# Theory Ordinal

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theory Ordinal
imports WF
`(*  Title:      ZF/Ordinal.thy    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory    Copyright   1994  University of Cambridge*)header{*Transitive Sets and Ordinals*}theory Ordinal imports WF Bool equalities begindefinition  Memrel        :: "i=>i"  where    "Memrel(A)   == {z∈A*A . ∃x y. z=<x,y> & x∈y }"definition  Transset  :: "i=>o"  where    "Transset(i) == ∀x∈i. x<=i"definition  Ord  :: "i=>o"  where    "Ord(i)      == Transset(i) & (∀x∈i. Transset(x))"definition  lt        :: "[i,i] => o"  (infixl "<" 50)   (*less-than on ordinals*)  where    "i<j         == i∈j & Ord(j)"definition  Limit         :: "i=>o"  where    "Limit(i)    == Ord(i) & 0<i & (∀y. y<i --> succ(y)<i)"abbreviation  le  (infixl "le" 50) where  "x le y == x < succ(y)"notation (xsymbols)  le  (infixl "≤" 50)notation (HTML output)  le  (infixl "≤" 50)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)donelemma 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)donelemma 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 ∪ j)"by (unfold Transset_def, blast)lemma Transset_Int:    "[| Transset(i);  Transset(j) |] ==> Transset(i ∩ 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 ∈ A ==> Transset(B(x))) ==> Transset (\<Union>x∈A. B(x))"by (rule Transset_Union_family, auto)lemma Transset_INT:     "(!!x. x ∈ A ==> Transset(B(x))) ==> Transset (\<Inter>x∈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 ∪ j)"apply (unfold Ord_def)apply (blast intro!: Transset_Un)donelemma Ord_Int [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i ∩ j)"apply (unfold Ord_def)apply (blast intro!: Transset_Int)donetext{*There is no set of all ordinals, for then it would contain itself*}lemma ON_class: "~ (∀i. i∈X <-> Ord(i))"proof (rule notI)  assume X: "∀i. i ∈ X <-> Ord(i)"  have "∀x y. x∈X --> y∈x --> y∈X"    by (simp add: X, blast intro: Ord_in_Ord)  hence "Transset(X)"     by (auto simp add: Transset_def)  moreover have "!!x. x ∈ X ==> Transset(x)"     by (simp add: X Ord_def)  ultimately have "Ord(X)" by (rule OrdI)  hence "X ∈ X" by (simp add: X)  thus "False" by (rule mem_irrefl)qedsubsection{*< 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)donelemma 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)(* @{term"ja ≤ j ==> Ord(j)"} *)lemmas le_Ord2 = lt_Ord2 [THEN Ord_succD](* i<0 ==> R *)lemmas lt0E = not_lt0 [THEN notE, elim!]lemma lt_trans [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)donetext{* Recall that  @{term"i ≤ j"}  abbreviates  @{term"i<succ(j)"} !! *}lemma le_iff: "i ≤ j <-> i<j | (i=j & Ord(j))"by (unfold lt_def, blast)(*Equivalently, i<j ==> i < succ(j)*)lemma leI: "i<j ==> i ≤ j"by (simp add: le_iff)lemma le_eqI: "[| i=j;  Ord(j) |] ==> i ≤ j"by (simp add: le_iff)lemmas le_refl = refl [THEN le_eqI]lemma le_refl_iff [iff]: "i ≤ i <-> Ord(i)"by (simp (no_asm_simp) add: lt_not_refl le_iff)lemma leCI: "(~ (i=j & Ord(j)) ==> i<j) ==> i ≤ j"by (simp add: le_iff, blast)lemma leE:    "[| i ≤ j;  i<j ==> P;  [| i=j;  Ord(j) |] ==> P |] ==> P"by (simp add: le_iff, blast)lemma le_anti_sym: "[| i ≤ j;  j ≤ i |] ==> i=j"apply (simp add: le_iff)apply (blast elim: lt_asym)donelemma le0_iff [simp]: "i ≤ 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 