src/ZF/Ordinal.thy
author paulson
Mon Jun 24 11:58:21 2002 +0200 (2002-06-24)
changeset 13243 ba53d07d32d5
parent 13203 fac77a839aa2
child 13269 3ba9be497c33
permissions -rw-r--r--
new lemmas
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(*  Title:      ZF/Ordinal.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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Ordinals in Zermelo-Fraenkel Set Theory 
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*)
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theory Ordinal = WF + Bool + equalities:
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constdefs
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  Memrel        :: "i=>i"
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    "Memrel(A)   == {z: A*A . EX x y. z=<x,y> & x:y }"
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  Transset  :: "i=>o"
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    "Transset(i) == ALL x:i. x<=i"
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  Ord  :: "i=>o"
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    "Ord(i)      == Transset(i) & (ALL x:i. Transset(x))"
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  lt        :: "[i,i] => o"  (infixl "<" 50)   (*less-than on ordinals*)
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    "i<j         == i:j & Ord(j)"
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  Limit         :: "i=>o"
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    "Limit(i)    == Ord(i) & 0<i & (ALL y. y<i --> succ(y)<i)"
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syntax
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  "le"          :: "[i,i] => o"  (infixl 50)   (*less-than or equals*)
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translations
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  "x le y"      == "x < succ(y)"
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syntax (xsymbols)
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  "op le"       :: "[i,i] => o"  (infixl "\<le>" 50)  (*less-than or equals*)
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(*** Rules for Transset ***)
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(** Three neat characterisations of Transset **)
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lemma Transset_iff_Pow: "Transset(A) <-> A<=Pow(A)"
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by (unfold Transset_def, blast)
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lemma Transset_iff_Union_succ: "Transset(A) <-> Union(succ(A)) = A"
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apply (unfold Transset_def)
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apply (blast elim!: equalityE)
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done
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lemma Transset_iff_Union_subset: "Transset(A) <-> Union(A) <= A"
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by (unfold Transset_def, blast)
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(** Consequences of downwards closure **)
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lemma Transset_doubleton_D: 
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    "[| Transset(C); {a,b}: C |] ==> a:C & b: C"
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by (unfold Transset_def, blast)
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lemma Transset_Pair_D:
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    "[| Transset(C); <a,b>: C |] ==> a:C & b: C"
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apply (simp add: Pair_def)
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apply (blast dest: Transset_doubleton_D)
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done
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lemma Transset_includes_domain:
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    "[| Transset(C); A*B <= C; b: B |] ==> A <= C"
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by (blast dest: Transset_Pair_D)
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lemma Transset_includes_range:
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    "[| Transset(C); A*B <= C; a: A |] ==> B <= C"
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by (blast dest: Transset_Pair_D)
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(** Closure properties **)
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lemma Transset_0: "Transset(0)"
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by (unfold Transset_def, blast)
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lemma Transset_Un: 
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    "[| Transset(i);  Transset(j) |] ==> Transset(i Un j)"
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by (unfold Transset_def, blast)
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lemma Transset_Int: 
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    "[| Transset(i);  Transset(j) |] ==> Transset(i Int j)"
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by (unfold Transset_def, blast)
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lemma Transset_succ: "Transset(i) ==> Transset(succ(i))"
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by (unfold Transset_def, blast)
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lemma Transset_Pow: "Transset(i) ==> Transset(Pow(i))"
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by (unfold Transset_def, blast)
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lemma Transset_Union: "Transset(A) ==> Transset(Union(A))"
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by (unfold Transset_def, blast)
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lemma Transset_Union_family: 
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    "[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))"
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by (unfold Transset_def, blast)
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lemma Transset_Inter_family: 
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    "[| !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))"
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by (unfold Inter_def Transset_def, blast)
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lemma Transset_UN:
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     "(!!x. x \<in> A ==> Transset(B(x))) ==> Transset (UN x:A. B(x))"
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by (rule Transset_Union_family, auto) 
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lemma Transset_INT:
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     "(!!x. x \<in> A ==> Transset(B(x))) ==> Transset (INT x:A. B(x))"
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by (rule Transset_Inter_family, auto) 
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(*** Natural Deduction rules for Ord ***)
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lemma OrdI:
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    "[| Transset(i);  !!x. x:i ==> Transset(x) |]  ==>  Ord(i)"
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by (simp add: Ord_def) 
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lemma Ord_is_Transset: "Ord(i) ==> Transset(i)"
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by (simp add: Ord_def) 
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lemma Ord_contains_Transset: 
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    "[| Ord(i);  j:i |] ==> Transset(j) "
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by (unfold Ord_def, blast)
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(*** Lemmas for ordinals ***)
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lemma Ord_in_Ord: "[| Ord(i);  j:i |] ==> Ord(j)"
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by (unfold Ord_def Transset_def, blast)
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(*suitable for rewriting PROVIDED i has been fixed*)
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lemma Ord_in_Ord': "[| j:i; Ord(i) |] ==> Ord(j)"
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by (blast intro: Ord_in_Ord)
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(* Ord(succ(j)) ==> Ord(j) *)
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lemmas Ord_succD = Ord_in_Ord [OF _ succI1]
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lemma Ord_subset_Ord: "[| Ord(i);  Transset(j);  j<=i |] ==> Ord(j)"
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by (simp add: Ord_def Transset_def, blast)
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lemma OrdmemD: "[| j:i;  Ord(i) |] ==> j<=i"
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by (unfold Ord_def Transset_def, blast)
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lemma Ord_trans: "[| i:j;  j:k;  Ord(k) |] ==> i:k"
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by (blast dest: OrdmemD)
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lemma Ord_succ_subsetI: "[| i:j;  Ord(j) |] ==> succ(i) <= j"
