succ(i))"

 by blast

 lemma Limit_le_succD: "[| Limit(i);  i le succ(j) |] ==> i le j"

 by (blast elim!: leE)

 subsubsection{*Traditional 3-Way Case Analysis on Ordinals*}

 lemma Ord_cases_disj: "Ord(i) ==> i=0 | (EX j. Ord(j) & i=succ(j)) | Limit(i)"

 by (blast intro!: non_succ_LimitI Ord_0_lt)

 lemma Ord_cases:

     "[| Ord(i);                  

         i=0                          ==> P;      

         !!j. [| Ord(j); i=succ(j) |] ==> P;      

         Limit(i)                     ==> P       

      |] ==> P"

 by (drule Ord_cases_disj, blast)  

 lemma trans_induct3 [case_names 0 succ limit, consumes 1]:

      "[| Ord(i);                 

          P(0);                   

          !!x. [| Ord(x);  P(x) |] ==> P(succ(x));        

          !!x. [| Limit(x);  ALL y:x. P(y) |] ==> P(x)    

       |] ==> P(i)"

 apply (erule trans_induct)

 apply (erule Ord_cases, blast+)

 done

 lemmas trans_induct3_rule = trans_induct3 [rule_format, case_names 0 succ limit, consumes 1]

 text{*A set of ordinals is either empty, contains its own union, or its

 union is a limit ordinal.*}

 lemma Ord_set_cases:

    "\<forall>i\<in>I. Ord(i) ==> 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

 text{*If the union of a set of ordinals is a successor, then it is

 an element of that set.*}

 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)

 lemma Limit_Union [rule_format]: "[| I \<noteq> 0;  \<forall>i\<in>I. Limit(i) |] ==> Limit(\<Union>I)"

 apply (simp add: Limit_def lt_def)

 apply (blast intro!: equalityI)

 done

 ML 

 {*

 val Memrel_def = thm "Memrel_def";

 val Transset_def = thm "Transset_def";

 val Ord_def = thm "Ord_def";

 val lt_def = thm "lt_def";

 val Limit_def = thm "Limit_def";

 val Transset_iff_Pow = thm "Transset_iff_Pow";

 val Transset_iff_Union_succ = thm "Transset_iff_Union_succ";

 val Transset_iff_Union_subset = thm "Transset_iff_Union_subset";

 val Transset_doubleton_D = thm "Transset_doubleton_D";

 val Transset_Pair_D = thm "Transset_Pair_D";

 val Transset_includes_domain = thm "Transset_includes_domain";

 val Transset_includes_range = thm "Transset_includes_range";

 val Transset_0 = thm "Transset_0";

 val Transset_Un = thm "Transset_Un";

 val Transset_Int = thm "Transset_Int";

 val Transset_succ = thm "Transset_succ";

 val Transset_Pow = thm "Transset_Pow";

 val Transset_Union = thm "Transset_Union";

 val Transset_Union_family = thm "Transset_Union_family";

 val Transset_Inter_family = thm "Transset_Inter_family";

 val OrdI = thm "OrdI";

 val Ord_is_Transset = thm "Ord_is_Transset";

 val Ord_contains_Transset = thm "Ord_contains_Transset";

 val Ord_in_Ord = thm "Ord_in_Ord";

 val Ord_succD = thm "Ord_succD";

 val Ord_subset_Ord = thm "Ord_subset_Ord";

 val OrdmemD = thm "OrdmemD";

 val Ord_trans = thm "Ord_trans";

 val Ord_succ_subsetI = thm "Ord_succ_subsetI";

 val Ord_0 = thm "Ord_0";

 val Ord_succ = thm "Ord_succ";

 val Ord_1 = thm "Ord_1";

 val Ord_succ_iff = thm "Ord_succ_iff";

 val Ord_Un = thm "Ord_Un";

 val Ord_Int = thm "Ord_Int";

 val Ord_Inter = thm "Ord_Inter";

 val Ord_INT = thm "Ord_INT";

 val ON_class = thm "ON_class";

 val ltI = thm "ltI";

 val ltE = thm "ltE";

 val ltD = thm "ltD";

 val not_lt0 = thm "not_lt0";

 val lt_Ord = thm "lt_Ord";

 val lt_Ord2 = thm "lt_Ord2";

 val le_Ord2 = thm "le_Ord2";

 val lt0E = thm "lt0E";

 val lt_trans = thm "lt_trans";