autolemma MemrelE [elim!]:    "[| <a,b> ∈ Memrel(A);        [| a ∈ A;  b ∈ A;  a∈b |]  ==> P |]     ==> P"by autolemma 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)(*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)donetext{*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:    "∀j∈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;  ∀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 [rule_format, OF _ Ord_is_Transset](*Induction over the class of ordinals -- a useful corollary of Ord_induct*)lemma trans_induct [rule_format, consumes 1, case_names step]:    "[| Ord(i);        !!x.[| Ord(x);  ∀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])donesection{*Fundamental properties of the epsilon ordering (< on ordinals)*}subsubsection{*Proving That < is a Linear Ordering on the Ordinals*}lemma Ord_linear:     "Ord(i) ==> Ord(j) ==> i∈j | i=j | j∈i"proof (induct i arbitrary: j rule: trans_induct)  case (step i)  note step_i = step  show ?case using `Ord(j)`    proof (induct j rule: trans_induct)      case (step j)      thus ?case using step_i        by (blast dest: Ord_trans)    qedqedtext{*The trichotomy law for ordinals*}lemma Ord_linear_lt: assumes o: "Ord(i)" "Ord(j)" obtains (lt) "i<j" | (eq) "i=j" | (gt) "j<i"apply (simp add: lt_def)apply (rule_tac i1=i and j1=j in Ord_linear [THEN disjE])apply (blast intro: o)+donelemma Ord_linear2: assumes o: "Ord(i)" "Ord(j)" obtains (lt) "i<j" | (ge) "j ≤ i"apply (rule_tac i = i and j = j in Ord_linear_lt)apply (blast intro: leI le_eqI sym o) +donelemma Ord_linear_le: assumes o: "Ord(i)" "Ord(j)" obtains (le) "i ≤ j" | (ge) "j ≤ i"apply (rule_tac i = i and j = j in Ord_linear_lt)apply (blast intro: leI le_eqI o) +donelemma le_imp_not_lt: "j ≤ i ==> ~ i<j"by (blast elim!: leE elim: lt_asym)lemma not_lt_imp_le: "[| ~ i<j;  Ord(i);  Ord(j) |] ==> j ≤ 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 ≤ i"by (blast dest: le_imp_not_lt not_lt_imp_le)lemma not_le_iff_lt: "[| Ord(i);  Ord(j) |] ==> ~ i ≤ 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 ≤ 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)donelemma 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, @{term"j⊆i"} implies @{term"j ≤ i"} (less-than or equals) **)lemma zero_le_succ_iff [iff]: "0 ≤ succ(x) <-> Ord(x)"by (blast intro: Ord_0_le elim: ltE)lemma subset_imp_le: "[| j<=i;  Ord(i);  Ord(j) |] ==> j ≤ i"apply (rule not_lt_iff_le [THEN iffD1], assumption+)apply (blast elim: ltE mem_irrefl)donelemma le_imp_subset: "i ≤ j ==> i<=j"by (blast dest: OrdmemD elim: ltE leE)lemma le_subset_iff: "j ≤ i <-> j<=i & Ord(i) & Ord(j)"by (blast dest: subset_imp_le le_imp_subset elim: ltE)lemma le_succ_iff: "i ≤ succ(j) <-> i ≤ j | i=succ(j) & Ord(i)"apply (simp (no_asm) add: le_iff)apply blastdone(*Just a variant of subset_imp_le*)lemma all_lt_imp_le: "[| Ord(i);  Ord(j);  !!x. x<j ==> x<i |] ==> j ≤ i"by (blast intro: not_lt_imp_le dest: lt_irrefl)subsubsection{*Transitivity Laws*}lemma lt_trans1: "[| i ≤ j;  j<k |] ==> i<k"by (blast elim!: leE intro: lt_trans)lemma lt_trans2: "[| i<j;  j ≤ k |] ==> i<k"by (blast elim!: leE intro: lt_trans)lemma le_trans: "[| i ≤ j;  j ≤ k |] ==> i ≤ k"by (blast intro: lt_trans1)lemma succ_leI: "i<j ==> succ(i) ≤ 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) ≤ j ==> i<j"apply (rule not_le_iff_lt [THEN iffD1])apply (blast elim: ltE leE lt_asym)+donelemma succ_le_iff [iff]: "succ(i) ≤ j <-> i<j"by (blast intro: succ_leI succ_leE)lemma succ_le_imp_le: "succ(i) ≤ succ(j) ==> i ≤ 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) +donelemma 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) ≠ j"apply autoapply (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 blastapply (simp add: lt_Ord lt_Ord2 le_iff)apply (blast dest: lt_asym)donelemma Ord_succ_mem_iff: "Ord(j) ==> succ(i) ∈ succ(j) <-> i∈j"apply (insert succ_le_iff [of i j])apply (simp add: lt_def)donesubsubsection{*Union and Intersection*}lemma Un_upper1_le: "[| Ord(i); Ord(j) |] ==> i ≤ i ∪ j"by (rule Un_upper1 [THEN subset_imp_le], auto)lemma Un_upper2_le: "[| Ord(i); Ord(j) |] ==> j ≤ i ∪ 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 ∪ 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)donelemma Un_least_lt_iff: "[| Ord(i); Ord(j) |] ==> i ∪ 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)donelemma Un_least_mem_iff:    "[| Ord(i); Ord(j); Ord(k) |] ==> i ∪ 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 ∩ 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)donelemma Ord_Un_if:     "[| Ord(i); Ord(j) |] ==> i ∪ 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 ∪ j) = succ(i) ∪ succ(j)"by (simp add: Ord_Un_if lt_Ord le_Ord2)lemma lt_Un_iff:     "[| Ord(i); Ord(j) |] ==> k < i ∪ j <-> k < i | k < j"apply (simp add: Ord_Un_if not_lt_iff_le)apply (blast intro: leI lt_trans2)+donelemma le_Un_iff:     "[| Ord(i); Ord(j) |] ==> k ≤ i ∪ j <-> k ≤ i | k ≤ j"by (simp add: succ_Un_distrib lt_Un_iff [symmetric])lemma Un_upper1_lt: "[|k < i; Ord(j)|] ==> k < i ∪ j"by (simp add: lt_Un_iff lt_Ord2)lemma Un_upper2_lt: "[|k < j; Ord(i)|] ==> k < i ∪ 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)+donelemma Ord_UN [intro,simp,TC]:     "[| !!x. x∈A ==> Ord(B(x)) |] ==> Ord(\<Union>x∈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)donelemma Ord_INT [intro,simp,TC]:    "[| !!x. x∈A ==> Ord(B(x)) |] ==> Ord(\<Inter>x∈A. B(x))"by (rule Ord_Inter, blast)(* No < version of this theorem: consider that @{term"(\<Union>i∈nat.i)=nat"}! *)lemma UN_least_le:    "[| Ord(i);  !!x. x∈A ==> b(x) ≤ i |] ==> (\<Union>x∈A. b(x)) ≤ i"apply (rule le_imp_subset [THEN UN_least, THEN subset_imp_le])apply (blast intro: Ord_UN elim: ltE)+donelemma UN_succ_least_lt:    "[| j<i;  !!x. x∈A ==> b(x)<j |] ==> (\<Union>x∈A. succ(b(x))) < i"apply (rule ltE, assumption)apply (rule UN_least_le [THEN lt_trans2])apply (blast intro: succ_leI)+donelemma UN_upper_lt:     "[| a∈A;  i < b(a);  Ord(\<Union>x∈A. b(x)) |] ==> i < (\<Union>x∈A. b(x))"by (unfold lt_def, blast)lemma UN_upper_le:     "[| a ∈ A;  i ≤ b(a);  Ord(\<Union>x∈A. b(x)) |] ==> i ≤ (\<Union>x∈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)+donelemma lt_Union_iff: "∀i∈A. Ord(i) ==> (j < \<Union>(A)) <-> (∃i∈A. j<i)"by (auto simp: lt_def Ord_Union)lemma Union_upper_le:     "[| j ∈ J;  i≤j;  Ord(\<Union>(J)) |] ==> i ≤ \<Union>J"apply (subst Union_eq_UN)apply (rule UN_upper_le, auto)donelemma le_implies_UN_le_UN:    "[| !!x. x∈A ==> c(x) ≤ d(x) |] ==> (\<Union>x∈A. c(x)) ≤ (\<Union>x∈A. d(x))"apply (rule UN_least_le)apply (rule_tac [2] UN_upper_le)apply (blast intro: