(* 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 imports WF Bool equalities begin
definition
Memrel :: "i=>i" where
"Memrel(A) == {z: A*A . EX x y. z=<x,y> & x:y }"
definition
Transset :: "i=>o" where
"Transset(i) == ALL x:i. x<=i"
definition
Ord :: "i=>o" where
"Ord(i) == Transset(i) & (ALL 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 & (ALL y. y<i --> succ(y)<i)"
abbreviation
le (infixl "le" 50) where
"x le y == x < succ(y)"
notation (xsymbols)
le (infixl "\<le>" 50)
notation (HTML output)
le (infixl "\<le>" 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)
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)
(*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
end