# Theory Ordinal

```(*  Title:      ZF/Ordinal.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section‹Transitive Sets and Ordinals›

theory Ordinal imports WF Bool equalities begin

definition
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 ‹≤› 50) where
"x ≤ y ≡ x < succ(y)"

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) <-> ⋃(succ(A)) = A"
unfolding Transset_def
apply (blast elim!: equalityE)
done

lemma Transset_iff_Union_subset: "Transset(A) <-> ⋃(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 (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 ∪ 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(⋃(A))"
by (unfold Transset_def, blast)

lemma Transset_Union_family:
"⟦⋀i. i∈A ⟹ Transset(i)⟧ ⟹ Transset(⋃(A))"
by (unfold Transset_def, blast)

lemma Transset_Inter_family:
"⟦⋀i. i∈A ⟹ Transset(i)⟧ ⟹ Transset(⋂(A))"
by (unfold Inter_def Transset_def, blast)

lemma Transset_UN:
"(⋀x. x ∈ A ⟹ Transset(B(x))) ⟹ Transset (⋃x∈A. B(x))"
by (rule Transset_Union_family, auto)

lemma Transset_INT:
"(⋀x. x ∈ A ⟹ Transset(B(x))) ⟹ Transset (⋂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)"

lemma Ord_is_Transset: "Ord(i) ⟹ Transset(i)"

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)"
unfolding Ord_def
apply (blast intro!: Transset_Un)
done

lemma Ord_Int [TC]: "⟦Ord(i); Ord(j)⟧ ⟹ Ord(i ∩ j)"
unfolding Ord_def
apply (blast intro!: Transset_Int)
done

text‹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)"
moreover have "⋀x. x ∈ X ⟹ Transset(x)"
ultimately have "Ord(X)" by (rule OrdI)
hence "X ∈ X" by (simp add: X)
thus "False" by (rule mem_irrefl)
qed

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"
unfolding 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)

(* @{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)"
unfolding 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

text‹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"

lemma le_eqI: "⟦i=j;  Ord(j)⟧ ⟹ i ≤ j"

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"

lemma leE:
"⟦i ≤ j;  i<j ⟹ P;  ⟦i=j;  Ord(j)⟧ ⟹ P⟧ ⟹ P"

lemma le_anti_sym: "⟦i ≤ j;  j ≤ i⟧ ⟹ i=j"
apply (blast elim: lt_asym)
done

lemma 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 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))"

(*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))"
unfolding 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:
"∀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 (erule wf_Memrel [THEN wf_induct2], blast+)
done

(*Induction over an ordinal*)
lemma Ord_induct [consumes 2]:
"i ∈ k ⟹ Ord(k) ⟹ (⋀x. x ∈ k ⟹ (⋀y. y ∈ x ⟹ P(y)) ⟹ P(x)) ⟹ P(i)"
using Transset_induct [OF _ Ord_is_Transset, of i k P] by simp

(*Induction over the class of ordinals -- a useful corollary of Ord_induct*)
lemma trans_induct [consumes 1, case_names step]:
"Ord(i) ⟹ (⋀x. Ord(x) ⟹ (⋀y. 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

section‹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)
qed
qed

text‹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 (rule_tac i1=i and j1=j in Ord_linear [THEN disjE])
apply (blast intro: o)+
done

lemma 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) +
done

lemma 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) +
done

lemma 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 ‹<›, ‹≤››

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)
done

lemma Ord_0_lt_iff: "Ord(i) ⟹ i≠0 <-> 0<i"
by (blast intro: Ord_0_lt)

(** 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)
done

lemma 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 blast
done

(*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)+
done

lemma 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) +
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) ≠ 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) ∈ succ(j) <-> i∈j"