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by (blast dest: OrdmemD)
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(*** The construction of ordinals: 0, succ, Union ***)
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lemma Ord_0 [iff,TC]: "Ord(0)"
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by (blast intro: OrdI Transset_0)
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lemma Ord_succ [TC]: "Ord(i) ==> Ord(succ(i))"
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by (blast intro: OrdI Transset_succ Ord_is_Transset Ord_contains_Transset)
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lemmas Ord_1 = Ord_0 [THEN Ord_succ]
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lemma Ord_succ_iff [iff]: "Ord(succ(i)) <-> Ord(i)"
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by (blast intro: Ord_succ dest!: Ord_succD)
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lemma Ord_Un [intro,simp,TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Un j)"
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apply (unfold Ord_def)
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apply (blast intro!: Transset_Un)
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done
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lemma Ord_Int [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Int j)"
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apply (unfold Ord_def)
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apply (blast intro!: Transset_Int)
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done
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(*There is no set of all ordinals, for then it would contain itself*)
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lemma ON_class: "~ (ALL i. i:X <-> Ord(i))"
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apply (rule notI)
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apply (frule_tac x = "X" in spec)
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apply (safe elim!: mem_irrefl)
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apply (erule swap, rule OrdI [OF _ Ord_is_Transset])
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apply (simp add: Transset_def)
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apply (blast intro: Ord_in_Ord)+
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done
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(*** < is 'less than' for ordinals ***)
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lemma ltI: "[| i:j;  Ord(j) |] ==> i<j"
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by (unfold lt_def, blast)
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lemma ltE:
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    "[| i<j;  [| i:j;  Ord(i);  Ord(j) |] ==> P |] ==> P"
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apply (unfold lt_def)
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apply (blast intro: Ord_in_Ord)
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done
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lemma ltD: "i<j ==> i:j"
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by (erule ltE, assumption)
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lemma not_lt0 [simp]: "~ i<0"
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by (unfold lt_def, blast)
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lemma lt_Ord: "j<i ==> Ord(j)"
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by (erule ltE, assumption)
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lemma lt_Ord2: "j<i ==> Ord(i)"
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by (erule ltE, assumption)
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(* "ja le j ==> Ord(j)" *)
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lemmas le_Ord2 = lt_Ord2 [THEN Ord_succD]
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(* i<0 ==> R *)
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lemmas lt0E = not_lt0 [THEN notE, elim!]
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lemma lt_trans: "[| i<j;  j<k |] ==> i<k"
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by (blast intro!: ltI elim!: ltE intro: Ord_trans)
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lemma lt_not_sym: "i<j ==> ~ (j<i)"
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apply (unfold lt_def)
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apply (blast elim: mem_asym)
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done
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(* [| i<j;  ~P ==> j<i |] ==> P *)
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lemmas lt_asym = lt_not_sym [THEN swap]
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lemma lt_irrefl [elim!]: "i<i ==> P"
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by (blast intro: lt_asym)
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lemma lt_not_refl: "~ i<i"
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apply (rule notI)
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apply (erule lt_irrefl)
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done
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(** le is less than or equals;  recall  i le j  abbrevs  i<succ(j) !! **)
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lemma le_iff: "i le j <-> i<j | (i=j & Ord(j))"
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by (unfold lt_def, blast)
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(*Equivalently, i<j ==> i < succ(j)*)
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lemma leI: "i<j ==> i le j"
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by (simp (no_asm_simp) add: le_iff)
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lemma le_eqI: "[| i=j;  Ord(j) |] ==> i le j"
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by (simp (no_asm_simp) add: le_iff)
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lemmas le_refl = refl [THEN le_eqI]
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lemma le_refl_iff [iff]: "i le i <-> Ord(i)"
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by (simp (no_asm_simp) add: lt_not_refl le_iff)
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lemma leCI: "(~ (i=j & Ord(j)) ==> i<j) ==> i le j"
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by (simp add: le_iff, blast)
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lemma leE:
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    "[| i le j;  i<j ==> P;  [| i=j;  Ord(j) |] ==> P |] ==> P"
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by (simp add: le_iff, blast)
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lemma le_anti_sym: "[| i le j;  j le i |] ==> i=j"
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apply (simp add: le_iff)
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apply (blast elim: lt_asym)
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done
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lemma le0_iff [simp]: "i le 0 <-> i=0"
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by (blast elim!: leE)
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lemmas le0D = le0_iff [THEN iffD1, dest!]
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(*** Natural Deduction rules for Memrel ***)
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(*The lemmas MemrelI/E give better speed than [iff] here*)
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lemma Memrel_iff [simp]: "<a,b> : Memrel(A) <-> a:b & a:A & b:A"
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by (unfold Memrel_def, blast)
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lemma MemrelI [intro!]: "[| a: b;  a: A;  b: A |] ==> <a,b> : Memrel(A)"
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by auto
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lemma MemrelE [elim!]:
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    "[| <a,b> : Memrel(A);   
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        [| a: A;  b: A;  a:b |]  ==> P |]  
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     ==> P"
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by auto
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lemma Memrel_type: "Memrel(A) <= A*A"
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by (unfold Memrel_def, blast)
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lemma Memrel_mono: "A<=B ==> Memrel(A) <= Memrel(B)"
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by (unfold Memrel_def, blast)
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lemma Memrel_0 [simp]: "Memrel(0) = 0"
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by (unfold Memrel_def, blast)
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lemma Memrel_1 [simp]: "Memrel(1) = 0"
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by (unfold Memrel_def, blast)
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(*The membership relation (as a set) is well-founded.
