--- a/src/ZF/OrdQuant.thy Thu Jul 04 16:48:21 2002 +0200
+++ b/src/ZF/OrdQuant.thy Thu Jul 04 16:59:54 2002 +0200
@@ -10,18 +10,18 @@
subsection {*Quantifiers and union operator for ordinals*}
constdefs
-
+
(* Ordinal Quantifiers *)
oall :: "[i, i => o] => o"
"oall(A, P) == ALL x. x<A --> P(x)"
-
+
oex :: "[i, i => o] => o"
"oex(A, P) == EX x. x<A & P(x)"
(* Ordinal Union *)
OUnion :: "[i, i => i] => i"
"OUnion(i,B) == {z: UN x:i. B(x). Ord(i)}"
-
+
syntax
"@oall" :: "[idt, i, o] => o" ("(3ALL _<_./ _)" 10)
"@oex" :: "[idt, i, o] => o" ("(3EX _<_./ _)" 10)
@@ -42,29 +42,29 @@
(*MOST IMPORTANT that this is added to the simpset BEFORE Ord_atomize
- is proved. Ord_atomize would convert this rule to
+ is proved. Ord_atomize would convert this rule to
x < 0 ==> P(x) == True, which causes dire effects!*)
lemma [simp]: "(ALL x<0. P(x))"
-by (simp add: oall_def)
+by (simp add: oall_def)
lemma [simp]: "~(EX x<0. P(x))"
-by (simp add: oex_def)
+by (simp add: oex_def)
lemma [simp]: "(ALL x<succ(i). P(x)) <-> (Ord(i) --> P(i) & (ALL x<i. P(x)))"
-apply (simp add: oall_def le_iff)
-apply (blast intro: lt_Ord2)
+apply (simp add: oall_def le_iff)
+apply (blast intro: lt_Ord2)
done
lemma [simp]: "(EX x<succ(i). P(x)) <-> (Ord(i) & (P(i) | (EX x<i. P(x))))"
-apply (simp add: oex_def le_iff)
-apply (blast intro: lt_Ord2)
+apply (simp add: oex_def le_iff)
+apply (blast intro: lt_Ord2)
done
(** Union over ordinals **)
lemma Ord_OUN [intro,simp]:
"[| !!x. x<A ==> Ord(B(x)) |] ==> Ord(\<Union>x<A. B(x))"
-by (simp add: OUnion_def ltI Ord_UN)
+by (simp add: OUnion_def ltI Ord_UN)
lemma OUN_upper_lt:
"[| a<A; i < b(a); Ord(\<Union>x<A. b(x)) |] ==> i < (\<Union>x<A. b(x))"
@@ -74,7 +74,7 @@
"[| a<A; i\<le>b(a); Ord(\<Union>x<A. b(x)) |] ==> i \<le> (\<Union>x<A. b(x))"
apply (unfold OUnion_def, auto)
apply (rule UN_upper_le )
-apply (auto simp add: lt_def)
+apply (auto simp add: lt_def)
done
lemma Limit_OUN_eq: "Limit(i) ==> (UN x<i. x) = i"
@@ -97,12 +97,12 @@
lemma OUN_UN_eq:
"(!!x. x:A ==> Ord(B(x)))
==> (UN z < (UN x:A. B(x)). C(z)) = (UN x:A. UN z < B(x). C(z))"
-by (simp add: OUnion_def)
+by (simp add: OUnion_def)
lemma OUN_Union_eq:
"(!!x. x:X ==> Ord(x))
==> (UN z < Union(X). C(z)) = (UN x:X. UN z < x. C(z))"
-by (simp add: OUnion_def)
+by (simp add: OUnion_def)
(*So that rule_format will get rid of ALL x<A...*)
lemma atomize_oall [symmetric, rulify]:
@@ -113,20 +113,18 @@
lemma oallI [intro!]:
"[| !!x. x<A ==> P(x) |] ==> ALL x<A. P(x)"
-by (simp add: oall_def)
+by (simp add: oall_def)
lemma ospec: "[| ALL x<A. P(x); x<A |] ==> P(x)"
-by (simp add: oall_def)
+by (simp add: oall_def)
lemma oallE:
"[| ALL x<A. P(x); P(x) ==> Q; ~x<A ==> Q |] ==> Q"
-apply (simp add: oall_def, blast)
-done
+by (simp add: oall_def, blast)
lemma rev_oallE [elim]:
"[| ALL x<A. P(x); ~x<A ==> Q; P(x) ==> Q |] ==> Q"
-apply (simp add: oall_def, blast)
-done
+by (simp add: oall_def, blast)
(*Trival rewrite rule; (ALL x<a.P)<->P holds only if a is not 0!*)
@@ -135,7 +133,7 @@
(*Congruence rule for rewriting*)
lemma oall_cong [cong]:
- "[| a=a'; !!x. x<a' ==> P(x) <-> P'(x) |]
+ "[| a=a'; !!x. x<a' ==> P(x) <-> P'(x) |]
==> oall(a, %x. P(x)) <-> oall(a', %x. P'(x))"
by (simp add: oall_def)
@@ -144,22 +142,22 @@
lemma oexI [intro]:
"[| P(x); x<A |] ==> EX x<A. P(x)"
-apply (simp add: oex_def, blast)
+apply (simp add: oex_def, blast)
done
(*Not of the general form for such rules; ~EX has become ALL~ *)
lemma oexCI:
"[| ALL x<A. ~P(x) ==> P(a); a<A |] ==> EX x<A. P(x)"
-apply (simp add: oex_def, blast)
+apply (simp add: oex_def, blast)
