(* Title: ZF/Ordinal.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
*)
section\<open>Transitive Sets and Ordinals\<close>
theory Ordinal imports WF Bool equalities begin
definition
Memrel :: "i=>i" where
"Memrel(A) == {z\A*A . \x y. z= & 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 \<open><\<close> 50) (*less-than on ordinals*) where
"ij & Ord(j)"
definition
Limit :: "i=>o" where
"Limit(i) == Ord(i) & 0y. y succ(y)
abbreviation
le (infixl \<open>\<le>\<close> 50) where
"x \ y == x < succ(y)"
subsection\<open>Rules for Transset\<close>
subsubsection\<open>Three Neat Characterisations of Transset\<close>
lemma Transset_iff_Pow: "Transset(A) <-> A<=Pow(A)"
by (unfold Transset_def, blast)
lemma Transset_iff_Union_succ: "Transset(A) <-> \(succ(A)) = A"
apply (unfold Transset_def)
apply (blast elim!: equalityE)
done
lemma Transset_iff_Union_subset: "Transset(A) <-> \(A) \ A"
by (unfold Transset_def, blast)
subsubsection\<open>Consequences of Downwards Closure\<close>
lemma Transset_doubleton_D:
"[| Transset(C); {a,b}: C |] ==> a\C & b\C"
by (unfold Transset_def, blast)
lemma Transset_Pair_D:
"[| Transset(C); \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\<open>Closure Properties\<close>
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\<open>Lemmas for Ordinals\<close>
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\<open>The Construction of Ordinals: 0, succ, Union\<close>
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)
done
lemma Ord_Int [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i \ j)"
apply (unfold Ord_def)
apply (blast intro!: Transset_Int)
done
text\<open>There is no set of all ordinals, for then it would contain itself\<close>
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)
qed
subsection\<open>< is 'less Than' for Ordinals\<close>
lemma ltI: "[| i\j; Ord(j) |] ==> i
by (unfold lt_def, blast)
lemma ltE:
"[| ij; Ord(i); Ord(j) |] ==> P |] ==> P"
apply (unfold lt_def)
apply (blast intro: Ord_in_Ord)
done
lemma ltD: "i i\j"
by (erule ltE, assumption)
lemma not_lt0 [simp]: "~ i<0"
by (unfold lt_def, blast)
lemma lt_Ord: "j Ord(j)"
by (erule ltE, assumption)
lemma lt_Ord2: "j Ord(i)"
by (erule ltE, assumption)
(* @{term"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 [trans]: "[| i i
by (blast intro!: ltI elim!: ltE intro: Ord_trans)
lemma lt_not_sym: "i ~ (j
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 P"
by (blast intro: lt_asym)
lemma lt_not_refl: "~ i
apply (rule notI)
apply (erule lt_irrefl)
done
text\<open>Recall that \<^term>\<open>i \<le> j\<close> abbreviates \<^term>\<open>i<succ(j)\<close> !!\<close>
lemma le_iff: "i \ j <-> i
by (unfold lt_def, blast)
(*Equivalently, i<j ==> i < succ(j)*)
lemma leI: "i 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 i \ j"
by (simp add: le_iff, blast)
lemma leE:
"[| i \ j; i 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)
done
lemma le0_iff [simp]: "i \ 0 <-> i=0"
by (blast elim!: leE)
lemmas le0D = le0_iff [THEN iffD1, dest!]
