--- a/src/HOL/Library/Zorn.thy Thu May 06 12:43:00 2004 +0200
+++ b/src/HOL/Library/Zorn.thy Thu May 06 14:14:18 2004 +0200
@@ -1,39 +1,40 @@
-(* Title : Zorn.thy
+(* Title : HOL/Library/Zorn.thy
ID : $Id$
Author : Jacques D. Fleuriot
- Description : Zorn's Lemma -- See Larry Paulson's Zorn.thy in ZF
-*)
+ Description : Zorn's Lemma -- see Larry Paulson's Zorn.thy in ZF
+*)
-header {*Zorn's Lemma*}
+header {* Zorn's Lemma *}
theory Zorn = Main:
-text{*The lemma and section numbers refer to an unpublished article ``Towards
-the Mechanization of the Proofs of Some Classical Theorems of Set Theory,'' by
-Abrial and Laffitte. *}
+text{*
+ The lemma and section numbers refer to an unpublished article
+ \cite{Abrial-Laffitte}.
+*}
constdefs
chain :: "'a set set => 'a set set set"
- "chain S == {F. F \<subseteq> S & (\<forall>x \<in> F. \<forall>y \<in> F. x \<subseteq> y | y \<subseteq> x)}"
+ "chain S == {F. F \<subseteq> S & (\<forall>x \<in> F. \<forall>y \<in> F. x \<subseteq> y | y \<subseteq> x)}"
super :: "['a set set,'a set set] => 'a set set set"
- "super S c == {d. d \<in> chain(S) & c < d}"
+ "super S c == {d. d \<in> chain S & c \<subset> d}"
maxchain :: "'a set set => 'a set set set"
- "maxchain S == {c. c \<in> chain S & super S c = {}}"
+ "maxchain S == {c. c \<in> chain S & super S c = {}}"
succ :: "['a set set,'a set set] => 'a set set"
- "succ S c == if (c \<notin> chain S| c \<in> maxchain S)
- then c else (@c'. c': (super S c))"
+ "succ S c ==
+ if c \<notin> chain S | c \<in> maxchain S
+ then c else SOME c'. c' \<in> super S c"
-consts
- "TFin" :: "'a set set => 'a set set set"
+consts
+ TFin :: "'a set set => 'a set set set"
-inductive "TFin(S)"
+inductive "TFin S"
intros
succI: "x \<in> TFin S ==> succ S x \<in> TFin S"
Pow_UnionI: "Y \<in> Pow(TFin S) ==> Union(Y) \<in> TFin S"
-
monos Pow_mono
@@ -54,26 +55,26 @@
lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
-lemma TFin_induct:
- "[| n \<in> TFin S;
- !!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x);
- !!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |]
+lemma TFin_induct:
+ "[| n \<in> TFin S;
+ !!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x);
+ !!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |]
==> P(n)"
apply (erule TFin.induct, blast+)
done
lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"
-apply (erule subset_trans)
-apply (rule Abrial_axiom1)
+apply (erule subset_trans)
+apply (rule Abrial_axiom1)
done
text{*Lemma 1 of section 3.1*}
lemma TFin_linear_lemma1:
- "[| n \<in> TFin S; m \<in> TFin S;
- \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
+ "[| n \<in> TFin S; m \<in> TFin S;
+ \<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
|] ==> n \<subseteq> m | succ S m \<subseteq> n"
apply (erule TFin_induct)
-apply (erule_tac [2] Union_lemma0) txt{*or just Blast_tac*}
+apply (erule_tac [2] Union_lemma0) (*or just blast*)
apply (blast del: subsetI intro: succ_trans)
done
@@ -82,20 +83,20 @@
"m \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> m"
apply (erule TFin_induct)
apply (rule impI [THEN ballI])
-txt{*case split using TFin_linear_lemma1*}
-apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
+txt{*case split using @{text TFin_linear_lemma1}*}
+apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
assumption+)
apply (drule_tac x = n in bspec, assumption)
-apply (blast del: subsetI intro: succ_trans, blast)
+apply (blast del: subsetI intro: succ_trans, blast)
txt{*second induction step*}
apply (rule impI [THEN ballI])
apply (rule Union_lemma0 [THEN disjE])
apply (rule_tac [3] disjI2)
- prefer 2 apply blast
+ prefer 2 apply blast
