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src/HOL/Hahn_Banach/Zorn_Lemma.thy

author | wenzelm |

Sat, 07 Apr 2012 16:41:59 +0200 | |

changeset 47389 | e8552cba702d |

parent 44887 | 7ca82df6e951 |

child 52183 | 667961fa6a60 |

permissions | -rw-r--r-- |

explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions;
tuned;

(* Title: HOL/Hahn_Banach/Zorn_Lemma.thy Author: Gertrud Bauer, TU Munich *) header {* Zorn's Lemma *} theory Zorn_Lemma imports "~~/src/HOL/Library/Zorn" begin text {* Zorn's Lemmas states: if every linear ordered subset of an ordered set @{text S} has an upper bound in @{text S}, then there exists a maximal element in @{text S}. In our application, @{text S} is a set of sets ordered by set inclusion. Since the union of a chain of sets is an upper bound for all elements of the chain, the conditions of Zorn's lemma can be modified: if @{text S} is non-empty, it suffices to show that for every non-empty chain @{text c} in @{text S} the union of @{text c} also lies in @{text S}. *} theorem Zorn's_Lemma: assumes r: "\<And>c. c \<in> chain S \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> \<Union>c \<in> S" and aS: "a \<in> S" shows "\<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z \<longrightarrow> y = z" proof (rule Zorn_Lemma2) show "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y" proof fix c assume "c \<in> chain S" show "\<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y" proof cases txt {* If @{text c} is an empty chain, then every element in @{text S} is an upper bound of @{text c}. *} assume "c = {}" with aS show ?thesis by fast txt {* If @{text c} is non-empty, then @{text "\<Union>c"} is an upper bound of @{text c}, lying in @{text S}. *} next assume "c \<noteq> {}" show ?thesis proof show "\<forall>z \<in> c. z \<subseteq> \<Union>c" by fast show "\<Union>c \<in> S" proof (rule r) from `c \<noteq> {}` show "\<exists>x. x \<in> c" by fast show "c \<in> chain S" by fact qed qed qed qed qed end