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  Proof idea: show A<=B by applying the foundation axiom to A-B *)
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lemma wf_Memrel: "wf(Memrel(A))"
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apply (unfold wf_def)
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apply (rule foundation [THEN disjE, THEN allI], erule disjI1, blast) 
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done
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(*Transset(i) does not suffice, though ALL j:i.Transset(j) does*)
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lemma trans_Memrel: 
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    "Ord(i) ==> trans(Memrel(i))"
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by (unfold Ord_def Transset_def trans_def, blast)
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(*If Transset(A) then Memrel(A) internalizes the membership relation below A*)
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lemma Transset_Memrel_iff: 
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    "Transset(A) ==> <a,b> : Memrel(A) <-> a:b & b:A"
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by (unfold Transset_def, blast)
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(*** Transfinite induction ***)
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(*Epsilon induction over a transitive set*)
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lemma Transset_induct: 
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    "[| i: k;  Transset(k);                           
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        !!x.[| x: k;  ALL y:x. P(y) |] ==> P(x) |]
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     ==>  P(i)"
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apply (simp add: Transset_def) 
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apply (erule wf_Memrel [THEN wf_induct2], blast)
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apply blast 
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done
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(*Induction over an ordinal*)
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lemmas Ord_induct = Transset_induct [OF _ Ord_is_Transset]
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(*Induction over the class of ordinals -- a useful corollary of Ord_induct*)
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lemma trans_induct:
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    "[| Ord(i);  
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        !!x.[| Ord(x);  ALL y:x. P(y) |] ==> P(x) |]
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     ==>  P(i)"
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apply (rule Ord_succ [THEN succI1 [THEN Ord_induct]], assumption)
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apply (blast intro: Ord_succ [THEN Ord_in_Ord]) 
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done
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(*** Fundamental properties of the epsilon ordering (< on ordinals) ***)
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(** Proving that < is a linear ordering on the ordinals **)
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lemma Ord_linear [rule_format]:
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     "Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)"
paulson@13155
   344
apply (erule trans_induct)
paulson@13155
   345
apply (rule impI [THEN allI])
paulson@13155
   346
apply (erule_tac i=j in trans_induct) 
paulson@13155
   347
apply (blast dest: Ord_trans) 
paulson@13155
   348
done
paulson@13155
   349
paulson@13155
   350
(*The trichotomy law for ordinals!*)
paulson@13155
   351
lemma Ord_linear_lt:
paulson@13155
   352
    "[| Ord(i);  Ord(j);  i<j ==> P;  i=j ==> P;  j<i ==> P |] ==> P"
paulson@13155
   353
apply (simp add: lt_def) 
paulson@13155
   354
apply (rule_tac i1=i and j1=j in Ord_linear [THEN disjE], blast+)
paulson@13155
   355
done
paulson@13155
   356
paulson@13155
   357
lemma Ord_linear2:
paulson@13155
   358
    "[| Ord(i);  Ord(j);  i<j ==> P;  j le i ==> P |]  ==> P"
paulson@13155
   359
apply (rule_tac i = "i" and j = "j" in Ord_linear_lt)
paulson@13155
   360
apply (blast intro: leI le_eqI sym ) +
paulson@13155
   361
done
paulson@13155
   362
paulson@13155
   363
lemma Ord_linear_le:
paulson@13155
   364
    "[| Ord(i);  Ord(j);  i le j ==> P;  j le i ==> P |]  ==> P"
paulson@13155
   365
apply (rule_tac i = "i" and j = "j" in Ord_linear_lt)
paulson@13155
   366
apply (blast intro: leI le_eqI ) +
paulson@13155
   367
done
paulson@13155
   368
paulson@13155
   369
lemma le_imp_not_lt: "j le i ==> ~ i<j"
paulson@13155
   370
by (blast elim!: leE elim: lt_asym)
paulson@13155
   371
paulson@13155
   372
lemma not_lt_imp_le: "[| ~ i<j;  Ord(i);  Ord(j) |] ==> j le i"
paulson@13155
   373
by (rule_tac i = "i" and j = "j" in Ord_linear2, auto)
paulson@13155
   374
paulson@13155
   375
(** Some rewrite rules for <, le **)
paulson@13155
   376