apply (rule ballI)
-apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
- assumption+, auto)
-apply (blast intro!: Abrial_axiom1 [THEN subsetD])
+apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
+ assumption+, auto)
+apply (blast intro!: Abrial_axiom1 [THEN subsetD])
done
text{*Re-ordering the premises of Lemma 2*}
@@ -107,10 +108,10 @@
text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*}
lemma TFin_subset_linear: "[| m \<in> TFin S; n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> n"
-apply (rule disjE)
+apply (rule disjE)
apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
apply (assumption+, erule disjI2)
-apply (blast del: subsetI
+apply (blast del: subsetI
intro: subsetI Abrial_axiom1 [THEN subset_trans])
done
@@ -130,12 +131,12 @@
apply (erule ssubst)
apply (rule Abrial_axiom1 [THEN equalityI])
apply (blast del: subsetI
- intro: subsetI TFin_UnionI TFin.succI)
+ intro: subsetI TFin_UnionI TFin.succI)
done
subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*}
-text{*NB: We assume the partial ordering is @{text "\<subseteq>"},
+text{*NB: We assume the partial ordering is @{text "\<subseteq>"},
the subset relation!*}
lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S"
@@ -150,13 +151,13 @@
lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> ? d. d \<in> super S c"
by (unfold super_def maxchain_def, auto)
-lemma select_super: "c \<in> chain S - maxchain S ==>
+lemma select_super: "c \<in> chain S - maxchain S ==>
(@c'. c': super S c): super S c"
apply (erule mem_super_Ex [THEN exE])
apply (rule someI2, auto)
done
-lemma select_not_equals: "c \<in> chain S - maxchain S ==>
+lemma select_not_equals: "c \<in> chain S - maxchain S ==>
(@c'. c': super S c) \<noteq> c"
apply (rule notI)
apply (drule select_super)
@@ -180,26 +181,26 @@
apply (unfold chain_def)
apply (rule CollectI, safe)
apply (drule bspec, assumption)
-apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE],
+apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE],
blast+)
done
-
+
theorem Hausdorff: "\<exists>c. (c :: 'a set set): maxchain S"
apply (rule_tac x = "Union (TFin S) " in exI)
apply (rule classical)
apply (subgoal_tac "succ S (Union (TFin S)) = Union (TFin S) ")
prefer 2
- apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric])
+ apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric])
apply (cut_tac subset_refl [THEN TFin_UnionI, THEN TFin_chain_lemma4])
apply (drule DiffI [THEN succ_not_equals], blast+)
done
-subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then
+subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then
There Is a Maximal Element*}
-lemma chain_extend:
- "[| c \<in> chain S; z \<in> S;
+lemma chain_extend:
+ "[| c \<in> chain S; z \<in> S;
\<forall>x \<in> c. x<=(z:: 'a set) |] ==> {z} Un c \<in> chain S"
by (unfold chain_def, blast)
@@ -237,16 +238,16 @@
"\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y
==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> x --> y = x"
apply (cut_tac Hausdorff maxchain_subset_chain)
-apply (erule exE)
-apply (drule subsetD, assumption)
-apply (drule bspec, assumption, erule bexE)
+apply (erule exE)
+apply (drule subsetD, assumption)
+apply (drule bspec, assumption, erule bexE)
apply (rule_tac x = y in bexI)
prefer 2 apply assumption
-apply clarify
-apply (rule ccontr)
+apply clarify
+apply (rule ccontr)
apply (frule_tac z = x in chain_extend)
apply (assumption, blast)
-apply (unfold maxchain_def super_def psubset_def)
+apply (unfold maxchain_def super_def psubset_def)
apply (blast elim!: equalityCE)
done
@@ -259,4 +260,3 @@
by (unfold chain_def, blast)
end
-