done
lemma oexE [elim!]:
"[| EX x<A. P(x); !!x. [| x<A; P(x) |] ==> Q |] ==> Q"
-apply (simp add: oex_def, blast)
+apply (simp add: oex_def, blast)
done
lemma oex_cong [cong]:
- "[| a=a'; !!x. x<a' ==> P(x) <-> P'(x) |]
+ "[| a=a'; !!x. x<a' ==> P(x) <-> P'(x) |]
==> oex(a, %x. P(x)) <-> oex(a', %x. P'(x))"
apply (simp add: oex_def cong add: conj_cong)
done
@@ -182,11 +180,11 @@
"[| i=j; !!x. x<j ==> C(x)=D(x) |] ==> (UN x<i. C(x)) = (UN x<j. D(x))"
by (simp add: OUnion_def lt_def OUN_iff)
-lemma lt_induct:
+lemma lt_induct:
"[| i<k; !!x.[| x<k; ALL y<x. P(y) |] ==> P(x) |] ==> P(i)"
apply (simp add: lt_def oall_def)
-apply (erule conjE)
-apply (erule Ord_induct, assumption, blast)
+apply (erule conjE)
+apply (erule Ord_induct, assumption, blast)
done
@@ -211,7 +209,8 @@
"ALL x[M]. P" == "rall(M, %x. P)"
"EX x[M]. P" == "rex(M, %x. P)"
-(*** Relativized universal quantifier ***)
+
+subsubsection{*Relativized universal quantifier*}
lemma rallI [intro!]: "[| !!x. M(x) ==> P(x) |] ==> ALL x[M]. P(x)"
by (simp add: rall_def)
@@ -220,9 +219,9 @@
by (simp add: rall_def)
(*Instantiates x first: better for automatic theorem proving?*)
-lemma rev_rallE [elim]:
+lemma rev_rallE [elim]:
"[| ALL x[M]. P(x); ~ M(x) ==> Q; P(x) ==> Q |] ==> Q"
-by (simp add: rall_def, blast)
+by (simp add: rall_def, blast)
lemma rallE: "[| ALL x[M]. P(x); P(x) ==> Q; ~ M(x) ==> Q |] ==> Q"
by blast
@@ -233,11 +232,12 @@
(*Congruence rule for rewriting*)
lemma rall_cong [cong]:
- "(!!x. M(x) ==> P(x) <-> P'(x))
+ "(!!x. M(x) ==> P(x) <-> P'(x))
==> rall(M, %x. P(x)) <-> rall(M, %x. P'(x))"
by (simp add: rall_def)
-(*** Relativized existential quantifier ***)
+
+subsubsection{*Relativized existential quantifier*}
lemma rexI [intro]: "[| P(x); M(x) |] ==> EX x[M]. P(x)"
by (simp add: rex_def, blast)
@@ -258,7 +258,7 @@
by (simp add: rex_def)
lemma rex_cong [cong]:
- "(!!x. M(x) ==> P(x) <-> P'(x))
+ "(!!x. M(x) ==> P(x) <-> P'(x))
==> rex(M, %x. P(x)) <-> rex(M, %x. P'(x))"
by (simp add: rex_def cong: conj_cong)
@@ -277,7 +277,7 @@
"(ALL x[M]. P(x) & Q) <-> (ALL x[M]. P(x)) & ((ALL x[M]. False) | Q)"
"(ALL x[M]. P(x) | Q) <-> ((ALL x[M]. P(x)) | Q)"
"(ALL x[M]. P(x) --> Q) <-> ((EX x[M]. P(x)) --> Q)"
- "(~(ALL x[M]. P(x))) <-> (EX x[M]. ~P(x))"
+ "(~(ALL x[M]. P(x))) <-> (EX x[M]. ~P(x))"
by blast+
lemma rall_simps2:
@@ -312,7 +312,7 @@
by blast
-(** One-point rule for bounded quantifiers: see HOL/Set.ML **)
+subsubsection{*One-point rule for bounded quantifiers*}
lemma rex_triv_one_point1 [simp]: "(EX x[M]. x=a) <-> ( M(a))"
by blast
@@ -333,6 +333,20 @@
by blast
+subsubsection{*Sets as Classes*}
+
+constdefs setclass :: "[i,i] => o" ("**_")
+ "setclass(A,x) == x : A"
+
+declare setclass_def [simp]
+
+lemma rall_setclass_is_ball [simp]: "(\<forall>x[**A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
+by auto
+
+lemma rex_setclass_is_bex [simp]: "(\<exists>x[**A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
+by auto
+
+
ML
{*
val oall_def = thm "oall_def"
@@ -370,7 +384,7 @@
val Ord_atomize =
atomize ([("OrdQuant.oall", [ospec]),("OrdQuant.rall", [rspec])]@
- ZF_conn_pairs,
+ ZF_conn_pairs,
ZF_mem_pairs);
simpset_ref() := simpset() setmksimps (map mk_eq o Ord_atomize o gen_all);
*}
@@ -391,7 +405,7 @@
val all_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
("ALL x[M]. P(x) --> Q(x)", FOLogic.oT)
-val prove_rall_tac = rewtac rall_def THEN
+val prove_rall_tac = rewtac rall_def THEN
Quantifier1.prove_one_point_all_tac;
val rearrange_ball = Quantifier1.rearrange_ball prove_rall_tac;