paulson@13155
   377
lemma Ord_mem_iff_lt: "Ord(j) ==> i:j <-> i<j"
paulson@13155
   378
by (unfold lt_def, blast)
paulson@13155
   379
paulson@13155
   380
lemma not_lt_iff_le: "[| Ord(i);  Ord(j) |] ==> ~ i<j <-> j le i"
paulson@13155
   381
by (blast dest: le_imp_not_lt not_lt_imp_le)
wenzelm@2540
   382
paulson@13155
   383
lemma not_le_iff_lt: "[| Ord(i);  Ord(j) |] ==> ~ i le j <-> j<i"
paulson@13155
   384
by (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym])
paulson@13155
   385
paulson@13155
   386
(*This is identical to 0<succ(i) *)
paulson@13155
   387
lemma Ord_0_le: "Ord(i) ==> 0 le i"
paulson@13155
   388
by (erule not_lt_iff_le [THEN iffD1], auto)
paulson@13155
   389
paulson@13155
   390
lemma Ord_0_lt: "[| Ord(i);  i~=0 |] ==> 0<i"
paulson@13155
   391
apply (erule not_le_iff_lt [THEN iffD1])
paulson@13155
   392
apply (rule Ord_0, blast)
paulson@13155
   393
done
paulson@13155
   394
paulson@13155
   395
lemma Ord_0_lt_iff: "Ord(i) ==> i~=0 <-> 0<i"
paulson@13155
   396
by (blast intro: Ord_0_lt)
paulson@13155
   397
paulson@13155
   398
paulson@13155
   399
(*** Results about less-than or equals ***)
paulson@13155
   400
paulson@13155
   401
(** For ordinals, j<=i (subset) implies j le i (less-than or equals) **)
paulson@13155
   402
paulson@13155
   403
lemma zero_le_succ_iff [iff]: "0 le succ(x) <-> Ord(x)"
paulson@13155
   404
by (blast intro: Ord_0_le elim: ltE)
paulson@13155
   405
paulson@13155
   406
lemma subset_imp_le: "[| j<=i;  Ord(i);  Ord(j) |] ==> j le i"
paulson@13155
   407
apply (rule not_lt_iff_le [THEN iffD1], assumption)
paulson@13155
   408
apply assumption
paulson@13155
   409
apply (blast elim: ltE mem_irrefl)
paulson@13155
   410
done
paulson@13155
   411
paulson@13155
   412
lemma le_imp_subset: "i le j ==> i<=j"
paulson@13155
   413
by (blast dest: OrdmemD elim: ltE leE)
paulson@13155
   414
paulson@13155
   415
lemma le_subset_iff: "j le i <-> j<=i & Ord(i) & Ord(j)"
paulson@13155
   416
by (blast dest: subset_imp_le le_imp_subset elim: ltE)
paulson@13155
   417
paulson@13155
   418
lemma le_succ_iff: "i le succ(j) <-> i le j | i=succ(j) & Ord(i)"
paulson@13155
   419
apply (simp (no_asm) add: le_iff)
paulson@13155
   420
apply blast
paulson@13155
   421
done
paulson@13155
   422
paulson@13155
   423
(*Just a variant of subset_imp_le*)
paulson@13155
   424
lemma all_lt_imp_le: "[| Ord(i);  Ord(j);  !!x. x<j ==> x<i |] ==> j le i"
paulson@13155
   425
by (blast intro: not_lt_imp_le dest: lt_irrefl)
paulson@13155
   426
paulson@13155
   427
(** Transitive laws **)
paulson@13155
   428
paulson@13155
   429
lemma lt_trans1: "[| i le j;  j<k |] ==> i<k"
paulson@13155
   430
by (blast elim!: leE intro: lt_trans)
paulson@13155
   431
paulson@13155
   432
lemma lt_trans2: "[| i<j;  j le k |] ==> i<k"
paulson@13155
   433
by (blast elim!: leE intro: lt_trans)
paulson@13155
   434
paulson@13155
   435
lemma le_trans: "[| i le j;  j le k |] ==> i le k"
paulson@13155
   436
by (blast intro: lt_trans1)
paulson@13155
   437
paulson@13155
   438
lemma succ_leI: "i<j ==> succ(i) le j"
paulson@13155
   439
apply (rule not_lt_iff_le [THEN iffD1]) 
paulson@13155
   440
apply (blast elim: ltE leE lt_asym)+
paulson@13155
   441
done
paulson@13155
   442
paulson@13155
   443
(*Identical to  succ(i) < succ(j) ==> i<j  *)
paulson@13155
   444
lemma succ_leE: "succ(i) le j ==> i<j"
paulson@13155
   445
apply (rule not_le_iff_lt [THEN iffD1])
paulson@13155
   446
apply (blast elim: ltE leE lt_asym)+
paulson@13155
   447
done
paulson@13155
   448
paulson@13155
   449
lemma succ_le_iff [iff]: "succ(i) le j <-> i<j"
paulson@13155
   450
by (blast intro: succ_leI succ_leE)
paulson@13155
   451
paulson@13155
   452
lemma succ_le_imp_le: "succ(i) le succ(j) ==> i le j"
paulson@13155
   453
by (blast dest!: succ_leE)
paulson@13155
   454
paulson@13155
   455
lemma lt_subset_trans: "[| i <= j;  j<k;  Ord(i) |] ==> i<k"
paulson@13155
   456
apply (rule subset_imp_le [THEN lt_trans1]) 
paulson@13155
   457
apply (blast intro: elim: ltE) +
paulson@13155
   458
done
paulson@13155
   459
paulson@13172
   460
lemma lt_imp_0_lt: "j<i ==> 0<i"
paulson@13172
   461
by (blast intro: lt_trans1 Ord_0_le [OF lt_Ord]) 
paulson@13172
   462
paulson@13243
   463
lemma succ_lt_iff: "succ(i) < j <-> i<j & succ(i) \<noteq> j"
paulson@13162
   464
apply auto 
paulson@13162
   465
apply (blast intro: lt_trans le_refl dest: lt_Ord) 
paulson@13162
   466
apply (frule lt_Ord) 
paulson@13162
   467
apply (rule not_le_iff_lt [THEN iffD1]) 
paulson@13162
   468
  apply (blast intro: lt_Ord2)
paulson@13162
   469
 apply blast  
paulson@13162
   470
apply (simp add: lt_Ord lt_Ord2 le_iff) 
paulson@13162
   471
apply (blast dest: lt_asym) 
paulson@13162
   472
done
paulson@13162
   473
paulson@13243
   474
lemma Ord_succ_mem_iff: "Ord(j) ==> succ(i) \<in> succ(j) <-> i\<in>j"
paulson@13243
   475
apply (insert succ_le_iff [of i j]) 
paulson@13243
   476
apply (simp add: lt_def) 
paulson@13243
   477
done
paulson@13243
   478
paulson@13155
   479
(** Union and Intersection **)
paulson@13155
   480
paulson@13155
   481