apply (insert succ_le_iff [of i j])
done

subsubsection‹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)
done

lemma 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)
done

lemma 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])
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)
done

lemma 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 (blast intro: leI lt_trans2)+
done

lemma 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"

lemma Un_upper2_lt: "⟦k < j; Ord(i)⟧ ⟹ k < i ∪ j"

lemma Ord_Union_succ_eq: "Ord(i) ⟹ ⋃(succ(i)) = i"
by (blast intro: Ord_trans)

lemma Ord_Union [intro,simp,TC]: "⟦⋀i. i∈A ⟹ Ord(i)⟧ ⟹ Ord(⋃(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(⋃x∈A. B(x))"
by (rule Ord_Union, blast)

lemma Ord_Inter [intro,simp,TC]:
"⟦⋀i. i∈A ⟹ Ord(i)⟧ ⟹ Ord(⋂(A))"
apply (rule Transset_Inter_family [THEN OrdI])
apply (blast intro: Ord_is_Transset)
apply (blast intro: Ord_contains_Transset)
done

lemma Ord_INT [intro,simp,TC]:
"⟦⋀x. x∈A ⟹ Ord(B(x))⟧ ⟹ Ord(⋂x∈A. B(x))"
by (rule Ord_Inter, blast)

(* No < version of this theorem: consider that @{term"(⋃i∈nat.i)=nat"}! *)
lemma UN_least_le:
"⟦Ord(i);  ⋀x. x∈A ⟹ b(x) ≤ i⟧ ⟹ (⋃x∈A. b(x)) ≤ 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⟧ ⟹ (⋃x∈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∈A;  i < b(a);  Ord(⋃x∈A. b(x))⟧ ⟹ i < (⋃x∈A. b(x))"
by (unfold lt_def, blast)

lemma UN_upper_le:
"⟦a ∈ A;  i ≤ b(a);  Ord(⋃x∈A. b(x))⟧ ⟹ i ≤ (⋃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)+
done

lemma lt_Union_iff: "∀i∈A. Ord(i) ⟹ (j < ⋃(A)) <-> (∃i∈A. j<i)"
by (auto simp: lt_def Ord_Union)

lemma Union_upper_le:
"⟦j ∈ J;  i≤j;  Ord(⋃(J))⟧ ⟹ i ≤ ⋃J"
apply (subst Union_eq_UN)
apply (rule UN_upper_le, auto)
done

lemma le_implies_UN_le_UN:
"⟦⋀x. x∈A ⟹ c(x) ≤ d(x)⟧ ⟹ (⋃x∈A. c(x)) ≤ (⋃x∈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) ⟹ (⋃y∈i. succ(y)) = i"
by (blast intro: Ord_trans)

(*Holds for all transitive sets, not just ordinals*)
lemma Ord_Union_subset: "Ord(i) ⟹ ⋃(i) ⊆ i"
by (blast intro: Ord_trans)

subsection‹Limit Ordinals -- General Properties›

lemma Limit_Union_eq: "Limit(i) ⟹ ⋃(i) = i"
unfolding Limit_def
apply (fast intro!: ltI elim!: ltE elim: Ord_trans)
done

lemma Limit_is_Ord: "Limit(i) ⟹ Ord(i)"
unfolding Limit_def
apply (erule conjunct1)
done

lemma Limit_has_0: "Limit(i) ⟹ 0 < i"
unfolding 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)"

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)
done

lemma 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)
qed

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 ≤ 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+)
done

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) ⟹ (⋀y. y ∈ x ⟹ P(y)) ⟹ P(x)) ⟹ P(i)"
using trans_induct3_raw [of i P] by simp

text‹A set of ordinals is either empty, contains its own union, or its
union is a limit ordinal.›

lemma Union_le: "⟦⋀x. x∈I ⟹ x≤j; Ord(j)⟧ ⟹ ⋃(I) ≤ j"
by (auto simp add: le_subset_iff Union_least)

lemma Ord_set_cases:
assumes I: "∀i∈I. Ord(i)"
shows "I=0 ∨ ⋃(I) ∈ I ∨ (⋃(I) ∉ I ∧ Limit(⋃(I)))"
proof (cases "⋃(I)" rule: Ord_cases)
show "Ord(⋃I)" using I by (blast intro: Ord_Union)
next
assume "⋃I = 0" thus ?thesis by (simp, blast intro: subst_elem)
next
fix j
assume j: "Ord(j)" and UIj:"⋃(I) = succ(j)"
{ assume "∀i∈I. i≤j"
hence "⋃(I) ≤ 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 "⋃(I) ≤ succ(j)" using UIj j by auto
hence "i ≤ succ(j)" using i
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(⋃I)" thus ?thesis by auto
qed

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: "⟦∀x∈X. Ord(x);  ⋃X = succ(j)⟧ ⟹ succ(j) ∈ X"
by (drule Ord_set_cases, auto)

lemma Limit_Union [rule_format]: "⟦I ≠ 0;  (⋀i. i∈I ⟹ Limit(i))⟧ ⟹ Limit(⋃I)"