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src/HOL/Hahn_Banach/Zorn_Lemma.thy

author | blanchet |

Thu Jan 16 16:33:19 2014 +0100 (2014-01-16) | |

changeset 55018 | 2a526bd279ed |

parent 52183 | 667961fa6a60 |

child 58744 | c434e37f290e |

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

moved 'Zorn' into 'Main', since it's a BNF dependency

1 (* Title: HOL/Hahn_Banach/Zorn_Lemma.thy

2 Author: Gertrud Bauer, TU Munich

3 *)

5 header {* Zorn's Lemma *}

7 theory Zorn_Lemma

8 imports Main

9 begin

11 text {*

12 Zorn's Lemmas states: if every linear ordered subset of an ordered

13 set @{text S} has an upper bound in @{text S}, then there exists a

14 maximal element in @{text S}. In our application, @{text S} is a

15 set of sets ordered by set inclusion. Since the union of a chain of

16 sets is an upper bound for all elements of the chain, the conditions

17 of Zorn's lemma can be modified: if @{text S} is non-empty, it

18 suffices to show that for every non-empty chain @{text c} in @{text

19 S} the union of @{text c} also lies in @{text S}.

20 *}

22 theorem Zorn's_Lemma:

23 assumes r: "\<And>c. c \<in> chains S \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> \<Union>c \<in> S"

24 and aS: "a \<in> S"

25 shows "\<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z \<longrightarrow> z = y"

26 proof (rule Zorn_Lemma2)

27 show "\<forall>c \<in> chains S. \<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"

28 proof

29 fix c assume "c \<in> chains S"

30 show "\<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"

31 proof cases

33 txt {* If @{text c} is an empty chain, then every element in

34 @{text S} is an upper bound of @{text c}. *}

36 assume "c = {}"

37 with aS show ?thesis by fast

39 txt {* If @{text c} is non-empty, then @{text "\<Union>c"} is an upper

40 bound of @{text c}, lying in @{text S}. *}

42 next

43 assume "c \<noteq> {}"

44 show ?thesis

45 proof

46 show "\<forall>z \<in> c. z \<subseteq> \<Union>c" by fast

47 show "\<Union>c \<in> S"

48 proof (rule r)

49 from `c \<noteq> {}` show "\<exists>x. x \<in> c" by fast

50 show "c \<in> chains S" by fact

51 qed

52 qed

53 qed

54 qed

55 qed

57 end