lemma Un_upper1_le: "[| Ord(i); Ord(j) |] ==> i le i Un j"
paulson@13155
   482
by (rule Un_upper1 [THEN subset_imp_le], auto)
paulson@13155
   483
paulson@13155
   484
lemma Un_upper2_le: "[| Ord(i); Ord(j) |] ==> j le i Un j"
paulson@13155
   485
by (rule Un_upper2 [THEN subset_imp_le], auto)
paulson@13155
   486
paulson@13155
   487
(*Replacing k by succ(k') yields the similar rule for le!*)
paulson@13155
   488
lemma Un_least_lt: "[| i<k;  j<k |] ==> i Un j < k"
paulson@13155
   489
apply (rule_tac i = "i" and j = "j" in Ord_linear_le)
paulson@13155
   490
apply (auto simp add: Un_commute le_subset_iff subset_Un_iff lt_Ord) 
paulson@13155
   491
done
paulson@13155
   492
paulson@13155
   493
lemma Un_least_lt_iff: "[| Ord(i); Ord(j) |] ==> i Un j < k  <->  i<k & j<k"
paulson@13155
   494
apply (safe intro!: Un_least_lt)
paulson@13155
   495
apply (rule_tac [2] Un_upper2_le [THEN lt_trans1])
paulson@13155
   496
apply (rule Un_upper1_le [THEN lt_trans1], auto) 
paulson@13155
   497
done
paulson@13155
   498
paulson@13155
   499
lemma Un_least_mem_iff:
paulson@13155
   500
    "[| Ord(i); Ord(j); Ord(k) |] ==> i Un j : k  <->  i:k & j:k"
paulson@13155
   501
apply (insert Un_least_lt_iff [of i j k]) 
paulson@13155
   502
apply (simp add: lt_def)
paulson@13155
   503
done
paulson@13155
   504
paulson@13155
   505
(*Replacing k by succ(k') yields the similar rule for le!*)
paulson@13155
   506
lemma Int_greatest_lt: "[| i<k;  j<k |] ==> i Int j < k"
paulson@13155
   507
apply (rule_tac i = "i" and j = "j" in Ord_linear_le)
paulson@13155
   508
apply (auto simp add: Int_commute le_subset_iff subset_Int_iff lt_Ord) 
paulson@13155
   509
done
paulson@13155
   510
paulson@13162
   511
lemma Ord_Un_if:
paulson@13162
   512
     "[| Ord(i); Ord(j) |] ==> i \<union> j = (if j<i then i else j)"
paulson@13162
   513
by (simp add: not_lt_iff_le le_imp_subset leI
paulson@13162
   514
              subset_Un_iff [symmetric]  subset_Un_iff2 [symmetric]) 
paulson@13162
   515
paulson@13162
   516
lemma succ_Un_distrib:
paulson@13162
   517
     "[| Ord(i); Ord(j) |] ==> succ(i \<union> j) = succ(i) \<union> succ(j)"
paulson@13162
   518
by (simp add: Ord_Un_if lt_Ord le_Ord2) 
paulson@13162
   519
paulson@13162
   520
lemma lt_Un_iff:
paulson@13162
   521
     "[| Ord(i); Ord(j) |] ==> k < i \<union> j <-> k < i | k < j";
paulson@13162
   522
apply (simp add: Ord_Un_if not_lt_iff_le) 
paulson@13162
   523
apply (blast intro: leI lt_trans2)+ 
paulson@13162
   524
done
paulson@13162
   525
paulson@13162
   526
lemma le_Un_iff:
paulson@13162
   527
     "[| Ord(i); Ord(j) |] ==> k \<le> i \<union> j <-> k \<le> i | k \<le> j";
paulson@13162
   528
by (simp add: succ_Un_distrib lt_Un_iff [symmetric]) 
paulson@13162
   529
paulson@13172
   530
lemma Un_upper1_lt: "[|k < i; Ord(j)|] ==> k < i Un j"
paulson@13172
   531
by (simp add: lt_Un_iff lt_Ord2) 
paulson@13172
   532
paulson@13172
   533
lemma Un_upper2_lt: "[|k < j; Ord(i)|] ==> k < i Un j"
paulson@13172
   534
by (simp add: lt_Un_iff lt_Ord2) 
paulson@13172
   535
paulson@13172
   536
(*See also Transset_iff_Union_succ*)
paulson@13172
   537
lemma Ord_Union_succ_eq: "Ord(i) ==> \<Union>(succ(i)) = i"
paulson@13172
   538
by (blast intro: Ord_trans)
paulson@13172
   539
paulson@13162
   540
paulson@13155
   541
(*** Results about limits ***)
paulson@13155
   542
paulson@13172
   543
lemma Ord_Union [intro,simp,TC]: "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))"
paulson@13155
   544
apply (rule Ord_is_Transset [THEN Transset_Union_family, THEN OrdI])
paulson@13155
   545
apply (blast intro: Ord_contains_Transset)+
paulson@13155
   546
done
paulson@13155
   547
paulson@13172
   548
lemma Ord_UN [intro,simp,TC]:
paulson@13172
   549
     "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(UN x:A. B(x))"
paulson@13155
   550
by (rule Ord_Union, blast)
paulson@13155
   551
paulson@13203
   552
lemma Ord_Inter [intro,simp,TC]:
paulson@13203
   553
    "[| !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))" 
paulson@13203
   554
apply (rule Transset_Inter_family [THEN OrdI])
paulson@13203
   555
apply (blast intro: Ord_is_Transset) 
paulson@13203
   556
apply (simp add: Inter_def) 
paulson@13203
   557
apply (blast intro: Ord_contains_Transset) 
paulson@13203
   558
done
paulson@13203
   559
paulson@13203
   560
lemma Ord_INT [intro,simp,TC]:
paulson@13203
   561
    "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(INT x:A. B(x))"
paulson@13203
   562
by (rule Ord_Inter, blast) 
paulson@13203
   563
paulson@13203
   564
paulson@13155
   565
(* No < version; consider (UN i:nat.i)=nat *)
paulson@13155
   566
lemma UN_least_le:
paulson@13155
   567
    "[| Ord(i);  !!x. x:A ==> b(x) le i |] ==> (UN x:A. b(x)) le i"
paulson@13155
   568
apply (rule le_imp_subset [THEN UN_least, THEN subset_imp_le])
paulson@13155
   569
apply (blast intro: Ord_UN elim: ltE)+
paulson@13155
   570
done
paulson@13155
   571
paulson@13155
   572
lemma UN_succ_least_lt:
paulson@13155
   573
    "[| j<i;  !!x. x:A ==> b(x)<j |] ==> (UN x:A. succ(b(x))) < i"
paulson@13155
   574