Ord_UN le_Ord2)+donelemma Ord_equality: "Ord(i) ==> (\<Union>y∈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)donelemma Limit_is_Ord: "Limit(i) ==> Ord(i)"apply (unfold Limit_def)apply (erule conjunct1)donelemma Limit_has_0: "Limit(i) ==> 0 < i"apply (unfold Limit_def)apply (erule conjunct2 [THEN conjunct1])donelemma 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)donelemma 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; ∀x∈l. ∃y∈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)donelemma non_succ_LimitI:  assumes i: "0<i" and nsucc: "!!y. succ(y) ≠ i"  shows "Limit(i)"proof -  have Oi: "Ord(i)" using i by (simp add: lt_def)  { fix y    assume yi: "y<i"    hence Osy: "Ord(succ(y))" by (simp add: lt_Ord Ord_succ)    have "~ i ≤ y" using yi by (blast dest: le_imp_not_lt)    hence "succ(y) < i" using nsucc [of y]      by (blast intro: Ord_linear_lt [OF Osy Oi]) }  thus ?thesis using i Oi by (auto simp add: Limit_def)qedlemma 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])donelemma not_succ_Limit [simp]: "~ Limit(succ(i))"by blastlemma Limit_le_succD: "[| Limit(i);  i ≤ succ(j) |] ==> i ≤ j"by (blast elim!: leE)subsubsection{*Traditional 3-Way Case Analysis on Ordinals*}lemma Ord_cases_disj: "Ord(i) ==> i=0 | (∃j. Ord(j) & i=succ(j)) | Limit(i)"by (blast intro!: non_succ_LimitI Ord_0_lt)lemma Ord_cases: assumes i: "Ord(i)" obtains ("0") "i=0" | (succ) j where "Ord(j)" "i=succ(j)" | (limit) "Limit(i)"by (insert Ord_cases_disj [OF i], auto)lemma trans_induct3_raw:     "[| Ord(i);         P(0);         !!x. [| Ord(x);  P(x) |] ==> P(succ(x));         !!x. [| Limit(x);  ∀y∈x. P(y) |] ==> P(x)      |] ==> P(i)"apply (erule trans_induct)apply (erule Ord_cases, blast+)donelemmas trans_induct3 = trans_induct3_raw [rule_format, case_names 0 succ limit, consumes 1]text{*A set of ordinals is either empty, contains its own union, or itsunion is a limit ordinal.*}lemma Union_le: "[| !!x. x∈I ==> x≤j; Ord(j) |] ==> \<Union>(I) ≤ j"  by (auto simp add: le_subset_iff Union_least)lemma Ord_set_cases:  assumes I: "∀i∈I. Ord(i)"  shows "I=0 ∨ \<Union>(I) ∈ I ∨ (\<Union>(I) ∉ I ∧ Limit(\<Union>(I)))"proof (cases "\<Union>(I)" rule: Ord_cases)  show "Ord(\<Union>I)" using I by (blast intro: Ord_Union)next  assume "\<Union>I = 0" thus ?thesis by (simp, blast intro: subst_elem)next  fix j  assume j: "Ord(j)" and UIj:"\<Union>(I) = succ(j)"  { assume "∀i∈I. i≤j"    hence "\<Union>(I) ≤ j"      by (simp add: Union_le j)    hence False      by (simp add: UIj lt_not_refl) }  then obtain i where i: "i ∈ I" "succ(j) ≤ i" using I j    by (atomize, auto simp add: not_le_iff_lt)  have "\<Union>(I) ≤ succ(j)" using UIj j by auto  hence "i ≤ succ(j)" using i    by (simp add: le_subset_iff Union_subset_iff)  hence "succ(j) = i" using i    by (blast intro: le_anti_sym)  hence "succ(j) ∈ I" by (simp add: i)  thus ?thesis by (simp add: UIj)next  assume "Limit(\<Union>I)" thus ?thesis by autoqedtext{*If the union of a set of ordinals is a successor, then it is an element of that set.*}lemma Ord_Union_eq_succD: "[|∀x∈X. Ord(x);  \<Union>X = succ(j)|] ==> succ(j) ∈ X"  by (drule Ord_set_cases, auto)lemma Limit_Union [rule_format]: "[| I ≠ 0;  ∀i∈I. Limit(i) |] ==> Limit(\<Union>I)"apply (simp add: Limit_def lt_def)apply (blast intro!: equalityI)doneend`