 val lt_not_sym = thm "lt_not_sym";

 val lt_asym = thm "lt_asym";

 val lt_irrefl = thm "lt_irrefl";

 val lt_not_refl = thm "lt_not_refl";

 val le_iff = thm "le_iff";

 val leI = thm "leI";

 val le_eqI = thm "le_eqI";

 val le_refl = thm "le_refl";

 val le_refl_iff = thm "le_refl_iff";

 val leCI = thm "leCI";

 val leE = thm "leE";

 val le_anti_sym = thm "le_anti_sym";

 val le0_iff = thm "le0_iff";

 val le0D = thm "le0D";

 val Memrel_iff = thm "Memrel_iff";

 val MemrelI = thm "MemrelI";

 val MemrelE = thm "MemrelE";

 val Memrel_type = thm "Memrel_type";

 val Memrel_mono = thm "Memrel_mono";

 val Memrel_0 = thm "Memrel_0";

 val Memrel_1 = thm "Memrel_1";

 val wf_Memrel = thm "wf_Memrel";

 val trans_Memrel = thm "trans_Memrel";

 val Transset_Memrel_iff = thm "Transset_Memrel_iff";

 val Transset_induct = thm "Transset_induct";

 val Ord_induct = thm "Ord_induct";

 val trans_induct = thm "trans_induct";

 val Ord_linear = thm "Ord_linear";

 val Ord_linear_lt = thm "Ord_linear_lt";

 val Ord_linear2 = thm "Ord_linear2";

 val Ord_linear_le = thm "Ord_linear_le";

 val le_imp_not_lt = thm "le_imp_not_lt";

 val not_lt_imp_le = thm "not_lt_imp_le";

 val Ord_mem_iff_lt = thm "Ord_mem_iff_lt";

 val not_lt_iff_le = thm "not_lt_iff_le";

 val not_le_iff_lt = thm "not_le_iff_lt";

 val Ord_0_le = thm "Ord_0_le";

 val Ord_0_lt = thm "Ord_0_lt";

 val Ord_0_lt_iff = thm "Ord_0_lt_iff";

 val zero_le_succ_iff = thm "zero_le_succ_iff";

 val subset_imp_le = thm "subset_imp_le";

 val le_imp_subset = thm "le_imp_subset";

 val le_subset_iff = thm "le_subset_iff";

 val le_succ_iff = thm "le_succ_iff";

 val all_lt_imp_le = thm "all_lt_imp_le";

 val lt_trans1 = thm "lt_trans1";

 val lt_trans2 = thm "lt_trans2";

 val le_trans = thm "le_trans";

 val succ_leI = thm "succ_leI";

 val succ_leE = thm "succ_leE";

 val succ_le_iff = thm "succ_le_iff";

 val succ_le_imp_le = thm "succ_le_imp_le";

 val lt_subset_trans = thm "lt_subset_trans";

 val Un_upper1_le = thm "Un_upper1_le";

 val Un_upper2_le = thm "Un_upper2_le";

 val Un_least_lt = thm "Un_least_lt";

 val Un_least_lt_iff = thm "Un_least_lt_iff";

 val Un_least_mem_iff = thm "Un_least_mem_iff";

 val Int_greatest_lt = thm "Int_greatest_lt";

 val Ord_Union = thm "Ord_Union";

 val Ord_UN = thm "Ord_UN";

 val UN_least_le = thm "UN_least_le";

 val UN_succ_least_lt = thm "UN_succ_least_lt";

 val UN_upper_le = thm "UN_upper_le";

 val le_implies_UN_le_UN = thm "le_implies_UN_le_UN";

 val Ord_equality = thm "Ord_equality";

 val Ord_Union_subset = thm "Ord_Union_subset";

 val Limit_Union_eq = thm "Limit_Union_eq";

 val Limit_is_Ord = thm "Limit_is_Ord";

 val Limit_has_0 = thm "Limit_has_0";

 val Limit_has_succ = thm "Limit_has_succ";

 val non_succ_LimitI = thm "non_succ_LimitI";

 val succ_LimitE = thm "succ_LimitE";

 val not_succ_Limit = thm "not_succ_Limit";

 val Limit_le_succD = thm "Limit_le_succD";

 val Ord_cases_disj = thm "Ord_cases_disj";

 val Ord_cases = thm "Ord_cases";

 val trans_induct3 = thm "trans_induct3";

 *}

 end