apply (rule ltE, assumption)
paulson@13155
   575
apply (rule UN_least_le [THEN lt_trans2])
paulson@13155
   576
apply (blast intro: succ_leI)+
paulson@13155
   577
done
paulson@13155
   578
paulson@13172
   579
lemma UN_upper_lt:
paulson@13172
   580
     "[| a\<in>A;  i < b(a);  Ord(\<Union>x\<in>A. b(x)) |] ==> i < (\<Union>x\<in>A. b(x))"
paulson@13172
   581
by (unfold lt_def, blast) 
paulson@13172
   582
paulson@13155
   583
lemma UN_upper_le:
paulson@13155
   584
     "[| a: A;  i le b(a);  Ord(UN x:A. b(x)) |] ==> i le (UN x:A. b(x))"
paulson@13155
   585
apply (frule ltD)
paulson@13155
   586
apply (rule le_imp_subset [THEN subset_trans, THEN subset_imp_le])
paulson@13155
   587
apply (blast intro: lt_Ord UN_upper)+
paulson@13155
   588
done
paulson@13155
   589
paulson@13172
   590
lemma lt_Union_iff: "\<forall>i\<in>A. Ord(i) ==> (j < \<Union>(A)) <-> (\<exists>i\<in>A. j<i)"
paulson@13172
   591
by (auto simp: lt_def Ord_Union)
paulson@13172
   592
paulson@13172
   593
lemma Union_upper_le:
paulson@13172
   594
     "[| j: J;  i\<le>j;  Ord(\<Union>(J)) |] ==> i \<le> \<Union>J"
paulson@13172
   595
apply (subst Union_eq_UN)  
paulson@13172
   596
apply (rule UN_upper_le, auto)
paulson@13172
   597
done
paulson@13172
   598
paulson@13155
   599
lemma le_implies_UN_le_UN:
paulson@13155
   600
    "[| !!x. x:A ==> c(x) le d(x) |] ==> (UN x:A. c(x)) le (UN x:A. d(x))"
paulson@13155
   601
apply (rule UN_least_le)
paulson@13155
   602
apply (rule_tac [2] UN_upper_le)
paulson@13155
   603
apply (blast intro: Ord_UN le_Ord2)+ 
paulson@13155
   604
done
paulson@13155
   605
paulson@13155
   606
lemma Ord_equality: "Ord(i) ==> (UN y:i. succ(y)) = i"
paulson@13155
   607
by (blast intro: Ord_trans)
paulson@13155
   608
paulson@13155
   609
(*Holds for all transitive sets, not just ordinals*)
paulson@13155
   610
lemma Ord_Union_subset: "Ord(i) ==> Union(i) <= i"
paulson@13155
   611
by (blast intro: Ord_trans)
paulson@13155
   612
paulson@13155
   613
paulson@13155
   614
(*** Limit ordinals -- general properties ***)
paulson@13155
   615
paulson@13155
   616
lemma Limit_Union_eq: "Limit(i) ==> Union(i) = i"
paulson@13155
   617
apply (unfold Limit_def)
paulson@13155
   618
apply (fast intro!: ltI elim!: ltE elim: Ord_trans)
paulson@13155
   619
done
paulson@13155
   620
paulson@13155
   621
lemma Limit_is_Ord: "Limit(i) ==> Ord(i)"
paulson@13155
   622
apply (unfold Limit_def)
paulson@13155
   623
apply (erule conjunct1)
paulson@13155
   624
done
paulson@13155
   625
paulson@13155
   626
lemma Limit_has_0: "Limit(i) ==> 0 < i"
paulson@13155
   627
apply (unfold Limit_def)
paulson@13155
   628
apply (erule conjunct2 [THEN conjunct1])
paulson@13155
   629
done
paulson@13155
   630
paulson@13155
   631
lemma Limit_has_succ: "[| Limit(i);  j<i |] ==> succ(j) < i"
paulson@13155
   632
by (unfold Limit_def, blast)
paulson@13155
   633
paulson@13172
   634
lemma zero_not_Limit [iff]: "~ Limit(0)"
paulson@13172
   635
by (simp add: Limit_def)
paulson@13172
   636
paulson@13172
   637
lemma Limit_has_1: "Limit(i) ==> 1 < i"
paulson@13172
   638
by (blast intro: Limit_has_0 Limit_has_succ)
paulson@13172
   639
paulson@13172
   640
lemma increasing_LimitI: "[| 0<l; \<forall>x\<in>l. \<exists>y\<in>l. x<y |] ==> Limit(l)"
paulson@13172
   641
apply (simp add: Limit_def lt_Ord2, clarify)
paulson@13172
   642
apply (drule_tac i=y in ltD) 
paulson@13172
   643
apply (blast intro: lt_trans1 [OF _ ltI] lt_Ord2)
paulson@13172
   644
done
paulson@13172
   645
paulson@13155
   646
lemma non_succ_LimitI: 
paulson@13155
   647
    "[| 0<i;  ALL y. succ(y) ~= i |] ==> Limit(i)"
paulson@13155
   648
apply (unfold Limit_def)
paulson@13155
   649
apply (safe del: subsetI)
paulson@13155
   650
apply (rule_tac [2] not_le_iff_lt [THEN iffD1])
paulson@13155
   651
apply (simp_all add: lt_Ord lt_Ord2) 
paulson@13155
   652
apply (blast elim: leE lt_asym)
paulson@13155
   653
done
paulson@13155
   654
paulson@13155
   655
lemma succ_LimitE [elim!]: "Limit(succ(i)) ==> P"
paulson@13155
   656
apply (rule lt_irrefl)
paulson@13155
   657
apply (rule Limit_has_succ, assumption)
paulson@13155
   658
apply (erule Limit_is_Ord [THEN Ord_succD, THEN le_refl])
paulson@13155
   659
done
paulson@13155
   660
paulson@13155
   661
lemma not_succ_Limit [simp]: "~ Limit(succ(i))"
paulson@13155
   662
by blast
paulson@13155
   663
paulson@13155
   664
lemma Limit_le_succD: "[| Limit(i);  i le succ(j) |] ==> i le j"
paulson@13155
   665
by (blast elim!: leE)
paulson@13155
   666
paulson@13172
   667
paulson@13155
   668
(** Traditional 3-way case analysis on ordinals **)
paulson@13155
   669
paulson@13155
   670
lemma Ord_cases_disj: "Ord(i) ==> i=0 | (EX j. Ord(j) & i=succ(j)) | Limit(i)"
paulson@13155
   671
by (blast intro!: non_succ_LimitI Ord_0_lt)
paulson@13155
   672
paulson@13155
   673
lemma Ord_cases:
paulson@13155
   674
    "[| Ord(i);                  
paulson@13155
   675
        i=0                          ==> P;      
paulson@13155
   676
        !!j. [| Ord(j); i=succ(j) |] ==> P;      
paulson@13155
   677
        Limit(i)                     ==> P       