subsection\<open>Natural Deduction Rules for Memrel\<close>
(*The lemmas MemrelI/E give better speed than [iff] here*)
lemma Memrel_iff [simp]: " \ Memrel(A) <-> a\b & a\A & b\A"
by (unfold Memrel_def, blast)
lemma MemrelI [intro!]: "[| a \ b; a \ A; b \ A |] ==> \ Memrel(A)"
by auto
lemma MemrelE [elim!]:
"[| \ Memrel(A);
[| a \<in> A; b \<in> A; a\<in>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)
(*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\<open>The premise \<^term>\<open>Ord(i)\<close> does not suffice.\<close>
lemma trans_Memrel:
"Ord(i) ==> trans(Memrel(i))"
by (unfold Ord_def Transset_def trans_def, blast)
text\<open>However, the following premise is strong enough.\<close>
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) ==> \ Memrel(A) <-> a\b & b\A"
by (unfold Transset_def, blast)
subsection\<open>Transfinite Induction\<close>
(*Epsilon induction over a transitive set*)
lemma Transset_induct:
"[| i \ k; Transset(k);
!!x.[| x \<in> k; \<forall>y\<in>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*)
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\<open>Fundamental properties of the epsilon ordering (< on ordinals)\<close>
subsubsection\<open>Proving That < is a Linear Ordering on the Ordinals\<close>
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 \<open>Ord(j)\<close>
proof (induct j rule: trans_induct)
case (step j)
thus ?case using step_i
by (blast dest: Ord_trans)
qed
qed
text\<open>The trichotomy law for ordinals\<close>
lemma Ord_linear_lt:
assumes o: "Ord(i)" "Ord(j)"
obtains (lt) "i | (eq) "i=j" | (gt) "j
apply (simp add: lt_def)
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 | (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
by (blast elim!: leE elim: lt_asym)
lemma not_lt_imp_le: "[| ~ i j \ i"
by (rule_tac i = i and j = j in Ord_linear2, auto)
subsubsection \<open>Some Rewrite Rules for \<open><\<close>, \<open>\<le>\<close>\<close>
lemma Ord_mem_iff_lt: "Ord(j) ==> i\j <-> i
by (unfold lt_def, blast)
lemma not_lt_iff_le: "[| Ord(i); Ord(j) |] ==> ~ i 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
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
apply (erule not_le_iff_lt [THEN iffD1])
apply (rule Ord_0, blast)
done
lemma Ord_0_lt_iff: "Ord(i) ==> i\0 <-> 0
by (blast intro: Ord_0_lt)
subsection\<open>Results about Less-Than or Equals\<close>
(** For ordinals, @{term"j\<subseteq>i"} implies @{term"j \<le> 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 (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 x j \ i"
by (blast intro: not_lt_imp_le dest: lt_irrefl)
subsubsection\<open>Transitivity Laws\<close>
lemma lt_trans1: "[| i \ j; j i
by (blast elim!: leE intro: lt_trans)
lemma lt_trans2: "[| i k |] ==> i
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 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
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
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 i
apply (rule subset_imp_le [THEN lt_trans1])
apply (blast intro: elim: ltE) +
done
lemma lt_imp_0_lt: "j 0
by (blast intro: lt_trans1 Ord_0_le [OF lt_Ord])
lemma succ_lt_iff: "succ(i) < j <-> 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])
apply (simp add: lt_def)
done
subsubsection\<open>Union and Intersection\<close>
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 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
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])
apply (simp add: lt_def)
done
(*Replacing k by succ(k') yields the similar rule for le!*)
lemma Int_greatest_lt: "[| i 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
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)+
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"
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) ==> \(succ(i)) = i"
by (blast intro: Ord_trans)
subsection\<open>Results about Limits\<close>
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 (simp add: Inter_def)
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"(\<Union>i\<in>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:
"[| jA ==> b(x) (\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
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\<open>Limit Ordinals -- General Properties\<close>
lemma Limit_Union_eq: "Limit(i) ==> \(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 succ(j) < i"
by (unfold Limit_def, blast)
lemma Limit_succ_lt_iff [simp]: "Limit(i) ==> succ(j) < i <-> (j
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: "[| 0x\l. \y\l. x 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 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
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\<open>Traditional 3-Way Case Analysis on Ordinals\<close>
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); \<forall>y\<in>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\<open>A set of ordinals is either empty, contains its own union, or its
union is a limit ordinal.\<close>
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"
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 "\(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(\I)" thus ?thesis by auto
qed
text\<open>If the union of a set of ordinals is a successor, then it is an element of that set.\<close>
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)"
apply (simp add: Limit_def lt_def)
apply (blast intro!: equalityI)
done
end
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