--- a/src/ZF/Ordinal.thy Wed May 15 13:50:38 2002 +0200
+++ b/src/ZF/Ordinal.thy Thu May 16 09:16:22 2002 +0200
@@ -6,27 +6,728 @@
Ordinals in Zermelo-Fraenkel Set Theory
*)
-Ordinal = WF + Bool + equalities +
-consts
- Memrel :: i=>i
- Transset,Ord :: i=>o
- "<" :: [i,i] => o (infixl 50) (*less than on ordinals*)
- Limit :: i=>o
+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*)
+ "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*)
+ "op le" :: "[i,i] => o" (infixl "\<le>" 50) (*less-than or equals*)
+
+
+(*** Rules for Transset ***)
+
+(** 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)
+
+(** 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)
+
+(** 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:
+ "[| j:A; !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))"
+by (unfold Transset_def, blast)
+
+(*** Natural Deduction rules for Ord ***)
+
+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)
+
+(*** Lemmas for ordinals ***)
+
+lemma Ord_in_Ord: "[| Ord(i); j:i |] ==> Ord(j)"
+by (unfold Ord_def Transset_def, blast)
+
+(* 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)
+
+
+(*** 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 [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
+
+
+lemma Ord_Inter:
+ "[| j:A; !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))"
+apply (rule Transset_Inter_family [THEN OrdI], assumption)
+apply (blast intro: Ord_is_Transset)
+apply (blast intro: Ord_contains_Transset)
+done
+
+lemma Ord_INT:
+ "[| j:A; !!x. x:A ==> Ord(B(x)) |] ==> Ord(INT x:A. B(x))"
+by (rule RepFunI [THEN Ord_Inter], assumption, blast)
+
+(*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
+
+(*** < 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!]
+
+(*** 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)
+
+(*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
+
+(*Transset(i) does not suffice, though ALL j:i.Transset(j) does*)
+lemma trans_Memrel:
+ "Ord(i) ==> trans(Memrel(i))"
+by (unfold Ord_def 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)
+
+
+(*** 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)
+apply blast
+done
+
+(*Induction over an ordinal*)
+lemmas Ord_induct = Transset_induct [OF _ Ord_is_Transset]
+
+(*Induction over the class of ordinals -- a useful corollary of Ord_induct*)
+
+lemma trans_induct:
+ "[| 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
+
+
+(*** Fundamental properties of the epsilon ordering (< on ordinals) ***)
+
+
+(** 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)
+
+(** 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)
-defs
- Memrel_def "Memrel(A) == {z: A*A . EX x y. z=<x,y> & x:y }"
- Transset_def "Transset(i) == ALL x:i. x<=i"
- Ord_def "Ord(i) == Transset(i) & (ALL x:i. Transset(x))"
- lt_def "i<j == i:j & Ord(j)"
- Limit_def "Limit(i) == Ord(i) & 0<i & (ALL y. y<i --> succ(y)<i)"
+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)
+
+
+(*** 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 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)
+
+(** Transitive 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
+
+(** 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
+
+(*FIXME: the Intersection duals are missing!*)
+
+(*** Results about limits ***)
+
+lemma Ord_Union: "[| !!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: "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(UN x:A. B(x))"
+by (rule Ord_Union, blast)
+
+(* No < version; consider (UN i:nat.i)=nat *)
+lemma UN_least_le:
+ "[| Ord(i); !!x. x:A ==> b(x) le i |] ==> (UN x: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 |] ==> (UN 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_le:
+ "[| a: A; i le b(a); Ord(UN x:A. b(x)) |] ==> i le (UN 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 le_implies_UN_le_UN:
+ "[| !!x. x:A ==> c(x) le d(x) |] ==> (UN x:A. c(x)) le (UN 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) ==> (UN 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)
+
+
+(*** 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_has_succ: "[| Limit(i); j<i |] ==> succ(j) < i"
+by (unfold Limit_def, blast)
+
+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)
+
+(** 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:
+ "[| 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
+
+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