paulson@13155
   678
     |] ==> P"
paulson@13155
   679
by (drule Ord_cases_disj, blast)  
paulson@13155
   680
paulson@13155
   681
lemma trans_induct3:
paulson@13155
   682
     "[| Ord(i);                 
paulson@13155
   683
         P(0);                   
paulson@13155
   684
         !!x. [| Ord(x);  P(x) |] ==> P(succ(x));        
paulson@13155
   685
         !!x. [| Limit(x);  ALL y:x. P(y) |] ==> P(x)    
paulson@13155
   686
      |] ==> P(i)"
paulson@13155
   687
apply (erule trans_induct)
paulson@13155
   688
apply (erule Ord_cases, blast+)
paulson@13155
   689
done
paulson@13155
   690
paulson@13172
   691
text{*A set of ordinals is either empty, contains its own union, or its
paulson@13172
   692
union is a limit ordinal.*}
paulson@13172
   693
lemma Ord_set_cases:
paulson@13172
   694
   "\<forall>i\<in>I. Ord(i) ==> I=0 \<or> \<Union>(I) \<in> I \<or> (\<Union>(I) \<notin> I \<and> Limit(\<Union>(I)))"
paulson@13172
   695
apply (clarify elim!: not_emptyE) 
paulson@13172
   696
apply (cases "\<Union>(I)" rule: Ord_cases) 
paulson@13172
   697
   apply (blast intro: Ord_Union)
paulson@13172
   698
  apply (blast intro: subst_elem)
paulson@13172
   699
 apply auto 
paulson@13172
   700
apply (clarify elim!: equalityE succ_subsetE)
paulson@13172
   701
apply (simp add: Union_subset_iff)
paulson@13172
   702
apply (subgoal_tac "B = succ(j)", blast)
paulson@13172
   703
apply (rule le_anti_sym) 
paulson@13172
   704
 apply (simp add: le_subset_iff) 
paulson@13172
   705
apply (simp add: ltI)
paulson@13172
   706
done
paulson@13172
   707
paulson@13172
   708
text{*If the union of a set of ordinals is a successor, then it is
paulson@13172
   709
an element of that set.*}
paulson@13172
   710
lemma Ord_Union_eq_succD: "[|\<forall>x\<in>X. Ord(x);  \<Union>X = succ(j)|] ==> succ(j) \<in> X"
paulson@13172
   711
by (drule Ord_set_cases, auto)
paulson@13172
   712
paulson@13172
   713
lemma Limit_Union [rule_format]: "[| I \<noteq> 0;  \<forall>i\<in>I. Limit(i) |] ==> Limit(\<Union>I)"
paulson@13172
   714
apply (simp add: Limit_def lt_def)
paulson@13172
   715
apply (blast intro!: equalityI)
paulson@13172
   716
done
paulson@13172
   717
paulson@13162
   718
(*special induction rules for the "induct" method*)
paulson@13162
   719
lemmas Ord_induct = Ord_induct [consumes 2]
paulson@13162
   720
  and Ord_induct_rule = Ord_induct [rule_format, consumes 2]
paulson@13162
   721
  and trans_induct = trans_induct [consumes 1]
paulson@13162
   722
  and trans_induct_rule = trans_induct [rule_format, consumes 1]
paulson@13162
   723
  and trans_induct3 = trans_induct3 [case_names 0 succ limit, consumes 1]
paulson@13162
   724
  and trans_induct3_rule = trans_induct3 [rule_format, case_names 0 succ limit, consumes 1]
paulson@13162
   725
paulson@13155
   726
ML 
paulson@13155
   727
{*
paulson@13155
   728
val Memrel_def = thm "Memrel_def";
paulson@13155
   729
val Transset_def = thm "Transset_def";
paulson@13155
   730
val Ord_def = thm "Ord_def";
paulson@13155
   731
val lt_def = thm "lt_def";
paulson@13155
   732
val Limit_def = thm "Limit_def";
paulson@13155
   733
paulson@13155
   734
val Transset_iff_Pow = thm "Transset_iff_Pow";
paulson@13155
   735
val Transset_iff_Union_succ = thm "Transset_iff_Union_succ";
paulson@13155
   736
val Transset_iff_Union_subset = thm "Transset_iff_Union_subset";
paulson@13155
   737
val Transset_doubleton_D = thm "Transset_doubleton_D";
paulson@13155
   738
val Transset_Pair_D = thm "Transset_Pair_D";
paulson@13155
   739
val Transset_includes_domain = thm "Transset_includes_domain";
paulson@13155
   740
val Transset_includes_range = thm "Transset_includes_range";
paulson@13155
   741
val Transset_0 = thm "Transset_0";
paulson@13155
   742
val Transset_Un = thm "Transset_Un";
paulson@13155
   743
val Transset_Int = thm "Transset_Int";
paulson@13155
   744
val Transset_succ = thm "Transset_succ";
paulson@13155
   745
val Transset_Pow = thm "Transset_Pow";
paulson@13155
   746
val Transset_Union = thm "Transset_Union";
paulson@13155
   747
val Transset_Union_family = thm "Transset_Union_family";
paulson@13155
   748
val Transset_Inter_family = thm "Transset_Inter_family";
paulson@13155
   749
val OrdI = thm "OrdI";
paulson@13155
   750
val Ord_is_Transset = thm "Ord_is_Transset";
paulson@13155
   751
val Ord_contains_Transset = thm "Ord_contains_Transset";
paulson@13155
   752
val Ord_in_Ord = thm "Ord_in_Ord";
paulson@13155
   753
val Ord_succD = thm "Ord_succD";
paulson@13155
   754
val Ord_subset_Ord = thm "Ord_subset_Ord";
paulson@13155
   755
val OrdmemD = thm "OrdmemD";
paulson@13155
   756
val Ord_trans = thm "Ord_trans";
paulson@13155
   757
val Ord_succ_subsetI = thm "Ord_succ_subsetI";
paulson@13155
   758
val Ord_0 = thm "Ord_0";
paulson@13155
   759
val Ord_succ = thm "Ord_succ";
paulson@13155
   760
val Ord_1 = thm "Ord_1";
paulson@13155
   761
val Ord_succ_iff = thm "Ord_succ_iff";
paulson@13155
   762
val Ord_Un = thm "Ord_Un";
paulson@13155
   763
val Ord_Int = thm "Ord_Int";
paulson@13155
   764
val Ord_Inter = thm "Ord_Inter";
paulson@13155
   765
val Ord_INT = thm "Ord_INT";
paulson@13155
   766
val ON_class = thm "ON_class";
paulson@13155
   767
val ltI = thm "ltI";
paulson@13155
   768
val ltE = thm "ltE";
paulson@13155
   769
val ltD = thm "ltD";
paulson@13155
   770
val not_lt0 = thm "not_lt0";
paulson@13155
   771
val lt_Ord = thm "lt_Ord";
paulson@13155
   772
val lt_Ord2 = thm "lt_Ord2";
paulson@13155
   773
val le_Ord2 = thm "le_Ord2";
paulson@13155
   774
val lt0E = thm "lt0E";
paulson@13155
   775
val lt_trans = thm "lt_trans";
paulson@13155
   776
val lt_not_sym = thm "lt_not_sym";
paulson@13155
   777
val lt_asym = thm "lt_asym";
paulson@13155
   778
val lt_irrefl = thm "lt_irrefl";
paulson@13155
   779
val lt_not_refl = thm "lt_not_refl";
paulson@13155
   780
val le_iff = thm "le_iff";
paulson@13155
   781
val leI = thm "leI";
paulson@13155
   782
val le_eqI = thm "le_eqI";
paulson@13155
   783
val le_refl = thm "le_refl";
paulson@13155
   784
val le_refl_iff = thm "le_refl_iff";
paulson@13155
   785
val leCI = thm "leCI";
paulson@13155
   786
val leE = thm "leE";
paulson@13155
   787
val le_anti_sym = thm "le_anti_sym";
paulson@13155
   788
val le0_iff = thm "le0_iff";
paulson@13155
   789
val le0D = thm "le0D";
paulson@13155
   790
val Memrel_iff = thm "Memrel_iff";
paulson@13155
   791
val MemrelI = thm "MemrelI";
paulson@13155
   792
val MemrelE = thm "MemrelE";
paulson@13155
   793
val Memrel_type = thm "Memrel_type";
paulson@13155
   794
val Memrel_mono = thm "Memrel_mono";
paulson@13155
   795
val Memrel_0 = thm "Memrel_0";
paulson@13155
   796
val Memrel_1 = thm "Memrel_1";
paulson@13155
   797
val wf_Memrel = thm "wf_Memrel";
paulson@13155
   798
val trans_Memrel = thm "trans_Memrel";
paulson@13155
   799
val Transset_Memrel_iff = thm "Transset_Memrel_iff";
paulson@13155
   800
val Transset_induct = thm "Transset_induct";
paulson@13155
   801
val Ord_induct = thm "Ord_induct";
paulson@13155
   802
val trans_induct = thm "trans_induct";
paulson@13155
   803
val Ord_linear = thm "Ord_linear";
paulson@13155
   804
val Ord_linear_lt = thm "Ord_linear_lt";
paulson@13155
   805
val Ord_linear2 = thm "Ord_linear2";
paulson@13155
   806
val Ord_linear_le = thm "Ord_linear_le";
paulson@13155
   807
val le_imp_not_lt = thm "le_imp_not_lt";
paulson@13155
   808
val not_lt_imp_le = thm "not_lt_imp_le";
paulson@13155
   809
val Ord_mem_iff_lt = thm "Ord_mem_iff_lt";
paulson@13155
   810
val not_lt_iff_le = thm "not_lt_iff_le";
paulson@13155
   811
val not_le_iff_lt = thm "not_le_iff_lt";
paulson@13155
   812
val Ord_0_le = thm "Ord_0_le";
paulson@13155
   813
val Ord_0_lt = thm "Ord_0_lt";
paulson@13155
   814
val Ord_0_lt_iff = thm "Ord_0_lt_iff";
paulson@13155
   815
val zero_le_succ_iff = thm "zero_le_succ_iff";
paulson@13155
   816
val subset_imp_le = thm "subset_imp_le";
paulson@13155
   817
val le_imp_subset = thm "le_imp_subset";
paulson@13155
   818
val le_subset_iff = thm "le_subset_iff";
paulson@13155
   819
val le_succ_iff = thm "le_succ_iff";
paulson@13155
   820
val all_lt_imp_le = thm "all_lt_imp_le";
paulson@13155
   821
val lt_trans1 = thm "lt_trans1";
paulson@13155
   822
val lt_trans2 = thm "lt_trans2";
paulson@13155
   823
val le_trans = thm "le_trans";
paulson@13155
   824
val succ_leI = thm "succ_leI";
paulson@13155
   825
val succ_leE = thm "succ_leE";
paulson@13155
   826
val succ_le_iff = thm "succ_le_iff";
paulson@13155
   827
val succ_le_imp_le = thm "succ_le_imp_le";
paulson@13155
   828
val lt_subset_trans = thm "lt_subset_trans";
paulson@13155
   829
val Un_upper1_le = thm "Un_upper1_le";
paulson@13155
   830
val Un_upper2_le = thm "Un_upper2_le";
paulson@13155
   831
val Un_least_lt = thm "Un_least_lt";
paulson@13155
   832
val Un_least_lt_iff = thm "Un_least_lt_iff";
paulson@13155
   833
val Un_least_mem_iff = thm "Un_least_mem_iff";
paulson@13155
   834
val Int_greatest_lt = thm "Int_greatest_lt";
paulson@13155
   835
val Ord_Union = thm "Ord_Union";
paulson@13155
   836
val Ord_UN = thm "Ord_UN";
paulson@13155
   837
val UN_least_le = thm "UN_least_le";
paulson@13155
   838
val UN_succ_least_lt = thm "UN_succ_least_lt";
paulson@13155
   839
val UN_upper_le = thm "UN_upper_le";
paulson@13155
   840
val le_implies_UN_le_UN = thm "le_implies_UN_le_UN";
paulson@13155
   841
val Ord_equality = thm "Ord_equality";
paulson@13155
   842
val Ord_Union_subset = thm "Ord_Union_subset";
paulson@13155
   843
val Limit_Union_eq = thm "Limit_Union_eq";
paulson@13155
   844
val Limit_is_Ord = thm "Limit_is_Ord";
paulson@13155
   845
val Limit_has_0 = thm "Limit_has_0";
paulson@13155
   846
val Limit_has_succ = thm "Limit_has_succ";
paulson@13155
   847
val non_succ_LimitI = thm "non_succ_LimitI";
paulson@13155
   848
val succ_LimitE = thm "succ_LimitE";
paulson@13155
   849
val not_succ_Limit = thm "not_succ_Limit";
paulson@13155
   850
val Limit_le_succD = thm "Limit_le_succD";
paulson@13155
   851
val Ord_cases_disj = thm "Ord_cases_disj";
paulson@13155
   852
val Ord_cases = thm "Ord_cases";
paulson@13155
   853
val trans_induct3 = thm "trans_induct3";
paulson@13155
   854
*}
lcp@435
   855
lcp@435
   856
end